Termination w.r.t. Q proof of /tmp/tmpd03dq8/PEANO_complete

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
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__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U22(X)) → A__U22(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U41(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__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U13(X)) → A__U13(mark(X))
A__U41(tt, M, N) → MARK(N)
MARK(U22(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
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__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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__U21(tt, V1) → A__U22(a__isNat(V1))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
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__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U22(X)) → A__U22(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U41(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__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U12(tt, V2) → A__U13(a__isNat(V2))
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U13(X)) → A__U13(mark(X))
A__U41(tt, M, N) → MARK(N)
MARK(U22(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
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__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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(s(X)) → MARK(X)
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
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__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U41(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__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
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__PLUS(N, 0) → A__ISNAT(N)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, M, N) → MARK(N)
MARK(U22(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
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__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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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(s(X)) → MARK(X)
A__PLUS(N, s(M)) → A__AND(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__PLUS(N, s(M)) → A__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNatKind(M))

The remaining pairs can at least be oriented weakly.

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
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__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
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__PLUS(N, 0) → A__ISNAT(N)
A__U11(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, M, N) → MARK(N)
MARK(U22(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__PLUS(x₁, x₂)) = x₁ + x₂
POL(A__U11(x₁, x₂, x₃)) = 0
POL(A__U12(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = x₂
POL(A__U41(x₁, x₂, x₃)) = x₂ + x₃
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
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)) = x₁ + x₂
POL(a__U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented:

mark(U22(X)) → a__U22(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
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__and(tt, X) → mark(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U31(tt, N) → mark(N)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNat(X)) → a__isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
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__U21(tt, V1) → a__U22(a__isNat(V1))
a__U13(tt) → tt
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__isNatKind(0) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U22(tt) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, M, N) → A__PLUS(mark(N), mark(M))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
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__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
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__U11(tt, V1, V2) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__U41(tt, M, N) → MARK(N)
MARK(U22(X)) → MARK(X)
A__U41(tt, M, N) → MARK(M)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__PLUS(N, 0) → A__ISNAT(N)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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.

A__PLUS(N, 0) → A__AND(a__isNat(N), isNatKind(N))
A__PLUS(N, 0) → A__U31(a__and(a__isNat(N), isNatKind(N)), N)
A__PLUS(N, 0) → A__ISNAT(N)

The remaining pairs can at least be oriented weakly.

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(X)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__PLUS(x₁, x₂)) = x₁ + x₂
POL(A__U11(x₁, x₂, x₃)) = 0
POL(A__U12(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂)) = x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U41(x₁, x₂, x₃)) = x₁
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)) = x₁ + x₂
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(mark(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 0
POL(tt) = 0

The following usable rules [17] were oriented:

mark(U22(X)) → a__U22(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
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__and(tt, X) → mark(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__U31(tt, N) → mark(N)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNat(X)) → a__isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
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__U21(tt, V1) → a__U22(a__isNat(V1))
a__U13(tt) → tt
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__isNatKind(0) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(0) → tt
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U22(tt) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(isNat(X)) → A__ISNAT(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

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

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
A__U12(tt, V2) → A__ISNAT(V2)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
MARK(plus(X1, X2)) → MARK(X1)
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__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(isNat(X)) → A__ISNAT(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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
A__ISNATKIND(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNAT(plus(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X1)
A__U11(tt, V1, V2) → A__U12(a__isNat(V1), V2)
A__U11(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(isNat(X)) → A__ISNAT(X)
A__U21(tt, V1) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.

A__U31(tt, N) → MARK(N)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__AND(tt, X) → MARK(X)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
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)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1 + x₂
POL(A__ISNAT(x₁)) = x₁
POL(A__ISNATKIND(x₁)) = x₁
POL(A__U11(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U12(x₁, x₂)) = x₂
POL(A__U21(x₁, x₂)) = 1 + x₂
POL(A__U31(x₁, x₂)) = 1 + x₂
POL(MARK(x₁)) = 1 + x₁
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂)) = 1 + x₁ + x₂
POL(U13(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U41(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)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__plus(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(mark(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

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

A__U31(tt, N) → MARK(N)
A__U12(tt, V2) → A__ISNAT(V2)
A__ISNATKIND(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(plus(V1, V2)) → A__AND(a__isNatKind(V1), isNatKind(V2))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(plus(V1, V2)) → A__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
A__AND(tt, X) → MARK(X)

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, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
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__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(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)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(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) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(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 0 SCCs with 7 less nodes.