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, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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:
Combined order from the following AFS and order.
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x1, x2)
tt  =  tt
U12(x1)  =  U12(x1)
isNeList(x1)  =  isNeList(x1)
activate(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22(x1, x2)
isList(x1)  =  isList(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  x1
isQid(x1)  =  isQid(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1)  =  x1
isNePal(x1)  =  isNePal(x1)
and(x1, x2)  =  and(x1, x2)
isPalListKind(x1)  =  isPalListKind(x1)
n__nil  =  n__nil
n____(x1, x2)  =  n____(x1, x2)
n__isPalListKind(x1)  =  n__isPalListKind(x1)
n__and(x1, x2)  =  n__and(x1, x2)
n__isPal(x1)  =  n__isPal(x1)
isPal(x1)  =  isPal(x1)
n__a  =  n__a
n__e  =  n__e
n__i  =  n__i
n__o  =  n__o
n__u  =  n__u
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u

Recursive path order with status [RPO].
Quasi-Precedence:

[₂, n₂, n_isPal1, isPal₁] > U21₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U12₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U21₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U31₂ > isQid₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U21₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > [isPalListKind₁, n_{isPalListKind1}] > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U21₃ > [U22₂, isList₁, U52₂] > [and₂, n_and2]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U12₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U31₂ > isQid₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > [isPalListKind₁, n_{isPalListKind1}] > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > [U22₂, isList₁, U52₂] > [and₂, n_and2]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > U42₂ > [U11₂, isNeList₁] > U12₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > U42₂ > [U11₂, isNeList₁] > U31₂ > isQid₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U41₃ > U42₂ > [U11₂, isNeList₁] > [isPalListKind₁, n_{isPalListKind1}] > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U51₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U12₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U51₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > U31₂ > isQid₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U51₃ > [U22₂, isList₁, U52₂] > [U11₂, isNeList₁] > [isPalListKind₁, n_{isPalListKind1}] > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U51₃ > [U22₂, isList₁, U52₂] > [and₂, n_and2]
[₂, n₂, n_isPal1, isPal₁] > U71₂ > isNePal₁ > U61₂ > isQid₁ > [tt, n_a, n_o, a, o]
[₂, n₂, n_isPal1, isPal₁] > U71₂ > isNePal₁ > [isPalListKind₁, n_{isPalListKind1}] > [tt, n_a, n_o, a, o]
[nil, n_nil] > [tt, n_a, n_o, a, o]
[n_e, e] > [tt, n_a, n_o, a, o]
[n_i, i] > [tt, n_a, n_o, a, o]
[n_u, u] > [tt, n_a, n_o, a, o]

Status:

i: multiset
U52₂: [2,1]
_₂: [1,2]
n_u: multiset
U41₃: multiset
n_i: multiset
and₂: multiset
n_nil: multiset
U71₂: [2,1]
n_a: multiset
tt: multiset
isPalListKind₁: multiset
U12₁: multiset
U21₃: multiset
U51₃: [1,3,2]
nil: multiset
a: multiset
isList₁: [1]
U31₂: multiset
U22₂: [2,1]
U42₂: [1,2]
n_isPal1: [1]
e: multiset
n_e: multiset
isNePal₁: [1]
U11₂: [2,1]
n₂: [1,2]
o: multiset
isQid₁: multiset
n_and2: multiset
isPal₁: [1]
n_o: multiset
U61₂: multiset
u: multiset
isNeList₁: [1]
n_{isPalListKind1}: multiset

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, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U31(tt, V) → U32(isQid(activate(V)))
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U61(tt, V) → U62(isQid(activate(V)))
U71(tt, V) → U72(isNePal(activate(V)))
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__i) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n__u) → tt
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
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:

U23(tt) → tt
U32(tt) → tt
U43(tt) → tt
U53(tt) → tt
U62(tt) → tt
U72(tt) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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(U23(x₁)) = 1 + x₁   
POL(U32(x₁)) = 1 + x₁   
POL(U43(x₁)) = 1 + x₁   
POL(U53(x₁)) = 1 + x₁   
POL(U62(x₁)) = 1 + x₁   
POL(U72(x₁)) = 1 + x₁   
POL(__(x₁, x₂)) = x₁ + x₂   
POL(a) = 0   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(e) = 0   
POL(i) = 0   
POL(isPal(x₁)) = x₁   
POL(isPalListKind(x₁)) = x₁   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__a) = 0   
POL(n__and(x₁, x₂)) = x₁ + x₂   
POL(n__e) = 0   
POL(n__i) = 0   
POL(n__isPal(x₁)) = x₁   
POL(n__isPalListKind(x₁)) = x₁   
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:

U23(tt) → tt
U32(tt) → tt
U43(tt) → tt
U53(tt) → tt
U62(tt) → tt
U72(tt) → tt

(4) Obligation:

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
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)) = 2 + 2·x₁ + x₂   
POL(a) = 2   
POL(activate(x₁)) = 1 + 2·x₁   
POL(and(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(e) = 2   
POL(i) = 2   
POL(isPal(x₁)) = 2 + 2·x₁   
POL(isPalListKind(x₁)) = 2 + 2·x₁   
POL(n____(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(n__a) = 1   
POL(n__and(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(n__e) = 1   
POL(n__i) = 1   
POL(n__isPal(x₁)) = 1 + 2·x₁   
POL(n__isPalListKind(x₁)) = 1 + x₁   
POL(n__nil) = 2   
POL(n__o) = 1   
POL(n__u) = 2   
POL(nil) = 2   
POL(o) = 2   
POL(u) = 2   

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

isPalListKind(X) → n__isPalListKind(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
activate(n__nil) → nil
activate(n__isPalListKind(X)) → isPalListKind(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

(6) Obligation:

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

nil → n__nil
__(X1, X2) → n____(X1, X2)
and(X1, X2) → n__and(X1, X2)
u → n__u
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = x₁ + x₂   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__and(x₁, x₂)) = x₁ + x₂   
POL(n__nil) = 0   
POL(n__u) = 0   
POL(nil) = 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
u → n__u

(8) Obligation:

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

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

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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

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

activate(n__and(X1, X2)) → and(activate(X1), X2)

(10) Obligation:

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

__(X1, X2) → n____(X1, X2)
and(X1, X2) → n__and(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)) = x₁ + x₂   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = 1 + x₁ + x₂   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__and(x₁, x₂)) = x₁ + x₂   

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

and(X1, X2) → n__and(X1, X2)

(12) 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.

(13) 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))

(14) Obligation:

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

__(X1, X2) → n____(X1, X2)

Q is empty.

(15) 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)

(16) Obligation:

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

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) RisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE