Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 RisEmptyProof (⇔)
↳6 TRUE

(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
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__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] > U212 > U221 > [tt, activate1, isQid1]
[2, n2] > U212 > isList1 > U111 > [tt, activate1, isQid1]
[2, n2] > U212 > isList1 > isNeList1 > U311 > [tt, activate1, isQid1]
[2, n2] > U412 > U421 > [tt, activate1, isQid1]
[2, n2] > U412 > isNeList1 > U311 > [tt, activate1, isQid1]
[2, n2] > U512 > isList1 > U111 > [tt, activate1, isQid1]
[2, n2] > U512 > isList1 > isNeList1 > U311 > [tt, activate1, isQid1]
[2, n2] > U512 > U521 > [tt, activate1, isQid1]
[2, n2] > U712 > U721 > [tt, activate1, isQid1]
[2, n2] > U712 > isPal1 > U811 > [tt, activate1, isQid1]
[2, n2] > U712 > isPal1 > isNePal1 > U611 > [tt, activate1, isQid1]
[nil, nnil] > [tt, activate1, isQid1]
[na, a] > [tt, activate1, isQid1]
[ne, e] > [tt, activate1, isQid1]
[ni, i] > [tt, activate1, isQid1]
[no, o] > [tt, activate1, isQid1]
[nu, u] > [tt, activate1, isQid1]

Status:
_2: [1,2]
nil: []
U111: [1]
tt: []
U212: [2,1]
U221: [1]
isList1: [1]
activate1: [1]
U311: [1]
U412: [2,1]
U421: [1]
isNeList1: [1]
U512: [2,1]
U521: [1]
U611: [1]
U712: [2,1]
U721: [1]
isPal1: [1]
U811: [1]
nnil: []
n2: [1,2]
isQid1: [1]
isNePal1: [1]
na: []
ne: []
ni: []
no: []
nu: []
a: []
e: []
i: []
o: []
u: []

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) → 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
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:

nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
nil > nnil
_2 > n2
a > na
e > ne
i > ni
o > no
u > nu

Status:
nil: []
nnil: []
_2: [1,2]
n2: [1,2]
a: []
na: []
e: []
ne: []
i: []
ni: []
o: []
no: []
u: []
nu: []

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u


(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