Termination w.r.t. Q proof of /tmp/tmprwr-n8/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:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(and(tt, X)) → mark(X)
__(mark(X1), X2) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
top(mark(X)) → top(proper(X))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(isNePal(x₁)) = x₁
POL(mark(x₁)) = 2 + x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(top(x₁)) = x₁
POL(tt) = 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(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:

active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
proper(nil) → ok(nil)
proper(isNePal(X)) → isNePal(proper(X))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNePal(x₁)) = 2 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 2·x₁
POL(top(x₁)) = x₁
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(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:

proper(and(X1, X2)) → and(proper(X1), proper(X2))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(isNePal(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 2·x₁
POL(top(x₁)) = 2·x₁
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(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:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(ok(X)) → top(active(X))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + x₁ + x₂
POL(active(x₁)) = 1 + 2·x₁
POL(and(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(isNePal(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 2 + 2·x₁
POL(proper(x₁)) = 2 + 2·x₁
POL(top(x₁)) = x₁
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:

active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))

Q is empty.

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

active(isNePal(X)) → isNePal(active(X))
__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))

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

active(isNePal(X)) → isNePal(active(X))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + 2·x₂
POL(active(x₁)) = 2·x₁
POL(isNePal(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 2·x₁
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))

Q is empty.

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)
__(ok(X1), ok(X2)) → ok(__(X1, X2))

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

proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(tt) → ok(tt)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + x₁ + x₂
POL(isNePal(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 1 + 2·x₁
POL(tt) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))

Q is empty.

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))

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

__(ok(X1), ok(X2)) → ok(__(X1, X2))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(isNePal(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 2 + 2·x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))

Q is empty.

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

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(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:

__(X1, mark(X2)) → mark(__(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(isNePal(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = 1 + x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ 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.