Termination w.r.t. Q proof of /tmp/tmpnjabfm/PEANO_nokinds

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, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))
A__PLUS(N, 0) → A__U11(a__isNat(N), N)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__ISNAT(N)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))
A__PLUS(N, 0) → A__U11(a__isNat(N), N)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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__ISNAT(N)
A__PLUS(N, 0) → A__U11(a__isNat(N), N)

The remaining pairs can at least be oriented weakly.

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = x_1 + x_2
POL(A__PLUS(x₁, x₂)) = (1/4)x_1 + (1/4)x_2
POL(A__U11(x₁, x₂)) = (1/4)x_2
POL(A__U21(x₁, x₂, x₃)) = (1/4)x_2 + (1/4)x_3
POL(A__ISNAT(x₁)) = 0
POL(a__U11(x₁, x₂)) = x_1 + x_2
POL(a__and(x₁, x₂)) = (4)x_1 + x_2
POL(a__U21(x₁, x₂, x₃)) = (4)x_1 + x_2 + x_3
POL(mark(x₁)) = x_1
POL(and(x₁, x₂)) = (4)x_1 + x_2
POL(U11(x₁, x₂)) = x_1 + x_2
POL(0) = 1/2
POL(a__plus(x₁, x₂)) = x_1 + x_2
POL(A__AND(x₁, x₂)) = (1/4)x_2
POL(MARK(x₁)) = (1/4)x_1
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = (4)x_1 + x_2 + x_3

The value of delta used in the strict ordering is 1/8.
The following usable rules [17] were oriented:

a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__plus(N, 0) → a__U11(a__isNat(N), N)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(V1)) → a__isNat(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__U11(tt, N) → mark(N)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U21(X1, X2, X3)) → MARK(X1)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)

The remaining pairs can at least be oriented weakly.

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(A__PLUS(x₁, x₂)) = (1/4)x_1 + x_2
POL(A__U11(x₁, x₂)) = x_2
POL(A__U21(x₁, x₂, x₃)) = x_2 + (1/4)x_3
POL(A__ISNAT(x₁)) = 0
POL(a__U11(x₁, x₂)) = 1/4 + (4)x_1 + (4)x_2
POL(a__and(x₁, x₂)) = (2)x_1 + (4)x_2
POL(a__U21(x₁, x₂, x₃)) = (4)x_1 + (4)x_2 + (4)x_3
POL(mark(x₁)) = x_1
POL(and(x₁, x₂)) = (2)x_1 + (4)x_2
POL(U11(x₁, x₂)) = 1/4 + (4)x_1 + (4)x_2
POL(0) = 1/4
POL(a__plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(A__AND(x₁, x₂)) = x_2
POL(MARK(x₁)) = (1/4)x_1
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = (4)x_1 + (4)x_2 + (4)x_3

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__plus(N, 0) → a__U11(a__isNat(N), N)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(V1)) → a__isNat(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__U11(tt, N) → mark(N)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U11(tt, N) → MARK(N)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
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), isNat(N))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))

The remaining pairs can at least be oriented weakly.

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNat(N))

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = 4 + (2)x_1 + (4)x_2
POL(A__PLUS(x₁, x₂)) = (1/4)x_1 + (1/2)x_2
POL(A__U21(x₁, x₂, x₃)) = (1/2)x_2 + (1/4)x_3
POL(A__ISNAT(x₁)) = 0
POL(a__U11(x₁, x₂)) = x_2
POL(a__and(x₁, x₂)) = (4)x_1 + (4)x_2
POL(a__U21(x₁, x₂, x₃)) = 4 + x_1 + (4)x_2 + (2)x_3
POL(mark(x₁)) = x_1
POL(U11(x₁, x₂)) = x_2
POL(and(x₁, x₂)) = (4)x_1 + (4)x_2
POL(0) = 0
POL(a__plus(x₁, x₂)) = 4 + (2)x_1 + (4)x_2
POL(A__AND(x₁, x₂)) = x_2
POL(MARK(x₁)) = (1/4)x_1
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 4 + x_1 + (4)x_2 + (2)x_3

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__plus(N, 0) → a__U11(a__isNat(N), N)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(V1)) → a__isNat(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__U11(tt, N) → mark(N)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
MARK(s(X)) → MARK(X)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U21(tt, M, N) → MARK(M)
A__AND(tt, X) → MARK(X)
A__U21(tt, M, N) → MARK(N)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)
MARK(and(X1, X2)) → MARK(X1)
A__PLUS(N, s(M)) → A__ISNAT(M)
MARK(isNat(X)) → A__ISNAT(X)
A__PLUS(N, s(M)) → A__AND(a__isNat(M), isNat(N))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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)

The remaining pairs can at least be oriented weakly.

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__AND(tt, X) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = 0
POL(A__ISNAT(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(mark(x₁)) = 0
POL(a__U21(x₁, x₂, x₃)) = 0
POL(U11(x₁, x₂)) = 0
POL(and(x₁, x₂)) = (4)x_1 + (4)x_2
POL(0) = 0
POL(a__plus(x₁, x₂)) = 0
POL(A__AND(x₁, x₂)) = x_2
POL(MARK(x₁)) = (1/4)x_1
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = 1/4 + (2)x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(plus(V1, V2)) → A__AND(a__isNat(V1), isNat(V2))
A__ISNAT(s(V1)) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__AND(tt, X) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = 1/4 + (4)x_1 + (4)x_2
POL(A__ISNAT(x₁)) = (1/4)x_1
POL(a__U11(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(mark(x₁)) = 0
POL(a__U21(x₁, x₂, x₃)) = 0
POL(U11(x₁, x₂)) = 0
POL(and(x₁, x₂)) = (4)x_1 + x_2
POL(0) = 0
POL(a__plus(x₁, x₂)) = 0
POL(A__AND(x₁, x₂)) = x_2
POL(MARK(x₁)) = x_1
POL(tt) = 0
POL(s(x₁)) = 1 + x_1
POL(a__isNat(x₁)) = 0
POL(isNat(x₁)) = (1/4)x_1
POL(U21(x₁, x₂, x₃)) = 0

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                          ↳ QDP

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

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                          ↳ QDP

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

MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__AND(tt, X) → MARK(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(mark(x₁)) = 0
POL(a__U21(x₁, x₂, x₃)) = 0
POL(U11(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(0) = 0
POL(a__plus(x₁, x₂)) = 0
POL(A__AND(x₁, x₂)) = 4 + (4)x_2
POL(MARK(x₁)) = 2 + (4)x_1
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = 0
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 0

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ PisEmptyProof

                          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPOrderProof

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

A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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, s(M)) → A__U21(a__and(a__isNat(M), isNat(N)), M, N)

The remaining pairs can at least be oriented weakly.

A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))

Used ordering: Polynomial interpretation [25,35]:

POL(plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(A__PLUS(x₁, x₂)) = (4)x_2
POL(A__U21(x₁, x₂, x₃)) = (4)x_2
POL(a__U11(x₁, x₂)) = (2)x_2
POL(a__and(x₁, x₂)) = x_2
POL(mark(x₁)) = x_1
POL(a__U21(x₁, x₂, x₃)) = 1/2 + (4)x_2 + (4)x_3
POL(and(x₁, x₂)) = x_2
POL(U11(x₁, x₂)) = (2)x_2
POL(0) = 0
POL(a__plus(x₁, x₂)) = (4)x_1 + (4)x_2
POL(tt) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = 1/2 + x_1
POL(isNat(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = 1/2 + (4)x_2 + (4)x_3

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
a__plus(N, 0) → a__U11(a__isNat(N), N)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(V1)) → a__isNat(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__U11(tt, N) → mark(N)
mark(tt) → tt
mark(s(X)) → s(mark(X))
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

A__U21(tt, M, N) → A__PLUS(mark(N), mark(M))

The TRS R consists of the following rules:

a__U11(tt, N) → mark(N)
a__U21(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__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(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 1 less node.