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, n____(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)) → __(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) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
__¹(__(X, Y), Z) → __¹(Y, Z)
U11¹(tt, V) → U12¹(isPalListKind(activate(V)), activate(V))
U11¹(tt, V) → ISPALLISTKIND(activate(V))
U11¹(tt, V) → ACTIVATE(V)
U12¹(tt, V) → U13¹(isNeList(activate(V)))
U12¹(tt, V) → ISNELIST(activate(V))
U12¹(tt, V) → ACTIVATE(V)
U21¹(tt, V1, V2) → U22¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U21¹(tt, V1, V2) → ISPALLISTKIND(activate(V1))
U21¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V1, V2) → U23¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U22¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U22¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V1, V2) → ACTIVATE(V1)
U23¹(tt, V1, V2) → U24¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U23¹(tt, V1, V2) → ACTIVATE(V2)
U23¹(tt, V1, V2) → ACTIVATE(V1)
U24¹(tt, V1, V2) → U25¹(isList(activate(V1)), activate(V2))
U24¹(tt, V1, V2) → ISLIST(activate(V1))
U24¹(tt, V1, V2) → ACTIVATE(V1)
U24¹(tt, V1, V2) → ACTIVATE(V2)
U25¹(tt, V2) → U26¹(isList(activate(V2)))
U25¹(tt, V2) → ISLIST(activate(V2))
U25¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, V) → U32¹(isPalListKind(activate(V)), activate(V))
U31¹(tt, V) → ISPALLISTKIND(activate(V))
U31¹(tt, V) → ACTIVATE(V)
U32¹(tt, V) → U33¹(isQid(activate(V)))
U32¹(tt, V) → ISQID(activate(V))
U32¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → U42¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → ISPALLISTKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → U43¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U43¹(tt, V1, V2) → U44¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → U45¹(isList(activate(V1)), activate(V2))
U44¹(tt, V1, V2) → ISLIST(activate(V1))
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U45¹(tt, V2) → U46¹(isNeList(activate(V2)))
U45¹(tt, V2) → ISNELIST(activate(V2))
U45¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → U52¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U51¹(tt, V1, V2) → ISPALLISTKIND(activate(V1))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V1, V2) → U53¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U52¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U52¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V1, V2) → ACTIVATE(V1)
U53¹(tt, V1, V2) → U54¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53¹(tt, V1, V2) → ISPALLISTKIND(activate(V2))
U53¹(tt, V1, V2) → ACTIVATE(V2)
U53¹(tt, V1, V2) → ACTIVATE(V1)
U54¹(tt, V1, V2) → U55¹(isNeList(activate(V1)), activate(V2))
U54¹(tt, V1, V2) → ISNELIST(activate(V1))
U54¹(tt, V1, V2) → ACTIVATE(V1)
U54¹(tt, V1, V2) → ACTIVATE(V2)
U55¹(tt, V2) → U56¹(isList(activate(V2)))
U55¹(tt, V2) → ISLIST(activate(V2))
U55¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V) → U62¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ISPALLISTKIND(activate(V))
U61¹(tt, V) → ACTIVATE(V)
U62¹(tt, V) → U63¹(isQid(activate(V)))
U62¹(tt, V) → ISQID(activate(V))
U62¹(tt, V) → ACTIVATE(V)
U71¹(tt, I, P) → U72¹(isPalListKind(activate(I)), activate(P))
U71¹(tt, I, P) → ISPALLISTKIND(activate(I))
U71¹(tt, I, P) → ACTIVATE(I)
U71¹(tt, I, P) → ACTIVATE(P)
U72¹(tt, P) → U73¹(isPal(activate(P)), activate(P))
U72¹(tt, P) → ISPAL(activate(P))
U72¹(tt, P) → ACTIVATE(P)
U73¹(tt, P) → U74¹(isPalListKind(activate(P)))
U73¹(tt, P) → ISPALLISTKIND(activate(P))
U73¹(tt, P) → ACTIVATE(P)
U81¹(tt, V) → U82¹(isPalListKind(activate(V)), activate(V))
U81¹(tt, V) → ISPALLISTKIND(activate(V))
U81¹(tt, V) → ACTIVATE(V)
U82¹(tt, V) → U83¹(isNePal(activate(V)))
U82¹(tt, V) → ISNEPAL(activate(V))
U82¹(tt, V) → ACTIVATE(V)
U91¹(tt, V2) → U92¹(isPalListKind(activate(V2)))
U91¹(tt, V2) → ISPALLISTKIND(activate(V2))
U91¹(tt, V2) → ACTIVATE(V2)
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U21¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
ISLIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → U31¹(isPalListKind(activate(V)), activate(V))
ISNELIST(V) → ISPALLISTKIND(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → U41¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
ISNELIST(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → U51¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → U71¹(isQid(activate(I)), activate(I), activate(P))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → U81¹(isPalListKind(activate(V)), activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → U91¹(isPalListKind(activate(V1)), activate(V2))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__a) → A
ACTIVATE(n__e) → E
ACTIVATE(n__i) → I
ACTIVATE(n__o) → O
ACTIVATE(n__u) → U

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 97 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2)  =  x1
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x1, x2)
tt  =  tt
U12(x1, x2)  =  U12(x1, x2)
isPalListKind(x1)  =  isPalListKind
activate(x1)  =  x1
U13(x1)  =  x1
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x1, x3)
U22(x1, x2, x3)  =  U22(x1, x3)
U23(x1, x2, x3)  =  U23(x1, x3)
U24(x1, x2, x3)  =  U24(x1, x3)
U25(x1, x2)  =  U25(x2)
isList(x1)  =  isList(x1)
U26(x1)  =  x1
U31(x1, x2)  =  U31
U32(x1, x2)  =  x1
U33(x1)  =  x1
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  x1
U42(x1, x2, x3)  =  x1
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2, x3)  =  U52(x2, x3)
U53(x1, x2, x3)  =  U53(x1, x2, x3)
U54(x1, x2, x3)  =  U54(x1, x2, x3)
U55(x1, x2)  =  U55
U56(x1)  =  U56
U61(x1, x2)  =  U61(x1, x2)
U62(x1, x2)  =  x1
U63(x1)  =  x1
U71(x1, x2, x3)  =  U71(x1)
U72(x1, x2)  =  U72
U73(x1, x2)  =  U73
isPal(x1)  =  isPal
U74(x1)  =  U74
U81(x1, x2)  =  U81
U82(x1, x2)  =  U82
U83(x1)  =  U83
isNePal(x1)  =  isNePal(x1)
U91(x1, x2)  =  x1
U92(x1)  =  U92
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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[₂, n₂] > [U21₂, U22₂, U23₂] > [U24₂, U25₁] > isList₁ > [U11₂, U12₂, isNeList₁, U31] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[₂, n₂] > U51₃ > U52₂ > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[₂, n₂] > U71₁ > U72 > [U73, U74] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[nil, n_nil] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[isPal, U81] > U82 > U83 > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
isNePal₁ > U61₂ > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
isNePal₁ > U71₁ > U72 > [U73, U74] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[n_i, i] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃
[n_o, o]
[n_u, u] > [tt, isPalListKind, isQid, U43, U44, U45, U46, U55, U56, U92, n_a, n_e, a, e] > U53₃ > U54₃

Status:

_₂: [1,2]
nil: []
U11₂: [2,1]
tt: []
U12₂: [2,1]
isPalListKind: []
isNeList₁: [1]
U21₂: [1,2]
U22₂: [1,2]
U23₂: [1,2]
U24₂: [2,1]
U25₁: [1]
isList₁: [1]
U31: []
isQid: []
U43: []
U44: []
U45: []
U46: []
U51₃: [2,3,1]
U52₂: [1,2]
U53₃: [2,1,3]
U54₃: [3,2,1]
U55: []
U56: []
U61₂: [1,2]
U71₁: [1]
U72: []
U73: []
isPal: []
U74: []
U81: []
U82: []
U83: []
isNePal₁: [1]
U92: []
n_nil: []
n₂: [1,2]
n_a: []
n_e: []
n_i: []
n_o: []
n_u: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

__(__(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, n____(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)) → __(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

(7) Obligation:

Q DP problem:
P is empty.
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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1)  =  ACTIVATE(x1)
n____(x1, x2)  =  n____(x1, x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x1, x2)
isPalListKind(x1)  =  isPalListKind
activate(x1)  =  x1
U13(x1)  =  U13
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x2)
U22(x1, x2, x3)  =  U22
U23(x1, x2, x3)  =  U23
U24(x1, x2, x3)  =  U24
U25(x1, x2)  =  U25
isList(x1)  =  isList(x1)
U26(x1)  =  U26
U31(x1, x2)  =  U31(x1, x2)
U32(x1, x2)  =  U32(x1)
U33(x1)  =  x1
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  x3
U42(x1, x2, x3)  =  x3
U43(x1, x2, x3)  =  x3
U44(x1, x2, x3)  =  x3
U45(x1, x2)  =  x2
U46(x1)  =  U46
U51(x1, x2, x3)  =  U51(x3)
U52(x1, x2, x3)  =  U52(x3)
U53(x1, x2, x3)  =  U53(x3)
U54(x1, x2, x3)  =  U54(x3)
U55(x1, x2)  =  U55(x2)
U56(x1)  =  x1
U61(x1, x2)  =  U61
U62(x1, x2)  =  U62
U63(x1)  =  x1
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
U73(x1, x2)  =  U73
isPal(x1)  =  isPal
U74(x1)  =  U74
U81(x1, x2)  =  U81
U82(x1, x2)  =  U82
U83(x1)  =  U83
isNePal(x1)  =  isNePal
U91(x1, x2)  =  U91
U92(x1)  =  U92
n__nil  =  n__nil
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

Lexicographic path order with status [LPO].
Quasi-Precedence:

ACTIVATE₁ > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > U21₁ > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U12₂ > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > U21₁ > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U26 > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > U51₁ > [U52₁, U53₁, U54₁] > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U12₂ > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > U51₁ > [U52₁, U53₁, U54₁] > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U26 > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > [U71, isNePal] > U61 > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n₂, ₂] > [U71, isNePal] > U72 > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[nil, n_nil] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
isNeList₁ > [U31₂, U32₁] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
isNeList₁ > U51₁ > [U52₁, U53₁, U54₁] > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U12₂ > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
isNeList₁ > U51₁ > [U52₁, U53₁, U54₁] > [U11₁, U22, U23, U24, U25, isList₁, U55₁] > U26 > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n_a, a] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n_e, e] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n_i, i] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n_o, o] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]
[n_u, u] > [tt, isPalListKind, U13, isQid, U46, U62, U73, isPal, U74, U81, U82, U83, U91, U92]

Status:

ACTIVATE₁: [1]
n₂: [1,2]
_₂: [1,2]
nil: []
U11₁: [1]
tt: []
U12₂: [2,1]
isPalListKind: []
U13: []
isNeList₁: [1]
U21₁: [1]
U22: []
U23: []
U24: []
U25: []
isList₁: [1]
U26: []
U31₂: [1,2]
U32₁: [1]
isQid: []
U46: []
U51₁: [1]
U52₁: [1]
U53₁: [1]
U54₁: [1]
U55₁: [1]
U61: []
U62: []
U71: []
U72: []
U73: []
isPal: []
U74: []
U81: []
U82: []
U83: []
isNePal: []
U91: []
U92: []
n_nil: []
n_a: []
n_e: []
n_i: []
n_o: []
n_u: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

__(__(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, n____(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)) → __(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

(12) Obligation:

Q DP problem:
P is empty.
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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U91¹(tt, V2) → ISPALLISTKIND(activate(V2))
ISPALLISTKIND(n____(V1, V2)) → U91¹(isPalListKind(activate(V1)), activate(V2))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(tt, V2) → ISPALLISTKIND(activate(V2))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x1, x2)
tt  =  tt
ISPALLISTKIND(x1)  =  x1
activate(x1)  =  x1
n____(x1, x2)  =  n____(x1, x2)
isPalListKind(x1)  =  x1
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  U11
U12(x1, x2)  =  U12
U13(x1)  =  U13
isNeList(x1)  =  isNeList
U21(x1, x2, x3)  =  U21
U22(x1, x2, x3)  =  U22
U23(x1, x2, x3)  =  U23
U24(x1, x2, x3)  =  U24
U25(x1, x2)  =  U25
isList(x1)  =  isList
U26(x1)  =  U26
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
isQid(x1)  =  x1
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  x1
U51(x1, x2, x3)  =  U51
U52(x1, x2, x3)  =  U52
U53(x1, x2, x3)  =  U53
U54(x1, x2, x3)  =  U54
U55(x1, x2)  =  U55
U56(x1)  =  U56
U61(x1, x2)  =  U61
U62(x1, x2)  =  U62
U63(x1)  =  U63
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
U73(x1, x2)  =  U73
isPal(x1)  =  isPal(x1)
U74(x1)  =  U74
U81(x1, x2)  =  U81(x2)
U82(x1, x2)  =  x1
U83(x1)  =  U83
isNePal(x1)  =  isNePal
U91(x1, x2)  =  x2
U92(x1)  =  x1
n__nil  =  n__nil
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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[nil, n_nil] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[U11, U12, U13, isList] > [U91^1₂, n₂, ₂, U21] > [isNeList, U41, U42, U43, U44, U45] > [U31, U32, U33] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[U11, U12, U13, isList] > [U91^1₂, n₂, ₂, U21] > [U22, U23] > [U24, U25] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[U11, U12, U13, isList] > [U91^1₂, n₂, ₂, U21] > [U71, U72, U73, U74] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[U61, U62, isNePal] > U63 > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[U61, U62, isNePal] > [U71, U72, U73, U74] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
isPal₁ > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
isPal₁ > U81₁
[n_e, e] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[n_i, i] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[n_o, o] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]
[n_u, u] > [tt, U26, U51, U52, U53, U54, U55, U56, U83, n_a, a]

Status:

U91^1₂: [1,2]
tt: []
n₂: [1,2]
_₂: [1,2]
nil: []
U11: []
U12: []
U13: []
isNeList: []
U21: []
U22: []
U23: []
U24: []
U25: []
isList: []
U26: []
U31: []
U32: []
U33: []
U41: []
U42: []
U43: []
U44: []
U45: []
U51: []
U52: []
U53: []
U54: []
U55: []
U56: []
U61: []
U62: []
U63: []
U71: []
U72: []
U73: []
isPal₁: [1]
U74: []
U81₁: [1]
U83: []
isNePal: []
n_nil: []
n_a: []
n_e: []
n_i: []
n_o: []
n_u: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

__(__(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, n____(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)) → __(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

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISPALLISTKIND(n____(V1, V2)) → U91¹(isPalListKind(activate(V1)), activate(V2))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(19) TRUE

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNEPAL(n____(I, n____(P, I))) → U71¹(isQid(activate(I)), activate(I), activate(P))
U71¹(tt, I, P) → U72¹(isPalListKind(activate(I)), activate(P))
U72¹(tt, P) → ISPAL(activate(P))
ISPAL(V) → U81¹(isPalListKind(activate(V)), activate(V))
U81¹(tt, V) → U82¹(isPalListKind(activate(V)), activate(V))
U82¹(tt, V) → ISNEPAL(activate(V))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNEPAL(n____(I, n____(P, I))) → U71¹(isQid(activate(I)), activate(I), activate(P))
U71¹(tt, I, P) → U72¹(isPalListKind(activate(I)), activate(P))
ISPAL(V) → U81¹(isPalListKind(activate(V)), activate(V))
U81¹(tt, V) → U82¹(isPalListKind(activate(V)), activate(V))
U82¹(tt, V) → ISNEPAL(activate(V))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1)  =  ISNEPAL(x1)
n____(x1, x2)  =  n____(x1, x2)
U71¹(x1, x2, x3)  =  U71¹(x1, x3)
isQid(x1)  =  isQid(x1)
activate(x1)  =  x1
tt  =  tt
U72¹(x1, x2)  =  U72¹(x2)
isPalListKind(x1)  =  x1
ISPAL(x1)  =  ISPAL(x1)
U81¹(x1, x2)  =  U81¹(x1, x2)
U82¹(x1, x2)  =  U82¹(x1, x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  x1
U12(x1, x2)  =  U12
U13(x1)  =  U13
isNeList(x1)  =  isNeList
U21(x1, x2, x3)  =  x2
U22(x1, x2, x3)  =  x2
U23(x1, x2, x3)  =  x2
U24(x1, x2, x3)  =  x2
U25(x1, x2)  =  x1
isList(x1)  =  x1
U26(x1)  =  U26
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
U51(x1, x2, x3)  =  U51
U52(x1, x2, x3)  =  U52
U53(x1, x2, x3)  =  U53
U54(x1, x2, x3)  =  U54
U55(x1, x2)  =  U55
U56(x1)  =  U56
U61(x1, x2)  =  U61(x2)
U62(x1, x2)  =  U62(x2)
U63(x1)  =  U63
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U72(x1, x2)  =  U72
U73(x1, x2)  =  U73
isPal(x1)  =  isPal(x1)
U74(x1)  =  U74
U81(x1, x2)  =  U81(x2)
U82(x1, x2)  =  U82(x2)
U83(x1)  =  x1
isNePal(x1)  =  isNePal(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92
n__nil  =  n__nil
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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n₂, isQid₁, ₂, U71₃, U72, U91₂] > U71^1₂ > [U72^1₁, ISPAL₁] > U81^1₂ > U82^1₂ > ISNEPAL₁
[n₂, isQid₁, ₂, U71₃, U72, U91₂] > [U41, U42] > U43 > [U44, U45] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n₂, isQid₁, ₂, U71₃, U72, U91₂] > [U51, U52] > [U53, U54, U55] > U56 > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n₂, isQid₁, ₂, U71₃, U72, U91₂] > U73 > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[nil, n_nil] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[isNeList, U31] > U32 > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[isNeList, U31] > [U41, U42] > U43 > [U44, U45] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[isNeList, U31] > [U51, U52] > [U53, U54, U55] > U56 > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
isPal₁ > U81₁ > U82₁ > [U61₁, isNePal₁] > U62₁ > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n_a, a] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n_i, i] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n_o, o] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]
[n_u, u] > [tt, U12, U13, U26, U33, U46, U63, U74, U92, n_e, e]

Status:

ISNEPAL₁: [1]
n₂: [1,2]
U71^1₂: [2,1]
isQid₁: [1]
tt: []
U72^1₁: [1]
ISPAL₁: [1]
U81^1₂: [2,1]
U82^1₂: [1,2]
_₂: [1,2]
nil: []
U12: []
U13: []
isNeList: []
U26: []
U31: []
U32: []
U33: []
U41: []
U42: []
U43: []
U44: []
U45: []
U46: []
U51: []
U52: []
U53: []
U54: []
U55: []
U56: []
U61₁: [1]
U62₁: [1]
U63: []
U71₃: [2,3,1]
U72: []
U73: []
isPal₁: [1]
U74: []
U81₁: [1]
U82₁: [1]
isNePal₁: [1]
U91₂: [1,2]
U92: []
n_nil: []
n_a: []
n_e: []
n_i: []
n_o: []
n_u: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

__(__(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, n____(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)) → __(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

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U72¹(tt, P) → ISPAL(activate(P))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(24) TRUE

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, V) → U12¹(isPalListKind(activate(V)), activate(V))
U12¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isList(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52¹(tt, V1, V2) → U53¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53¹(tt, V1, V2) → U54¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54¹(tt, V1, V2) → U55¹(isNeList(activate(V1)), activate(V2))
U55¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(n____(V1, V2)) → U21¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22¹(tt, V1, V2) → U23¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23¹(tt, V1, V2) → U24¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24¹(tt, V1, V2) → U25¹(isList(activate(V1)), activate(V2))
U25¹(tt, V2) → ISLIST(activate(V2))
U24¹(tt, V1, V2) → ISLIST(activate(V1))
U54¹(tt, V1, V2) → ISNELIST(activate(V1))
U44¹(tt, V1, V2) → ISLIST(activate(V1))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, V) → U12¹(isPalListKind(activate(V)), activate(V))
U12¹(tt, V) → ISNELIST(activate(V))
U41¹(tt, V1, V2) → U42¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isList(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U55¹(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U21¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U24¹(tt, V1, V2) → U25¹(isList(activate(V1)), activate(V2))
U24¹(tt, V1, V2) → ISLIST(activate(V1))
U54¹(tt, V1, V2) → ISNELIST(activate(V1))
U44¹(tt, V1, V2) → ISLIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  U11¹(x2)
tt  =  tt
U12¹(x1, x2)  =  U12¹(x2)
isPalListKind(x1)  =  isPalListKind
activate(x1)  =  x1
ISNELIST(x1)  =  x1
n____(x1, x2)  =  n____(x1, x2)
U41¹(x1, x2, x3)  =  U41¹(x2, x3)
U42¹(x1, x2, x3)  =  U42¹(x1, x2, x3)
U43¹(x1, x2, x3)  =  U43¹(x1, x2, x3)
U44¹(x1, x2, x3)  =  U44¹(x2, x3)
U45¹(x1, x2)  =  U45¹(x1, x2)
isList(x1)  =  x1
U51¹(x1, x2, x3)  =  U51¹(x2, x3)
U52¹(x1, x2, x3)  =  U52¹(x2, x3)
U53¹(x1, x2, x3)  =  U53¹(x2, x3)
U54¹(x1, x2, x3)  =  U54¹(x2, x3)
U55¹(x1, x2)  =  U55¹(x1, x2)
isNeList(x1)  =  x1
ISLIST(x1)  =  ISLIST(x1)
U21¹(x1, x2, x3)  =  U21¹(x2, x3)
U22¹(x1, x2, x3)  =  U22¹(x2, x3)
U23¹(x1, x2, x3)  =  U23¹(x2, x3)
U24¹(x1, x2, x3)  =  U24¹(x2, x3)
U25¹(x1, x2)  =  U25¹(x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1, x2)  =  x2
U12(x1, x2)  =  x2
U13(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2, x3)  =  U22(x3)
U23(x1, x2, x3)  =  U23(x3)
U24(x1, x2, x3)  =  U24(x3)
U25(x1, x2)  =  x2
U26(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1, x2)  =  x2
U33(x1)  =  x1
isQid(x1)  =  x1
U41(x1, x2, x3)  =  x2
U42(x1, x2, x3)  =  x2
U43(x1, x2, x3)  =  x2
U44(x1, x2, x3)  =  x2
U45(x1, x2)  =  x1
U46(x1)  =  U46
U51(x1, x2, x3)  =  U51(x2, x3)
U52(x1, x2, x3)  =  U52(x2, x3)
U53(x1, x2, x3)  =  U53(x2, x3)
U54(x1, x2, x3)  =  U54(x2, x3)
U55(x1, x2)  =  x2
U56(x1)  =  x1
U61(x1, x2)  =  U61
U62(x1, x2)  =  U62(x1)
U63(x1)  =  U63
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
U73(x1, x2)  =  U73
isPal(x1)  =  isPal(x1)
U74(x1)  =  U74
U81(x1, x2)  =  U81
U82(x1, x2)  =  U82(x1)
U83(x1)  =  x1
isNePal(x1)  =  isNePal
U91(x1, x2)  =  U91
U92(x1)  =  x1
n__nil  =  n__nil
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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n₂, U41^1₂, U21^1₂, ₂] > [U51^1₂, U52^1₂, U53^1₂, U54^1₂, U55^1₂] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > U45^1₂
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > U45^1₂
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n₂, U41^1₂, U21^1₂, ₂] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [isPalListKind, U42^1₃, U91] > U43^1₃ > U44^1₂ > U45^1₂
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[isPal₁, U81] > [U61, U71, U82₁, isNePal] > [U72, U73] > U74 > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n_a, a] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n_a, a] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n_a, a] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n_a, a] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n_a, a] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n_e, e] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n_e, e] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n_e, e] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n_e, e] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n_e, e] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n_i, i] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n_i, i] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n_i, i] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n_i, i] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n_i, i] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n_o, o] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n_o, o] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n_o, o] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n_o, o] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n_o, o] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁
[n_u, u] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U11^1₁, ISLIST₁, U25^1₁] > U12^1₁
[n_u, u] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U45^1₂
[n_u, u] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U21₃, U22₁, U23₁] > U24₁
[n_u, u] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > [U51₂, U52₂, U53₂, U54₂]
[n_u, u] > [tt, U22^1₂, U23^1₂, U24^1₂, nil, U46, U63, n_nil] > U62₁

Status:

U11^1₁: [1]
tt: []
U12^1₁: [1]
isPalListKind: []
n₂: [1,2]
U41^1₂: [1,2]
U42^1₃: [1,2,3]
U43^1₃: [1,2,3]
U44^1₂: [2,1]
U45^1₂: [1,2]
U51^1₂: [1,2]
U52^1₂: [1,2]
U53^1₂: [1,2]
U54^1₂: [1,2]
U55^1₂: [1,2]
ISLIST₁: [1]
U21^1₂: [1,2]
U22^1₂: [1,2]
U23^1₂: [1,2]
U24^1₂: [1,2]
U25^1₁: [1]
_₂: [1,2]
nil: []
U21₃: [3,1,2]
U22₁: [1]
U23₁: [1]
U24₁: [1]
U46: []
U51₂: [2,1]
U52₂: [2,1]
U53₂: [2,1]
U54₂: [2,1]
U61: []
U62₁: [1]
U63: []
U71: []
U72: []
U73: []
isPal₁: [1]
U74: []
U81: []
U82₁: [1]
isNePal: []
U91: []
n_nil: []
n_a: []
n_e: []
n_i: []
n_o: []
n_u: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

__(__(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, n____(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)) → __(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

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNELIST(n____(V1, V2)) → U41¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52¹(tt, V1, V2) → U53¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53¹(tt, V1, V2) → U54¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54¹(tt, V1, V2) → U55¹(isNeList(activate(V1)), activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U22¹(tt, V1, V2) → U23¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23¹(tt, V1, V2) → U24¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
U25¹(tt, V2) → ISLIST(activate(V2))

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, n____(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)) → __(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.
We have to consider all minimal (P,Q,R)-chains.

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 9 less nodes.

(29) TRUE