Termination w.r.t. Q proof of /tmp/aprove_benchmark_example

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U32(tt, V2) → A__U33(a__isNat(V2))
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U13(X)) → A__U13(mark(X))
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
A__X(N, 0) → A__U61(a__and(a__isNat(N), isNatKind(N)))
MARK(U33(X)) → A__U33(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U22(X)) → A__U22(mark(X))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U61(X)) → A__U61(mark(X))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U32(tt, V2) → A__U33(a__isNat(V2))
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U13(X)) → A__U13(mark(X))
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
A__X(N, 0) → A__U61(a__and(a__isNat(N), isNatKind(N)))
MARK(U33(X)) → A__U33(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U22(X)) → A__U22(mark(X))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U61(X)) → A__U61(mark(X))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2) at position [0] we obtained the following new rules:

A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U11(tt, x0, y1) → A__U12(isNat(x0), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(x(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U71(X1, X2, X3)) → MARK(X1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
A__U11(tt, x0, y1) → A__U12(isNat(x0), y1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(plus(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(x(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U71(X1, X2, X3)) → MARK(X1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(plus(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(x(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNATKIND(plus(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNATKIND(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(x(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(x(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
A__ISNATKIND(x(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U22(X)) → MARK(X)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U61(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U33(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3) at position [0] we obtained the following new rules:

MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(U11(s(x0), y1, y2)) → A__U11(s(mark(x0)), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U11(0, y1, y2)) → A__U11(0, y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U11(s(x0), y1, y2)) → A__U11(s(mark(x0)), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(0, y1, y2)) → A__U11(0, y1, y2)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U12(X1, X2)) → A__U12(mark(X1), X2) at position [0] we obtained the following new rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(U12(s(x0), y1)) → A__U12(s(mark(x0)), y1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U12(0, y1)) → A__U12(0, y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U12(0, y1)) → A__U12(0, y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U12(s(x0), y1)) → A__U12(s(mark(x0)), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U31(tt, V1, V2) → A__U32(a__isNat(V1), V2) at position [0] we obtained the following new rules:

A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__U31(tt, x0, y1) → A__U32(isNat(x0), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__U71(tt, M, N) → MARK(M)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__U31(tt, x0, y1) → A__U32(isNat(x0), y1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNAT(x(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2)) at position [0] we obtained the following new rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(x(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__ISNAT(x(x0, y1)) → A__AND(isNatKind(x0), isNatKind(y1))
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U21(X1, X2)) → A__U21(mark(X1), X2) at position [0] we obtained the following new rules:

MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U21(s(x0), y1)) → A__U21(s(mark(x0)), y1)
MARK(U21(0, y1)) → A__U21(0, y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(U21(s(x0), y1)) → A__U21(s(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U21(0, y1)) → A__U21(0, y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3) at position [0] we obtained the following new rules:

MARK(U31(s(x0), y1, y2)) → A__U31(s(mark(x0)), y1, y2)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U31(0, y1, y2)) → A__U31(0, y1, y2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
A__U71(tt, M, N) → MARK(M)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U31(s(x0), y1, y2)) → A__U31(s(mark(x0)), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(0, y1, y2)) → A__U31(0, y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U32(X1, X2)) → A__U32(mark(X1), X2) at position [0] we obtained the following new rules:

MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(U32(0, y1)) → A__U32(0, y1)
MARK(U32(s(x0), y1)) → A__U32(s(mark(x0)), y1)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
A__U71(tt, M, N) → MARK(M)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(U32(0, y1)) → A__U32(0, y1)
MARK(plus(X1, X2)) → MARK(X1)
MARK(U32(s(x0), y1)) → A__U32(s(mark(x0)), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, N) → MARK(N)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → MARK(M)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
A__U41(tt, N) → MARK(N)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U41(X1, X2)) → A__U41(mark(X1), X2) at position [0] we obtained the following new rules:

MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(U41(0, y1)) → A__U41(0, y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U41(s(x0), y1)) → A__U41(s(mark(x0)), y1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(0, y1)) → A__U41(0, y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U41(s(x0), y1)) → A__U41(s(mark(x0)), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3) at position [0] we obtained the following new rules:

MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U51(0, y1, y2)) → A__U51(0, y1, y2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U51(s(x0), y1, y2)) → A__U51(s(mark(x0)), y1, y2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U51(s(x0), y1, y2)) → A__U51(s(mark(x0)), y1, y2)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U51(0, y1, y2)) → A__U51(0, y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2)) at position [1] we obtained the following new rules:

MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(plus(y0, tt)) → A__PLUS(mark(y0), tt)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, tt)) → A__PLUS(mark(y0), tt)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3) at position [0] we obtained the following new rules:

MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U71(s(x0), y1, y2)) → A__U71(s(mark(x0)), y1, y2)
MARK(U71(0, y1, y2)) → A__U71(0, y1, y2)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U71(s(x0), y1, y2)) → A__U71(s(mark(x0)), y1, y2)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U71(0, y1, y2)) → A__U71(0, y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
A__U51(tt, M, N) → MARK(N)
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(x(X1, X2)) → A__X(mark(X1), mark(X2)) at position [1] we obtained the following new rules:

MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(x(y0, tt)) → A__X(mark(y0), tt)
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
MARK(x(y0, tt)) → A__X(mark(y0), tt)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(and(X1, X2)) → A__AND(mark(X1), X2) at position [0] we obtained the following new rules:

MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(and(0, y1)) → A__AND(0, y1)
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(s(x0), y1)) → A__AND(s(mark(x0)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(and(s(x0), y1)) → A__AND(s(mark(x0)), y1)
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(0, y1)) → A__AND(0, y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__U21(tt, V1) → A__ISNAT(V1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(s(X)) → MARK(X)
A__U71(tt, M, N) → MARK(N)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(x(y0, isNat(x0))) → A__X(mark(y0), a__isNat(x0))
MARK(x(y0, U32(x0, x1))) → A__X(mark(y0), a__U32(mark(x0), x1))
MARK(x(y0, U21(x0, x1))) → A__X(mark(y0), a__U21(mark(x0), x1))
MARK(x(y0, U33(x0))) → A__X(mark(y0), a__U33(mark(x0)))
MARK(x(y0, U11(x0, x1, x2))) → A__X(mark(y0), a__U11(mark(x0), x1, x2))
MARK(x(y0, U31(x0, x1, x2))) → A__X(mark(y0), a__U31(mark(x0), x1, x2))
MARK(x(y0, U22(x0))) → A__X(mark(y0), a__U22(mark(x0)))

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = x₂
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 1
POL(U61(x₁)) = 1
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂, x₃)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(s(x₁)) = 1
POL(tt) = 0
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(x(y0, U13(x0))) → A__X(mark(y0), a__U13(mark(x0)))
MARK(x(y0, U12(x0, x1))) → A__X(mark(y0), a__U12(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = x₂
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂, x₃)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = x₂
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(U51(x0, x1, x2), y1)) → A__AND(a__U51(mark(x0), x1, x2), y1)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = x₁
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 1

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U51(U51(x0, x1, x2), y1, y2)) → A__U51(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(U61(x0), y1, y2)) → A__U51(a__U61(mark(x0)), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = x₁
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 1
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = 1
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = 1
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(plus(y0, U33(x0))) → A__PLUS(mark(y0), a__U33(mark(x0)))
MARK(plus(y0, U13(x0))) → A__PLUS(mark(y0), a__U13(mark(x0)))
MARK(plus(y0, U12(x0, x1))) → A__PLUS(mark(y0), a__U12(mark(x0), x1))
MARK(plus(y0, isNat(x0))) → A__PLUS(mark(y0), a__isNat(x0))
MARK(plus(y0, U11(x0, x1, x2))) → A__PLUS(mark(y0), a__U11(mark(x0), x1, x2))
MARK(plus(y0, U32(x0, x1))) → A__PLUS(mark(y0), a__U32(mark(x0), x1))
MARK(plus(y0, U31(x0, x1, x2))) → A__PLUS(mark(y0), a__U31(mark(x0), x1, x2))
MARK(plus(y0, U22(x0))) → A__PLUS(mark(y0), a__U22(mark(x0)))
MARK(plus(y0, U21(x0, x1))) → A__PLUS(mark(y0), a__U21(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = x₂
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U13(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂, x₃)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = x₂
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(x(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U31(U51(x0, x1, x2), y1, y2)) → A__U31(a__U51(mark(x0), x1, x2), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = x₁
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = x₁
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = 1
POL(a__U41(x₁, x₂)) = x₁
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = x₁
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 1

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U41(U51(x0, x1, x2), y1)) → A__U41(a__U51(mark(x0), x1, x2), y1)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = x₁
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 1
POL(U31(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 1
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 1
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = 1
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = 1
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U12(U61(x0), y1)) → A__U12(a__U61(mark(x0)), y1)
MARK(U71(U61(x0), y1, y2)) → A__U71(a__U61(mark(x0)), y1, y2)
MARK(U41(U61(x0), y1)) → A__U41(a__U61(mark(x0)), y1)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = x₁
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = x₁
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 1
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = x₁
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = x₁
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = x₁
POL(tt) = 1
POL(x(x₁, x₂)) = 1

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U71(U51(x0, x1, x2), y1, y2)) → A__U71(a__U51(mark(x0), x1, x2), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = x₁
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 1

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U11(U51(x0, x1, x2), y1, y2)) → A__U11(a__U51(mark(x0), x1, x2), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = x₁
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = x₁
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = x₁
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = 1
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U32(U51(x0, x1, x2), y1)) → A__U32(a__U51(mark(x0), x1, x2), y1)
MARK(U21(U51(x0, x1, x2), y1)) → A__U21(a__U51(mark(x0), x1, x2), y1)
MARK(U12(U51(x0, x1, x2), y1)) → A__U12(a__U51(mark(x0), x1, x2), y1)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = x₁
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = x₁
POL(A__U21(x₁, x₂)) = x₁
POL(A__U31(x₁, x₂, x₃)) = x₁
POL(A__U32(x₁, x₂)) = x₁
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = x₁
POL(A__U71(x₁, x₂, x₃)) = x₁
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = x₁
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 1
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U61(x₁)) = 1
POL(a__U71(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = x₂
POL(a__x(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = x₂
POL(s(x₁)) = 0
POL(tt) = 1
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(and(U61(x0), y1)) → A__AND(a__U61(mark(x0)), y1)
MARK(U21(U61(x0), y1)) → A__U21(a__U61(mark(x0)), y1)
MARK(U31(U61(x0), y1, y2)) → A__U31(a__U61(mark(x0)), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₁
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = x₁
POL(A__U21(x₁, x₂)) = x₁
POL(A__U31(x₁, x₂, x₃)) = x₁
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = x₁
POL(A__U51(x₁, x₂, x₃)) = x₁
POL(A__U71(x₁, x₂, x₃)) = x₁
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = x₁
POL(a__U11(x₁, x₂, x₃)) = x₁
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = x₂
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 1
POL(tt) = 1
POL(x(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U32(U61(x0), y1)) → A__U32(a__U61(mark(x0)), y1)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = 1
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = x₁
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = x₁
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 0
POL(U32(x₁, x₂)) = 1
POL(U33(x₁)) = 1
POL(U41(x₁, x₂)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = x₁
POL(a__U13(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂, x₃)) = 1
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = 1
POL(a__U41(x₁, x₂)) = 1
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = x₂
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = 1
POL(tt) = 1
POL(x(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U11(U61(x0), y1, y2)) → A__U11(a__U61(mark(x0)), y1, y2)

The remaining pairs can at least be oriented weakly.

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__X(N, 0) → A__ISNAT(N)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
A__U41(tt, N) → MARK(N)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(x(X1, X2)) → MARK(X1)
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U22(X)) → MARK(X)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1
POL(A__ISNAT(x₁)) = 1
POL(A__ISNATKIND(x₁)) = 1
POL(A__PLUS(x₁, x₂)) = 1
POL(A__U11(x₁, x₂, x₃)) = x₁
POL(A__U12(x₁, x₂)) = 1
POL(A__U21(x₁, x₂)) = 1
POL(A__U31(x₁, x₂, x₃)) = 1
POL(A__U32(x₁, x₂)) = 1
POL(A__U41(x₁, x₂)) = 1
POL(A__U51(x₁, x₂, x₃)) = 1
POL(A__U71(x₁, x₂, x₃)) = 1
POL(A__X(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 1
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = x₁
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁
POL(U51(x₁, x₂, x₃)) = 1
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
POL(a__U11(x₁, x₂, x₃)) = 1
POL(a__U12(x₁, x₂)) = 1
POL(a__U13(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂, x₃)) = x₁
POL(a__U32(x₁, x₂)) = 1
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U61(x₁)) = 0
POL(a__U71(x₁, x₂, x₃)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__plus(x₁, x₂)) = 1
POL(a__x(x₁, x₂)) = x₂
POL(and(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(mark(x₁)) = 1
POL(plus(x₁, x₂)) = 1
POL(s(x₁)) = 1
POL(tt) = 1
POL(x(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U33(tt) → tt
a__U61(tt) → 0
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__and(tt, X) → mark(X)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U41(tt, N) → mark(N)
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__U41(X1, X2) → U41(X1, X2)
a__U33(X) → U33(X)
a__U32(X1, X2) → U32(X1, X2)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U13(X) → U13(X)
a__isNat(X) → isNat(X)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__isNatKind(X) → isNatKind(X)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                        ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MARK(plus(y0, isNatKind(x0))) → A__PLUS(mark(y0), a__isNatKind(x0))
A__ISNAT(x(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(U41(plus(x0, x1), y1)) → A__U41(a__plus(mark(x0), mark(x1)), y1)
MARK(plus(y0, x(x0, x1))) → A__PLUS(mark(y0), a__x(mark(x0), mark(x1)))
A__U51(tt, M, N) → MARK(M)
MARK(U12(U21(x0, x1), y1)) → A__U12(a__U21(mark(x0), x1), y1)
A__ISNATKIND(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U22(x0), y1)) → A__AND(a__U22(mark(x0)), y1)
A__ISNAT(x(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(U11(x0, x1, x2), y1)) → A__U41(a__U11(mark(x0), x1, x2), y1)
A__ISNAT(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(and(U41(x0, x1), y1)) → A__AND(a__U41(mark(x0), x1), y1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U12(x0, x1), y1)) → A__U41(a__U12(mark(x0), x1), y1)
MARK(U71(and(x0, x1), y1, y2)) → A__U71(a__and(mark(x0), x1), y1, y2)
A__ISNATKIND(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U11(U33(x0), y1, y2)) → A__U11(a__U33(mark(x0)), y1, y2)
MARK(plus(y0, U71(x0, x1, x2))) → A__PLUS(mark(y0), a__U71(mark(x0), x1, x2))
A__U31(tt, s(x0), y1) → A__U32(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(tt, y1)) → A__U21(tt, y1)
MARK(x(y0, and(x0, x1))) → A__X(mark(y0), a__and(mark(x0), x1))
MARK(U32(x(x0, x1), y1)) → A__U32(a__x(mark(x0), mark(x1)), y1)
MARK(U41(X1, X2)) → MARK(X1)
A__X(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(x(y0, 0)) → A__X(mark(y0), 0)
MARK(U31(U11(x0, x1, x2), y1, y2)) → A__U31(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U11(U31(x0, x1, x2), y1, y2)) → A__U11(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U11(U41(x0, x1), y1, y2)) → A__U11(a__U41(mark(x0), x1), y1, y2)
MARK(U21(x(x0, x1), y1)) → A__U21(a__x(mark(x0), mark(x1)), y1)
MARK(U71(tt, y1, y2)) → A__U71(tt, y1, y2)
MARK(U12(and(x0, x1), y1)) → A__U12(a__and(mark(x0), x1), y1)
MARK(and(U71(x0, x1, x2), y1)) → A__AND(a__U71(mark(x0), x1, x2), y1)
MARK(U51(U22(x0), y1, y2)) → A__U51(a__U22(mark(x0)), y1, y2)
MARK(U31(plus(x0, x1), y1, y2)) → A__U31(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U13(X)) → MARK(X)
MARK(U51(tt, y1, y2)) → A__U51(tt, y1, y2)
MARK(U11(isNat(x0), y1, y2)) → A__U11(a__isNat(x0), y1, y2)
MARK(U71(U22(x0), y1, y2)) → A__U71(a__U22(mark(x0)), y1, y2)
A__U71(tt, M, N) → MARK(M)
MARK(U41(U41(x0, x1), y1)) → A__U41(a__U41(mark(x0), x1), y1)
MARK(U21(U31(x0, x1, x2), y1)) → A__U21(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U11(x0, x1, x2), y1)) → A__U21(a__U11(mark(x0), x1, x2), y1)
MARK(U12(isNatKind(x0), y1)) → A__U12(a__isNatKind(x0), y1)
A__ISNATKIND(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(plus(y0, U51(x0, x1, x2))) → A__PLUS(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U71(U12(x0, x1), y1, y2)) → A__U71(a__U12(mark(x0), x1), y1, y2)
A__U51(tt, M, N) → MARK(N)
MARK(x(y0, U71(x0, x1, x2))) → A__X(mark(y0), a__U71(mark(x0), x1, x2))
MARK(U11(and(x0, x1), y1, y2)) → A__U11(a__and(mark(x0), x1), y1, y2)
MARK(and(U12(x0, x1), y1)) → A__AND(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(0, y1)) → A__AND(tt, isNatKind(y1))
MARK(and(U21(x0, x1), y1)) → A__AND(a__U21(mark(x0), x1), y1)
MARK(U11(isNatKind(x0), y1, y2)) → A__U11(a__isNatKind(x0), y1, y2)
MARK(U41(U21(x0, x1), y1)) → A__U41(a__U21(mark(x0), x1), y1)
A__ISNATKIND(x(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(x(y0, x(x0, x1))) → A__X(mark(y0), a__x(mark(x0), mark(x1)))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U41(and(x0, x1), y1)) → A__U41(a__and(mark(x0), x1), y1)
A__ISNATKIND(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), x1), y1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U21(U41(x0, x1), y1)) → A__U21(a__U41(mark(x0), x1), y1)
MARK(U32(U22(x0), y1)) → A__U32(a__U22(mark(x0)), y1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U51(x(x0, x1), y1, y2)) → A__U51(a__x(mark(x0), mark(x1)), y1, y2)
A__X(N, 0) → A__ISNAT(N)
A__ISNAT(x(V1, V2)) → A__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
MARK(U41(U31(x0, x1, x2), y1)) → A__U41(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U33(x0), y1)) → A__U12(a__U33(mark(x0)), y1)
A__U31(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(U32(x0, x1), y1)) → A__U41(a__U32(mark(x0), x1), y1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(and(U33(x0), y1)) → A__AND(a__U33(mark(x0)), y1)
MARK(U41(U71(x0, x1, x2), y1)) → A__U41(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U31(x0, x1, x2), y1, y2)) → A__U71(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U41(isNatKind(x0), y1)) → A__U41(a__isNatKind(x0), y1)
MARK(plus(y0, U61(x0))) → A__PLUS(mark(y0), a__U61(mark(x0)))
A__U41(tt, N) → MARK(N)
MARK(U32(isNat(x0), y1)) → A__U32(a__isNat(x0), y1)
MARK(U71(plus(x0, x1), y1, y2)) → A__U71(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(U32(tt, y1)) → A__U32(tt, y1)
MARK(U12(U31(x0, x1, x2), y1)) → A__U12(a__U31(mark(x0), x1, x2), y1)
MARK(U31(U13(x0), y1, y2)) → A__U31(a__U13(mark(x0)), y1, y2)
MARK(U41(U13(x0), y1)) → A__U41(a__U13(mark(x0)), y1)
MARK(U32(plus(x0, x1), y1)) → A__U32(a__plus(mark(x0), mark(x1)), y1)
MARK(U11(x(x0, x1), y1, y2)) → A__U11(a__x(mark(x0), mark(x1)), y1, y2)
A__U71(tt, M, N) → A__X(mark(N), mark(M))
A__U31(tt, 0, y1) → A__U32(tt, y1)
MARK(U51(U32(x0, x1), y1, y2)) → A__U51(a__U32(mark(x0), x1), y1, y2)
MARK(U71(isNatKind(x0), y1, y2)) → A__U71(a__isNatKind(x0), y1, y2)
MARK(U12(U41(x0, x1), y1)) → A__U12(a__U41(mark(x0), x1), y1)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(and(x(x0, x1), y1)) → A__AND(a__x(mark(x0), mark(x1)), y1)
MARK(U41(tt, y1)) → A__U41(tt, y1)
MARK(and(tt, y1)) → A__AND(tt, y1)
MARK(U21(isNatKind(x0), y1)) → A__U21(a__isNatKind(x0), y1)
MARK(U51(U33(x0), y1, y2)) → A__U51(a__U33(mark(x0)), y1, y2)
MARK(U31(tt, y1, y2)) → A__U31(tt, y1, y2)
A__ISNATKIND(x(0, y1)) → A__AND(tt, isNatKind(y1))
A__ISNATKIND(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(U21(isNat(x0), y1)) → A__U21(a__isNat(x0), y1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U32(U31(x0, x1, x2), y1)) → A__U32(a__U31(mark(x0), x1, x2), y1)
MARK(U12(U13(x0), y1)) → A__U12(a__U13(mark(x0)), y1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U31(U71(x0, x1, x2), y1, y2)) → A__U31(a__U71(mark(x0), x1, x2), y1, y2)
MARK(x(y0, U41(x0, x1))) → A__X(mark(y0), a__U41(mark(x0), x1))
MARK(U32(isNatKind(x0), y1)) → A__U32(a__isNatKind(x0), y1)
MARK(U31(U21(x0, x1), y1, y2)) → A__U31(a__U21(mark(x0), x1), y1, y2)
MARK(U41(isNat(x0), y1)) → A__U41(a__isNat(x0), y1)
MARK(U71(x(x0, x1), y1, y2)) → A__U71(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U32(U11(x0, x1, x2), y1)) → A__U32(a__U11(mark(x0), x1, x2), y1)
MARK(x(y0, U51(x0, x1, x2))) → A__X(mark(y0), a__U51(mark(x0), x1, x2))
MARK(U31(and(x0, x1), y1, y2)) → A__U31(a__and(mark(x0), x1), y1, y2)
MARK(U11(U32(x0, x1), y1, y2)) → A__U11(a__U32(mark(x0), x1), y1, y2)
A__ISNAT(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U51(U31(x0, x1, x2), y1, y2)) → A__U51(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U33(x0), y1)) → A__U32(a__U33(mark(x0)), y1)
MARK(U51(isNat(x0), y1, y2)) → A__U51(a__isNat(x0), y1, y2)
A__U11(tt, s(x0), y1) → A__U12(a__U21(a__isNatKind(x0), x0), y1)
MARK(U21(U33(x0), y1)) → A__U21(a__U33(mark(x0)), y1)
A__U21(tt, V1) → A__ISNAT(V1)
A__U31(tt, x(x0, x1), y1) → A__U32(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U21(U71(x0, x1, x2), y1)) → A__U21(a__U71(mark(x0), x1, x2), y1)
MARK(U71(U71(x0, x1, x2), y1, y2)) → A__U71(a__U71(mark(x0), x1, x2), y1, y2)
A__U71(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
MARK(and(U11(x0, x1, x2), y1)) → A__AND(a__U11(mark(x0), x1, x2), y1)
MARK(U12(U12(x0, x1), y1)) → A__U12(a__U12(mark(x0), x1), y1)
MARK(U31(U22(x0), y1, y2)) → A__U31(a__U22(mark(x0)), y1, y2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(isNatKind(x0), y1, y2)) → A__U51(a__isNatKind(x0), y1, y2)
A__U11(tt, plus(x0, x1), y1) → A__U12(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), y1)
MARK(U31(U12(x0, x1), y1, y2)) → A__U31(a__U12(mark(x0), x1), y1, y2)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(x(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
MARK(plus(y0, s(x0))) → A__PLUS(mark(y0), s(mark(x0)))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U31(U31(x0, x1, x2), y1, y2)) → A__U31(a__U31(mark(x0), x1, x2), y1, y2)
MARK(U32(U32(x0, x1), y1)) → A__U32(a__U32(mark(x0), x1), y1)
MARK(U41(U22(x0), y1)) → A__U41(a__U22(mark(x0)), y1)
MARK(U32(U41(x0, x1), y1)) → A__U32(a__U41(mark(x0), x1), y1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U51(U11(x0, x1, x2), y1, y2)) → A__U51(a__U11(mark(x0), x1, x2), y1, y2)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
A__U11(tt, x(x0, x1), y1) → A__U12(a__U31(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U11(U13(x0), y1, y2)) → A__U11(a__U13(mark(x0)), y1, y2)
A__ISNATKIND(x(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
A__X(N, s(M)) → A__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
A__U71(tt, M, N) → MARK(N)
MARK(s(X)) → MARK(X)
MARK(U51(U21(x0, x1), y1, y2)) → A__U51(a__U21(mark(x0), x1), y1, y2)
MARK(U32(and(x0, x1), y1)) → A__U32(a__and(mark(x0), x1), y1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U12(U22(x0), y1)) → A__U12(a__U22(mark(x0)), y1)
MARK(U11(U11(x0, x1, x2), y1, y2)) → A__U11(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U21(and(x0, x1), y1)) → A__U21(a__and(mark(x0), x1), y1)
MARK(U32(U21(x0, x1), y1)) → A__U32(a__U21(mark(x0), x1), y1)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(s(x0), y1)) → A__AND(a__isNatKind(x0), isNatKind(y1))
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U51(U12(x0, x1), y1, y2)) → A__U51(a__U12(mark(x0), x1), y1, y2)
MARK(U11(U22(x0), y1, y2)) → A__U11(a__U22(mark(x0)), y1, y2)
MARK(U21(U12(x0, x1), y1)) → A__U21(a__U12(mark(x0), x1), y1)
MARK(U31(x(x0, x1), y1, y2)) → A__U31(a__x(mark(x0), mark(x1)), y1, y2)
MARK(U11(U71(x0, x1, x2), y1, y2)) → A__U11(a__U71(mark(x0), x1, x2), y1, y2)
A__U32(tt, V2) → A__ISNAT(V2)
MARK(U41(U33(x0), y1)) → A__U41(a__U33(mark(x0)), y1)
MARK(U51(U13(x0), y1, y2)) → A__U51(a__U13(mark(x0)), y1, y2)
MARK(plus(X1, X2)) → MARK(X1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U71(U13(x0), y1, y2)) → A__U71(a__U13(mark(x0)), y1, y2)
MARK(U61(X)) → MARK(X)
A__U31(tt, plus(x0, x1), y1) → A__U32(a__U11(a__and(a__isNatKind(x0), isNatKind(x1)), x0, x1), y1)
MARK(U71(U33(x0), y1, y2)) → A__U71(a__U33(mark(x0)), y1, y2)
MARK(U31(isNatKind(x0), y1, y2)) → A__U31(a__isNatKind(x0), y1, y2)
MARK(U33(X)) → MARK(X)
MARK(U32(U12(x0, x1), y1)) → A__U32(a__U12(mark(x0), x1), y1)
A__ISNAT(plus(x(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U32(U13(x0), y1)) → A__U32(a__U13(mark(x0)), y1)
A__U11(tt, 0, y1) → A__U12(tt, y1)
A__ISNATKIND(plus(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(and(U31(x0, x1, x2), y1)) → A__AND(a__U31(mark(x0), x1, x2), y1)
MARK(U21(U21(x0, x1), y1)) → A__U21(a__U21(mark(x0), x1), y1)
MARK(plus(y0, plus(x0, x1))) → A__PLUS(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(U31(isNat(x0), y1, y2)) → A__U31(a__isNat(x0), y1, y2)
MARK(x(y0, U61(x0))) → A__X(mark(y0), a__U61(mark(x0)))
MARK(x(y0, plus(x0, x1))) → A__X(mark(y0), a__plus(mark(x0), mark(x1)))
MARK(and(plus(x0, x1), y1)) → A__AND(a__plus(mark(x0), mark(x1)), y1)
MARK(x(X1, X2)) → MARK(X1)
MARK(and(U13(x0), y1)) → A__AND(a__U13(mark(x0)), y1)
MARK(U12(U32(x0, x1), y1)) → A__U12(a__U32(mark(x0), x1), y1)
MARK(U11(tt, y1, y2)) → A__U11(tt, y1, y2)
A__U51(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(U71(U21(x0, x1), y1, y2)) → A__U71(a__U21(mark(x0), x1), y1, y2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(plus(y0, U41(x0, x1))) → A__PLUS(mark(y0), a__U41(mark(x0), x1))
MARK(x(y0, s(x0))) → A__X(mark(y0), s(mark(x0)))
MARK(U11(U21(x0, x1), y1, y2)) → A__U11(a__U21(mark(x0), x1), y1, y2)
A__ISNAT(x(plus(x0, x1), y1)) → A__AND(a__and(a__isNatKind(x0), isNatKind(x1)), isNatKind(y1))
MARK(U31(U41(x0, x1), y1, y2)) → A__U31(a__U41(mark(x0), x1), y1, y2)
MARK(U12(U71(x0, x1, x2), y1)) → A__U12(a__U71(mark(x0), x1, x2), y1)
MARK(U11(plus(x0, x1), y1, y2)) → A__U11(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(plus(y0, and(x0, x1))) → A__PLUS(mark(y0), a__and(mark(x0), x1))
MARK(U22(X)) → MARK(X)
MARK(U12(U11(x0, x1, x2), y1)) → A__U12(a__U11(mark(x0), x1, x2), y1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U12(isNat(x0), y1)) → A__U12(a__isNat(x0), y1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U31(U32(x0, x1), y1, y2)) → A__U31(a__U32(mark(x0), x1), y1, y2)
MARK(U12(plus(x0, x1), y1)) → A__U12(a__plus(mark(x0), mark(x1)), y1)
MARK(U31(U33(x0), y1, y2)) → A__U31(a__U33(mark(x0)), y1, y2)
A__PLUS(N, 0) → A__U41(a__and(a__isNat(N), isNatKind(N)), N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U51(and(x0, x1), y1, y2)) → A__U51(a__and(mark(x0), x1), y1, y2)
MARK(U12(tt, y1)) → A__U12(tt, y1)
MARK(U21(U32(x0, x1), y1)) → A__U21(a__U32(mark(x0), x1), y1)
MARK(U21(U22(x0), y1)) → A__U21(a__U22(mark(x0)), y1)
MARK(and(U32(x0, x1), y1)) → A__AND(a__U32(mark(x0), x1), y1)
A__X(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U51(U71(x0, x1, x2), y1, y2)) → A__U51(a__U71(mark(x0), x1, x2), y1, y2)
A__X(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))
MARK(U71(U11(x0, x1, x2), y1, y2)) → A__U71(a__U11(mark(x0), x1, x2), y1, y2)
MARK(U12(x(x0, x1), y1)) → A__U12(a__x(mark(x0), mark(x1)), y1)
MARK(U32(U71(x0, x1, x2), y1)) → A__U32(a__U71(mark(x0), x1, x2), y1)
MARK(plus(y0, 0)) → A__PLUS(mark(y0), 0)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U51(U41(x0, x1), y1, y2)) → A__U51(a__U41(mark(x0), x1), y1, y2)
MARK(and(isNatKind(x0), y1)) → A__AND(a__isNatKind(x0), y1)
MARK(U41(x(x0, x1), y1)) → A__U41(a__x(mark(x0), mark(x1)), y1)
MARK(U21(U13(x0), y1)) → A__U21(a__U13(mark(x0)), y1)
MARK(U11(U12(x0, x1), y1, y2)) → A__U11(a__U12(mark(x0), x1), y1, y2)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U21(plus(x0, x1), y1)) → A__U21(a__plus(mark(x0), mark(x1)), y1)
MARK(U71(U41(x0, x1), y1, y2)) → A__U71(a__U41(mark(x0), x1), y1, y2)
MARK(U71(isNat(x0), y1, y2)) → A__U71(a__isNat(x0), y1, y2)
MARK(U51(plus(x0, x1), y1, y2)) → A__U51(a__plus(mark(x0), mark(x1)), y1, y2)
MARK(x(y0, isNatKind(x0))) → A__X(mark(y0), a__isNatKind(x0))
MARK(U71(U32(x0, x1), y1, y2)) → A__U71(a__U32(mark(x0), x1), y1, y2)

The TRS R consists of the following rules:

a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V1, V2) → a__U32(a__isNat(V1), V2)
a__U32(tt, V2) → a__U33(a__isNat(V2))
a__U33(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(x(V1, V2)) → a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatKind(x(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__plus(N, 0) → a__U41(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
a__x(N, 0) → a__U61(a__and(a__isNat(N), isNatKind(N)))
a__x(N, s(M)) → a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.