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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 RisEmptyProof (⇔)
↳8 TRUE

(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, __(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)) → __(X1, 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:
Lexicographic Path Order [LPO].
Precedence:
[2, n2] > and2 > activate1 > [nil, nnil] > tt
[2, n2] > and2 > activate1 > [ni, i]
[2, n2] > and2 > activate1 > e > ne > tt
[2, n2] > and2 > activate1 > u > nu > tt
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > activate1 > [nil, nnil] > tt
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > activate1 > [ni, i]
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > activate1 > e > ne > tt
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > activate1 > u > nu > tt
[2, n2] > [isList1, nisList1] > [isNeList1, nisNeList1] > isQid1 > tt
[2, n2] > [nisPal1, isPal1] > isNePal1 > activate1 > [nil, nnil] > tt
[2, n2] > [nisPal1, isPal1] > isNePal1 > activate1 > [ni, i]
[2, n2] > [nisPal1, isPal1] > isNePal1 > activate1 > e > ne > tt
[2, n2] > [nisPal1, isPal1] > isNePal1 > activate1 > u > nu > tt
[2, n2] > [nisPal1, isPal1] > isNePal1 > isQid1 > tt
[na, a] > tt
[no, o] > tt

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
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, __(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
e → n__e
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, 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


(2) 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
i → n__i
o → n__o

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:
[nil, nnil] > n2
_2 > n2
[isList1, nisList1] > n2
isNeList1 > nisNeList1 > n2
[isPal1, nisPal1] > n2
[a, na] > n2
[i, ni] > n2
[o, no] > n2

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)
isNeList(X) → n__isNeList(X)


(4) Obligation:

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

nil → n__nil
isList(X) → n__isList(X)
isPal(X) → n__isPal(X)
a → n__a
i → n__i
o → n__o

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:
nil > nnil > nisList1
isList1 > nisList1
isPal1 > nisPal1 > nisList1
a > na > nisList1
i > ni > nisList1
o > no > nisList1

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

nil → n__nil
isList(X) → n__isList(X)
isPal(X) → n__isPal(X)
a → n__a
i → n__i
o → n__o


(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