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, n____(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)) → __(activate(X1), activate(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:
Polynomial interpretation [POLO]:

POL(U11(x₁)) = x₁   
POL(U21(x₁, x₂)) = 4 + x₁ + x₂   
POL(U22(x₁)) = x₁   
POL(U31(x₁)) = 1 + x₁   
POL(U41(x₁, x₂)) = 4 + x₁ + x₂   
POL(U42(x₁)) = 1 + x₁   
POL(U51(x₁, x₂)) = 5 + x₁ + x₂   
POL(U52(x₁)) = 1 + x₁   
POL(U61(x₁)) = 1 + x₁   
POL(U71(x₁, x₂)) = 6 + x₁ + x₂   
POL(U72(x₁)) = 1 + x₁   
POL(U81(x₁)) = 1 + x₁   
POL(__(x₁, x₂)) = 6 + x₁ + x₂   
POL(a) = 1   
POL(activate(x₁)) = x₁   
POL(e) = 1   
POL(i) = 1   
POL(isList(x₁)) = 3 + x₁   
POL(isNeList(x₁)) = 2 + x₁   
POL(isNePal(x₁)) = 2 + x₁   
POL(isPal(x₁)) = 4 + x₁   
POL(isQid(x₁)) = x₁   
POL(n____(x₁, x₂)) = 6 + x₁ + x₂   
POL(n__a) = 1   
POL(n__e) = 1   
POL(n__i) = 1   
POL(n__nil) = 0   
POL(n__o) = 1   
POL(n__u) = 1   
POL(nil) = 0   
POL(o) = 1   
POL(tt) = 0   
POL(u) = 1   

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

__(X, nil) → X
__(nil, X) → X
U21(tt, V2) → U22(isList(activate(V2)))
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, n____(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

(2) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → tt
U22(tt) → 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)) → __(activate(X1), activate(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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁)) = 1 + x₁   
POL(U22(x₁)) = 1 + x₁   
POL(__(x₁, x₂)) = x₁ + x₂   
POL(a) = 0   
POL(activate(x₁)) = x₁   
POL(e) = 0   
POL(i) = 0   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__a) = 0   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__nil) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

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

U11(tt) → tt
U22(tt) → tt

(4) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
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)) → __(activate(X1), activate(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.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(a) = 1   
POL(activate(x₁)) = 1 + 2·x₁   
POL(e) = 1   
POL(i) = 0   
POL(n____(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(n__a) = 0   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__nil) = 1   
POL(n__o) = 0   
POL(n__u) = 2   
POL(nil) = 2   
POL(o) = 0   
POL(u) = 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))
nil → n__nil
a → n__a
e → n__e
activate(n__nil) → nil
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

(6) Obligation:

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

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

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = x₁ + x₂   
POL(a) = 0   
POL(activate(x₁)) = x₁   
POL(e) = 0   
POL(i) = 1   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__a) = 1   
POL(n__e) = 1   
POL(n__i) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(o) = 1   
POL(u) = 1   

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

i → n__i
o → n__o
u → n__u
activate(n__a) → a
activate(n__e) → e

(8) Obligation:

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

__(X1, X2) → n____(X1, X2)
activate(n____(X1, X2)) → __(activate(X1), activate(X2))

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(activate(x₁)) = 2·x₁   
POL(n____(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   

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

activate(n____(X1, X2)) → __(activate(X1), activate(X2))

(10) Obligation:

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

__(X1, X2) → n____(X1, X2)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + x₁ + x₂   
POL(n____(x₁, x₂)) = x₁ + x₂   

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)

(12) Obligation:

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

(13) RisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE