Termination w.r.t. Q proof of /tmp/tmpRal0KU/PALINDROME_nosorts

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____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(isNePal(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNePal(X)) → A__ISNEPAL(mark(X))
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(isNePal(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNePal(X)) → A__ISNEPAL(mark(X))
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(isNePal(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)

The remaining pairs can at least be oriented weakly.

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(isNePal(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25,35]:

POL(a____(x₁, x₂)) = x_1 + x_2
POL(A__AND(x₁, x₂)) = (4)x_2
POL(A____(x₁, x₂)) = (4)x_1 + (4)x_2
POL(MARK(x₁)) = (4)x_1
POL(__(x₁, x₂)) = x_1 + x_2
POL(tt) = 0
POL(a__and(x₁, x₂)) = (4)x_1 + (2)x_2
POL(a__isNePal(x₁)) = (2)x_1
POL(mark(x₁)) = x_1
POL(isNePal(x₁)) = (2)x_1
POL(and(x₁, x₂)) = (4)x_1 + (2)x_2
POL(nil) = 2

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(isNePal(X)) → a__isNePal(mark(X))
mark(tt) → tt
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isNePal(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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(isNePal(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(__(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(a____(x₁, x₂)) = x_1 + x_2
POL(A__AND(x₁, x₂)) = (2)x_2
POL(A____(x₁, x₂)) = x_1 + x_2
POL(MARK(x₁)) = x_1
POL(__(x₁, x₂)) = x_1 + x_2
POL(tt) = 0
POL(a__and(x₁, x₂)) = 2 + x_1 + (2)x_2
POL(a__isNePal(x₁)) = 1 + (3)x_1
POL(mark(x₁)) = x_1
POL(isNePal(x₁)) = 1 + (3)x_1
POL(and(x₁, x₂)) = 2 + x_1 + (2)x_2
POL(nil) = 2

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(isNePal(X)) → a__isNePal(mark(X))
mark(tt) → tt
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(__(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ QDPOrderProof

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(__(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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(__(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))

Used ordering: Polynomial interpretation [25,35]:

POL(a____(x₁, x₂)) = 1 + x_1 + x_2
POL(A____(x₁, x₂)) = 2 + (4)x_1 + (4)x_2
POL(MARK(x₁)) = 4 + (4)x_1
POL(__(x₁, x₂)) = 1 + x_1 + x_2
POL(tt) = 1
POL(a__and(x₁, x₂)) = (2)x_1 + x_2
POL(a__isNePal(x₁)) = 1
POL(mark(x₁)) = x_1
POL(isNePal(x₁)) = 1
POL(and(x₁, x₂)) = (2)x_1 + x_2
POL(nil) = 0

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(isNePal(X)) → a__isNePal(mark(X))
mark(tt) → tt
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))

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

POL(a____(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(A____(x₁, x₂)) = (4)x_1
POL(__(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(tt) = 0
POL(a__and(x₁, x₂)) = 3 + (3)x_1 + (4)x_2
POL(a__isNePal(x₁)) = 0
POL(mark(x₁)) = x_1
POL(isNePal(x₁)) = 0
POL(and(x₁, x₂)) = 3 + (3)x_1 + (4)x_2
POL(nil) = 0

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

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(isNePal(X)) → a__isNePal(mark(X))
mark(tt) → tt
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a____(X1, X2) → __(X1, X2)
a__isNePal(X) → isNePal(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ PisEmptyProof

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isNePal(__(I, __(P, I))) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(mark(X))
mark(nil) → nil
mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(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.