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

(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 > isList₁ > isNeList₁ > [activate₁, i, o, u] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U21₂ > isList₁ > isNeList₁ > U31₁ > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U41₂ > U42₁ > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U41₂ > isNeList₁ > [activate₁, i, o, u] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U41₂ > isNeList₁ > U31₁ > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U51₂ > isList₁ > isNeList₁ > [activate₁, i, o, u] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U51₂ > isList₁ > isNeList₁ > U31₁ > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U71₂ > isPal₁ > U81₁ > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[₂, n₂] > U71₂ > isPal₁ > isNePal₁ > [activate₁, i, o, u] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[nil, n_nil] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]
[n_a, a] > [U11₁, tt, U22₁, U52₁, U61₁, U72₁, isQid₁, n_e, n_i, n_o, n_u, e]

Status:

i: []
_₂: [1,2]
n_u: []
U81₁: [1]
n_i: []
activate₁: [1]
n_nil: []
U21₂: [1,2]
U51₂: [1,2]
U71₂: [2,1]
n_a: []
tt: []
U52₁: [1]
U72₁: [1]
U11₁: [1]
nil: []
a: []
isList₁: [1]
U61₁: [1]
e: []
n_e: []
isNePal₁: [1]
n₂: [1,2]
o: []
isQid₁: [1]
isPal₁: [1]
U22₁: [1]
n_o: []
U31₁: [1]
U41₂: [1,2]
u: []
U42₁: [1]
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
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
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

(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

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

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

nil > n_nil > n₂
_₂ > n₂
[a, n_a] > n₂
e > n_e > n₂

Status:

a: []
_₂: [2,1]
e: []
n_e: []
n_nil: []
n₂: [1,2]
nil: []
n_a: []

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)
e → n__e

(4) Obligation:

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

a → n__a

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

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

a > n_a

Status:

a: []
n_a: []

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

a → n__a

(6) Obligation:

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

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE