Termination w.r.t. Q proof of /tmp/tmp5QAiV5/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

  ↳ 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__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.

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__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A____(__(X, Y), Z) → MARK(Z)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISNELIST(V) → A__ISQID(V)
A__ISNEPAL(V) → A__ISQID(V)
MARK(isQid(X)) → A__ISQID(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(Y)
A__ISPAL(V) → A__ISNEPAL(V)
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__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.
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__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A____(__(X, Y), Z) → MARK(Z)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
A____(__(X, Y), Z) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISNELIST(V) → A__ISQID(V)
A__ISNEPAL(V) → A__ISQID(V)
MARK(isQid(X)) → A__ISQID(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(Y)
A__ISPAL(V) → A__ISNEPAL(V)
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__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.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

A____(__(X, Y), Z) → MARK(Z)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
MARK(and(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(Y)
A__ISPAL(V) → A__ISNEPAL(V)
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__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.
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) → MARK(Z)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
MARK(and(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(Y)
A__ISPAL(V) → A__ISNEPAL(V)
MARK(__(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 0
POL(A____(x₁, x₂)) = (2)x_1 + (2)x_2
POL(__(x₁, x₂)) = x_1 + x_2
POL(a__isNePal(x₁)) = 0
POL(mark(x₁)) = x_1
POL(A__ISNELIST(x₁)) = 0
POL(and(x₁, x₂)) = (4)x_1 + (4)x_2
POL(A__AND(x₁, x₂)) = (3)x_2
POL(a__isQid(x₁)) = 0
POL(tt) = 0
POL(a__isList(x₁)) = 0
POL(nil) = 1
POL(a____(x₁, x₂)) = x_1 + x_2
POL(A__ISPAL(x₁)) = 0
POL(a) = 1
POL(isList(x₁)) = 0
POL(a__and(x₁, x₂)) = (4)x_1 + (4)x_2
POL(a__isNeList(x₁)) = 0
POL(e) = 0
POL(A__ISNEPAL(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(o) = 0
POL(isQid(x₁)) = 0
POL(isPal(x₁)) = 0
POL(MARK(x₁)) = (2)x_1
POL(u) = 1
POL(isNeList(x₁)) = 0
POL(a__isPal(x₁)) = 0
POL(A__ISLIST(x₁)) = 0

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

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

↳ 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)))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
MARK(and(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
A__ISPAL(V) → A__ISNEPAL(V)
MARK(__(X1, X2)) → MARK(X2)
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))

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.
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)))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
MARK(and(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))

The remaining pairs can at least be oriented weakly.

MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__ISNEPAL(V)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 4
POL(A____(x₁, x₂)) = (4)x_1 + x_2
POL(__(x₁, x₂)) = 2 + (4)x_1 + x_2
POL(a__isNePal(x₁)) = (2)x_1
POL(mark(x₁)) = x_1
POL(and(x₁, x₂)) = 2 + (2)x_1 + x_2
POL(A__ISNELIST(x₁)) = (4)x_1
POL(A__AND(x₁, x₂)) = 1 + x_1 + x_2
POL(a__isQid(x₁)) = (2)x_1
POL(tt) = 0
POL(a__isList(x₁)) = 2 + (4)x_1
POL(nil) = 0
POL(A__ISPAL(x₁)) = (2)x_1
POL(a____(x₁, x₂)) = 2 + (4)x_1 + x_2
POL(a) = 0
POL(isList(x₁)) = 2 + (4)x_1
POL(a__and(x₁, x₂)) = 2 + (2)x_1 + x_2
POL(a__isNeList(x₁)) = (4)x_1
POL(e) = 0
POL(A__ISNEPAL(x₁)) = (2)x_1
POL(isNePal(x₁)) = (2)x_1
POL(o) = 0
POL(isQid(x₁)) = (2)x_1
POL(isPal(x₁)) = (2)x_1
POL(MARK(x₁)) = x_1
POL(u) = 4
POL(isNeList(x₁)) = (4)x_1
POL(a__isPal(x₁)) = (2)x_1
POL(A__ISLIST(x₁)) = 1 + (4)x_1

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISPAL(V) → A__ISNEPAL(V)
MARK(isPal(X)) → A__ISPAL(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__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.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.