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

POL(__(x₁, x₂)) = 3 + x₁ + x₂   
POL(a) = 2   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(e) = 3   
POL(i) = 2   
POL(isList(x₁)) = 3 + x₁   
POL(isNeList(x₁)) = 3 + x₁   
POL(isNePal(x₁)) = 1 + x₁   
POL(isPal(x₁)) = 2 + x₁   
POL(isQid(x₁)) = x₁   
POL(n____(x₁, x₂)) = 3 + x₁ + x₂   
POL(n__a) = 2   
POL(n__e) = 3   
POL(n__i) = 2   
POL(n__isList(x₁)) = 3 + x₁   
POL(n__isNeList(x₁)) = 3 + x₁   
POL(n__isPal(x₁)) = 2 + x₁   
POL(n__nil) = 0   
POL(n__o) = 2   
POL(n__u) = 3   
POL(nil) = 0   
POL(o) = 2   
POL(tt) = 2   
POL(u) = 3   

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

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 2 + x₁ + x₂   
POL(a) = 1   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(e) = 0   
POL(i) = 1   
POL(isList(x₁)) = 1 + x₁   
POL(isNeList(x₁)) = x₁   
POL(isPal(x₁)) = 1 + x₁   
POL(isQid(x₁)) = x₁   
POL(n____(x₁, x₂)) = 2 + x₁ + x₂   
POL(n__a) = 1   
POL(n__e) = 0   
POL(n__i) = 1   
POL(n__isList(x₁)) = 1 + x₁   
POL(n__isNeList(x₁)) = x₁   
POL(n__isPal(x₁)) = 1 + x₁   
POL(n__nil) = 0   
POL(n__o) = 1   
POL(n__u) = 0   
POL(nil) = 0   
POL(o) = 1   
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:

isList(V) → isNeList(activate(V))
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__i) → tt
isQid(n__o) → 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)
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)) → __(activate(X1), activate(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.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(a) = 0   
POL(activate(x₁)) = x₁   
POL(e) = 0   
POL(i) = 0   
POL(isList(x₁)) = x₁   
POL(isNeList(x₁)) = x₁   
POL(isPal(x₁)) = x₁   
POL(n____(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(n__a) = 0   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__isList(x₁)) = x₁   
POL(n__isNeList(x₁)) = x₁   
POL(n__isPal(x₁)) = x₁   
POL(n__nil) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(nil) = 0   
POL(o) = 0   
POL(u) = 0   

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))

(6) 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)
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__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.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(a) = 2   
POL(activate(x₁)) = 1 + 2·x₁   
POL(e) = 2   
POL(i) = 2   
POL(isList(x₁)) = 1 + 2·x₁   
POL(isNeList(x₁)) = 2 + 2·x₁   
POL(isPal(x₁)) = 2·x₁   
POL(n____(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(n__a) = 1   
POL(n__e) = 1   
POL(n__i) = 2   
POL(n__isList(x₁)) = x₁   
POL(n__isNeList(x₁)) = 1 + x₁   
POL(n__isPal(x₁)) = 2·x₁   
POL(n__nil) = 2   
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:

isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
a → n__a
e → n__e
activate(n__nil) → nil
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

(8) Obligation:

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
isPal(X) → n__isPal(X)
i → n__i
o → n__o
u → n__u
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = x₁ + x₂   
POL(activate(x₁)) = x₁   
POL(i) = 1   
POL(isList(x₁)) = x₁   
POL(isPal(x₁)) = 1 + x₁   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__i) = 0   
POL(n__isList(x₁)) = 1 + x₁   
POL(n__isPal(x₁)) = x₁   
POL(n__nil) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(nil) = 1   
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:

nil → n__nil
isPal(X) → n__isPal(X)
i → n__i
o → n__o
u → n__u
activate(n__isList(X)) → isList(X)

(10) 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.

(11) 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))

(12) Obligation:

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

__(X1, X2) → n____(X1, X2)

Q is empty.

(13) 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)

(14) Obligation:

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

(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