(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
Q is empty.