/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 342 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 116 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 124 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 83 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 150 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) QDP (16) QReductionProof [EQUIVALENT, 2 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QReductionProof [EQUIVALENT, 0 ms] (24) QDP (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] (26) YES (27) QDP (28) UsableRulesProof [EQUIVALENT, 0 ms] (29) QDP (30) QReductionProof [EQUIVALENT, 0 ms] (31) QDP (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] (33) YES (34) QDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) QDP (37) QReductionProof [EQUIVALENT, 0 ms] (38) QDP (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] (40) YES (41) QDP (42) UsableRulesProof [EQUIVALENT, 0 ms] (43) QDP (44) QReductionProof [EQUIVALENT, 0 ms] (45) QDP (46) QDPSizeChangeProof [EQUIVALENT, 0 ms] (47) YES (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) QReductionProof [EQUIVALENT, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) QDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) QDP (58) QReductionProof [EQUIVALENT, 0 ms] (59) QDP (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] (61) YES (62) QDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) QDP (65) QReductionProof [EQUIVALENT, 0 ms] (66) QDP (67) QDPSizeChangeProof [EQUIVALENT, 0 ms] (68) YES (69) QDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) QDP (72) QReductionProof [EQUIVALENT, 0 ms] (73) QDP (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] (75) YES (76) QDP (77) UsableRulesProof [EQUIVALENT, 0 ms] (78) QDP (79) QReductionProof [EQUIVALENT, 0 ms] (80) QDP (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] (82) YES (83) QDP (84) UsableRulesProof [EQUIVALENT, 0 ms] (85) QDP (86) QReductionProof [EQUIVALENT, 0 ms] (87) QDP (88) QDPSizeChangeProof [EQUIVALENT, 0 ms] (89) YES (90) QDP (91) UsableRulesProof [EQUIVALENT, 0 ms] (92) QDP (93) QReductionProof [EQUIVALENT, 0 ms] (94) QDP (95) QDPSizeChangeProof [EQUIVALENT, 0 ms] (96) YES (97) QDP (98) UsableRulesProof [EQUIVALENT, 0 ms] (99) QDP (100) QReductionProof [EQUIVALENT, 0 ms] (101) QDP (102) QDPSizeChangeProof [EQUIVALENT, 0 ms] (103) YES (104) QDP (105) UsableRulesProof [EQUIVALENT, 0 ms] (106) QDP (107) QReductionProof [EQUIVALENT, 0 ms] (108) QDP (109) QDPSizeChangeProof [EQUIVALENT, 0 ms] (110) YES (111) QDP (112) UsableRulesProof [EQUIVALENT, 0 ms] (113) QDP (114) QReductionProof [EQUIVALENT, 0 ms] (115) QDP (116) QDPSizeChangeProof [EQUIVALENT, 0 ms] (117) YES (118) QDP (119) UsableRulesProof [EQUIVALENT, 0 ms] (120) QDP (121) MRRProof [EQUIVALENT, 349 ms] (122) QDP (123) MRRProof [EQUIVALENT, 230 ms] (124) QDP (125) MRRProof [EQUIVALENT, 132 ms] (126) QDP (127) MRRProof [EQUIVALENT, 273 ms] (128) QDP (129) QDPQMonotonicMRRProof [EQUIVALENT, 322 ms] (130) QDP (131) QDPQMonotonicMRRProof [EQUIVALENT, 249 ms] (132) QDP (133) DependencyGraphProof [EQUIVALENT, 0 ms] (134) QDP (135) QDPQMonotonicMRRProof [EQUIVALENT, 204 ms] (136) QDP (137) QDPOrderProof [EQUIVALENT, 373 ms] (138) QDP (139) QDPOrderProof [EQUIVALENT, 290 ms] (140) QDP (141) DependencyGraphProof [EQUIVALENT, 0 ms] (142) QDP (143) QDPOrderProof [EQUIVALENT, 313 ms] (144) QDP (145) QDPOrderProof [EQUIVALENT, 428 ms] (146) QDP (147) DependencyGraphProof [EQUIVALENT, 0 ms] (148) QDP (149) QDPQMonotonicMRRProof [EQUIVALENT, 263 ms] (150) QDP (151) QDPQMonotonicMRRProof [EQUIVALENT, 114 ms] (152) QDP (153) DependencyGraphProof [EQUIVALENT, 0 ms] (154) QDP (155) QDPQMonotonicMRRProof [EQUIVALENT, 100 ms] (156) QDP (157) DependencyGraphProof [EQUIVALENT, 0 ms] (158) QDP (159) MRRProof [EQUIVALENT, 117 ms] (160) QDP (161) QDPOrderProof [EQUIVALENT, 0 ms] (162) QDP (163) UsableRulesProof [EQUIVALENT, 0 ms] (164) QDP (165) QReductionProof [EQUIVALENT, 0 ms] (166) QDP (167) QDPSizeChangeProof [EQUIVALENT, 0 ms] (168) YES ---------------------------------------- (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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) 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(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 1 + 2*x_1 POL(isNatList(x_1)) = 2*x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 2 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) ---------------------------------------- (2) 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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 1 + x_1 POL(U41(x_1, x_2)) = x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(U31(tt)) -> mark(tt) ---------------------------------------- (4) 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(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = 2*x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 2*x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(isNat(length(V1))) -> mark(U11(isNatList(V1))) ---------------------------------------- (6) 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(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(U11(x_1)) = 1 + x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(U11(tt)) -> mark(tt) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(zeros) -> CONS(0, zeros) ACTIVE(U21(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, L, N)) -> MARK(U62(isNat(N), L)) ACTIVE(U61(tt, L, N)) -> U62^1(isNat(N), L) ACTIVE(U61(tt, L, N)) -> ISNAT(N) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) ACTIVE(U62(tt, L)) -> S(length(L)) ACTIVE(U62(tt, L)) -> LENGTH(L) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1)) ACTIVE(isNat(s(V1))) -> ISNAT(V1) 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(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(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) ACTIVE(length(cons(N, L))) -> U61^1(isNatList(L), L, N) ACTIVE(length(cons(N, L))) -> ISNATLIST(L) 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, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> U61^1(mark(X1), X2, X3) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) MARK(U62(X1, X2)) -> U62^1(mark(X1), X2) MARK(U62(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(nil) -> ACTIVE(nil) 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, X3) -> U61^1(X1, X2, X3) U61^1(X1, mark(X2), X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, mark(X3)) -> U61^1(X1, X2, X3) U61^1(active(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, active(X2), X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, active(X3)) -> U61^1(X1, X2, X3) U62^1(mark(X1), X2) -> U62^1(X1, X2) U62^1(X1, mark(X2)) -> U62^1(X1, X2) U62^1(active(X1), X2) -> U62^1(X1, X2) U62^1(X1, active(X2)) -> U62^1(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) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 36 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (15) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (17) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) 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: *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 ---------------------------------------- (19) YES ---------------------------------------- (20) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (22) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (24) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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 ---------------------------------------- (26) YES ---------------------------------------- (27) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (29) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (31) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) 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 ---------------------------------------- (33) YES ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(X1, mark(X2)) -> U62^1(X1, X2) U62^1(mark(X1), X2) -> U62^1(X1, X2) U62^1(active(X1), X2) -> U62^1(X1, X2) U62^1(X1, active(X2)) -> U62^1(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(X1, mark(X2)) -> U62^1(X1, X2) U62^1(mark(X1), X2) -> U62^1(X1, X2) U62^1(active(X1), X2) -> U62^1(X1, X2) U62^1(X1, active(X2)) -> U62^1(X1, X2) R is empty. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(X1, mark(X2)) -> U62^1(X1, X2) U62^1(mark(X1), X2) -> U62^1(X1, X2) U62^1(active(X1), X2) -> U62^1(X1, X2) U62^1(X1, active(X2)) -> U62^1(X1, X2) R is empty. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) 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(X1, mark(X2)) -> U62^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U62^1(mark(X1), X2) -> U62^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U62^1(active(X1), X2) -> U62^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U62^1(X1, active(X2)) -> U62^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (40) YES ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2), X3) -> U61^1(X1, X2, X3) U61^1(mark(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, mark(X3)) -> U61^1(X1, X2, X3) U61^1(active(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, active(X2), X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, active(X3)) -> U61^1(X1, X2, X3) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2), X3) -> U61^1(X1, X2, X3) U61^1(mark(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, mark(X3)) -> U61^1(X1, X2, X3) U61^1(active(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, active(X2), X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, active(X3)) -> U61^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2), X3) -> U61^1(X1, X2, X3) U61^1(mark(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, mark(X3)) -> U61^1(X1, X2, X3) U61^1(active(X1), X2, X3) -> U61^1(X1, X2, X3) U61^1(X1, active(X2), X3) -> U61^1(X1, X2, X3) U61^1(X1, X2, active(X3)) -> U61^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) 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), X3) -> U61^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U61^1(mark(X1), X2, X3) -> U61^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U61^1(X1, X2, mark(X3)) -> U61^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U61^1(active(X1), X2, X3) -> U61^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U61^1(X1, active(X2), X3) -> U61^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U61^1(X1, X2, active(X3)) -> U61^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (47) YES ---------------------------------------- (48) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (50) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (52) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) 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: *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 ---------------------------------------- (54) YES ---------------------------------------- (55) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (57) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (59) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) 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 ---------------------------------------- (61) YES ---------------------------------------- (62) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (64) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (66) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) 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 ---------------------------------------- (68) YES ---------------------------------------- (69) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (71) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (72) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (73) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) 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 ---------------------------------------- (75) YES ---------------------------------------- (76) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (78) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (80) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) 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 ---------------------------------------- (82) YES ---------------------------------------- (83) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (85) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (87) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) 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: *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 ---------------------------------------- (89) YES ---------------------------------------- (90) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (92) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (94) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) 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 ---------------------------------------- (96) YES ---------------------------------------- (97) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (99) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (101) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (102) 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 ---------------------------------------- (103) YES ---------------------------------------- (104) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (106) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (108) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) 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 ---------------------------------------- (110) YES ---------------------------------------- (111) 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(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (112) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (113) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (114) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (115) 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. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) 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 ---------------------------------------- (117) YES ---------------------------------------- (118) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U62(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U21(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, L, N)) -> mark(U62(isNat(N), L)) active(U62(tt, L)) -> mark(s(length(L))) active(isNat(0)) -> mark(tt) 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(U61(isNatList(L), L, 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, X3)) -> active(U61(mark(X1), X2, X3)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) 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, X3) -> U61(X1, X2, X3) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(mark(X1), X2) -> U62(X1, X2) U62(X1, mark(X2)) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(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) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (120) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U62(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(U31(X)) -> MARK(X) 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)) = 2 + x_1 POL(U41(x_1, x_2)) = x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 2*x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 2*x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (122) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U62(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(U11(X)) -> MARK(X) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2*x_1 POL(U11(x_1)) = 2 + x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 2*x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = 2*x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (124) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U62(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(U41(X1, X2)) -> MARK(X1) 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)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = 1 + 2*x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (126) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U62(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(U61(X1, X2, X3)) -> MARK(X1) MARK(U62(X1, X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2*x_1 POL(U11(x_1)) = 2*x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_1 + x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (128) 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(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U31(X)) -> ACTIVE(U31(mark(X))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 1 POL(U11(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = 1 POL(U42(x_1)) = 0 POL(U51(x_1, x_2)) = 1 POL(U52(x_1)) = 0 POL(U61(x_1, x_2, x_3)) = 1 POL(U62(x_1, x_2)) = 1 POL(active(x_1)) = 0 POL(cons(x_1, x_2)) = 0 POL(isNat(x_1)) = 1 POL(isNatIList(x_1)) = 1 POL(isNatList(x_1)) = 1 POL(length(x_1)) = 1 POL(mark(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(tt) = 0 POL(zeros) = 1 ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(zeros) -> ACTIVE(zeros) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 0 POL(MARK(x_1)) = x_1 POL(U11(x_1)) = 1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = x_1 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 0 POL(U62(x_1, x_2)) = 0 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 0 POL(mark(x_1)) = 2*x_1 POL(nil) = 2 POL(s(x_1)) = 2*x_1 POL(tt) = 0 POL(zeros) = 2 ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), 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(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(cons(X1, X2)) -> MARK(X1) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 0 POL(MARK(x_1)) = x_1 POL(U11(x_1)) = x_1 POL(U21(x_1)) = 2*x_1 POL(U31(x_1)) = 1 POL(U41(x_1, x_2)) = 0 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 2*x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 0 POL(U62(x_1, x_2)) = 0 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 1 + 2*x_1 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 0 POL(mark(x_1)) = 2*x_1 POL(nil) = 2 POL(s(x_1)) = 2*x_1 POL(tt) = 0 POL(zeros) = 2 ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) 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) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = x_1 + 2 POL( U61_3(x_1, ..., x_3) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 POL( zeros ) = 0 POL( active_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( 0 ) = 2 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 POL( U21_1(x_1) ) = 2x_1 POL( U11_1(x_1) ) = 2 POL( U31_1(x_1) ) = 0 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: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) 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) 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(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) 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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) 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) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_3(x_1, ..., x_3) ) = 0 POL( U62_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = max{0, -2} POL( zeros ) = 0 POL( active_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 + 2 POL( isNatIList_1(x_1) ) = 0 POL( U52_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 + 2 POL( U21_1(x_1) ) = x_1 + 2 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U31_1(x_1) ) = 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) 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(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) length(active(X)) -> length(X) length(mark(X)) -> length(X) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (142) 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(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) 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( U61_3(x_1, ..., x_3) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = 1 POL( zeros ) = 0 POL( active_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 + 1 POL( U21_1(x_1) ) = x_1 + 2 POL( U11_1(x_1) ) = x_1 + 2 POL( U31_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: none ---------------------------------------- (144) 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(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) 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)) 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) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2x_2 + 2 POL( U51_2(x_1, x_2) ) = 2x_2 + 2 POL( U61_3(x_1, ..., x_3) ) = max{0, -2} POL( U62_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( zeros ) = 0 POL( active_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = 2x_2 + 2 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = 2x_1 + 1 POL( U21_1(x_1) ) = 2x_1 + 2 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U31_1(x_1) ) = 2x_1 + 2 POL( nil ) = 2 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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) ---------------------------------------- (146) 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(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (148) 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) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(s(X)) -> MARK(X) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: ACTIVE(U62(tt, L)) -> MARK(s(length(L))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2*x_1 POL(U11(x_1)) = 0 POL(U21(x_1)) = 1 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = 0 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 0 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 POL(U62(x_1, x_2)) = 2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(isNat(x_1)) = 1 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 0 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = 2*x_1 POL(tt) = 1 POL(zeros) = 0 ---------------------------------------- (150) 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) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 1 POL(U11(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = 1 POL(U42(x_1)) = 0 POL(U51(x_1, x_2)) = 1 POL(U52(x_1)) = 0 POL(U61(x_1, x_2, x_3)) = 1 POL(U62(x_1, x_2)) = 0 POL(active(x_1)) = 0 POL(cons(x_1, x_2)) = 0 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 1 POL(mark(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (152) 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) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(s(X)) -> MARK(X) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 1 POL(U11(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1)) = 0 POL(U41(x_1, x_2)) = 1 POL(U42(x_1)) = 0 POL(U51(x_1, x_2)) = 1 POL(U52(x_1)) = 0 POL(U61(x_1, x_2, x_3)) = 0 POL(U62(x_1, x_2)) = 0 POL(active(x_1)) = 0 POL(cons(x_1, x_2)) = 0 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 1 POL(mark(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (157) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (158) 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(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: MARK(length(X)) -> ACTIVE(length(mark(X))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 2*x_1 POL(U11(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = 1 + 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 0 POL(zeros) = 0 ---------------------------------------- (160) 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(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) 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)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 U41(x1, x2) = U41 ACTIVE(x1) = ACTIVE U42(x1) = x1 isNatIList(x1) = isNatIList U51(x1, x2) = U51 U52(x1) = x1 isNatList(x1) = isNatList s(x1) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: isNatIList=1 U41=4 U51=4 ACTIVE=3 isNatList=2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (162) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(U42(X)) -> active(U42(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U52(X)) -> active(U52(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U62(X1, X2)) -> active(U62(mark(X1), X2)) active(U62(tt, L)) -> mark(s(length(L))) mark(s(X)) -> active(s(mark(X))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N)) mark(U61(X1, X2, X3)) -> active(U61(mark(X1), X2, X3)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U31(X)) -> active(U31(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(length(X)) -> active(length(mark(X))) mark(nil) -> active(nil) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(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) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(X1, mark(X2)) -> U62(X1, X2) U62(mark(X1), X2) -> U62(X1, X2) U62(active(X1), X2) -> U62(X1, X2) U62(X1, active(X2)) -> U62(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) 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) U61(X1, mark(X2), X3) -> U61(X1, X2, X3) U61(mark(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, X2, mark(X3)) -> U61(X1, X2, X3) U61(active(X1), X2, X3) -> U61(X1, X2, X3) U61(X1, active(X2), X3) -> U61(X1, X2, X3) U61(X1, X2, active(X3)) -> U61(X1, X2, X3) The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (164) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(s(X)) -> MARK(X) R is empty. The set Q consists of the following terms: active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) U42(mark(x0)) U42(active(x0)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) U52(mark(x0)) U52(active(x0)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) s(mark(x0)) s(active(x0)) length(mark(x0)) length(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (165) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. active(zeros) active(U11(tt)) active(U21(tt)) active(U31(tt)) active(U41(tt, x0)) active(U42(tt)) active(U51(tt, x0)) active(U52(tt)) active(U61(tt, x0, x1)) active(U62(tt, x0)) active(isNat(0)) active(isNat(length(x0))) active(isNat(s(x0))) active(isNatIList(x0)) active(isNatList(nil)) active(isNatList(cons(x0, x1))) active(length(nil)) active(length(cons(x0, x1))) mark(zeros) mark(cons(x0, x1)) mark(0) mark(U11(x0)) mark(tt) mark(U21(x0)) mark(U31(x0)) mark(U41(x0, x1)) mark(U42(x0)) mark(isNatIList(x0)) mark(U51(x0, x1)) mark(U52(x0)) mark(isNatList(x0)) mark(U61(x0, x1, x2)) mark(U62(x0, x1)) mark(isNat(x0)) mark(s(x0)) mark(length(x0)) mark(nil) cons(mark(x0), x1) cons(x0, mark(x1)) cons(active(x0), x1) cons(x0, active(x1)) U11(mark(x0)) U11(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0)) U31(active(x0)) U41(mark(x0), x1) U41(x0, mark(x1)) U41(active(x0), x1) U41(x0, active(x1)) isNatIList(mark(x0)) isNatIList(active(x0)) U51(mark(x0), x1) U51(x0, mark(x1)) U51(active(x0), x1) U51(x0, active(x1)) isNatList(mark(x0)) isNatList(active(x0)) U61(mark(x0), x1, x2) U61(x0, mark(x1), x2) U61(x0, x1, mark(x2)) U61(active(x0), x1, x2) U61(x0, active(x1), x2) U61(x0, x1, active(x2)) U62(mark(x0), x1) U62(x0, mark(x1)) U62(active(x0), x1) U62(x0, active(x1)) isNat(mark(x0)) isNat(active(x0)) length(mark(x0)) length(active(x0)) ---------------------------------------- (166) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U52(X)) -> MARK(X) MARK(s(X)) -> MARK(X) R is empty. The set Q consists of the following terms: U42(mark(x0)) U42(active(x0)) U52(mark(x0)) U52(active(x0)) s(mark(x0)) s(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) 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: *MARK(U42(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U52(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(s(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (168) YES