Termination w.r.t. Q proof of /tmp/tmp4MWTpo/PALINDROME_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____(__(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__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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)
a__isList(nil) → tt
a__isPal(nil) → tt
a__isQid(o) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a) = 0
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(o) = 1
POL(tt) = 0
POL(u) = 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__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isQid(e) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a) = 0
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = 2·x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(e) = 1
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 2·x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 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)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(i) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isQid(i) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a) = 0
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = 2·x₁
POL(a__isNeList(x₁)) = 2·x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 1
POL(i) = 1
POL(isList(x₁)) = 2·x₁
POL(isNeList(x₁)) = 2·x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

↳ 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)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(a) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isQid(a) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a) = 1
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = 2·x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 2·x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 2
POL(tt) = 0
POL(u) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ 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)))
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isList(V) → a__isNeList(V)
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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)))
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isQid(u) → tt

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a) = 0
POL(a____(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 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:

a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

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

a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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)) = 2·x₁ + x₂
POL(a) = 0
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = x₁
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(o) = 0
POL(tt) = 2
POL(u) = 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:

a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.

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

a__isList(V) → a__isNeList(V)
a__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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__isNeList(V) → a__isQid(V)
a__isNePal(V) → a__isQid(V)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 2
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = 2 + 2·x₁
POL(a__isNePal(x₁)) = 2 + x₁
POL(a__isPal(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = 1 + x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 1
POL(i) = 2
POL(isList(x₁)) = 2 + x₁
POL(isNeList(x₁)) = 2 + x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1 + x₁
POL(isQid(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 2
POL(o) = 0
POL(tt) = 1
POL(u) = 0

↳ 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:

a__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(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__isList(V) → a__isNeList(V)
a__isPal(V) → a__isNePal(V)
mark(isNePal(X)) → a__isNePal(X)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = x₁ + 2·x₂
POL(a____(x₁, x₂)) = x₁ + 2·x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isList(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = 1 + x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = 1 + 2·x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + 2·x₁
POL(isPal(x₁)) = 2 + 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(o) = 0
POL(u) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ 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:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)

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(isPal(X)) → a__isPal(X)
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(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:

mark(isPal(X)) → a__isPal(X)
mark(u) → u

Used ordering:
Polynomial interpretation [25]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a____(x₁, x₂)) = 2·x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isPal(x₁)) = 2·x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(isPal(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = 2·x₁
POL(o) = 0
POL(u) = 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ 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:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(o) → o
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)

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(o) → o
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(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:

mark(o) → o
a__and(X1, X2) → and(X1, X2)

Used ordering:
Polynomial interpretation [25]:

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ 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:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
a____(X1, X2) → __(X1, X2)

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)
a____(X1, X2) → __(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:

mark(and(X1, X2)) → a__and(mark(X1), X2)
a____(X1, X2) → __(X1, X2)

Used ordering:
Polynomial interpretation [25]:

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ 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:

mark(__(X1, X2)) → a____(mark(X1), mark(X2))

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))

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))

Used ordering:
Polynomial interpretation [25]:

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ 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.