Termination w.r.t. Q proof of /tmp/tmpCQ9hF_/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:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNELIST(V) → ACTIVATE(V)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)
ACTIVATE(n__o) → O
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ACTIVATE(n__a) → A
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISNELIST(V) → ISQID(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
__¹(__(X, Y), Z) → __¹(Y, Z)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n__u) → U
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ISPAL(V) → ISNEPAL(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ACTIVATE(n__nil) → NIL
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISPAL(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__i) → I
ACTIVATE(n__e) → E
ISNEPAL(V) → ISQID(activate(V))
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNELIST(V) → ACTIVATE(V)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)
ACTIVATE(n__o) → O
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ACTIVATE(n__a) → A
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISNELIST(V) → ISQID(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
__¹(__(X, Y), Z) → __¹(Y, Z)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n__u) → U
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ISPAL(V) → ISNEPAL(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ACTIVATE(n__nil) → NIL
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISPAL(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__i) → I
ACTIVATE(n__e) → E
ISNEPAL(V) → ISQID(activate(V))
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 10 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

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

POL(__(x₁, x₂)) = 4 + (2)x_1 + (2)x_2
POL(__¹(x₁, x₂)) = (9/4)x_1
POL(n____(x₁, x₂)) = 7/2 + (13/4)x_1 + (4)x_2
POL(nil) = 0

The value of delta used in the strict ordering is 9.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISPAL(V) → ISNEPAL(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISPAL(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)

The remaining pairs can at least be oriented weakly.

ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISPAL(V) → ISNEPAL(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 1
POL(n__u) = 3/4
POL(__(x₁, x₂)) = (4)x_1 + x_2
POL(n__i) = 1
POL(activate(x₁)) = x_1
POL(and(x₁, x₂)) = x_2
POL(n__nil) = 0
POL(n__a) = 4
POL(tt) = 0
POL(ISPAL(x₁)) = 1 + (3)x_1
POL(AND(x₁, x₂)) = x_2
POL(n__isList(x₁)) = (2)x_1
POL(nil) = 0
POL(ACTIVATE(x₁)) = x_1
POL(a) = 4
POL(isList(x₁)) = (2)x_1
POL(ISNELIST(x₁)) = (2)x_1
POL(n__isPal(x₁)) = 1 + (4)x_1
POL(e) = 0
POL(n__e) = 0
POL(isNePal(x₁)) = 1 + (4)x_1
POL(n____(x₁, x₂)) = (4)x_1 + x_2
POL(o) = 2
POL(isQid(x₁)) = 0
POL(n__isNeList(x₁)) = (2)x_1
POL(isPal(x₁)) = 1 + (4)x_1
POL(n__o) = 2
POL(u) = 3/4
POL(isNeList(x₁)) = (2)x_1
POL(ISLIST(x₁)) = (2)x_1
POL(ISNEPAL(x₁)) = 1 + (3)x_1

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

activate(n____(X1, X2)) → __(activate(X1), activate(X2))
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isPal(V) → isNePal(activate(V))
activate(n__isPal(X)) → isPal(X)
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n__u) → u
activate(X) → X
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
isList(n__nil) → tt
__(nil, X) → X
isNeList(V) → isQid(activate(V))
isPal(n__nil) → tt
isNePal(V) → isQid(activate(V))
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt
isList(X) → n__isList(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
isQid(n__u) → tt
e → n__e
a → n__a
isPal(X) → n__isPal(X)
isNeList(X) → n__isNeList(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISPAL(V) → ISNEPAL(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ISLIST(V) → ISNELIST(activate(V))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISPAL(V) → ISNEPAL(activate(V))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ACTIVATE(n__isList(X)) → ISLIST(X)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 0
POL(n__u) = 0
POL(__(x₁, x₂)) = 1 + (5/4)x_1 + x_2
POL(n__i) = 0
POL(activate(x₁)) = x_1
POL(and(x₁, x₂)) = x_2
POL(n__nil) = 4
POL(n__a) = 0
POL(tt) = 0
POL(ISPAL(x₁)) = 1/4 + (2)x_1
POL(AND(x₁, x₂)) = 2 + (1/2)x_2
POL(n__isList(x₁)) = 4 + (4)x_1
POL(nil) = 4
POL(ACTIVATE(x₁)) = 2 + (1/2)x_1
POL(a) = 0
POL(isList(x₁)) = 4 + (4)x_1
POL(ISNELIST(x₁)) = 2 + (2)x_1
POL(n__isPal(x₁)) = 1/4 + (4)x_1
POL(e) = 0
POL(n__e) = 0
POL(isNePal(x₁)) = (4)x_1
POL(n____(x₁, x₂)) = 1 + (5/4)x_1 + x_2
POL(o) = 1/4
POL(isQid(x₁)) = (4)x_1
POL(n__isNeList(x₁)) = (4)x_1
POL(isPal(x₁)) = 1/4 + (4)x_1
POL(n__o) = 1/4
POL(u) = 0
POL(isNeList(x₁)) = (4)x_1
POL(ISLIST(x₁)) = 4 + (2)x_1
POL(ISNEPAL(x₁)) = (2)x_1

The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented:

activate(n____(X1, X2)) → __(activate(X1), activate(X2))
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isPal(V) → isNePal(activate(V))
activate(n__isPal(X)) → isPal(X)
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n__u) → u
activate(X) → X
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
isList(n__nil) → tt
__(nil, X) → X
isNeList(V) → isQid(activate(V))
isPal(n__nil) → tt
isNePal(V) → isQid(activate(V))
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt
isList(X) → n__isList(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
isQid(n__u) → tt
e → n__e
a → n__a
isPal(X) → n__isPal(X)
isNeList(X) → n__isNeList(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

ACTIVATE(n__isList(X)) → ISLIST(X)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ACTIVATE(n__isNeList(X)) → ISNELIST(X)

The remaining pairs can at least be oriented weakly.

ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 5/4
POL(n__u) = 3
POL(__(x₁, x₂)) = 9/4 + x_1 + (1/4)x_2
POL(n__i) = 11/4
POL(activate(x₁)) = 0
POL(n__nil) = 2
POL(and(x₁, x₂)) = 4 + (3/4)x_1
POL(n__a) = 13/4
POL(tt) = 1/2
POL(AND(x₁, x₂)) = x_2
POL(n__isList(x₁)) = 0
POL(nil) = 7/4
POL(ACTIVATE(x₁)) = (1/4)x_1
POL(a) = 7/2
POL(isList(x₁)) = (1/4)x_1
POL(ISNELIST(x₁)) = (1/4)x_1
POL(n__isPal(x₁)) = 15/4 + (15/4)x_1
POL(e) = 7/4
POL(n__e) = 11/4
POL(isNePal(x₁)) = 3/2 + (13/4)x_1
POL(n____(x₁, x₂)) = (9/4)x_1 + (1/2)x_2
POL(o) = 5/4
POL(isQid(x₁)) = 13/4 + (5/4)x_1
POL(n__isNeList(x₁)) = 2 + x_1
POL(isPal(x₁)) = 15/4 + (4)x_1
POL(n__o) = 1
POL(u) = 5/4
POL(isNeList(x₁)) = 2 + (4)x_1

The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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 3 less nodes.