(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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
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__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__ISNATILISTKIND(V1)
A__U12(tt, V1) → A__U13(a__isNatList(V1))
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNATKIND(V1)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U22(tt, V1) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U31(tt, V) → A__ISNATILISTKIND(V)
A__U32(tt, V) → A__U33(a__isNatList(V))
A__U32(tt, V) → A__ISNATLIST(V)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U45(tt, V2) → A__U46(a__isNatIList(V2))
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V2) → A__U52(a__isNatIListKind(V2))
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__U81(tt, V1, V2) → A__U82(a__isNatKind(V1), V1, V2)
A__U81(tt, V1, V2) → A__ISNATKIND(V1)
A__U82(tt, V1, V2) → A__U83(a__isNatIListKind(V2), V1, V2)
A__U82(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U83(tt, V1, V2) → A__U84(a__isNatIListKind(V2), V1, V2)
A__U83(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U84(tt, V1, V2) → A__U85(a__isNat(V1), V2)
A__U84(tt, V1, V2) → A__ISNAT(V1)
A__U85(tt, V2) → A__U86(a__isNatList(V2))
A__U85(tt, V2) → A__ISNATLIST(V2)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U91(tt, L, N) → A__ISNATILISTKIND(L)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U92(tt, L, N) → A__ISNAT(N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U93(tt, L, N) → A__ISNATKIND(N)
A__U94(tt, L) → A__LENGTH(mark(L))
A__U94(tt, L) → MARK(L)
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__isNatKind(V1), V1, V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__U61(a__isNatIListKind(V1))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__U71(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__U81(a__isNatKind(V1), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
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(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(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(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(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(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
MARK(U33(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, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U46(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → A__U71(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → A__U81(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → A__U82(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → A__U83(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → A__U84(mark(X1), X2, X3)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → A__U85(mark(X1), X2)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → A__U86(mark(X))
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 61 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The TRS R consists of the following rules:
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
a__U71(tt) → tt
a__U71(X) → U71(X)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) QDPSizeChangeProof (EQUIVALENT transformation)
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__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
The graph contains the following edges 1 > 2
- A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1
- A__U51(tt, V2) → A__ISNATILISTKIND(V2)
The graph contains the following edges 2 >= 1
- A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
The graph contains the following edges 1 > 1
- A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U12(tt, V1) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__U81(a__isNatKind(V1), V1, V2)
A__U81(tt, V1, V2) → A__U82(a__isNatKind(V1), V1, V2)
A__U82(tt, V1, V2) → A__U83(a__isNatIListKind(V2), V1, V2)
A__U83(tt, V1, V2) → A__U84(a__isNatIListKind(V2), V1, V2)
A__U84(tt, V1, V2) → A__U85(a__isNat(V1), V2)
A__U85(tt, V2) → A__ISNATLIST(V2)
A__U84(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPSizeChangeProof (EQUIVALENT transformation)
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__ISNATLIST(cons(V1, V2)) → A__U81(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
- A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
The graph contains the following edges 2 >= 2
- A__U81(tt, V1, V2) → A__U82(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
- A__U82(tt, V1, V2) → A__U83(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
- A__U83(tt, V1, V2) → A__U84(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
- A__U85(tt, V2) → A__ISNATLIST(V2)
The graph contains the following edges 2 >= 1
- A__U12(tt, V1) → A__ISNATLIST(V1)
The graph contains the following edges 2 >= 1
- A__U84(tt, V1, V2) → A__U85(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
- A__U84(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
- A__U22(tt, V1) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
- A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
The graph contains the following edges 2 >= 2
- A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
The graph contains the following edges 1 > 2
- A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
The graph contains the following edges 1 > 2
(12) TRUE
(13) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) QDPSizeChangeProof (EQUIVALENT transformation)
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__U45(tt, V2) → A__ISNATILIST(V2)
The graph contains the following edges 2 >= 1
- A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
- A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
- A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
- A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
- A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
(15) TRUE
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U94(tt, L) → MARK(L)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__LENGTH(x1)) = x1
POL(A__U91(x1, x2, x3)) = x2 + x3
POL(A__U92(x1, x2, x3)) = x2 + x3
POL(A__U93(x1, x2, x3)) = x2 + x3
POL(A__U94(x1, x2)) = x2
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U12(x1, x2)) = x1
POL(U13(x1)) = x1
POL(U21(x1, x2)) = x1
POL(U22(x1, x2)) = x1
POL(U23(x1)) = x1
POL(U31(x1, x2)) = 1 + x1
POL(U32(x1, x2)) = 1 + x1
POL(U33(x1)) = 1 + x1
POL(U41(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U42(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U43(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U44(x1, x2, x3)) = 1 + x1 + x3
POL(U45(x1, x2)) = 1 + x1 + x2
POL(U46(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1)) = x1
POL(U71(x1)) = x1
POL(U81(x1, x2, x3)) = x1
POL(U82(x1, x2, x3)) = x1
POL(U83(x1, x2, x3)) = x1
POL(U84(x1, x2, x3)) = x1
POL(U85(x1, x2)) = x1
POL(U86(x1)) = x1
POL(U91(x1, x2, x3)) = x1 + x2 + x3
POL(U92(x1, x2, x3)) = x1 + x2 + x3
POL(U93(x1, x2, x3)) = x1 + x2 + x3
POL(U94(x1, x2)) = x1 + x2
POL(a__U11(x1, x2)) = x1
POL(a__U12(x1, x2)) = x1
POL(a__U13(x1)) = x1
POL(a__U21(x1, x2)) = x1
POL(a__U22(x1, x2)) = x1
POL(a__U23(x1)) = x1
POL(a__U31(x1, x2)) = 1 + x1
POL(a__U32(x1, x2)) = 1 + x1
POL(a__U33(x1)) = 1 + x1
POL(a__U41(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U42(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U43(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U44(x1, x2, x3)) = 1 + x1 + x3
POL(a__U45(x1, x2)) = 1 + x1 + x2
POL(a__U46(x1)) = x1
POL(a__U51(x1, x2)) = x1
POL(a__U52(x1)) = x1
POL(a__U61(x1)) = x1
POL(a__U71(x1)) = x1
POL(a__U81(x1, x2, x3)) = x1
POL(a__U82(x1, x2, x3)) = x1
POL(a__U83(x1, x2, x3)) = x1
POL(a__U84(x1, x2, x3)) = x1
POL(a__U85(x1, x2)) = x1
POL(a__U86(x1)) = x1
POL(a__U91(x1, x2, x3)) = x1 + x2 + x3
POL(a__U92(x1, x2, x3)) = x1 + x2 + x3
POL(a__U93(x1, x2, x3)) = x1 + x2 + x3
POL(a__U94(x1, x2)) = x1 + x2
POL(a__isNat(x1)) = 0
POL(a__isNatIList(x1)) = 1 + x1
POL(a__isNatIListKind(x1)) = 0
POL(a__isNatKind(x1)) = 0
POL(a__isNatList(x1)) = 0
POL(a__length(x1)) = x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 1 + x1
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__isNatIListKind(X) → isNatIListKind(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__isNat(X) → isNat(X)
a__U23(X) → U23(X)
a__isNatKind(X) → isNatKind(X)
a__U22(X1, X2) → U22(X1, X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U23(X)) → a__U23(mark(X))
mark(isNatKind(X)) → a__isNatKind(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U61(X)) → a__U61(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U46(X)) → a__U46(mark(X))
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U61(tt) → tt
a__U71(tt) → tt
a__U86(tt) → tt
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U94(tt, L) → s(a__length(mark(L)))
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIList(zeros) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U13(tt) → tt
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__length(X) → length(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U86(X) → U86(X)
a__U85(X1, X2) → U85(X1, X2)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U71(X) → U71(X)
a__U61(X) → U61(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U45(X1, X2) → U45(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U94(tt, L) → MARK(L)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__LENGTH(x1)) = x1
POL(A__U91(x1, x2, x3)) = x2 + x3
POL(A__U92(x1, x2, x3)) = x2 + x3
POL(A__U93(x1, x2, x3)) = x2
POL(A__U94(x1, x2)) = x2
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U12(x1, x2)) = x1
POL(U13(x1)) = x1
POL(U21(x1, x2)) = x1
POL(U22(x1, x2)) = x1
POL(U23(x1)) = x1
POL(U31(x1, x2)) = x1
POL(U32(x1, x2)) = x1
POL(U33(x1)) = x1
POL(U41(x1, x2, x3)) = 1
POL(U42(x1, x2, x3)) = 1
POL(U43(x1, x2, x3)) = 1
POL(U44(x1, x2, x3)) = 1
POL(U45(x1, x2)) = 1
POL(U46(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1)) = x1
POL(U71(x1)) = x1
POL(U81(x1, x2, x3)) = x1
POL(U82(x1, x2, x3)) = x1
POL(U83(x1, x2, x3)) = x1
POL(U84(x1, x2, x3)) = x1
POL(U85(x1, x2)) = x1
POL(U86(x1)) = x1
POL(U91(x1, x2, x3)) = x1 + x2 + x3
POL(U92(x1, x2, x3)) = x1 + x2 + x3
POL(U93(x1, x2, x3)) = x1 + x2 + x3
POL(U94(x1, x2)) = x1 + x2
POL(a__U11(x1, x2)) = x1
POL(a__U12(x1, x2)) = x1
POL(a__U13(x1)) = x1
POL(a__U21(x1, x2)) = x1
POL(a__U22(x1, x2)) = x1
POL(a__U23(x1)) = x1
POL(a__U31(x1, x2)) = x1
POL(a__U32(x1, x2)) = x1
POL(a__U33(x1)) = x1
POL(a__U41(x1, x2, x3)) = 1
POL(a__U42(x1, x2, x3)) = 1
POL(a__U43(x1, x2, x3)) = 1
POL(a__U44(x1, x2, x3)) = 1
POL(a__U45(x1, x2)) = 1
POL(a__U46(x1)) = x1
POL(a__U51(x1, x2)) = x1
POL(a__U52(x1)) = x1
POL(a__U61(x1)) = x1
POL(a__U71(x1)) = x1
POL(a__U81(x1, x2, x3)) = x1
POL(a__U82(x1, x2, x3)) = x1
POL(a__U83(x1, x2, x3)) = x1
POL(a__U84(x1, x2, x3)) = x1
POL(a__U85(x1, x2)) = x1
POL(a__U86(x1)) = x1
POL(a__U91(x1, x2, x3)) = x1 + x2 + x3
POL(a__U92(x1, x2, x3)) = x1 + x2 + x3
POL(a__U93(x1, x2, x3)) = x1 + x2 + x3
POL(a__U94(x1, x2)) = x1 + x2
POL(a__isNat(x1)) = 1
POL(a__isNatIList(x1)) = 1
POL(a__isNatIListKind(x1)) = 1
POL(a__isNatKind(x1)) = 1
POL(a__isNatList(x1)) = 1
POL(a__length(x1)) = 1 + x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatIListKind(x1)) = 1
POL(isNatKind(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 1
POL(s(x1)) = x1
POL(tt) = 1
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__isNatIListKind(X) → isNatIListKind(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__isNat(X) → isNat(X)
a__U23(X) → U23(X)
a__isNatKind(X) → isNatKind(X)
a__U22(X1, X2) → U22(X1, X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U23(X)) → a__U23(mark(X))
mark(isNatKind(X)) → a__isNatKind(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U61(X)) → a__U61(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U46(X)) → a__U46(mark(X))
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U61(tt) → tt
a__U71(tt) → tt
a__U86(tt) → tt
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U94(tt, L) → s(a__length(mark(L)))
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIList(zeros) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U13(tt) → tt
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__length(X) → length(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U86(X) → U86(X)
a__U85(X1, X2) → U85(X1, X2)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U71(X) → U71(X)
a__U61(X) → U61(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U45(X1, X2) → U45(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U94(tt, L) → MARK(L)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → A__U94(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__LENGTH(x1)) = x1
POL(A__U91(x1, x2, x3)) = x2 + x3
POL(A__U92(x1, x2, x3)) = x2 + x3
POL(A__U93(x1, x2, x3)) = x2 + x3
POL(A__U94(x1, x2)) = x2
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U12(x1, x2)) = x1
POL(U13(x1)) = x1
POL(U21(x1, x2)) = x1
POL(U22(x1, x2)) = x1
POL(U23(x1)) = x1
POL(U31(x1, x2)) = 0
POL(U32(x1, x2)) = 0
POL(U33(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U42(x1, x2, x3)) = 0
POL(U43(x1, x2, x3)) = 0
POL(U44(x1, x2, x3)) = 0
POL(U45(x1, x2)) = 0
POL(U46(x1)) = x1
POL(U51(x1, x2)) = x1
POL(U52(x1)) = x1
POL(U61(x1)) = x1
POL(U71(x1)) = x1
POL(U81(x1, x2, x3)) = x1
POL(U82(x1, x2, x3)) = x1
POL(U83(x1, x2, x3)) = x1
POL(U84(x1, x2, x3)) = x1
POL(U85(x1, x2)) = x1
POL(U86(x1)) = x1
POL(U91(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U92(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U93(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U94(x1, x2)) = 1 + x1 + x2
POL(a__U11(x1, x2)) = x1
POL(a__U12(x1, x2)) = x1
POL(a__U13(x1)) = x1
POL(a__U21(x1, x2)) = x1
POL(a__U22(x1, x2)) = x1
POL(a__U23(x1)) = x1
POL(a__U31(x1, x2)) = 0
POL(a__U32(x1, x2)) = 0
POL(a__U33(x1)) = 0
POL(a__U41(x1, x2, x3)) = 0
POL(a__U42(x1, x2, x3)) = 0
POL(a__U43(x1, x2, x3)) = 0
POL(a__U44(x1, x2, x3)) = 0
POL(a__U45(x1, x2)) = 0
POL(a__U46(x1)) = x1
POL(a__U51(x1, x2)) = x1
POL(a__U52(x1)) = x1
POL(a__U61(x1)) = x1
POL(a__U71(x1)) = x1
POL(a__U81(x1, x2, x3)) = x1
POL(a__U82(x1, x2, x3)) = x1
POL(a__U83(x1, x2, x3)) = x1
POL(a__U84(x1, x2, x3)) = x1
POL(a__U85(x1, x2)) = x1
POL(a__U86(x1)) = x1
POL(a__U91(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U92(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U93(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U94(x1, x2)) = 1 + x1 + x2
POL(a__isNat(x1)) = 0
POL(a__isNatIList(x1)) = 0
POL(a__isNatIListKind(x1)) = 0
POL(a__isNatKind(x1)) = 0
POL(a__isNatList(x1)) = 0
POL(a__length(x1)) = 1 + x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [FROCOS05] were oriented:
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__isNatIListKind(X) → isNatIListKind(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__isNat(X) → isNat(X)
a__U23(X) → U23(X)
a__isNatKind(X) → isNatKind(X)
a__U22(X1, X2) → U22(X1, X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U23(X)) → a__U23(mark(X))
mark(isNatKind(X)) → a__isNatKind(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U61(X)) → a__U61(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U46(X)) → a__U46(mark(X))
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U61(tt) → tt
a__U71(tt) → tt
a__U86(tt) → tt
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U94(tt, L) → s(a__length(mark(L)))
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIList(zeros) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U13(tt) → tt
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__length(X) → length(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U86(X) → U86(X)
a__U85(X1, X2) → U85(X1, X2)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U71(X) → U71(X)
a__U61(X) → U61(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U45(X1, X2) → U45(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U94(tt, L) → MARK(L)
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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(23) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(24) Complex Obligation (AND)
(25) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(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__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(26) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(27) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(28) QDPSizeChangeProof (EQUIVALENT transformation)
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(U12(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U11(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U13(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U21(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U22(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U23(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U46(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U51(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U52(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U61(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U71(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U81(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U82(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U83(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U84(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U85(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(U86(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(cons(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
- MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1
(29) TRUE
(30) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(31) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A__U91(tt, L, N) → A__U92(a__isNatIListKind(L), L, N)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(A__U94(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(A__LENGTH(x1)) = | | + | | · | x1 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(A__U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__isNatList(x1)) = | | + | | · | x1 |
POL(A__U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__isNatIListKind(x1)) = | | + | | · | x1 |
POL(A__U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__isNat(x1)) = | | + | | · | x1 |
POL(a__isNatKind(x1)) = | | + | | · | x1 |
POL(U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U91(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U84(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U84(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U85(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U85(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U82(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U82(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U83(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U83(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U81(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U81(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U94(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U94(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__length(x1)) = | | + | | · | x1 |
POL(U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U92(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U93(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U21(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatList(x1)) = | | + | | · | x1 |
POL(isNatIListKind(x1)) = | | + | | · | x1 |
POL(a__U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U12(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U11(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U41(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U32(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U31(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatKind(x1)) = | | + | | · | x1 |
POL(a__U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U22(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U42(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U51(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(isNatIList(x1)) = | | + | | · | x1 |
POL(a__isNatIList(x1)) = | | + | | · | x1 |
POL(U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U44(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(a__U43(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
POL(U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(a__U45(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(U94(X1, X2)) → a__U94(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__isNatIListKind(X) → isNatIListKind(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__zeros → zeros
mark(nil) → nil
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__isNat(X) → isNat(X)
a__U23(X) → U23(X)
a__isNatKind(X) → isNatKind(X)
a__U22(X1, X2) → U22(X1, X2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U23(X)) → a__U23(mark(X))
mark(isNatKind(X)) → a__isNatKind(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U61(X)) → a__U61(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U46(X)) → a__U46(mark(X))
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U61(tt) → tt
a__U71(tt) → tt
a__U86(tt) → tt
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U94(tt, L) → s(a__length(mark(L)))
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIList(zeros) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U13(tt) → tt
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__length(X) → length(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U86(X) → U86(X)
a__U85(X1, X2) → U85(X1, X2)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U71(X) → U71(X)
a__U61(X) → U61(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U45(X1, X2) → U45(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
(32) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U94(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U91(a__isNatList(L), L, N)
A__U92(tt, L, N) → A__U93(a__isNat(N), L, N)
A__U93(tt, L, N) → A__U94(a__isNatKind(N), L)
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt, V1, V2) → a__U82(a__isNatKind(V1), V1, V2)
a__U82(tt, V1, V2) → a__U83(a__isNatIListKind(V2), V1, V2)
a__U83(tt, V1, V2) → a__U84(a__isNatIListKind(V2), V1, V2)
a__U84(tt, V1, V2) → a__U85(a__isNat(V1), V2)
a__U85(tt, V2) → a__U86(a__isNatList(V2))
a__U86(tt) → tt
a__U91(tt, L, N) → a__U92(a__isNatIListKind(L), L, N)
a__U92(tt, L, N) → a__U93(a__isNat(N), L, N)
a__U93(tt, L, N) → a__U94(a__isNatKind(N), L)
a__U94(tt, L) → s(a__length(mark(L)))
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__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U61(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U71(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U81(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U91(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U23(X)) → a__U23(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(U83(X1, X2, X3)) → a__U83(mark(X1), X2, X3)
mark(U84(X1, X2, X3)) → a__U84(mark(X1), X2, X3)
mark(U85(X1, X2)) → a__U85(mark(X1), X2)
mark(U86(X)) → a__U86(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2)) → a__U94(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__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__isNatKind(X) → isNatKind(X)
a__U23(X) → U23(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__U83(X1, X2, X3) → U83(X1, X2, X3)
a__U84(X1, X2, X3) → U84(X1, X2, X3)
a__U85(X1, X2) → U85(X1, X2)
a__U86(X) → U86(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2) → U94(X1, X2)
a__length(X) → length(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(33) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
(34) TRUE