YES proof of Transformed_CSR_04_LengthOfFiniteLists_complete_GM.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 33 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 210 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) QDP (13) QDPOrderProof [EQUIVALENT, 93 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) AND (17) QDP (18) QDPOrderProof [EQUIVALENT, 104 ms] (19) QDP (20) DependencyGraphProof [EQUIVALENT, 0 ms] (21) TRUE (22) QDP (23) QDPOrderProof [EQUIVALENT, 103 ms] (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) DependencyGraphProof [EQUIVALENT, 0 ms] (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, V1) -> A__U12(a__isNatList(V1)) A__U11(tt, V1) -> A__ISNATLIST(V1) A__U21(tt, V1) -> A__U22(a__isNat(V1)) A__U21(tt, V1) -> A__ISNAT(V1) A__U31(tt, V) -> A__U32(a__isNatList(V)) A__U31(tt, V) -> A__ISNATLIST(V) A__U41(tt, V1, V2) -> A__U42(a__isNat(V1), V2) A__U41(tt, V1, V2) -> A__ISNAT(V1) A__U42(tt, V2) -> A__U43(a__isNatIList(V2)) A__U42(tt, V2) -> A__ISNATILIST(V2) A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U51(tt, V1, V2) -> A__ISNAT(V1) A__U52(tt, V2) -> A__U53(a__isNatList(V2)) A__U52(tt, V2) -> A__ISNATLIST(V2) A__U61(tt, L) -> A__LENGTH(mark(L)) A__U61(tt, L) -> MARK(L) A__AND(tt, X) -> MARK(X) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__ISNATILIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) A__LENGTH(cons(N, L)) -> A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) A__LENGTH(cons(N, L)) -> A__AND(a__isNatList(L), isNatIListKind(L)) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(zeros) -> A__ZEROS MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> A__U12(mark(X)) MARK(U12(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> A__U22(mark(X)) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> A__U32(mark(X)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2)) -> A__U42(mark(X1), X2) MARK(U42(X1, X2)) -> MARK(X1) MARK(U43(X)) -> A__U43(mark(X)) MARK(U43(X)) -> MARK(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> A__U53(mark(X)) MARK(U53(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) MARK(U61(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, V1) -> A__ISNATLIST(V1) A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) A__U31(tt, V) -> A__ISNATLIST(V) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) A__U41(tt, V1, V2) -> A__U42(a__isNat(V1), V2) A__U42(tt, V2) -> A__ISNATILIST(V2) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U41(tt, V1, V2) -> A__ISNAT(V1) A__ISNATILIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2)) -> A__U42(mark(X1), X2) MARK(U42(X1, X2)) -> MARK(X1) MARK(U43(X)) -> MARK(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) A__U51(tt, V1, V2) -> A__ISNAT(V1) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) A__U61(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) A__U61(tt, L) -> MARK(L) MARK(U61(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) A__LENGTH(cons(N, L)) -> A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) A__LENGTH(cons(N, L)) -> A__AND(a__isNatList(L), isNatIListKind(L)) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(length(X)) -> MARK(X) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U31(tt, V) -> A__ISNATLIST(V) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__U41(tt, V1, V2) -> A__ISNAT(V1) A__ISNATILIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = 2x_2 POL( A__LENGTH_1(x_1) ) = x_1 POL( A__U11_2(x_1, x_2) ) = max{0, -2} POL( A__U21_2(x_1, x_2) ) = max{0, -1} POL( A__U31_2(x_1, x_2) ) = x_2 + 2 POL( A__U41_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( A__U42_2(x_1, x_2) ) = 2x_2 + 2 POL( A__U51_3(x_1, ..., x_3) ) = 0 POL( A__U52_2(x_1, x_2) ) = max{0, -2} POL( A__U61_2(x_1, x_2) ) = 2x_2 POL( a__isNatKind_1(x_1) ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( mark_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( a__and_2(x_1, x_2) ) = x_1 + 2x_2 POL( isNatIListKind_1(x_1) ) = 0 POL( a__isNatIListKind_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( isNatKind_1(x_1) ) = 0 POL( length_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( a__isNat_1(x_1) ) = 0 POL( a__U11_2(x_1, x_2) ) = x_1 POL( a__U21_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U11_2(x_1, x_2) ) = x_1 POL( U12_1(x_1) ) = x_1 POL( a__U12_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( U21_2(x_1, x_2) ) = x_1 POL( U22_1(x_1) ) = x_1 POL( a__U22_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( a__U31_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( U32_1(x_1) ) = x_1 + 1 POL( a__U32_1(x_1) ) = x_1 + 1 POL( U41_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 1 POL( a__U41_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 1 POL( U42_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( a__U42_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( U43_1(x_1) ) = x_1 POL( a__U43_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( a__isNatIList_1(x_1) ) = 2x_1 + 1 POL( U51_3(x_1, ..., x_3) ) = x_1 POL( a__U51_3(x_1, ..., x_3) ) = x_1 POL( U52_2(x_1, x_2) ) = 2x_1 POL( a__U52_2(x_1, x_2) ) = 2x_1 POL( U53_1(x_1) ) = 2x_1 POL( a__U53_1(x_1) ) = 2x_1 POL( U61_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__U61_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__length_1(x_1) ) = x_1 POL( nil ) = 2 POL( A__ISNATLIST_1(x_1) ) = 0 POL( MARK_1(x_1) ) = 2x_1 POL( A__ISNATKIND_1(x_1) ) = 0 POL( A__ISNATILISTKIND_1(x_1) ) = 0 POL( A__ISNAT_1(x_1) ) = 0 POL( A__ISNATILIST_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a__isNatKind(0) -> tt mark(and(X1, X2)) -> a__and(mark(X1), X2) a__and(tt, X) -> mark(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) mark(isNatKind(X)) -> a__isNatKind(X) a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__and(X1, X2) -> and(X1, X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatList(X) -> isNatList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(X1, X2) -> U52(X1, X2) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__U31(tt, V) -> a__U32(a__isNatList(V)) a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) a__U61(tt, L) -> s(a__length(mark(L))) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: A__U11(tt, V1) -> A__ISNATLIST(V1) A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) A__U41(tt, V1, V2) -> A__U42(a__isNat(V1), V2) A__U42(tt, V2) -> A__ISNATILIST(V2) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) MARK(U42(X1, X2)) -> A__U42(mark(X1), X2) MARK(U43(X)) -> MARK(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) A__U51(tt, V1, V2) -> A__ISNAT(V1) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) A__U61(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) A__U61(tt, L) -> MARK(L) MARK(U61(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) A__LENGTH(cons(N, L)) -> A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) A__LENGTH(cons(N, L)) -> A__AND(a__isNatList(L), isNatIListKind(L)) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(length(X)) -> MARK(X) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: A__U42(tt, V2) -> A__ISNATILIST(V2) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U41(tt, V1, V2) -> A__U42(a__isNat(V1), V2) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) 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: *A__ISNATILIST(cons(V1, V2)) -> A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__U41(tt, V1, V2) -> A__U42(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 *A__U42(tt, V2) -> A__ISNATILIST(V2) The graph contains the following edges 2 >= 1 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) A__U11(tt, V1) -> A__ISNATLIST(V1) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U43(X)) -> MARK(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) A__U51(tt, V1, V2) -> A__ISNAT(V1) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) A__U61(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) A__U61(tt, L) -> MARK(L) MARK(U61(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) A__LENGTH(cons(N, L)) -> A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) A__LENGTH(cons(N, L)) -> A__AND(a__isNatList(L), isNatIListKind(L)) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(length(X)) -> MARK(X) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U11(tt, V1) -> A__ISNATLIST(V1) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__U61(tt, L) -> MARK(L) MARK(U61(X1, X2)) -> MARK(X1) A__LENGTH(cons(N, L)) -> A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) A__LENGTH(cons(N, L)) -> A__AND(a__isNatList(L), isNatIListKind(L)) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) MARK(length(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = x_2 POL( A__LENGTH_1(x_1) ) = x_1 + 2 POL( A__U11_2(x_1, x_2) ) = x_2 + 2 POL( A__U21_2(x_1, x_2) ) = 2x_2 POL( A__U51_3(x_1, ..., x_3) ) = 2x_2 + x_3 POL( A__U52_2(x_1, x_2) ) = x_1 + x_2 POL( A__U61_2(x_1, x_2) ) = x_2 + 2 POL( a__isNatKind_1(x_1) ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( a__and_2(x_1, x_2) ) = x_1 + x_2 POL( isNatIListKind_1(x_1) ) = 0 POL( a__isNatIListKind_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNatKind_1(x_1) ) = 0 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 POL( a__isNat_1(x_1) ) = 2x_1 POL( a__U11_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( a__U21_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNat_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U11_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( U12_1(x_1) ) = x_1 + 2 POL( a__U12_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = x_1 POL( a__isNatList_1(x_1) ) = x_1 POL( U21_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U22_1(x_1) ) = x_1 POL( a__U22_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = 0 POL( a__U31_2(x_1, x_2) ) = max{0, -1} POL( U32_1(x_1) ) = 0 POL( a__U32_1(x_1) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( a__U41_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U42_2(x_1, x_2) ) = 2x_2 + 2 POL( a__U42_2(x_1, x_2) ) = 2x_2 + 2 POL( U43_1(x_1) ) = x_1 POL( a__U43_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( a__isNatIList_1(x_1) ) = x_1 + 2 POL( U51_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 POL( a__U51_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 POL( U52_2(x_1, x_2) ) = x_1 + x_2 POL( a__U52_2(x_1, x_2) ) = x_1 + x_2 POL( U53_1(x_1) ) = x_1 POL( a__U53_1(x_1) ) = x_1 POL( U61_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( a__U61_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( a__length_1(x_1) ) = 2x_1 + 2 POL( nil ) = 0 POL( A__ISNATLIST_1(x_1) ) = x_1 POL( MARK_1(x_1) ) = x_1 POL( A__ISNATKIND_1(x_1) ) = 0 POL( A__ISNATILISTKIND_1(x_1) ) = 0 POL( A__ISNAT_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a__isNatKind(0) -> tt mark(and(X1, X2)) -> a__and(mark(X1), X2) a__and(tt, X) -> mark(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) mark(isNatKind(X)) -> a__isNatKind(X) a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatKind(X) -> isNatKind(X) a__and(X1, X2) -> and(X1, X2) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatList(X) -> isNatList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(X1, X2) -> U52(X1, X2) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__U31(tt, V) -> a__U32(a__isNatList(V)) a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) a__U61(tt, L) -> s(a__length(mark(L))) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U43(X)) -> MARK(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) A__U51(tt, V1, V2) -> A__ISNAT(V1) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) A__U61(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (16) Complex Obligation (AND) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) A__U61(tt, L) -> A__LENGTH(mark(L)) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__LENGTH(cons(N, L)) -> A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = x_1 + 2 POL( A__U61_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( a__isNatList_1(x_1) ) = x_1 + 1 POL( nil ) = 2 POL( tt ) = 2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( a__U51_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 POL( a__and_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__isNatKind_1(x_1) ) = 2 POL( isNatIListKind_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = x_1 POL( mark_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__isNatIListKind_1(x_1) ) = 2 POL( isNatKind_1(x_1) ) = 0 POL( length_1(x_1) ) = 1 POL( s_1(x_1) ) = 2 POL( zeros ) = 0 POL( a__zeros ) = 2 POL( U11_2(x_1, x_2) ) = 0 POL( a__U11_2(x_1, x_2) ) = 2 POL( U12_1(x_1) ) = 2 POL( a__U12_1(x_1) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( a__U21_2(x_1, x_2) ) = 2 POL( U22_1(x_1) ) = 1 POL( a__U22_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( a__isNat_1(x_1) ) = 2 POL( U31_2(x_1, x_2) ) = x_1 + x_2 POL( a__U31_2(x_1, x_2) ) = x_1 + x_2 POL( U32_1(x_1) ) = x_1 + 1 POL( a__U32_1(x_1) ) = x_1 + 1 POL( U41_3(x_1, ..., x_3) ) = x_2 POL( a__U41_3(x_1, ..., x_3) ) = x_2 + 2 POL( U42_2(x_1, x_2) ) = x_1 POL( a__U42_2(x_1, x_2) ) = x_1 POL( U43_1(x_1) ) = 0 POL( a__U43_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = x_1 POL( a__isNatIList_1(x_1) ) = x_1 + 2 POL( U51_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 POL( U52_2(x_1, x_2) ) = x_2 + 1 POL( a__U52_2(x_1, x_2) ) = x_2 + 1 POL( U53_1(x_1) ) = x_1 POL( a__U53_1(x_1) ) = x_1 POL( U61_2(x_1, x_2) ) = 1 POL( a__U61_2(x_1, x_2) ) = 2 POL( a__length_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatList(X) -> isNatList(X) mark(and(X1, X2)) -> a__and(mark(X1), X2) a__and(tt, X) -> mark(X) mark(isNatIListKind(X)) -> a__isNatIListKind(X) a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) mark(isNatKind(X)) -> a__isNatKind(X) a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__and(X1, X2) -> and(X1, X2) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__U11(X1, X2) -> U11(X1, X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__isNatKind(0) -> tt a__isNatKind(X) -> isNatKind(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__U52(X1, X2) -> U52(X1, X2) a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(X) -> isNatIListKind(X) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt) -> tt a__U22(X) -> U22(X) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt) -> tt a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(tt) -> tt a__U43(X) -> U43(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__U31(tt, V) -> a__U32(a__isNatList(V)) a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U53(tt) -> tt a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) a__U61(tt, L) -> s(a__length(mark(L))) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: A__U61(tt, L) -> A__LENGTH(mark(L)) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (21) TRUE ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__U51(tt, V1, V2) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U43(X)) -> MARK(X) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U52(X1, X2)) -> MARK(X1) MARK(U53(X)) -> MARK(X) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U51(tt, V1, V2) -> A__U52(a__isNat(V1), V2) A__U51(tt, V1, V2) -> A__ISNAT(V1) A__U21(tt, V1) -> A__ISNAT(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) MARK(isNatList(X)) -> A__ISNATLIST(X) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X)) -> MARK(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(U43(X)) -> MARK(X) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> A__AND(mark(X1), X2) MARK(and(X1, X2)) -> MARK(X1) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = x_2 POL( A__U21_2(x_1, x_2) ) = x_2 + 1 POL( A__U51_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( A__U52_2(x_1, x_2) ) = x_2 + 1 POL( a__isNat_1(x_1) ) = 2x_1 + 2 POL( 0 ) = 0 POL( tt ) = 0 POL( length_1(x_1) ) = 2x_1 + 1 POL( a__U11_2(x_1, x_2) ) = x_2 + 2 POL( a__isNatIListKind_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 + 1 POL( a__U21_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__isNatKind_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = x_1 + 1 POL( and_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( a__and_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( isNatIListKind_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( isNatKind_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( a__zeros ) = 1 POL( U11_2(x_1, x_2) ) = 2x_1 + x_2 + 1 POL( U12_1(x_1) ) = x_1 + 1 POL( a__U12_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( a__isNatList_1(x_1) ) = 2x_1 + 2 POL( U21_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( U22_1(x_1) ) = x_1 + 2 POL( a__U22_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = 1 POL( a__U31_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U32_1(x_1) ) = 2x_1 + 2 POL( a__U32_1(x_1) ) = max{0, -2} POL( U41_3(x_1, ..., x_3) ) = x_1 + x_3 + 2 POL( a__U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 POL( U42_2(x_1, x_2) ) = x_1 POL( a__U42_2(x_1, x_2) ) = 2x_2 + 2 POL( U43_1(x_1) ) = 2x_1 + 2 POL( a__U43_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2x_1 POL( a__isNatIList_1(x_1) ) = x_1 + 1 POL( U51_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2 POL( a__U51_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U52_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( a__U52_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( U53_1(x_1) ) = x_1 POL( a__U53_1(x_1) ) = x_1 + 2 POL( U61_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( a__U61_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( a__length_1(x_1) ) = 2x_1 + 2 POL( nil ) = 0 POL( A__ISNATLIST_1(x_1) ) = x_1 + 1 POL( A__ISNAT_1(x_1) ) = x_1 POL( A__ISNATKIND_1(x_1) ) = x_1 + 1 POL( A__ISNATILISTKIND_1(x_1) ) = 2x_1 + 2 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A__U52(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(cons(V1, V2)) -> A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__AND(tt, X) -> MARK(X) A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) MARK(U51(X1, X2, X3)) -> A__U51(mark(X1), X2, X3) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) MARK(U53(X)) -> MARK(X) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) A__AND(tt, X) -> MARK(X) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) A__U52(tt, V2) -> A__ISNATLIST(V2) MARK(U53(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule A__AND(tt, X) -> MARK(X) we obtained the following new rules [LPAR04]: (A__AND(tt, isNatIListKind(y_2)) -> MARK(isNatIListKind(y_2)),A__AND(tt, isNatIListKind(y_2)) -> MARK(isNatIListKind(y_2))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNATLIST(cons(V1, V2)) -> A__AND(a__isNatKind(V1), isNatIListKind(V2)) MARK(U52(X1, X2)) -> A__U52(mark(X1), X2) A__U52(tt, V2) -> A__ISNATLIST(V2) MARK(U53(X)) -> MARK(X) A__AND(tt, isNatIListKind(y_2)) -> MARK(isNatIListKind(y_2)) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U53(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, V1) -> a__U12(a__isNatList(V1)) a__U12(tt) -> tt a__U21(tt, V1) -> a__U22(a__isNat(V1)) a__U22(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatList(V)) a__U32(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) a__U43(tt) -> tt a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) a__U52(tt, V2) -> a__U53(a__isNatList(V2)) a__U53(tt) -> tt a__U61(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2)) -> a__U42(mark(X1), X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X) -> U32(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2) -> U42(X1, X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1, X2, X3) -> U51(X1, X2, X3) a__U52(X1, X2) -> U52(X1, X2) a__U53(X) -> U53(X) a__U61(X1, X2) -> U61(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U53(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MARK(U53(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (34) YES