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

  ↳ RRRPoloQTRSProof

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.

The following Q TRS is given: 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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a____(X, nil) → mark(X)
a____(nil, X) → mark(X)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + x₂
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isNePal(x₁)) = 2·x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNePal(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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__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.

The following Q TRS is given: 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__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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNePal(__(I, __(P, I))) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + x₂
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isNePal(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNePal(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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__and(tt, X) → mark(X)
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.

The following Q TRS is given: 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__and(tt, X) → mark(X)
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__and(tt, X) → mark(X)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + x₂
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isNePal(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNePal(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(tt) = 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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

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

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

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

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a____(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNePal(x₁)) = 2·x₁
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(isNePal(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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

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.

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

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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

mark(tt) → tt
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNePal(X) → isNePal(X)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a____(x₁, x₂)) = 2 + x₁ + x₂
POL(a__and(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(a__isNePal(x₁)) = 2 + x₁
POL(and(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(isNePal(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 0
POL(tt) = 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

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

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

Q is empty.

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

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

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

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

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a____(x₁, x₂)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = 1 + x₁ + x₂
POL(a__isNePal(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(isNePal(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = 2 + 2·x₁
POL(nil) = 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.