Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 PisEmptyProof (⇔)
    ↳9 TRUE
  ↳10 QDP

(0) Obligation:

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, __(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)) → __(X1, 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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

__1(__(X, Y), Z) → __1(X, __(Y, Z))
__1(__(X, Y), Z) → __1(Y, Z)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → ISNELIST(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → ISQID(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ISQID(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → ISNEPAL(activate(V))
ISPAL(V) → ACTIVATE(V)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __1(X1, 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

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, __(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)) → __(X1, 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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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, __(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)) → __(X1, 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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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.
Used ordering: Combined order from the following AFS and order.
__1(x1, x2)  =  x1
__(x1, x2)  =  __(x1, x2)
nil  =  nil
n____(x1, x2)  =  n____(x2)

Recursive Path Order [RPO].
Precedence:
nil > [2, n1]


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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, __(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)) → __(X1, 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.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

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

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