(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, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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)
U111(tt, V) → U121(isNeList(activate(V)))
U111(tt, V) → ISNELIST(activate(V))
U111(tt, V) → ACTIVATE(V)
U211(tt, V1, V2) → U221(isList(activate(V1)), activate(V2))
U211(tt, V1, V2) → ISLIST(activate(V1))
U211(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1, V2) → ACTIVATE(V2)
U221(tt, V2) → U231(isList(activate(V2)))
U221(tt, V2) → ISLIST(activate(V2))
U221(tt, V2) → ACTIVATE(V2)
U311(tt, V) → U321(isQid(activate(V)))
U311(tt, V) → ISQID(activate(V))
U311(tt, V) → ACTIVATE(V)
U411(tt, V1, V2) → U421(isList(activate(V1)), activate(V2))
U411(tt, V1, V2) → ISLIST(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V2) → U431(isNeList(activate(V2)))
U421(tt, V2) → ISNELIST(activate(V2))
U421(tt, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → U521(isNeList(activate(V1)), activate(V2))
U511(tt, V1, V2) → ISNELIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V1)
U511(tt, V1, V2) → ACTIVATE(V2)
U521(tt, V2) → U531(isList(activate(V2)))
U521(tt, V2) → ISLIST(activate(V2))
U521(tt, V2) → ACTIVATE(V2)
U611(tt, V) → U621(isQid(activate(V)))
U611(tt, V) → ISQID(activate(V))
U611(tt, V) → ACTIVATE(V)
U711(tt, V) → U721(isNePal(activate(V)))
U711(tt, V) → ISNEPAL(activate(V))
U711(tt, V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → U111(isPalListKind(activate(V)), activate(V))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U211(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISLIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISLIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → U311(isPalListKind(activate(V)), activate(V))
ISNELIST(V) → ISPALLISTKIND(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → U411(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNELIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISNELIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → U511(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNEPAL(V) → U611(isPalListKind(activate(V)), activate(V))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __1(activate(X1), activate(X2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
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
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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 3 SCCs with 41 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, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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
U11(x1, x2)  =  U11(x1)
tt  =  tt
U12(x1)  =  U12
isNeList(x1)  =  x1
activate(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2)
U22(x1, x2)  =  U22
isList(x1)  =  isList(x1)
U23(x1)  =  U23
U31(x1, x2)  =  x2
U32(x1)  =  U32
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x2, x3)
U42(x1, x2)  =  U42
U43(x1)  =  U43
U51(x1, x2, x3)  =  U51(x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  U53
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  x2
U72(x1)  =  U72
isNePal(x1)  =  isNePal(x1)
and(x1, x2)  =  and(x1, x2)
isPalListKind(x1)  =  x1
n__nil  =  n__nil
n____(x1, x2)  =  n____(x1, x2)
n__isPalListKind(x1)  =  x1
n__and(x1, x2)  =  n__and(x1, x2)
n__isPal(x1)  =  x1
isPal(x1)  =  x1
n__a  =  n__a
n__e  =  n__e
n__i  =  n__i
n__o  =  n__o
n__u  =  n__u
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u

Recursive path order with status [RPO].
Quasi-Precedence:
[nil, nnil] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[U111, isList1] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
isNePal1 > U612 > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[na, a] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[ne, e] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[ni, i] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[no, o] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]
[nu, u] > [2, tt, U12, U212, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, and2, n2, nand2]

Status:
_2: [1,2]
nil: multiset
U111: [1]
tt: multiset
U12: []
U212: multiset
U22: []
isList1: [1]
U23: []
U32: []
isQid: []
U412: [1,2]
U42: []
U43: []
U512: [1,2]
U522: [1,2]
U53: []
U612: multiset
U621: multiset
U72: []
isNePal1: multiset
and2: [1,2]
nnil: multiset
n2: [1,2]
nand2: [1,2]
na: multiset
ne: multiset
ni: multiset
no: multiset
nu: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset


The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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

(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, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
U711(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U611(isPalListKind(activate(V)), activate(V))
U611(tt, V) → ACTIVATE(V)
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
U711(tt, V) → ACTIVATE(V)
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1)  =  x1
n____(x1, x2)  =  n____(x1, x2)
n__isPalListKind(x1)  =  n__isPalListKind(x1)
ISPALLISTKIND(x1)  =  x1
AND(x1, x2)  =  AND(x2)
isPalListKind(x1)  =  isPalListKind(x1)
activate(x1)  =  x1
tt  =  tt
n__and(x1, x2)  =  n__and(x1, x2)
n__isPal(x1)  =  x1
ISPAL(x1)  =  x1
U711(x1, x2)  =  x2
ISNEPAL(x1)  =  x1
U611(x1, x2)  =  x2
and(x1, x2)  =  and(x1, x2)
isQid(x1)  =  isQid(x1)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x2)
U12(x1)  =  U12(x1)
isNeList(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2)
U22(x1, x2)  =  U22
isList(x1)  =  isList(x1)
U23(x1)  =  U23
U31(x1, x2)  =  x2
U32(x1)  =  U32
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x2)
U43(x1)  =  U43(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  U53
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  x2
U72(x1)  =  U72
isNePal(x1)  =  isNePal(x1)
n__nil  =  n__nil
isPal(x1)  =  x1
n__a  =  n__a
n__e  =  n__e
n__i  =  n__i
n__o  =  n__o
n__u  =  n__u
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u

Recursive path order with status [RPO].
Quasi-Precedence:
[n2, 2] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[nil, nnil] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[U111, isList1] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
isNePal1 > U612 > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[na, a] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[ne, e] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[ni, i] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[no, o] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]
[nu, u] > [nisPalListKind1, AND1, isPalListKind1, tt, nand2, and2, isQid1, U121, U212, U22, U23, U32, U413, U421, U431, U513, U522, U53, U621, U72]

Status:
n2: [1,2]
nisPalListKind1: multiset
AND1: multiset
isPalListKind1: multiset
tt: multiset
nand2: multiset
and2: multiset
isQid1: multiset
_2: [1,2]
nil: multiset
U111: multiset
U121: multiset
U212: [2,1]
U22: multiset
isList1: multiset
U23: multiset
U32: multiset
U413: multiset
U421: multiset
U431: multiset
U513: multiset
U522: multiset
U53: multiset
U612: multiset
U621: multiset
U72: multiset
isNePal1: multiset
nnil: multiset
na: multiset
ne: multiset
ni: multiset
no: multiset
nu: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset


The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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

(12) Obligation:

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

ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
U711(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U611(isPalListKind(activate(V)), activate(V))
U611(tt, V) → ACTIVATE(V)
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(V) → ACTIVATE(V)
U711(tt, V) → ACTIVATE(V)
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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 1 SCC with 2 less nodes.

(14) Obligation:

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

ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
U711(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U611(isPalListKind(activate(V)), activate(V))
U611(tt, V) → ACTIVATE(V)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
ISNEPAL(V) → ACTIVATE(V)
U711(tt, V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISPAL(V) → U711(isPalListKind(activate(V)), activate(V))
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISPAL(x1)  =  ISPAL(x1)
U711(x1, x2)  =  x2
isPalListKind(x1)  =  isPalListKind
activate(x1)  =  x1
tt  =  tt
ISNEPAL(x1)  =  x1
U611(x1, x2)  =  x2
ACTIVATE(x1)  =  x1
n__isPal(x1)  =  n__isPal(x1)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x1)
U12(x1)  =  U12
isNeList(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22
isList(x1)  =  isList(x1)
U23(x1)  =  U23
U31(x1, x2)  =  x1
U32(x1)  =  U32
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x1, x2)
U42(x1, x2)  =  U42
U43(x1)  =  U43
U51(x1, x2, x3)  =  U51(x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  U53
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1)  =  U72
isNePal(x1)  =  isNePal(x1)
and(x1, x2)  =  x2
n__nil  =  n__nil
n____(x1, x2)  =  n____(x1, x2)
n__isPalListKind(x1)  =  n__isPalListKind
n__and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
n__a  =  n__a
n__e  =  n__e
n__i  =  n__i
n__o  =  n__o
n__u  =  n__u
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u

Recursive path order with status [RPO].
Quasi-Precedence:
[nisPal1, isPal1] > ISPAL1 > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[nisPal1, isPal1] > U712 > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[nil, nnil] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[U111, isList1] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
isNePal1 > U612 > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[na, a] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[ne, e] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[ni, i] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[no, o] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]
[nu, u] > [isPalListKind, tt, 2, U12, U213, U22, U23, U32, isQid, U412, U42, U43, U512, U522, U53, U621, U72, n2, nisPalListKind]

Status:
ISPAL1: multiset
isPalListKind: multiset
tt: multiset
nisPal1: multiset
_2: [1,2]
nil: multiset
U111: [1]
U12: multiset
U213: [1,3,2]
U22: multiset
isList1: [1]
U23: multiset
U32: multiset
isQid: multiset
U412: [2,1]
U42: multiset
U43: multiset
U512: [1,2]
U522: [1,2]
U53: multiset
U612: multiset
U621: multiset
U712: multiset
U72: multiset
isNePal1: multiset
nnil: multiset
n2: [1,2]
nisPalListKind: multiset
isPal1: multiset
na: multiset
ne: multiset
ni: multiset
no: multiset
nu: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset


The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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

(16) Obligation:

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

U711(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U611(isPalListKind(activate(V)), activate(V))
U611(tt, V) → ACTIVATE(V)
ISNEPAL(V) → ACTIVATE(V)
U711(tt, V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(18) TRUE

(19) Obligation:

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

U111(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U411(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isList(activate(V1)), activate(V2))
U421(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U511(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNeList(activate(V1)), activate(V2))
U521(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U111(isPalListKind(activate(V)), activate(V))
ISLIST(n____(V1, V2)) → U211(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U211(tt, V1, V2) → U221(isList(activate(V1)), activate(V2))
U221(tt, V2) → ISLIST(activate(V2))
U211(tt, V1, V2) → ISLIST(activate(V1))
U511(tt, V1, V2) → ISNELIST(activate(V1))
U411(tt, 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
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U411(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isList(activate(V1)), activate(V2))
U421(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U511(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNeList(activate(V1)), activate(V2))
U521(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U211(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U211(tt, V1, V2) → U221(isList(activate(V1)), activate(V2))
U221(tt, V2) → ISLIST(activate(V2))
U211(tt, V1, V2) → ISLIST(activate(V1))
U511(tt, V1, V2) → ISNELIST(activate(V1))
U411(tt, V1, V2) → ISLIST(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x2)
tt  =  tt
ISNELIST(x1)  =  x1
activate(x1)  =  x1
n____(x1, x2)  =  n____(x1, x2)
U411(x1, x2, x3)  =  U411(x1, x2, x3)
and(x1, x2)  =  and(x2)
isPalListKind(x1)  =  x1
n__isPalListKind(x1)  =  x1
U421(x1, x2)  =  U421(x1, x2)
isList(x1)  =  isList(x1)
U511(x1, x2, x3)  =  U511(x1, x2, x3)
U521(x1, x2)  =  U521(x1, x2)
isNeList(x1)  =  x1
ISLIST(x1)  =  ISLIST(x1)
U211(x1, x2, x3)  =  U211(x2, x3)
U221(x1, x2)  =  U221(x1, x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  x2
U12(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x3)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  x1
isQid(x1)  =  x1
U41(x1, x2, x3)  =  U41(x3)
U42(x1, x2)  =  U42(x2)
U43(x1)  =  U43(x1)
U51(x1, x2, x3)  =  U51(x3)
U52(x1, x2)  =  U52(x2)
U53(x1)  =  U53(x1)
U61(x1, x2)  =  x2
U62(x1)  =  x1
U71(x1, x2)  =  U71(x2)
U72(x1)  =  U72(x1)
isNePal(x1)  =  x1
n__nil  =  n__nil
n__and(x1, x2)  =  n__and(x2)
n__isPal(x1)  =  n__isPal(x1)
isPal(x1)  =  isPal(x1)
n__a  =  n__a
n__e  =  n__e
n__i  =  n__i
n__o  =  n__o
n__u  =  n__u
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u

Recursive path order with status [RPO].
Quasi-Precedence:
[tt, nil, nnil, na, ne, ni, no, nu, a, e, i, o, u] > [U511, U521] > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]
[n2, 2] > U41^13 > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]
[n2, 2] > U51^13 > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]
[n2, 2] > U21^12 > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]
[n2, 2] > [U511, U521] > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]
[nisPal1, isPal1] > U711 > [U11^11, and1, U42^12, isList1, U52^12, ISLIST1, U22^12, U212, U221, U411, U421, U431, U531, U721, nand1]

Status:
U11^11: [1]
tt: multiset
n2: [1,2]
U41^13: multiset
and1: multiset
U42^12: multiset
isList1: multiset
U51^13: multiset
U52^12: [2,1]
ISLIST1: [1]
U21^12: multiset
U22^12: [2,1]
_2: [1,2]
nil: multiset
U212: multiset
U221: multiset
U411: [1]
U421: [1]
U431: [1]
U511: multiset
U521: multiset
U531: multiset
U711: multiset
U721: multiset
nnil: multiset
nand1: multiset
nisPal1: multiset
isPal1: multiset
na: multiset
ne: multiset
ni: multiset
no: multiset
nu: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset


The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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

(21) Obligation:

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

ISLIST(V) → U111(isPalListKind(activate(V)), activate(V))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isPal(X) → n__isPal(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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.

(22) DependencyGraphProof (EQUIVALENT transformation)

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

(23) TRUE