(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.