Termination w.r.t. Q proof of /tmp/tmpm2Gs_r/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) = x1
activate(x1) = x1
U13(x1) = U13
isNeList(x1) = isNeList(x1)
U21(x1, x2, x3) = x2
U22(x1, x2, x3) = x2
U23(x1, x2, x3) = x2
U24(x1, x2, x3) = x2
U25(x1, x2) = U25
isList(x1) = x1
U26(x1) = U26
U31(x1, x2) = U31(x2)
U32(x1, x2) = U32(x1, x2)
U33(x1) = U33
isQid(x1) = isQid
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x1, x2, x3)
U43(x1, x2, x3) = x3
U44(x1, x2, x3) = x1
U45(x1, x2) = U45
U46(x1) = U46
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) = x2
U62(x1, x2) = x1
U63(x1) = x1
U71(x1, x2, x3) = U71(x2)
U72(x1, x2) = U72
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) = U91(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

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

isNeList₁ > [_{_₂}, U41₃, U71₁, n_{_{_{_₂}}}] > __^1₁ > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
isNeList₁ > [_{_₂}, U41₃, U71₁, n_{_{_{_₂}}}] > U91₁ > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
isNeList₁ > U31₁ > [U32₂, U33] > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
isNePal₁ > [_{_₂}, U41₃, U71₁, n_{_{_{_₂}}}] > __^1₁ > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
isNePal₁ > [_{_₂}, U41₃, U71₁, n_{_{_{_₂}}}] > U91₁ > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
[n_{_o}, o] > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]
[n_{_u}, u] > [nil, tt, U13, U25, U26, isQid, U42₃, U45, U46, U51₁, U52, U53, U54, U55, U56, U72, isPal, U74, U81, U82, U83, U92, n_{_nil}, n_{_a}, n_{_e}, n_{_i}, a, e, i]

Status:

__^1₁: multiset
__₂: [1,2]
nil: multiset
tt: multiset
U13: multiset
isNeList₁: multiset
U25: multiset
U26: multiset
U31₁: [1]
U32₂: [2,1]
U33: []
isQid: multiset
U41₃: multiset
U42₃: [3,2,1]
U45: multiset
U46: multiset
U51₁: multiset
U52: multiset
U53: multiset
U54: multiset
U55: multiset
U56: multiset
U71₁: multiset
U72: multiset
isPal: multiset
U74: multiset
U81: multiset
U82: multiset
U83: multiset
isNePal₁: [1]
U91₁: multiset
U92: multiset
n_{_nil}: multiset
n_{_{_{_₂}}}: [1,2]
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

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) = x1
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
isPalListKind(x1) = isPalListKind
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11(x1, x2)
U12(x1, x2) = U12(x1, x2)
U13(x1) = U13
isNeList(x1) = isNeList
U21(x1, x2, x3) = U21(x1, x2)
U22(x1, x2, x3) = U22(x2)
U23(x1, x2, x3) = U23(x2)
U24(x1, x2, x3) = x1
U25(x1, x2) = U25
isList(x1) = isList(x1)
U26(x1) = U26
U31(x1, x2) = x1
U32(x1, x2) = x1
U33(x1) = U33
isQid(x1) = isQid
U41(x1, x2, x3) = x1
U42(x1, x2, x3) = x1
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(x1)
U54(x1, x2, x3) = U54
U55(x1, x2) = U55
U56(x1) = U56
U61(x1, x2) = x2
U62(x1, x2) = x1
U63(x1) = U63
U71(x1, x2, x3) = x3
U72(x1, x2) = x2
U73(x1, x2) = x2
isPal(x1) = x1
U74(x1) = x1
U81(x1, x2) = x1
U82(x1, x2) = x1
U83(x1) = U83
isNePal(x1) = 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

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

[n_{_{_{_₂}}}, _{_₂}] > U91^1₂ > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[nil, n_{_nil}] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[isNeList, U51] > U52 > U53₁ > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
isList₁ > [U11₂, U12₂, U13] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
isList₁ > U21₂ > [U22₁, U23₁] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[n_{_a}, a] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[n_{_i}, i] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[n_{_o}, o] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]
[n_{_u}, u] > [tt, isPalListKind, U25, U26, U33, isQid, U45, U46, U54, U55, U56, U63, U83, n_{_e}, e]

Status:

U91^1₂: multiset
tt: multiset
n_{_{_{_₂}}}: [1,2]
isPalListKind: []
__₂: [1,2]
nil: multiset
U11₂: multiset
U12₂: multiset
U13: multiset
isNeList: []
U21₂: [2,1]
U22₁: [1]
U23₁: [1]
U25: []
isList₁: multiset
U26: []
U33: []
isQid: []
U45: []
U46: []
U51: []
U52: multiset
U53₁: multiset
U54: []
U55: []
U56: []
U63: []
U83: []
n_{_nil}: multiset
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

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))
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) = ISNEPAL(x1)
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(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(x2)
U82¹(x1, x2) = U82¹(x2)
nil = nil
U11(x1, x2) = x1
U12(x1, x2) = U12
U13(x1) = U13
isNeList(x1) = x1
U21(x1, x2, x3) = U21(x2, x3)
U22(x1, x2, x3) = U22(x2, x3)
U23(x1, x2, x3) = x3
U24(x1, x2, x3) = x3
U25(x1, x2) = x2
isList(x1) = x1
U26(x1) = x1
U31(x1, x2) = x2
U32(x1, x2) = x1
U33(x1) = U33
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x2, x3)
U43(x1, x2, x3) = x2
U44(x1, x2, x3) = x2
U45(x1, x2) = x1
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(x2, x3)
U55(x1, x2) = U55(x1, x2)
U56(x1) = U56
U61(x1, x2) = U61(x2)
U62(x1, x2) = U62
U63(x1) = U63
U71(x1, x2, x3) = U71(x1, x2)
U72(x1, x2) = U72
U73(x1, x2) = U73
isPal(x1) = x1
U74(x1) = U74
U81(x1, x2) = x2
U82(x1, x2) = x1
U83(x1) = U83
isNePal(x1) = isNePal(x1)
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

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

[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U71^1₂ > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U21₂ > U22₂ > U55₂
[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U41₃ > U42₂ > U55₂
[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U74 > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U74 > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
[n_{_{_{_₂}}}, _{_₂}, U71₂, U72, U73] > U74 > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂
isNePal₁ > U61₁ > [U62, U63] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
isNePal₁ > U61₁ > [U62, U63] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
isNePal₁ > U61₁ > [U62, U63] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂
[n_{_a}, a] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_a}, a] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
[n_{_a}, a] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂
[n_{_i}, i] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_i}, i] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
[n_{_i}, i] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂
[n_{_o}, o] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_o}, o] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
[n_{_o}, o] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂
[n_{_u}, u] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U72^1₁ > [ISPAL₁, U81^1₁] > [ISNEPAL₁, U82^1₁] > U55₂
[n_{_u}, u] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > U22₂ > U55₂
[n_{_u}, u] > [isQid₁, tt, nil, U12, U13, U33, U46, U56, U83, n_{_nil}, n_{_e}, e] > [U51₃, U52₂] > U53₃ > U54₂ > U55₂

Status:

ISNEPAL₁: multiset
n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
U71^1₂: [1,2]
isQid₁: multiset
tt: multiset
U72^1₁: [1]
ISPAL₁: multiset
U81^1₁: multiset
U82^1₁: multiset
nil: multiset
U12: []
U13: []
U21₂: multiset
U22₂: multiset
U33: []
U41₃: multiset
U42₂: [1,2]
U46: []
U51₃: [2,3,1]
U52₂: [1,2]
U53₃: multiset
U54₂: [1,2]
U55₂: multiset
U56: []
U61₁: [1]
U62: []
U63: []
U71₂: [2,1]
U72: []
U73: []
U74: multiset
U83: []
isNePal₁: multiset
n_{_nil}: multiset
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

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:

ISPAL(V) → U81¹(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.

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(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))
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))
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))
U54¹(tt, V1, V2) → U55¹(isNeList(activate(V1)), activate(V2))
U55¹(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U21¹(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(x1, x2)
isPalListKind(x1) = x1
activate(x1) = x1
ISNELIST(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x1, x2, x3)
U42¹(x1, x2, x3) = U42¹(x2, x3)
U43¹(x1, x2, x3) = U43¹(x1, x2, x3)
U44¹(x1, x2, x3) = U44¹(x1, x2, x3)
U45¹(x1, x2) = x2
isList(x1) = isList(x1)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2, x3) = U52¹(x1, 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(x1, x2, x3)
U25¹(x1, x2) = U25¹(x1, x2)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11(x2)
U12(x1, x2) = x1
U13(x1) = U13
U21(x1, x2, x3) = U21(x3)
U22(x1, x2, x3) = x3
U23(x1, x2, x3) = x3
U24(x1, x2, x3) = x1
U25(x1, x2) = U25
U26(x1) = U26
U31(x1, x2) = x1
U32(x1, x2) = U32
U33(x1) = x1
isQid(x1) = isQid
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x2, x3)
U43(x1, x2, x3) = U43(x2, x3)
U44(x1, x2, x3) = U44(x1, x2)
U45(x1, x2) = U45(x1)
U46(x1) = U46
U51(x1, x2, x3) = U51(x1, x3)
U52(x1, x2, x3) = U52(x3)
U53(x1, x2, x3) = x1
U54(x1, x2, x3) = U54
U55(x1, x2) = U55
U56(x1) = U56
U61(x1, x2) = x2
U62(x1, x2) = x1
U63(x1) = x1
U71(x1, x2, x3) = U71(x1, x2, x3)
U72(x1, x2) = U72(x1, x2)
U73(x1, x2) = U73(x1, x2)
isPal(x1) = isPal(x1)
U74(x1) = x1
U81(x1, x2) = U81(x1, x2)
U82(x1, x2) = x2
U83(x1) = x1
isNePal(x1) = x1
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

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

[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > [U41^1₃, U42^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > [U52^1₃, U53^1₂, U54^1₂] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U24^1₃ > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U24^1₃ > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U24^1₃ > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U41₃ > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U41₃ > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U41₃ > U42₂ > [U43₂, U44₂] > U45₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U71₃ > U72₂ > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[nil, n_{_nil}] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_i}, i] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > U12^1₂
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U11₁
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > [U11^1₁, U44^1₃, ISLIST₁, U25^1₂] > isList₁ > U21₁
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U11₁
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > isList₁ > U21₁
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > U42₂ > [U43₂, U44₂] > U45₁
[n_{_u}, u] > [tt, U43^1₃, U55^1₂, U13, U25, U26, U32, isQid, U46, U52₁, U54, U55, U56, U73₂, n_{_a}, n_{_e}, n_{_o}, a, e, o] > isPal₁ > U81₂

Status:

U11^1₁: multiset
tt: multiset
U12^1₂: [1,2]
n_{_{_{_₂}}}: [1,2]
U41^1₃: multiset
U42^1₂: multiset
U43^1₃: [3,1,2]
U44^1₃: multiset
isList₁: [1]
U51^1₂: multiset
U52^1₃: multiset
U53^1₂: multiset
U54^1₂: multiset
U55^1₂: multiset
ISLIST₁: multiset
U21^1₂: [1,2]
U22^1₂: [1,2]
U23^1₂: [1,2]
U24^1₃: multiset
U25^1₂: multiset
__₂: [1,2]
nil: multiset
U11₁: multiset
U13: multiset
U21₁: [1]
U25: multiset
U26: multiset
U32: multiset
isQid: multiset
U41₃: multiset
U42₂: multiset
U43₂: [1,2]
U44₂: [2,1]
U45₁: [1]
U46: multiset
U51₂: [2,1]
U52₁: [1]
U54: multiset
U55: multiset
U56: multiset
U71₃: [1,2,3]
U72₂: multiset
U73₂: multiset
isPal₁: [1]
U81₂: multiset
n_{_nil}: multiset
n_{_a}: multiset
n_{_e}: multiset
n_{_i}: multiset
n_{_o}: multiset
n_{_u}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

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:

U45¹(tt, V2) → ISNELIST(activate(V2))
U53¹(tt, V1, V2) → U54¹(isPalListKind(activate(V2)), activate(V1), activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U21¹(tt, V1, V2) → U22¹(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 5 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