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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → 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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
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__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[2, n2] > and2 > [activate1, nu, a, o, u] > [nil, nnil]
[2, n2] > and2 > [activate1, nu, a, o, u] > na > [tt, isQid1]
[2, n2] > and2 > [activate1, nu, a, o, u] > no > [tt, isQid1]
[2, n2] > and2 > [activate1, nu, a, o, u] > e > ne
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > [activate1, nu, a, o, u] > [nil, nnil]
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > [activate1, nu, a, o, u] > na > [tt, isQid1]
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > [activate1, nu, a, o, u] > no > [tt, isQid1]
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > [activate1, nu, a, o, u] > e > ne
[2, n2] > [nisPal1, isPal1] > isNePal1 > [activate1, nu, a, o, u] > [nil, nnil]
[2, n2] > [nisPal1, isPal1] > isNePal1 > [activate1, nu, a, o, u] > na > [tt, isQid1]
[2, n2] > [nisPal1, isPal1] > isNePal1 > [activate1, nu, a, o, u] > no > [tt, isQid1]
[2, n2] > [nisPal1, isPal1] > isNePal1 > [activate1, nu, a, o, u] > e > ne
[ni, i] > [tt, isQid1]

Status:
i: []
nu: []
_2: [1,2]
ni: []
activate1: [1]
nnil: []
and2: [2,1]
na: []
tt: []
nisList1: [1]
nil: []
a: []
isList1: [1]
nisPal1: [1]
e: []
ne: []
isNePal1: [1]
o: []
n2: [1,2]
isQid1: [1]
nisNeList1: [1]
isPal1: [1]
no: []
u: []
isNeList1: [1]

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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → 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
a → n__a
e → n__e
o → n__o
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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


(2) Obligation:

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
i → n__i
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
nil > nnil > [n2, nisList1, nisNeList1, nisPal1]
_2 > [n2, nisList1, nisNeList1, nisPal1]
isList1 > [n2, nisList1, nisNeList1, nisPal1]
isNeList1 > [n2, nisList1, nisNeList1, nisPal1]
isPal1 > [n2, nisList1, nisNeList1, nisPal1]
i > ni > [n2, nisList1, nisNeList1, nisPal1]
u > nu > [n2, nisList1, nisNeList1, nisPal1]

Status:
i: []
isList1: [1]
_2: [2,1]
nu: []
nisPal1: [1]
ni: []
nnil: []
n2: [2,1]
nisNeList1: [1]
isPal1: [1]
u: []
isNeList1: [1]
nisList1: [1]
nil: []

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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
i → n__i
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