(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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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 > U222 > isList1 > U112 > U121
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > nil > nnil
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > tt
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > a > na
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > e > ne
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > i > ni
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > o > no
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > activate1 > u > nu
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > U321
[2, n2] > U213 > U222 > isList1 > U112 > isNeList1 > U312 > isQid1 > tt
[2, n2] > U213 > U222 > U231
[2, n2] > U413 > isList1 > U112 > U121
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > nil > nnil
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > tt
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > a > na
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > e > ne
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > i > ni
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > o > no
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > activate1 > u > nu
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > U321
[2, n2] > U413 > isList1 > U112 > isNeList1 > U312 > isQid1 > tt
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > nil > nnil
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > isPalListKind1 > tt
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > a > na
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > e > ne
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > i > ni
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > o > no
[2, n2] > U413 > U422 > isNeList1 > U312 > activate1 > u > nu
[2, n2] > U413 > U422 > isNeList1 > U312 > U321
[2, n2] > U413 > U422 > isNeList1 > U312 > isQid1 > tt
[2, n2] > U413 > U422 > U431
[2, n2] > U513 > U522 > isList1 > U112 > U121
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > nil > nnil
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > tt
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > a > na
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > e > ne
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > i > ni
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > o > no
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > activate1 > u > nu
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > U321
[2, n2] > U513 > U522 > isList1 > U112 > isNeList1 > U312 > isQid1 > tt
[2, n2] > U513 > U522 > U531
[2, n2] > isPal1 > U712 > U721
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > nil > nnil
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > isPalListKind1 > tt
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > a > na
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > e > ne
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > i > ni
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > o > no
[2, n2] > isPal1 > U712 > isNePal1 > U612 > activate1 > u > nu
[2, n2] > isPal1 > U712 > isNePal1 > U612 > isQid1 > tt
[2, n2] > isPal1 > U712 > isNePal1 > U612 > U621
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > nil > nnil
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > isPalListKind1 > tt
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > isPalListKind1 > nisPalListKind1
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > a > na
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > e > ne
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > i > ni
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > o > no
[2, n2] > isPal1 > U712 > isNePal1 > [and2, nand2] > activate1 > u > nu

Status:
nand2: [1,2]
_2: [1,2]
and2: [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(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, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(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
isPalListKind(X) → n__isPalListKind(X)
an__a
en__e
in__i
on__o
un__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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)
and(X1, X2) → n__and(X1, X2)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
_2 > n2
and2 > nand2

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)
and(X1, X2) → n__and(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