YES proof of Transformed_CSR_04_LengthOfFiniteLists_nokinds_noand_iGM.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 223 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 74 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 49 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QDPSizeChangeProof [EQUIVALENT, 0 ms] (30) YES (31) QDP (32) UsableRulesProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] (35) YES (36) QDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) QDP (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] (40) YES (41) QDP (42) UsableRulesProof [EQUIVALENT, 0 ms] (43) QDP (44) QDPSizeChangeProof [EQUIVALENT, 0 ms] (45) YES (46) QDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) QDP (49) QDPSizeChangeProof [EQUIVALENT, 0 ms] (50) YES (51) QDP (52) UsableRulesProof [EQUIVALENT, 0 ms] (53) QDP (54) QDPSizeChangeProof [EQUIVALENT, 0 ms] (55) YES (56) QDP (57) UsableRulesProof [EQUIVALENT, 0 ms] (58) QDP (59) QDPSizeChangeProof [EQUIVALENT, 0 ms] (60) YES (61) QDP (62) UsableRulesProof [EQUIVALENT, 0 ms] (63) QDP (64) QDPSizeChangeProof [EQUIVALENT, 0 ms] (65) YES (66) QDP (67) UsableRulesProof [EQUIVALENT, 0 ms] (68) QDP (69) QDPSizeChangeProof [EQUIVALENT, 0 ms] (70) YES (71) QDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) QDP (74) QDPSizeChangeProof [EQUIVALENT, 0 ms] (75) YES (76) QDP (77) UsableRulesProof [EQUIVALENT, 0 ms] (78) QDP (79) QDPSizeChangeProof [EQUIVALENT, 0 ms] (80) YES (81) QDP (82) UsableRulesProof [EQUIVALENT, 0 ms] (83) QDP (84) QDPSizeChangeProof [EQUIVALENT, 0 ms] (85) YES (86) QDP (87) MRRProof [EQUIVALENT, 51 ms] (88) QDP (89) MRRProof [EQUIVALENT, 34 ms] (90) QDP (91) QDPOrderProof [EQUIVALENT, 275 ms] (92) QDP (93) QDPOrderProof [EQUIVALENT, 232 ms] (94) QDP (95) QDPOrderProof [EQUIVALENT, 225 ms] (96) QDP (97) QDPOrderProof [EQUIVALENT, 304 ms] (98) QDP (99) QDPOrderProof [EQUIVALENT, 192 ms] (100) QDP (101) QDPOrderProof [EQUIVALENT, 320 ms] (102) QDP (103) QDPOrderProof [EQUIVALENT, 264 ms] (104) QDP (105) QDPOrderProof [EQUIVALENT, 185 ms] (106) QDP (107) QDPOrderProof [EQUIVALENT, 189 ms] (108) QDP (109) QDPOrderProof [EQUIVALENT, 239 ms] (110) QDP (111) QDPOrderProof [EQUIVALENT, 161 ms] (112) QDP (113) QDPOrderProof [EQUIVALENT, 130 ms] (114) QDP (115) QDPOrderProof [EQUIVALENT, 412 ms] (116) QDP (117) QDPOrderProof [EQUIVALENT, 110 ms] (118) QDP (119) QDPOrderProof [EQUIVALENT, 204 ms] (120) QDP (121) DependencyGraphProof [EQUIVALENT, 0 ms] (122) QDP (123) UsableRulesProof [EQUIVALENT, 0 ms] (124) QDP (125) QDPSizeChangeProof [EQUIVALENT, 0 ms] (126) 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)) = 2 + x_1 POL(U41(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 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)) = 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 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) active(isNatIList(zeros)) -> mark(tt) ---------------------------------------- (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(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(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. ---------------------------------------- (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 + 2*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 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 POL(U62(x_1, x_2)) = 2 + 2*x_1 + 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)) = x_1 POL(length(x_1)) = 2 + x_1 POL(mark(x_1)) = x_1 POL(nil) = 1 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))) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) ---------------------------------------- (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(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(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)) = 1 + x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 1 + 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)) = 1 + x_1 POL(isNatIList(x_1)) = 2 + 2*x_1 POL(isNatList(x_1)) = 1 + x_1 POL(length(x_1)) = 2 + 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 1 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) active(isNatIList(V)) -> mark(U31(isNatList(V))) ---------------------------------------- (6) 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. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) 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. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 36 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) 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. ---------------------------------------- (12) 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. ---------------------------------------- (13) 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. ---------------------------------------- (14) 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 ---------------------------------------- (15) YES ---------------------------------------- (16) 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. ---------------------------------------- (17) 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. ---------------------------------------- (18) 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. ---------------------------------------- (19) 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 ---------------------------------------- (20) YES ---------------------------------------- (21) 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. ---------------------------------------- (22) 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. ---------------------------------------- (23) 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. ---------------------------------------- (24) 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 ---------------------------------------- (25) YES ---------------------------------------- (26) 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. ---------------------------------------- (27) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) 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 ---------------------------------------- (30) YES ---------------------------------------- (31) 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. ---------------------------------------- (32) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) 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 ---------------------------------------- (35) YES ---------------------------------------- (36) 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. ---------------------------------------- (37) 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. ---------------------------------------- (38) 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. ---------------------------------------- (39) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (40) YES ---------------------------------------- (41) 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. ---------------------------------------- (42) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) 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 ---------------------------------------- (45) YES ---------------------------------------- (46) 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. ---------------------------------------- (47) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) 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 ---------------------------------------- (50) YES ---------------------------------------- (51) 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. ---------------------------------------- (52) 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. ---------------------------------------- (53) 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. ---------------------------------------- (54) 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 ---------------------------------------- (55) YES ---------------------------------------- (56) 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. ---------------------------------------- (57) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) 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 ---------------------------------------- (60) YES ---------------------------------------- (61) 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. ---------------------------------------- (62) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) 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 ---------------------------------------- (65) YES ---------------------------------------- (66) 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. ---------------------------------------- (67) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) 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 ---------------------------------------- (70) YES ---------------------------------------- (71) 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. ---------------------------------------- (72) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (75) YES ---------------------------------------- (76) 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. ---------------------------------------- (77) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) 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 ---------------------------------------- (80) YES ---------------------------------------- (81) 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. ---------------------------------------- (82) 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. ---------------------------------------- (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) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) 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 ---------------------------------------- (85) YES ---------------------------------------- (86) 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. ---------------------------------------- (87) 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) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U11(x_1)) = 1 + x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 1 + x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U62(x_1, x_2)) = 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)) = 1 + x_1 POL(isNatIList(x_1)) = 2 + 2*x_1 POL(isNatList(x_1)) = 1 + x_1 POL(length(x_1)) = 2 + 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(tt) = 1 POL(zeros) = 0 ---------------------------------------- (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(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))) 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(U61(X1, X2, X3)) -> MARK(X1) MARK(U62(X1, X2)) -> MARK(X1) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U11(x_1)) = 1 + x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = 1 + x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 POL(U62(x_1, x_2)) = 2 + x_1 + 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 + x_1 POL(isNatList(x_1)) = 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 ---------------------------------------- (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(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. ---------------------------------------- (91) 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) ) = x_1 + 1 POL( U11_1(x_1) ) = 1 POL( U21_1(x_1) ) = x_1 POL( U31_1(x_1) ) = max{0, -1} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U42_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = 2 POL( U62_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = x_1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 2 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) ---------------------------------------- (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(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. ---------------------------------------- (93) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U51(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 + 1 POL( U11_1(x_1) ) = 2 POL( U21_1(x_1) ) = x_1 POL( U31_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = x_1 + 2 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = 0 POL( U62_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) ---------------------------------------- (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(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(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. ---------------------------------------- (95) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U31(X)) -> ACTIVE(U31(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( U11_1(x_1) ) = 2x_1 + 1 POL( U21_1(x_1) ) = x_1 POL( U31_1(x_1) ) = 1 POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = 2x_1 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = max{0, -1} POL( U62_2(x_1, x_2) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, -1} POL( length_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 1 POL( 0 ) = 0 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( nil ) = 2 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) ---------------------------------------- (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))) 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) 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(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. ACTIVE(U61(tt, L, N)) -> MARK(U62(isNat(N), L)) 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) ) = x_1 + 1 POL( U21_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = 0 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = 2x_1 POL( U62_2(x_1, x_2) ) = 1 POL( cons_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 2 POL( tt ) = 2 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 2x_1 POL( U31_1(x_1) ) = 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) 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) ---------------------------------------- (98) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) 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(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. ---------------------------------------- (99) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(s(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( U21_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = max{0, -1} POL( U42_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = 0 POL( U52_1(x_1) ) = 2x_1 POL( U61_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U62_2(x_1, x_2) ) = 2x_1 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = x_1 + 1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 2 POL( tt ) = 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 1 POL( U31_1(x_1) ) = 2x_1 + 1 POL( nil ) = 2 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (100) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) 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(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(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. ACTIVE(length(cons(N, L))) -> MARK(U61(isNatList(L), L, N)) 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) ) = x_1 + 2 POL( U21_1(x_1) ) = 2x_1 POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = 2x_1 POL( U52_1(x_1) ) = 2x_1 POL( U61_3(x_1, ..., x_3) ) = 0 POL( U62_2(x_1, x_2) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = x_1 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = 0 POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) length(active(X)) -> length(X) length(mark(X)) -> length(X) active(U21(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (102) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> MARK(X) 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)) 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))) 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) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U21(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 + 2 POL( U21_1(x_1) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = x_2 + 2 POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = 2x_2 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = x_1 + 2x_2 POL( U62_2(x_1, x_2) ) = 2x_2 + 2 POL( cons_2(x_1, x_2) ) = 2x_2 POL( s_1(x_1) ) = x_1 + 1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 1 POL( tt ) = 2 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( U11_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 2x_1 POL( U31_1(x_1) ) = 2 POL( length_1(x_1) ) = 2x_1 POL( nil ) = 1 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) 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) length(active(X)) -> length(X) length(mark(X)) -> length(X) ---------------------------------------- (104) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) 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)) 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))) 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(U62(X1, X2)) -> ACTIVE(U62(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(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 + 2 POL( U21_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = 0 POL( U52_1(x_1) ) = 2x_1 POL( U61_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 POL( U62_2(x_1, x_2) ) = 2x_2 + 2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( s_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = x_1 + 2 POL( U31_1(x_1) ) = 2 POL( length_1(x_1) ) = x_1 POL( nil ) = 2 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) 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) length(active(X)) -> length(X) length(mark(X)) -> length(X) ---------------------------------------- (106) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) 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)) 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)) 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. ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2} POL( U21_1(x_1) ) = 1 POL( U41_2(x_1, x_2) ) = 1 POL( U42_1(x_1) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U52_1(x_1) ) = 2 POL( U61_3(x_1, ..., x_3) ) = 0 POL( cons_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 1 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = 0 POL( U62_2(x_1, x_2) ) = 2x_2 + 2 POL( isNat_1(x_1) ) = 2x_1 + 1 POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( s_1(x_1) ) = 1 POL( length_1(x_1) ) = 2x_1 + 1 POL( nil ) = 0 POL( MARK_1(x_1) ) = 0 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) 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) 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) ---------------------------------------- (108) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) 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)) 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. 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) ) = x_1 + 1 POL( U21_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( U61_3(x_1, ..., x_3) ) = 1 POL( cons_2(x_1, x_2) ) = max{0, -1} POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -1} POL( isNatList_1(x_1) ) = 0 POL( U62_2(x_1, x_2) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( length_1(x_1) ) = 2x_1 + 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) 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) ---------------------------------------- (110) 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))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) 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. ---------------------------------------- (111) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U21(X)) -> ACTIVE(U21(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 + 1 POL( U21_1(x_1) ) = x_1 + 1 POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = x_1 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U52_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = 0 POL( U61_3(x_1, ..., x_3) ) = 2x_1 POL( U62_2(x_1, x_2) ) = 2 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( U31_1(x_1) ) = 2 POL( s_1(x_1) ) = 2x_1 + 2 POL( length_1(x_1) ) = 0 POL( nil ) = 1 POL( MARK_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, 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) 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) 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) 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) 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) 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) 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(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) 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. ---------------------------------------- (113) 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)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 0 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U42_1(x_1) ) = x_1 + 1 POL( U51_2(x_1, x_2) ) = 1 POL( U52_1(x_1) ) = 2x_1 + 1 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 2 POL( 0 ) = 2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U61_3(x_1, ..., x_3) ) = max{0, -2} POL( U62_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 POL( U31_1(x_1) ) = max{0, x_1 - 1} POL( s_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -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: 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) 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) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) 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. ---------------------------------------- (115) 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: Combined order from the following AFS and order. ACTIVE(x1) = x1 U41(x1, x2) = U41 tt = tt MARK(x1) = MARK U42(x1) = U42 isNatIList(x1) = isNatIList U51(x1, x2) = U51 U52(x1) = U52 isNatList(x1) = isNatList mark(x1) = mark cons(x1, x2) = cons(x1, x2) isNat(x1) = isNat(x1) active(x1) = active zeros = zeros 0 = 0 U11(x1) = U11(x1) U21(x1) = U21(x1) U61(x1, x2, x3) = U61(x2, x3) U62(x1, x2) = U62(x1, x2) U31(x1) = U31(x1) s(x1) = s(x1) length(x1) = length(x1) nil = nil Recursive path order with status [RPO]. Quasi-Precedence: [mark, active, zeros, U11_1] > [U41, tt, MARK, isNatIList, U51, U52, isNatList, 0] > U42 > [U62_2, s_1] [mark, active, zeros, U11_1] > [U41, tt, MARK, isNatIList, U51, U52, isNatList, 0] > [isNat_1, U61_2, length_1] > U21_1 > [U62_2, s_1] [mark, active, zeros, U11_1] > cons_2 > [U62_2, s_1] [mark, active, zeros, U11_1] > U31_1 > [U62_2, s_1] [mark, active, zeros, U11_1] > nil > [U62_2, s_1] Status: U41: [] tt: multiset status MARK: [] U42: multiset status isNatIList: [] U51: [] U52: [] isNatList: [] mark: multiset status cons_2: multiset status isNat_1: multiset status active: multiset status zeros: multiset status 0: multiset status U11_1: multiset status U21_1: multiset status U61_2: multiset status U62_2: multiset status U31_1: multiset status s_1: multiset status length_1: multiset status nil: multiset status 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) 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) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) MARK(U52(X)) -> ACTIVE(U52(mark(X))) 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. ---------------------------------------- (117) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U52(X)) -> ACTIVE(U52(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) ) = 0 POL( isNatIList_1(x_1) ) = 1 POL( active_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = 1 POL( U51_2(x_1, x_2) ) = 1 POL( U52_1(x_1) ) = max{0, -1} POL( isNatList_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( zeros ) = 1 POL( 0 ) = 0 POL( tt ) = 0 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U61_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U62_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( isNat_1(x_1) ) = x_1 POL( U31_1(x_1) ) = max{0, 2x_1 - 2} POL( s_1(x_1) ) = x_1 + 1 POL( length_1(x_1) ) = 2x_1 POL( nil ) = 2 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) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(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) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) 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)) 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. ---------------------------------------- (119) 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)) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( MARK_1(x_1) ) = 2x_1 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = x_2 + 2 POL( U51_2(x_1, x_2) ) = x_2 + 2 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( zeros ) = 0 POL( 0 ) = 1 POL( tt ) = 2 POL( U11_1(x_1) ) = 0 POL( U21_1(x_1) ) = 0 POL( U61_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( U62_2(x_1, x_2) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, -2} POL( s_1(x_1) ) = 2 POL( length_1(x_1) ) = 2 POL( nil ) = 2 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) 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) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U42(X)) -> MARK(X) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U52(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. ---------------------------------------- (121) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (122) 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. ---------------------------------------- (123) 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. ---------------------------------------- (124) 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. ---------------------------------------- (125) 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 ---------------------------------------- (126) YES