Termination w.r.t. Q proof of /tmp/tmpD_l7Jv/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: SCNP Order with the following components:
Level mapping:
Top level AFS:
__¹(x0, x1, x2) = __¹(x1)

Tags:
__¹ has argument tags [2,0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Lexicographic path order with status [LPO].
Quasi-Precedence:

__₂ > [_{_^1₂}, n_{_{_{_₂}}}]
nil > [_{_^1₂}, n_{_{_{_₂}}}]

Status:

__^1₂: [1,2]
__₂: [1,2]
nil: []
n_{_{_{_₂}}}: [2,1]

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)

(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
U91¹(x0, x1, x2) = U91¹(x0)
ISPALLISTKIND(x0, x1) = ISPALLISTKIND(x1)

Tags:
U91¹ has argument tags [4,0,4] and root tag 0
ISPALLISTKIND has argument tags [0,4] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U91¹(x1, x2) = U91¹(x1, x2)
tt = tt
ISPALLISTKIND(x1) = ISPALLISTKIND
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
isPalListKind(x1) = x1
n__nil = n__nil
nil = nil
__(x1, x2) = __(x1, x2)
n__a = n__a
a = a
n__e = n__e
e = e
n__i = n__i
i = i
n__o = n__o
o = o
n__u = n__u
u = u
U91(x1, x2) = x2
U92(x1) = x1

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

ISPALLISTKIND > [U91^1₂, n_{_{_{_₂}}}, _{_₂}]
[n_{_a}, a] > [tt, n_{_nil}, nil]
[n_{_e}, e] > [tt, n_{_nil}, nil]
[n_{_i}, i] > [tt, n_{_nil}, nil]
[n_{_o}, o] > [tt, n_{_nil}, nil]
[n_{_u}, u] > [tt, n_{_nil}, nil]

Status:

U91^1₂: [1,2]
tt: []
ISPALLISTKIND: []
n_{_{_{_₂}}}: [1,2]
n_{_nil}: []
nil: []
__₂: [1,2]
n_{_a}: []
a: []
n_{_e}: []
e: []
n_{_i}: []
i: []
n_{_o}: []
o: []
n_{_u}: []
u: []

The following usable rules [FROCOS05] were oriented:

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
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))
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u

(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))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ISNEPAL(x0, x1) = ISNEPAL(x0, x1)
U71¹(x0, x1, x2, x3) = U71¹(x0)
U72¹(x0, x1, x2) = U72¹(x0, x1)
ISPAL(x0, x1) = ISPAL(x1)
U81¹(x0, x1, x2) = U81¹(x1, x2)
U82¹(x0, x1, x2) = U82¹(x2)

Tags:
ISNEPAL has argument tags [0,0] and root tag 4
U71¹ has argument tags [24,8,16,6] and root tag 5
U72¹ has argument tags [24,9,27] and root tag 0
ISPAL has argument tags [6,0] and root tag 4
U81¹ has argument tags [16,0,0] and root tag 4
U82¹ has argument tags [18,11,0] and root tag 4

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ISNEPAL(x1) = ISNEPAL
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
U71¹(x1, x2, x3) = U71¹(x2, x3)
isQid(x1) = isQid(x1)
activate(x1) = x1
tt = tt
U72¹(x1, x2) = U72¹(x1, x2)
isPalListKind(x1) = x1
ISPAL(x1) = ISPAL
U81¹(x1, x2) = U81¹(x1)
U82¹(x1, x2) = U82¹(x1, x2)
n__nil = n__nil
nil = nil
n__a = n__a
a = a
n__e = n__e
e = e
n__i = n__i
i = i
n__o = n__o
o = o
n__u = n__u
u = u
U91(x1, x2) = U91
U92(x1) = U92

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

[n_{_{_{_₂}}}, _{_₂}] > [U71^1₂, U72^1₂] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_{_{_₂}}}, _{_₂}] > [U91, U92] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
isQid₁ > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_nil}, nil] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_e}, e] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_i}, i] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_o}, o] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]
[n_{_u}, u] > [ISNEPAL, tt, ISPAL, U81^1₁, U82^1₂, n_{_a}, a]

Status:

ISNEPAL: []
n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
U71^1₂: [1,2]
isQid₁: [1]
tt: []
U72^1₂: [1,2]
ISPAL: []
U81^1₁: [1]
U82^1₂: [1,2]
n_{_nil}: []
nil: []
n_{_a}: []
a: []
n_{_e}: []
e: []
n_{_i}: []
i: []
n_{_o}: []
o: []
n_{_u}: []
u: []
U91: []
U92: []

The following usable rules [FROCOS05] were oriented:

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
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))
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u

(17) Obligation:

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

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.

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 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))
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))
U24¹(tt, V1, V2) → ISLIST(activate(V1))
U54¹(tt, V1, V2) → ISNELIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U11¹(x0, x1, x2) = U11¹(x0, x2)
U12¹(x0, x1, x2) = U12¹(x0, x2)
ISNELIST(x0, x1) = ISNELIST(x0, x1)
U41¹(x0, x1, x2, x3) = U41¹(x0)
U42¹(x0, x1, x2, x3) = U42¹(x0, x3)
U43¹(x0, x1, x2, x3) = U43¹(x1, x2, x3)
U44¹(x0, x1, x2, x3) = U44¹(x0, x2, x3)
U45¹(x0, x1, x2) = U45¹(x0, x2)
U51¹(x0, x1, x2, x3) = U51¹(x0, x2, x3)
U52¹(x0, x1, x2, x3) = U52¹(x0, x1)
U53¹(x0, x1, x2, x3) = U53¹(x0, x2, x3)
U54¹(x0, x1, x2, x3) = U54¹(x0, x2)
U55¹(x0, x1, x2) = U55¹(x0, x1, x2)
ISLIST(x0, x1) = ISLIST(x0, x1)
U21¹(x0, x1, x2, x3) = U21¹(x0, x1)
U22¹(x0, x1, x2, x3) = U22¹(x0, x1)
U23¹(x0, x1, x2, x3) = U23¹(x0, x1)
U24¹(x0, x1, x2, x3) = U24¹(x1, x2, x3)
U25¹(x0, x1, x2) = U25¹(x1, x2)

Tags:
U11¹ has argument tags [64,15,64] and root tag 10
U12¹ has argument tags [83,119,60] and root tag 28
ISNELIST has argument tags [93,60] and root tag 24
U41¹ has argument tags [45,33,2,0] and root tag 6
U42¹ has argument tags [5,2,66,78] and root tag 16
U43¹ has argument tags [22,49,71,72] and root tag 22
U44¹ has argument tags [49,5,71,72] and root tag 8
U45¹ has argument tags [123,85,67] and root tag 30
U51¹ has argument tags [66,64,32,0] and root tag 4
U52¹ has argument tags [116,96,120,0] and root tag 7
U53¹ has argument tags [0,1,0,31] and root tag 11
U54¹ has argument tags [13,3,93,0] and root tag 16
U55¹ has argument tags [10,33,71] and root tag 23
ISLIST has argument tags [45,71] and root tag 8
U21¹ has argument tags [120,101,0,3] and root tag 4
U22¹ has argument tags [112,26,0,124] and root tag 29
U23¹ has argument tags [72,125,4,0] and root tag 5
U24¹ has argument tags [22,75,76,71] and root tag 26
U25¹ has argument tags [32,48,71] and root tag 8

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹
tt = tt
U12¹(x1, x2) = U12¹
isPalListKind(x1) = isPalListKind
activate(x1) = x1
ISNELIST(x1) = ISNELIST
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x2, x3)
U42¹(x1, x2, x3) = U42¹(x1, x2)
U43¹(x1, x2, x3) = U43¹(x1)
U44¹(x1, x2, x3) = U44¹
U45¹(x1, x2) = U45¹
isList(x1) = isList
U51¹(x1, x2, x3) = U51¹(x1, x2, x3)
U52¹(x1, x2, x3) = U52¹(x1, x2, x3)
U53¹(x1, x2, x3) = U53¹(x1, x2, x3)
U54¹(x1, x2, x3) = U54¹(x3)
U55¹(x1, x2) = U55¹
isNeList(x1) = isNeList
ISLIST(x1) = ISLIST
U21¹(x1, x2, x3) = U21¹(x2, x3)
U22¹(x1, x2, x3) = U22¹(x2, x3)
U23¹(x1, x2, x3) = U23¹(x1, x2, x3)
U24¹(x1, x2, x3) = U24¹(x1, x2, x3)
U25¹(x1, x2) = U25¹(x1, x2)
n__nil = n__nil
nil = nil
__(x1, x2) = __(x1, x2)
n__a = n__a
a = a
n__e = n__e
e = e
n__i = n__i
i = i
n__o = n__o
o = o
n__u = n__u
u = u
U91(x1, x2) = U91
U11(x1, x2) = U11
U21(x1, x2, x3) = U21
U31(x1, x2) = U31
U41(x1, x2, x3) = U41
U51(x1, x2, x3) = U51
U42(x1, x2, x3) = U42
U12(x1, x2) = U12
U22(x1, x2, x3) = U22
U52(x1, x2, x3) = U52
U43(x1, x2, x3) = x1
U13(x1) = U13
U23(x1, x2, x3) = U23
U53(x1, x2, x3) = U53
U44(x1, x2, x3) = U44
U24(x1, x2, x3) = U24
U54(x1, x2, x3) = U54
U45(x1, x2) = x1
U25(x1, x2) = U25
U55(x1, x2) = U55
U46(x1) = U46
U26(x1) = U26
U56(x1) = U56
U32(x1, x2) = x1
U33(x1) = U33
isQid(x1) = isQid
U92(x1) = U92

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

[n_{_{_{_₂}}}, _{_₂}] > U41^1₂ > U42^1₂ > [isPalListKind, U54^1₁, U55^1, isNeList, U91, U31, U41, U51, U42, U52, U53] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[n_{_{_{_₂}}}, _{_₂}] > U51^1₃ > U52^1₃ > U53^1₃ > [isPalListKind, U54^1₁, U55^1, isNeList, U91, U31, U41, U51, U42, U52, U53] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[n_{_{_{_₂}}}, _{_₂}] > [U21^1₂, U22^1₂] > U23^1₃ > [isPalListKind, U54^1₁, U55^1, isNeList, U91, U31, U41, U51, U42, U52, U53] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
U43^1₁ > [isPalListKind, U54^1₁, U55^1, isNeList, U91, U31, U41, U51, U42, U52, U53] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[U24^1₃, U25^1₂] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[n_{_nil}, nil] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[n_{_a}, a]
[n_{_e}, e]
[n_{_i}, i]
[n_{_o}, o] > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]
[n_{_u}, u]
isQid > [tt, U44^1, isList, ISLIST, U11, U21, U12, U22, U13, U23, U44, U24, U54, U25, U55, U46, U26, U56, U33, U92] > U11^1 > U12^1 > [ISNELIST, U45^1]

Status:

U11^1: []
tt: []
U12^1: []
isPalListKind: []
ISNELIST: []
n_{_{_{_₂}}}: [1,2]
U41^1₂: [1,2]
U42^1₂: [2,1]
U43^1₁: [1]
U44^1: []
U45^1: []
isList: []
U51^1₃: [1,2,3]
U52^1₃: [2,1,3]
U53^1₃: [2,1,3]
U54^1₁: [1]
U55^1: []
isNeList: []
ISLIST: []
U21^1₂: [2,1]
U22^1₂: [2,1]
U23^1₃: [3,1,2]
U24^1₃: [2,3,1]
U25^1₂: [1,2]
n_{_nil}: []
nil: []
__₂: [1,2]
n_{_a}: []
a: []
n_{_e}: []
e: []
n_{_i}: []
i: []
n_{_o}: []
o: []
n_{_u}: []
u: []
U91: []
U11: []
U21: []
U31: []
U41: []
U51: []
U42: []
U12: []
U22: []
U52: []
U13: []
U23: []
U53: []
U44: []
U24: []
U54: []
U25: []
U55: []
U46: []
U26: []
U56: []
U33: []
isQid: []
U92: []

The following usable rules [FROCOS05] were oriented:

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
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))
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
U11(tt, V) → U12(isPalListKind(activate(V)), activate(V))
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V) → U13(isNeList(activate(V)))
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)))
U13(tt) → tt
U26(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u

(22) Obligation:

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

U25¹(tt, V2) → ISLIST(activate(V2))
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.

(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