/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 291 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 128 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 98 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 142 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 37 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) UsableRulesProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES (33) QDP (34) UsableRulesProof [EQUIVALENT, 0 ms] (35) QDP (36) QDPSizeChangeProof [EQUIVALENT, 0 ms] (37) YES (38) QDP (39) UsableRulesProof [EQUIVALENT, 0 ms] (40) QDP (41) QDPSizeChangeProof [EQUIVALENT, 0 ms] (42) YES (43) QDP (44) UsableRulesProof [EQUIVALENT, 0 ms] (45) QDP (46) QDPSizeChangeProof [EQUIVALENT, 0 ms] (47) YES (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) QDPSizeChangeProof [EQUIVALENT, 0 ms] (52) YES (53) QDP (54) UsableRulesProof [EQUIVALENT, 0 ms] (55) QDP (56) QDPSizeChangeProof [EQUIVALENT, 0 ms] (57) YES (58) QDP (59) UsableRulesProof [EQUIVALENT, 0 ms] (60) QDP (61) QDPSizeChangeProof [EQUIVALENT, 0 ms] (62) YES (63) QDP (64) UsableRulesProof [EQUIVALENT, 0 ms] (65) QDP (66) QDPSizeChangeProof [EQUIVALENT, 0 ms] (67) YES (68) QDP (69) UsableRulesProof [EQUIVALENT, 0 ms] (70) QDP (71) QDPSizeChangeProof [EQUIVALENT, 0 ms] (72) YES (73) QDP (74) UsableRulesProof [EQUIVALENT, 0 ms] (75) QDP (76) QDPSizeChangeProof [EQUIVALENT, 0 ms] (77) YES (78) QDP (79) UsableRulesProof [EQUIVALENT, 0 ms] (80) QDP (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] (82) YES (83) QDP (84) UsableRulesProof [EQUIVALENT, 0 ms] (85) QDP (86) QDPSizeChangeProof [EQUIVALENT, 0 ms] (87) YES (88) QDP (89) MRRProof [EQUIVALENT, 228 ms] (90) QDP (91) MRRProof [EQUIVALENT, 211 ms] (92) QDP (93) MRRProof [EQUIVALENT, 239 ms] (94) QDP (95) MRRProof [EQUIVALENT, 228 ms] (96) QDP (97) QDPOrderProof [EQUIVALENT, 358 ms] (98) QDP (99) QDPOrderProof [EQUIVALENT, 353 ms] (100) QDP (101) QDPOrderProof [EQUIVALENT, 372 ms] (102) QDP (103) DependencyGraphProof [EQUIVALENT, 0 ms] (104) QDP (105) QDPOrderProof [EQUIVALENT, 343 ms] (106) QDP (107) QDPOrderProof [EQUIVALENT, 504 ms] (108) QDP (109) QDPOrderProof [EQUIVALENT, 311 ms] (110) QDP (111) QDPOrderProof [EQUIVALENT, 242 ms] (112) QDP (113) QDPOrderProof [EQUIVALENT, 234 ms] (114) QDP (115) QDPOrderProof [EQUIVALENT, 239 ms] (116) QDP (117) QDPOrderProof [EQUIVALENT, 212 ms] (118) QDP (119) QDPOrderProof [EQUIVALENT, 252 ms] (120) QDP (121) QDPOrderProof [EQUIVALENT, 277 ms] (122) QDP (123) QDPOrderProof [EQUIVALENT, 232 ms] (124) QDP (125) QDPOrderProof [EQUIVALENT, 514 ms] (126) QDP (127) QDPOrderProof [EQUIVALENT, 248 ms] (128) QDP (129) QDPOrderProof [EQUIVALENT, 239 ms] (130) QDP (131) QDPOrderProof [EQUIVALENT, 202 ms] (132) QDP (133) QDPOrderProof [EQUIVALENT, 57 ms] (134) QDP (135) QDPOrderProof [EQUIVALENT, 41 ms] (136) QDP (137) DependencyGraphProof [EQUIVALENT, 0 ms] (138) QDP (139) UsableRulesProof [EQUIVALENT, 0 ms] (140) QDP (141) QDPSizeChangeProof [EQUIVALENT, 0 ms] (142) 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) Q is empty. ---------------------------------------- (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) Q is empty. ---------------------------------------- (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) Q is empty. ---------------------------------------- (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) Q is empty. ---------------------------------------- (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) Q is empty. ---------------------------------------- (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) Q is empty. 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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 ---------------------------------------- (17) YES ---------------------------------------- (18) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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 ---------------------------------------- (22) YES ---------------------------------------- (23) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) 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 ---------------------------------------- (27) YES ---------------------------------------- (28) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (30) 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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 ---------------------------------------- (32) YES ---------------------------------------- (33) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (35) 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) 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 ---------------------------------------- (37) YES ---------------------------------------- (38) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) 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 ---------------------------------------- (42) YES ---------------------------------------- (43) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U52^1(active(X)) -> U52^1(X) U52^1(mark(X)) -> U52^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (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: *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 ---------------------------------------- (47) YES ---------------------------------------- (48) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(X1, mark(X2)) -> U51^1(X1, X2) U51^1(mark(X1), X2) -> U51^1(X1, X2) U51^1(active(X1), X2) -> U51^1(X1, X2) U51^1(X1, active(X2)) -> U51^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) 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 ---------------------------------------- (52) YES ---------------------------------------- (53) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) 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 ---------------------------------------- (57) YES ---------------------------------------- (58) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(active(X)) -> U42^1(X) U42^1(mark(X)) -> U42^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) 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 ---------------------------------------- (62) YES ---------------------------------------- (63) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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 ---------------------------------------- (67) YES ---------------------------------------- (68) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(active(X)) -> U31^1(X) U31^1(mark(X)) -> U31^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) 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 ---------------------------------------- (72) YES ---------------------------------------- (73) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) 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 ---------------------------------------- (77) YES ---------------------------------------- (78) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(active(X)) -> U11^1(X) U11^1(mark(X)) -> U11^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (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: *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 ---------------------------------------- (82) YES ---------------------------------------- (83) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) 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 ---------------------------------------- (87) YES ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(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)) -> MARK(X1) MARK(U62(X1, X2)) -> ACTIVE(U62(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)) -> 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) 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)) = 1 + 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)) = 2*x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_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 ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(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)) -> MARK(X1) MARK(U62(X1, X2)) -> ACTIVE(U62(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)) -> 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) 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)) = 2*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 + 2*x_1 POL(U41(x_1, x_2)) = x_1 + 2*x_2 POL(U42(x_1)) = 2*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 ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(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)) -> MARK(X1) MARK(U62(X1, X2)) -> ACTIVE(U62(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)) -> 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) 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)) = 2*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 + 2*x_1 + 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)) = 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)) = 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 ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(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)) -> MARK(X1) MARK(U62(X1, X2)) -> ACTIVE(U62(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)) -> 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) 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 + 2*x_2 POL(U42(x_1)) = 2*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 ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(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(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U11_1(x_1) ) = max{0, -2} POL( U21_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = 1 POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U52_1(x_1) ) = 0 POL( U61_3(x_1, ..., x_3) ) = 2 POL( U62_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = 0 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(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) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> 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)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U42(X)) -> ACTIVE(U42(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( MARK_1(x_1) ) = 1 POL( U42_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 1 POL( active_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( U61_3(x_1, ..., x_3) ) = 1 POL( isNatList_1(x_1) ) = 1 POL( U41_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U62_2(x_1, x_2) ) = 1 POL( length_1(x_1) ) = 1 POL( U21_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 1 POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2} POL( zeros ) = 1 POL( 0 ) = 0 POL( tt ) = 0 POL( U11_1(x_1) ) = 2 POL( U31_1(x_1) ) = 0 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) length(active(X)) -> length(X) length(mark(X)) -> length(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) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> 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)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(zeros) -> ACTIVE(zeros) 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( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = x_1 + 1 POL( U51_2(x_1, x_2) ) = 2x_1 + 1 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( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 + 1 POL( cons_2(x_1, x_2) ) = 2x_1 + 1 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 1 POL( U11_1(x_1) ) = x_1 + 1 POL( U31_1(x_1) ) = x_1 + 2 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) 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) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U21(X)) -> MARK(X) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) 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(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> 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)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: 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))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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: 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> MARK(X1) 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) ) = 2x_1 + 1 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 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( active_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 1 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 2 POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = 2x_1 + 1 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) 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) s(active(X)) -> s(X) s(mark(X)) -> s(X) 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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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: 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) 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) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = 2x_1 POL( nil ) = 1 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U31(X)) -> active(U31(mark(X))) active(U62(tt, L)) -> mark(s(length(L))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U42(X)) -> active(U42(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(isNatIList(X)) -> active(isNatIList(X)) 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(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(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) 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) s(active(X)) -> s(X) s(mark(X)) -> s(X) 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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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: 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) 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))) MARK(s(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 0 POL( U61_3(x_1, ..., x_3) ) = 2x_1 POL( U62_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 0 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 1 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = 2 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = x_1 + 2 POL( U31_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 + 2 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(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, L, N)) -> mark(U62(isNat(N), L)) mark(U31(X)) -> active(U31(mark(X))) active(U62(tt, L)) -> mark(s(length(L))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U42(X)) -> active(U42(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(isNatIList(X)) -> active(isNatIList(X)) 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(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(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) 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) s(active(X)) -> s(X) s(mark(X)) -> s(X) 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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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: 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) 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) MARK(length(X)) -> ACTIVE(length(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 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) ) = 2 POL( mark_1(x_1) ) = x_1 + 2 POL( cons_2(x_1, x_2) ) = 2x_2 + 2 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 1 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 1 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U62(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) 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) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U61_3(x_1, ..., x_3) ) = 0 POL( U62_2(x_1, x_2) ) = 0 POL( mark_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = 2 POL( active_1(x_1) ) = 0 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 1 POL( U52_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = 2x_1 POL( length_1(x_1) ) = max{0, 2x_1 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) 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) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U61(X1, X2, X3)) -> ACTIVE(U61(mark(X1), X2, X3)) 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) ) = 0 POL( U61_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U62_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + 1 POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 2 POL( 0 ) = 1 POL( tt ) = 1 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, 2x_1 - 2} POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( s_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) 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) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) 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) ) = 1 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = 1 POL( mark_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 1 POL( U21_1(x_1) ) = max{0, 2x_1 - 2} POL( U61_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( s_1(x_1) ) = 0 POL( length_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) 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) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNat(X)) -> ACTIVE(isNat(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = x_2 + 2 POL( active_1(x_1) ) = x_1 POL( zeros ) = 1 POL( 0 ) = 1 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 + 2 POL( U61_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( isNat_1(x_1) ) = 2 POL( U31_1(x_1) ) = 1 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = max{0, -2} POL( nil ) = 2 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) 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) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 0 POL( U62_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2 POL( active_1(x_1) ) = 2x_1 + 1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 1} POL( U61_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 0 POL( s_1(x_1) ) = 0 POL( length_1(x_1) ) = x_1 + 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U62(tt, L)) -> MARK(s(length(L))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U62(tt, L)) -> MARK(s(length(L))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 1 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2 POL( U61_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( U62_2(x_1, x_2) ) = 2x_1 + 2 POL( isNat_1(x_1) ) = 2x_1 POL( U31_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = 2 POL( length_1(x_1) ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2x_2 + 2 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = x_2 + 2 POL( active_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U61_3(x_1, ..., x_3) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2 POL( active_1(x_1) ) = 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = 2 POL( U61_3(x_1, ..., x_3) ) = 2 POL( U62_2(x_1, x_2) ) = 2 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = x_2 + 2 POL( U51_2(x_1, x_2) ) = 0 POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2} POL( active_1(x_1) ) = 2x_1 POL( zeros ) = 1 POL( 0 ) = 1 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2 POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U61_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U62_2(x_1, x_2) ) = max{0, x_1 - 1} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( length_1(x_1) ) = 2x_1 + 2 POL( nil ) = 2 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( MARK_1(x_1) ) = max{0, -2} POL( U42_1(x_1) ) = 2 POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U51_2(x_1, x_2) ) = 1 POL( active_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 0 POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( U41_2(x_1, x_2) ) = 2 POL( tt ) = 0 POL( U11_1(x_1) ) = 2 POL( U21_1(x_1) ) = max{0, 2x_1 - 2} POL( U61_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U62_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 1 POL( s_1(x_1) ) = 0 POL( length_1(x_1) ) = 1 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 U42(x1) = x1 ACTIVE(x1) = x1 U51(x1, x2) = x2 U52(x1) = x1 isNatList(x1) = x1 cons(x1, x2) = cons(x2) active(x1) = x1 mark(x1) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: dummyConstant=1 cons_1=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) 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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 U42(x1) = x1 ACTIVE(x1) = x1 U51(x1, x2) = U51 U52(x1) = x1 isNatList(x1) = isNatList active(x1) = x1 mark(x1) = x1 Knuth-Bendix order [KBO] with precedence:U51 > isNatList and weight map: isNatList=1 U51=1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U52(X)) -> MARK(X) MARK(U42(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U52(X)) -> MARK(X) MARK(U42(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) 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(U52(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U42(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (142) YES