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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(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)) → U91(isPalListKind(activate(V1)), 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)
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:
Combined order from the following AFS and order.
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x1, x2)
tt  =  tt
U12(x1, x2)  =  U12(x1, x2)
isPalListKind(x1)  =  x1
activate(x1)  =  activate(x1)
U13(x1)  =  x1
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x1, x2, x3)
U23(x1, x2, x3)  =  U23(x1, x2, x3)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
U25(x1, x2)  =  U25(x1, x2)
isList(x1)  =  isList(x1)
U26(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  U33(x1)
isQid(x1)  =  isQid(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2, x3)  =  U42(x1, x2, x3)
U43(x1, x2, x3)  =  U43(x1, x2, x3)
U44(x1, x2, x3)  =  U44(x1, x2, x3)
U45(x1, x2)  =  U45(x1, x2)
U46(x1)  =  U46(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
U53(x1, x2, x3)  =  U53(x1, x2, x3)
U54(x1, x2, x3)  =  U54(x1, x2, x3)
U55(x1, x2)  =  U55(x1, x2)
U56(x1)  =  U56(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  U63(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
U73(x1, x2)  =  U73(x1, x2)
isPal(x1)  =  isPal(x1)
U74(x1)  =  U74(x1)
U81(x1, x2)  =  U81(x1, x2)
U82(x1, x2)  =  U82(x1, x2)
U83(x1)  =  x1
isNePal(x1)  =  isNePal(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
n__nil  =  n__nil
n____(x1, x2)  =  n____(x1, x2)
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] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U21₃ > U22₃ > U23₃ > U24₃ > U25₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U41₃ > U42₃ > U43₃ > U44₃ > U45₂ > U46₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > isList₁ > U11₂ > U12₂ > isNeList₁ > U31₂ > U32₂ > U33₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U51₃ > U52₃ > U53₃ > U54₃ > U55₂ > U56₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > U73₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > U73₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > U73₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U71₃ > U72₂ > isPal₁ > U81₂ > U82₂ > isNePal₁ > U61₂ > U62₂ > U63₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U91₂ > [activate₁, n_o, o] > a > n_a > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U91₂ > [activate₁, n_o, o] > e > n_e > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U91₂ > [activate₁, n_o, o] > i > [tt, isQid₁, U74₁, n_i, n_u, u]
[₂, n₂] > U91₂ > U92₁ > [tt, isQid₁, U74₁, n_i, n_u, u]
[nil, n_nil] > [tt, isQid₁, U74₁, n_i, n_u, u]

Status:

i: multiset
U62₂: multiset
U41₃: multiset
U32₂: [1,2]
U25₂: multiset
n_i: multiset
U24₃: multiset
U12₂: multiset
n_nil: multiset
U56₁: multiset
tt: multiset
U22₃: multiset
U21₃: multiset
U54₃: multiset
nil: multiset
U55₂: multiset
U31₂: [1,2]
e: multiset
isNePal₁: multiset
n₂: [1,2]
o: multiset
U53₃: multiset
n_o: multiset
U23₃: multiset
U81₂: multiset
U44₃: multiset
U33₁: multiset
U74₁: multiset
U71₃: multiset
U91₂: [2,1]
n_u: multiset
_₂: [1,2]
activate₁: [1]
U43₃: multiset
n_a: multiset
U42₃: multiset
U92₁: multiset
U45₂: [1,2]
U51₃: multiset
U73₂: multiset
a: multiset
isList₁: multiset
n_e: multiset
U72₂: multiset
U11₂: multiset
U63₁: multiset
isQid₁: multiset
U52₃: multiset
isPal₁: multiset
U82₂: multiset
U61₂: multiset
u: multiset
U46₁: multiset
isNeList₁: 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(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(I), activate(P))
isPal(V) → U81(isPalListKind(activate(V)), activate(V))
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isPalListKind(n__e) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__o) → tt
isPalListKind(n____(V1, V2)) → U91(isPalListKind(activate(V1)), activate(V2))
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
i → n__i
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:

U13(tt) → tt
U26(tt) → tt
U83(tt) → tt
isPalListKind(n__i) → tt
isPalListKind(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
o → n__o
u → n__u

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U13(x₁)) = 1 + x₁   
POL(U26(x₁)) = 1 + x₁   
POL(U83(x₁)) = 1 + x₁   
POL(__(x₁, x₂)) = 1 + x₁ + x₂   
POL(isPalListKind(x₁)) = 1 + x₁   
POL(n____(x₁, x₂)) = x₁ + x₂   
POL(n__i) = 0   
POL(n__nil) = 0   
POL(n__o) = 0   
POL(n__u) = 0   
POL(nil) = 1   
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:

U13(tt) → tt
U26(tt) → tt
U83(tt) → tt
isPalListKind(n__i) → tt
isPalListKind(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
o → n__o
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

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE