(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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(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)) → U91(isPalListKind(activate(V1)), 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)
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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > U131
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > isPalListKind1 > tt
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > nil > nnil
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > a > na
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > e > ne
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > i > ni
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > o > no
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > u > nu
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > U331
[2, n2] > U213 > U223 > U233 > U243 > U252 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > isQid1 > tt
[2, n2] > U213 > U223 > U233 > U243 > U252 > U261
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > U131
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > isPalListKind1 > tt
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > nil > nnil
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > a > na
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > e > ne
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > i > ni
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > o > no
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > u > nu
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > U331
[2, n2] > U413 > U423 > U433 > U443 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > isQid1 > tt
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > isPalListKind1 > tt
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > nil > nnil
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > a > na
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > e > ne
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > i > ni
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > o > no
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > activate1 > u > nu
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > U331
[2, n2] > U413 > U423 > U433 > U443 > U452 > isNeList1 > U312 > U322 > isQid1 > tt
[2, n2] > U413 > U423 > U433 > U443 > U452 > U461
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > U131
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > isPalListKind1 > tt
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > nil > nnil
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > a > na
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > e > ne
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > i > ni
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > o > no
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > activate1 > u > nu
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > U331
[2, n2] > U513 > U523 > U533 > U543 > U552 > isList1 > U112 > U122 > isNeList1 > U312 > U322 > isQid1 > tt
[2, n2] > U513 > U523 > U533 > U543 > U552 > U561
[2, n2] > U713 > U722 > U732 > isPalListKind1 > tt
[2, n2] > U713 > U722 > U732 > activate1 > nil > nnil
[2, n2] > U713 > U722 > U732 > activate1 > a > na
[2, n2] > U713 > U722 > U732 > activate1 > e > ne
[2, n2] > U713 > U722 > U732 > activate1 > i > ni
[2, n2] > U713 > U722 > U732 > activate1 > o > no
[2, n2] > U713 > U722 > U732 > activate1 > u > nu
[2, n2] > U713 > U722 > U732 > U741
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > U831
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > isPalListKind1 > tt
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > nil > nnil
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > a > na
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > e > ne
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > i > ni
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > o > no
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > activate1 > u > nu
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > isQid1 > tt
[2, n2] > U713 > U722 > isPal1 > U812 > U822 > isNePal1 > U612 > U622 > U631
[2, n2] > U912 > isPalListKind1 > tt
[2, n2] > U912 > activate1 > nil > nnil
[2, n2] > U912 > activate1 > a > na
[2, n2] > U912 > activate1 > e > ne
[2, n2] > U912 > activate1 > i > ni
[2, n2] > U912 > activate1 > o > no
[2, n2] > U912 > activate1 > u > nu
[2, n2] > U912 > U921

Status:
_2: [1,2]
n2: [1,2]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(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)) → U91(isPalListKind(activate(V1)), activate(V2))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
niln__nil
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


(2) Obligation:

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

__(X1, X2) → n____(X1, X2)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
_2 > n2

Status:
trivial

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

__(X1, X2) → n____(X1, X2)


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE