Termination w.r.t. Q proof of /tmp/tmpp3HMng/PALINDROME_complete-noand

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 PisEmptyProof (⇔)

    ↳14 TRUE

  ↳15 QDP

    ↳16 QDPOrderProof (⇔)

    ↳17 QDP

    ↳18 DependencyGraphProof (⇔)

    ↳19 TRUE

  ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 DependencyGraphProof (⇔)

    ↳24 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
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) 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, __(P, I))) → U71¹(isQid(activate(I)), activate(I), activate(P))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(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(X1, 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, __(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.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 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, __(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.
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) = __¹(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = x2
tt = tt
U12(x1, x2) = x2
isPalListKind(x1) = isPalListKind(x1)
activate(x1) = x1
U13(x1) = x1
isNeList(x1) = x1
U21(x1, x2, x3) = U21(x3)
U22(x1, x2, x3) = U22(x3)
U23(x1, x2, x3) = x1
U24(x1, x2, x3) = U24
U25(x1, x2) = U25
isList(x1) = x1
U26(x1) = U26
U31(x1, x2) = x2
U32(x1, x2) = x2
U33(x1) = x1
isQid(x1) = x1
U41(x1, x2, x3) = U41(x2, x3)
U42(x1, x2, x3) = U42(x2, x3)
U43(x1, x2, x3) = U43(x2, x3)
U44(x1, x2, x3) = U44(x3)
U45(x1, x2) = U45(x2)
U46(x1) = x1
U51(x1, x2, x3) = U51(x1)
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(x1, x2)
U62(x1, x2) = U62(x1, x2)
U63(x1) = x1
U71(x1, x2, x3) = U71(x2)
U72(x1, x2) = x1
U73(x1, x2) = x1
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}, n_{_{_{_₂}}}] > __^1₁ > [U45₁, U62₂]
[_{_₂}, n_{_{_{_₂}}}] > [isPalListKind₁, U21₁, U22₁, U51₁, U52, U53, U54, U71₁] > U55 > U56 > [nil, tt, U24, U25, U26, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_u}, a, u] > [U41₂, U42₂, U43₂, U44₁] > [U45₁, U62₂]
isNePal₁ > U61₂ > [isPalListKind₁, U21₁, U22₁, U51₁, U52, U53, U54, U71₁] > U55 > U56 > [nil, tt, U24, U25, U26, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_u}, a, u] > [U41₂, U42₂, U43₂, U44₁] > [U45₁, U62₂]
[n_{_e}, e] > [nil, tt, U24, U25, U26, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_u}, a, u] > [U41₂, U42₂, U43₂, U44₁] > [U45₁, U62₂]
[n_{_i}, i] > [nil, tt, U24, U25, U26, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_u}, a, u] > [U41₂, U42₂, U43₂, U44₁] > [U45₁, U62₂]
[n_{_o}, o] > [nil, tt, U24, U25, U26, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_u}, a, u] > [U41₂, U42₂, U43₂, U44₁] > [U45₁, U62₂]

Status:

__^1₁: [1]
__₂: [1,2]
nil: []
tt: []
isPalListKind₁: [1]
U21₁: [1]
U22₁: [1]
U24: []
U25: []
U26: []
U41₂: [2,1]
U42₂: [2,1]
U43₂: [2,1]
U44₁: [1]
U45₁: [1]
U51₁: [1]
U52: []
U53: []
U54: []
U55: []
U56: []
U61₂: [1,2]
U62₂: [1,2]
U71₁: [1]
isPal: []
U74: []
U81: []
U82: []
U83: []
isNePal₁: [1]
U92: []
n_{_nil}: []
n_{_{_{_₂}}}: [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, __(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

(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, __(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.
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:

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, __(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.
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.

U91¹(tt, V2) → ISPALLISTKIND(activate(V2))
ISPALLISTKIND(n____(V1, V2)) → U91¹(isPalListKind(activate(V1)), 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) = ISPALLISTKIND(x1)
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
isPalListKind(x1) = x1
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11(x1, x2)
U12(x1, x2) = U12
U13(x1) = U13
isNeList(x1) = isNeList
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2, x3) = U22(x2, x3)
U23(x1, x2, x3) = U23(x2)
U24(x1, x2, x3) = U24(x2)
U25(x1, x2) = x1
isList(x1) = isList(x1)
U26(x1) = U26
U31(x1, x2) = U31
U32(x1, x2) = U32
U33(x1) = U33
isQid(x1) = isQid(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) = 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) = x1
U62(x1, x2) = U62
U63(x1) = U63
U71(x1, x2, x3) = U71(x2)
U72(x1, x2) = U72(x1)
U73(x1, x2) = x1
isPal(x1) = isPal
U74(x1) = U74
U81(x1, x2) = U81
U82(x1, x2) = U82
U83(x1) = U83
isNePal(x1) = x1
U91(x1, x2) = x1
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:

[n_{_{_{_₂}}}, _{_₂}] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[n_{_{_{_₂}}}, _{_₂}] > [U71₁, U72₁]
[nil, n_{_nil}] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[nil, n_{_nil}] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[nil, n_{_nil}] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[isNeList, U31, U32, U33] > isQid₁ > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[isNeList, U31, U32, U33] > isQid₁ > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[isNeList, U31, U32, U33] > isQid₁ > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[isNeList, U31, U32, U33] > [U41, U42, U43] > [U44, U45] > U46 > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[isNeList, U31, U32, U33] > [U41, U42, U43] > [U44, U45] > U46 > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[isNeList, U31, U32, U33] > [U41, U42, U43] > [U44, U45] > U46 > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[isNeList, U31, U32, U33] > [U51, U52, U53] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[isNeList, U31, U32, U33] > [U51, U52, U53] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[isNeList, U31, U32, U33] > [U51, U52, U53] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[n_{_a}, a] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[n_{_a}, a] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[n_{_a}, a] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[n_{_e}, e] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[n_{_e}, e] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[n_{_e}, e] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]
[n_{_u}, u] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U91^1₂, ISPALLISTKIND₁]
[n_{_u}, u] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U21₃, U22₂, U23₁] > [U24₁, isList₁] > U11₂
[n_{_u}, u] > [tt, U12, U13, U26, U54, U55, U56, U62, U63, isPal, U74, U81, U82, U83, U92, n_{_i}, n_{_o}, i, o] > [U71₁, U72₁]

Status:

U91^1₂: [2,1]
tt: []
ISPALLISTKIND₁: [1]
n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
nil: []
U11₂: [2,1]
U12: []
U13: []
isNeList: []
U21₃: [2,3,1]
U22₂: [1,2]
U23₁: [1]
U24₁: [1]
isList₁: [1]
U26: []
U31: []
U32: []
U33: []
isQid₁: [1]
U41: []
U42: []
U43: []
U44: []
U45: []
U46: []
U51: []
U52: []
U53: []
U54: []
U55: []
U56: []
U62: []
U63: []
U71₁: [1]
U72₁: [1]
isPal: []
U74: []
U81: []
U82: []
U83: []
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, __(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

(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, __(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.
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:

ISNEPAL(n____(I, __(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, __(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.
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.

ISNEPAL(n____(I, __(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))

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

[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U71^1₃ > U72^1₁ > ISPAL₁ > U81^1₂ > [activate₁, U82^1₁, U32₁] > [nil, n_{_nil}] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U71^1₃ > U72^1₁ > ISPAL₁ > U81^1₂ > [activate₁, U82^1₁, U32₁] > [n_{_a}, a] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U71^1₃ > U72^1₁ > ISPAL₁ > U81^1₂ > [activate₁, U82^1₁, U32₁] > [n_{_e}, e] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U71^1₃ > U72^1₁ > ISPAL₁ > U81^1₂ > [activate₁, U82^1₁, U32₁] > [n_{_i}, i] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U71^1₃ > U72^1₁ > ISPAL₁ > U81^1₂ > [activate₁, U82^1₁, U32₁] > [n_{_o}, o] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [activate₁, U82^1₁, U32₁] > [nil, n_{_nil}] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [activate₁, U82^1₁, U32₁] > [n_{_a}, a] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [activate₁, U82^1₁, U32₁] > [n_{_e}, e] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [activate₁, U82^1₁, U32₁] > [n_{_i}, i] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [activate₁, U82^1₁, U32₁] > [n_{_o}, o] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [U22₃, U23] > U24 > [U11, isList₁] > U12₁ > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U21₃ > [U22₃, U23] > U24 > U25 > U26 > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > [U41, U42] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U61 > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U73 > [activate₁, U82^1₁, U32₁] > [nil, n_{_nil}] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U73 > [activate₁, U82^1₁, U32₁] > [n_{_a}, a] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U73 > [activate₁, U82^1₁, U32₁] > [n_{_e}, e] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U73 > [activate₁, U82^1₁, U32₁] > [n_{_i}, i] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > U73 > [activate₁, U82^1₁, U32₁] > [n_{_o}, o] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[n_{_{_{_₂}}}, _{_₂}, U71₃, U72₁, isNePal₁] > [isPal₁, U81, U82] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U31₁ > [activate₁, U82^1₁, U32₁] > [nil, n_{_nil}] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U31₁ > [activate₁, U82^1₁, U32₁] > [n_{_a}, a] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U31₁ > [activate₁, U82^1₁, U32₁] > [n_{_e}, e] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U31₁ > [activate₁, U82^1₁, U32₁] > [n_{_i}, i] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U31₁ > [activate₁, U82^1₁, U32₁] > [n_{_o}, o] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > [U41, U42] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [activate₁, U82^1₁, U32₁] > [nil, n_{_nil}] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [activate₁, U82^1₁, U32₁] > [n_{_a}, a] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [activate₁, U82^1₁, U32₁] > [n_{_e}, e] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [activate₁, U82^1₁, U32₁] > [n_{_i}, i] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [activate₁, U82^1₁, U32₁] > [n_{_o}, o] > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > [U11, isList₁] > U12₁ > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]
[isNeList₁, U51] > U52 > [U53, U54] > U55 > U56 > [tt, isPalListKind, U13, U45, U46, U62₁, U63, U83, n_{_u}, u]

Status:

n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
U71^1₃: [1,2,3]
activate₁: [1]
tt: []
U72^1₁: [1]
isPalListKind: []
ISPAL₁: [1]
U81^1₂: [1,2]
U82^1₁: [1]
nil: []
U11: []
U12₁: [1]
U13: []
isNeList₁: [1]
U21₃: [2,3,1]
U22₃: [1,2,3]
U23: []
U24: []
U25: []
isList₁: [1]
U26: []
U31₁: [1]
U32₁: [1]
U41: []
U42: []
U45: []
U46: []
U51: []
U52: []
U53: []
U54: []
U55: []
U56: []
U61: []
U62₁: [1]
U63: []
U71₃: [1,2,3]
U72₁: [1]
U73: []
isPal₁: [1]
U81: []
U82: []
U83: []
isNePal₁: [1]
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, __(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

(17) Obligation:

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

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, __(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.
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:

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, __(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.
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.

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))
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))
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 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) = ISNELIST(x1)
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x1, x2, x3)
U42¹(x1, x2, x3) = U42¹(x1, x2, x3)
U43¹(x1, x2, x3) = U43¹(x2, x3)
U44¹(x1, x2, x3) = U44¹(x1, x2, x3)
U45¹(x1, x2) = U45¹(x2)
isList(x1) = isList(x1)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2, x3) = U52¹(x2, x3)
U53¹(x1, x2, x3) = U53¹(x1, x2, x3)
U54¹(x1, x2, x3) = U54¹(x2, x3)
U55¹(x1, x2) = U55¹(x1, x2)
isNeList(x1) = isNeList(x1)
ISLIST(x1) = ISLIST(x1)
U21¹(x1, x2, x3) = U21¹(x1, x2, x3)
U22¹(x1, x2, x3) = U22¹(x1, x2, x3)
U23¹(x1, x2, x3) = U23¹(x1, x2, x3)
U24¹(x1, x2, x3) = U24¹(x1, x2, x3)
U25¹(x1, x2) = U25¹(x2)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11(x1)
U12(x1, x2) = U12
U13(x1) = U13
U21(x1, x2, x3) = U21(x2)
U22(x1, x2, x3) = x1
U23(x1, x2, x3) = U23
U24(x1, x2, x3) = x1
U25(x1, x2) = U25
U26(x1) = U26
U31(x1, x2) = U31(x2)
U32(x1, x2) = U32
U33(x1) = x1
isQid(x1) = isQid
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
U62(x1, x2) = U62
U63(x1) = U63
U71(x1, x2, x3) = U71(x2, x3)
U72(x1, x2) = U72(x1, x2)
U73(x1, x2) = U73(x1)
isPal(x1) = isPal(x1)
U74(x1) = U74
U81(x1, x2) = U81(x1, x2)
U82(x1, x2) = U82(x1, x2)
U83(x1) = U83
isNePal(x1) = isNePal(x1)
U91(x1, x2) = x1
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:

[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > [U41^1₃, U42^1₃] > U43^1₂ > U44^1₃ > U45^1₁ > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > [U41^1₃, U42^1₃] > U43^1₂ > U44^1₃ > [isList₁, U11₁, U12, U21₁, U55, U56] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > [U41^1₃, U42^1₃] > U43^1₂ > U44^1₃ > ISLIST₁ > U11^1₁ > U12^1₁ > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U52^1₂ > [U53^1₃, U54^1₂, isNeList₁] > U55^1₂ > ISLIST₁ > U11^1₁ > U12^1₁ > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U52^1₂ > [U53^1₃, U54^1₂, isNeList₁] > [U31₁, U32, isQid] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U52^1₂ > [U53^1₃, U54^1₂, isNeList₁] > U41 > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U52^1₂ > [U53^1₃, U54^1₂, isNeList₁] > U51 > [U52, U53, U54] > [isList₁, U11₁, U12, U21₁, U55, U56] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U21^1₃ > [U22^1₃, U23^1₃] > U24^1₃ > [isList₁, U11₁, U12, U21₁, U55, U56] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U21^1₃ > [U22^1₃, U23^1₃] > U24^1₃ > U25^1₁ > ISLIST₁ > U11^1₁ > U12^1₁ > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U71₂ > U72₂ > U73₁ > U74 > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_{_{_₂}}}, U51^1₂, _{_₂}] > U71₂ > U72₂ > isPal₁ > U81₂ > U82₂ > U83 > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[nil, n_{_nil}] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
isNePal₁ > [U31₁, U32, isQid] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
isNePal₁ > [U61, U62, U63] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
isNePal₁ > U71₂ > U72₂ > U73₁ > U74 > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
isNePal₁ > U71₂ > U72₂ > isPal₁ > U81₂ > U82₂ > U83 > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_a}, a] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_i}, i] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_o}, o] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]
[n_{_u}, u] > [tt, isPalListKind, ISNELIST₁, U13, U23, U25, U26, U42, U43, U44, U45, U46, U92, n_{_e}, e]

Status:

U11^1₁: [1]
tt: []
U12^1₁: [1]
isPalListKind: []
ISNELIST₁: [1]
n_{_{_{_₂}}}: [1,2]
U41^1₃: [2,1,3]
U42^1₃: [2,1,3]
U43^1₂: [2,1]
U44^1₃: [1,2,3]
U45^1₁: [1]
isList₁: [1]
U51^1₂: [1,2]
U52^1₂: [2,1]
U53^1₃: [2,3,1]
U54^1₂: [1,2]
U55^1₂: [2,1]
isNeList₁: [1]
ISLIST₁: [1]
U21^1₃: [1,2,3]
U22^1₃: [3,1,2]
U23^1₃: [3,1,2]
U24^1₃: [1,3,2]
U25^1₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
U12: []
U13: []
U21₁: [1]
U23: []
U25: []
U26: []
U31₁: [1]
U32: []
isQid: []
U41: []
U42: []
U43: []
U44: []
U45: []
U46: []
U51: []
U52: []
U53: []
U54: []
U55: []
U56: []
U61: []
U62: []
U63: []
U71₂: [2,1]
U72₂: [2,1]
U73₁: [1]
isPal₁: [1]
U74: []
U81₂: [2,1]
U82₂: [2,1]
U83: []
isNePal₁: [1]
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, __(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

(22) Obligation:

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

U41¹(tt, V1, V2) → U42¹(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22¹(tt, V1, V2) → U23¹(isPalListKind(activate(V2)), 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, __(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.
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 2 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE

(15) Obligation:

(16) QDPOrderProof (EQUIVALENT transformation)

(17) Obligation:

(18) DependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) TRUE