(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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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: SCNP Order with the following components:
Level mapping:
Top level AFS:
__1(x0, x1, x2)  =  __1(x0, x1)

Tags:
__1 has argument tags [0,1,2] and root tag 0

Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
__1(x1, x2)  =  __1
__(x1, x2)  =  __(x1, x2)
nil  =  nil
n____(x1, x2)  =  n____

Lexicographic path order with status [LPO].
Quasi-Precedence:
_^1 > _2
nil > _2
n > _2

Status:
_^1: []
_2: [2,1]
nil: []
n: []


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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(V) → ISNELIST(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 remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
ISLIST(x0, x1)  =  ISLIST(x0)
ISNELIST(x0, x1)  =  ISNELIST(x0, x1)
AND(x0, x1, x2)  =  AND(x0, x1, x2)
ISPAL(x0, x1)  =  ISPAL(x0, x1)
ISNEPAL(x0, x1)  =  ISNEPAL(x0, x1)

Tags:
ACTIVATE has argument tags [6,8] and root tag 2
ISLIST has argument tags [8,5] and root tag 1
ISNELIST has argument tags [8,8] and root tag 2
AND has argument tags [10,9,0] and root tag 4
ISPAL has argument tags [10,4] and root tag 0
ISNEPAL has argument tags [3,8] and root tag 4

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACTIVATE(x1)  =  x1
n__isList(x1)  =  n__isList(x1)
ISLIST(x1)  =  ISLIST(x1)
ISNELIST(x1)  =  ISNELIST
activate(x1)  =  activate(x1)
n__isNeList(x1)  =  n__isNeList(x1)
n____(x1, x2)  =  n____(x1, x2)
AND(x1, x2)  =  AND(x1, x2)
isList(x1)  =  isList(x1)
tt  =  tt
n__isPal(x1)  =  n__isPal(x1)
ISPAL(x1)  =  ISPAL(x1)
ISNEPAL(x1)  =  ISNEPAL
__(x1, x2)  =  __(x1, x2)
isQid(x1)  =  isQid
isNeList(x1)  =  isNeList(x1)
n__nil  =  n__nil
nil  =  nil
and(x1, x2)  =  and(x1, x2)
isPal(x1)  =  isPal(x1)
isNePal(x1)  =  isNePal(x1)
n__a  =  n__a
a  =  a
n__e  =  n__e
e  =  e
n__i  =  n__i
i  =  i
n__o  =  n__o
o  =  o
n__u  =  n__u
u  =  u

Lexicographic path order with status [LPO].
Quasi-Precedence:
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > nil > nnil
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > e > ne
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > i > ni
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > o > no
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > u > nu
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > nil > nnil
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > e > ne
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > i > ni
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > o > no
[n2, 2] > [nisList1, ISLIST1, isList1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > u > nu
[n2, 2] > [nisList1, ISLIST1, isList1] > AND2
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > nil > nnil
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > e > ne
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > i > ni
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > o > no
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > u > nu
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > nil > nnil
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > e > ne
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > i > ni
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > o > no
[n2, 2] > [nisPal1, isPal1] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > u > nu
[n2, 2] > [nisPal1, isPal1] > ISPAL1 > ISNEPAL > activate1 > nil > nnil
[n2, 2] > [nisPal1, isPal1] > ISPAL1 > ISNEPAL > activate1 > e > ne
[n2, 2] > [nisPal1, isPal1] > ISPAL1 > ISNEPAL > activate1 > i > ni
[n2, 2] > [nisPal1, isPal1] > ISPAL1 > ISNEPAL > activate1 > o > no
[n2, 2] > [nisPal1, isPal1] > ISPAL1 > ISNEPAL > activate1 > u > nu
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > nil > nnil
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > e > ne
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > i > ni
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > o > no
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > ISNELIST > activate1 > u > nu
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > nil > nnil
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > e > ne
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > i > ni
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > o > no
[na, a] > [nisNeList1, tt, isQid, isNeList1, isNePal1] > and2 > activate1 > u > nu

Status:
nisList1: [1]
ISLIST1: [1]
ISNELIST: []
activate1: [1]
nisNeList1: [1]
n2: [1,2]
AND2: [1,2]
isList1: [1]
tt: []
nisPal1: [1]
ISPAL1: [1]
ISNEPAL: []
_2: [1,2]
isQid: []
isNeList1: [1]
nnil: []
nil: []
and2: [2,1]
isPal1: [1]
isNePal1: [1]
na: []
a: []
ne: []
e: []
ni: []
i: []
no: []
o: []
nu: []
u: []


The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

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

ISNELIST(V) → ACTIVATE(V)

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
niln__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__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.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE