(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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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)
U211(tt, V2) → U221(isList(activate(V2)))
U211(tt, V2) → ISLIST(activate(V2))
U211(tt, V2) → ACTIVATE(V2)
U411(tt, V2) → U421(isNeList(activate(V2)))
U411(tt, V2) → ISNELIST(activate(V2))
U411(tt, V2) → ACTIVATE(V2)
U511(tt, V2) → U521(isList(activate(V2)))
U511(tt, V2) → ISLIST(activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U711(tt, P) → U721(isPal(activate(P)))
U711(tt, P) → ISPAL(activate(P))
U711(tt, P) → ACTIVATE(P)
ISLIST(V) → U111(isNeList(activate(V)))
ISLIST(V) → ISNELIST(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U211(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → U311(isQid(activate(V)))
ISNELIST(V) → ISQID(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → U411(isList(activate(V1)), 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)) → U511(isNeList(activate(V1)), activate(V2))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → U611(isQid(activate(V)))
ISNEPAL(V) → ISQID(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → U711(isQid(activate(I)), 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) → U811(isNePal(activate(V)))
ISPAL(V) → ISNEPAL(activate(V))
ISPAL(V) → ACTIVATE(V)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __1(X1, X2)
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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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 3 SCCs with 32 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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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: Recursive path order with status [RPO].
Quasi-Precedence:
_^12 > [2, n2]

Status:
_2: [1,2]
_^12: [1,2]
n2: [1,2]
nil: multiset


The following usable rules [FROCOS05] were oriented:

__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)
__(__(X, Y), Z) → __(X, __(Y, Z))

(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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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:

U711(tt, P) → ISPAL(activate(P))
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, __(P, I))) → U711(isQid(activate(I)), activate(P))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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.


U711(tt, P) → ISPAL(activate(P))
ISNEPAL(n____(I, __(P, I))) → U711(isQid(activate(I)), activate(P))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2)  =  U711(x2)
tt  =  tt
ISPAL(x1)  =  ISPAL(x1)
activate(x1)  =  activate(x1)
ISNEPAL(x1)  =  x1
n____(x1, x2)  =  n____(x1, x2)
__(x1, x2)  =  __(x1, x2)
isQid(x1)  =  isQid
u  =  u
n__u  =  n__u
o  =  o
n__o  =  n__o
i  =  i
n__i  =  n__i
nil  =  nil
e  =  e
n__e  =  n__e
a  =  a
n__a  =  n__a
n__nil  =  n__nil

Recursive path order with status [RPO].
Quasi-Precedence:
[n2, 2] > U71^11 > [tt, ISPAL1, activate1, i] > ni
[n2, 2] > isQid > [tt, ISPAL1, activate1, i] > ni
[u, nu] > [tt, ISPAL1, activate1, i] > ni
[o, no] > [tt, ISPAL1, activate1, i] > ni
[nil, nnil]
[e, ne] > [tt, ISPAL1, activate1, i] > ni
[a, na] > [tt, ISPAL1, activate1, i] > ni

Status:
i: multiset
a: multiset
_2: [1,2]
nu: multiset
e: multiset
ni: multiset
ne: multiset
activate1: [1]
nnil: multiset
n2: [1,2]
o: multiset
na: multiset
no: multiset
tt: multiset
ISPAL1: [1]
u: multiset
U71^11: multiset
isQid: multiset
nil: multiset


The following usable rules [FROCOS05] were oriented:

un__u
on__o
in__i
__(X, nil) → X
en__e
__(nil, X) → X
an__a
__(X1, X2) → n____(X1, X2)
__(__(X, Y), Z) → __(X, __(Y, Z))
niln__nil
activate(X) → X
activate(n__u) → u
activate(n__o) → o
activate(n__i) → i
activate(n__e) → e
activate(n__a) → a
activate(n____(X1, X2)) → __(X1, X2)
activate(n__nil) → nil

(12) Obligation:

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

ISPAL(V) → ISNEPAL(activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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

(15) Obligation:

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

U211(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U411(isList(activate(V1)), activate(V2))
U411(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → U211(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → U511(isNeList(activate(V1)), activate(V2))
U511(tt, V2) → 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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U411(isList(activate(V1)), activate(V2))
U411(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → U211(isList(activate(V1)), activate(V2))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → U511(isNeList(activate(V1)), activate(V2))
U511(tt, V2) → ISLIST(activate(V2))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x1, x2)
tt  =  tt
ISLIST(x1)  =  ISLIST(x1)
activate(x1)  =  activate(x1)
ISNELIST(x1)  =  ISNELIST(x1)
n____(x1, x2)  =  n____(x1, x2)
U411(x1, x2)  =  U411(x1, x2)
isList(x1)  =  isList
U511(x1, x2)  =  U511(x1, x2)
isNeList(x1)  =  isNeList
u  =  u
n__u  =  n__u
U11(x1)  =  U11(x1)
o  =  o
n__o  =  n__o
U21(x1, x2)  =  U21
U22(x1)  =  x1
i  =  i
n__i  =  n__i
__(x1, x2)  =  __(x1, x2)
nil  =  nil
e  =  e
n__e  =  n__e
a  =  a
n__a  =  n__a
n__nil  =  n__nil
isQid(x1)  =  isQid
U51(x1, x2)  =  U51
U52(x1)  =  U52
U41(x1, x2)  =  U41
U42(x1)  =  U42
U31(x1)  =  U31(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > isQid > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > U41 > U42 > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > U311 > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > u > nu > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > no > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > nil > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > ne > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [activate1, U41^12, o, e] > [a, na] > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > isQid > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > U41 > U42 > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > U311 > [tt, nnil, U52]
[n2, 2] > U21^12 > ISLIST1 > [isList, U21] > U111 > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > isQid > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > U41 > U42 > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > [ISNELIST1, isNeList, U51] > U311 > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > u > nu > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > no > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > nil > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > ne > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [activate1, U41^12, o, e] > [a, na] > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > isQid > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > U41 > U42 > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [isList, U21] > [ISNELIST1, isNeList, U51] > U311 > [tt, nnil, U52]
[n2, 2] > U51^12 > ISLIST1 > [isList, U21] > U111 > [tt, nnil, U52]
[i, ni] > [tt, nnil, U52]

Status:
i: multiset
U51^12: multiset
_2: [1,2]
nu: multiset
ni: multiset
isNeList: multiset
U42: multiset
U21^12: multiset
activate1: multiset
nnil: multiset
na: multiset
isList: []
tt: multiset
U41: []
U111: [1]
isQid: []
nil: multiset
U21: []
U51: multiset
a: multiset
ISNELIST1: multiset
e: multiset
U52: multiset
ne: multiset
o: multiset
n2: [1,2]
no: multiset
U41^12: multiset
U311: [1]
u: multiset
ISLIST1: multiset


The following usable rules [FROCOS05] were oriented:

un__u
U11(tt) → tt
on__o
U21(tt, V2) → U22(isList(activate(V2)))
in__i
__(X, nil) → X
en__e
__(nil, X) → X
an__a
__(X1, X2) → n____(X1, X2)
__(__(X, Y), Z) → __(X, __(Y, Z))
niln__nil
isQid(n__u) → tt
activate(X) → X
activate(n__u) → u
activate(n__o) → o
U51(tt, V2) → U52(isList(activate(V2)))
activate(n__i) → i
U52(tt) → tt
activate(n__e) → e
U41(tt, V2) → U42(isNeList(activate(V2)))
activate(n__a) → a
U42(tt) → tt
activate(n____(X1, X2)) → __(X1, X2)
U22(tt) → tt
U31(tt) → tt
activate(n__nil) → nil
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isList(n__nil) → tt
isList(V) → U11(isNeList(activate(V)))
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt

(17) 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
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
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.

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE