Termination w.r.t. Q proof of /tmp/tmpjGtzoL/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 DependencyGraphProof (⇔)

    ↳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, x2)
__(x1, x2) = __(x1, x2)
nil = nil
n____(x1, x2) = x1

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

__^1₂ > __₂
nil > __₂

Status:

__₂: [1,2]
__^1₂: [1,2]
nil: multiset

The following usable rules [FROCOS05] were oriented:

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

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) = x2
tt = tt
ISPALLISTKIND(x1) = x1
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
isPalListKind(x1) = isPalListKind
U92(x1) = x1
U91(x1, x2) = x1
n__u = n__u
u = u
n__o = n__o
o = o
n__i = n__i
i = i
__(x1, x2) = __(x1, x2)
nil = nil
n__nil = n__nil
a = a
n__a = n__a
n__e = n__e
e = e

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

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

Status:

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

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

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

U91¹(tt, V2) → ISPALLISTKIND(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.

(13) DependencyGraphProof (EQUIVALENT transformation)

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

(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))
U82¹(tt, V) → ISNEPAL(activate(V))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1) = x1
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
U71¹(x1, x2, x3) = U71¹(x3)
isQid(x1) = 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)
U92(x1) = U92
U91(x1, x2) = U91
n__u = n__u
u = u
n__o = n__o
o = o
n__i = n__i
i = i
nil = nil
n__nil = n__nil
a = a
n__a = n__a
n__e = n__e
e = e

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

[n_{_{_{_₂}}}, _{_₂}] > [U71^1₁, tt, U72^1₁, ISPAL₁, U81^1₁, U82^1₁, n_{_u}, u, n_{_o}, o]
[n_{_{_{_₂}}}, _{_₂}] > U91
U92 > [U71^1₁, tt, U72^1₁, ISPAL₁, U81^1₁, U82^1₁, n_{_u}, u, n_{_o}, o]
[n_{_i}, i]
[nil, n_{_nil}]
[a, n_{_a}] > [U71^1₁, tt, U72^1₁, ISPAL₁, U81^1₁, U82^1₁, n_{_u}, u, n_{_o}, o]
[n_{_e}, e]

Status:

i: multiset
a: multiset
__₂: [1,2]
n_{_u}: multiset
U92: multiset
U91: multiset
e: multiset
n_{_i}: multiset
n_{_e}: multiset
n_{_nil}: multiset
n_{_{_{_₂}}}: [1,2]
o: multiset
U72^1₁: [1]
n_{_a}: multiset
U81^1₁: [1]
n_{_o}: multiset
tt: multiset
ISPAL₁: [1]
u: multiset
U71^1₁: [1]
U82^1₁: [1]
nil: multiset

The following usable rules [FROCOS05] were oriented:

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

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

The 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 4 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.

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))
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))
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) = x2
tt = tt
U12¹(x1, x2) = x2
isPalListKind(x1) = isPalListKind
activate(x1) = x1
ISNELIST(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x2, x3)
U42¹(x1, x2, x3) = U42¹(x2, x3)
U43¹(x1, x2, x3) = U43¹(x2, x3)
U44¹(x1, x2, x3) = U44¹(x2, x3)
U45¹(x1, x2) = U45¹(x2)
isList(x1) = x1
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2, x3) = U52¹(x2, x3)
U53¹(x1, x2, x3) = U53¹(x2, x3)
U54¹(x1, x2, x3) = U54¹(x2, x3)
U55¹(x1, x2) = x2
isNeList(x1) = isNeList
ISLIST(x1) = x1
U21¹(x1, x2, x3) = U21¹(x2, x3)
U22¹(x1, x2, x3) = U22¹(x2, x3)
U23¹(x1, x2, x3) = U23¹(x2, x3)
U24¹(x1, x2, x3) = U24¹(x2, x3)
U25¹(x1, x2) = x2
U56(x1) = x1
U54(x1, x2, x3) = U54(x2)
U55(x1, x2) = x1
U92(x1) = U92(x1)
U11(x1, x2) = U11(x2)
U91(x1, x2) = U91
U41(x1, x2, x3) = U41(x2)
U42(x1, x2, x3) = U42(x1, x2)
U33(x1) = x1
U32(x1, x2) = U32
isQid(x1) = isQid
U31(x1, x2) = U31(x1, x2)
U26(x1) = x1
U25(x1, x2) = U25(x1, x2)
U24(x1, x2, x3) = x3
U23(x1, x2, x3) = U23(x3)
U53(x1, x2, x3) = U53(x1, x2)
U52(x1, x2, x3) = U52(x1, x2, x3)
U51(x1, x2, x3) = U51(x2, x3)
U46(x1) = U46
U45(x1, x2) = U45
U44(x1, x2, x3) = U44
U43(x1, x2, x3) = U43(x2, x3)
n__u = n__u
nil = nil
n__nil = n__nil
__(x1, x2) = __(x1, x2)
a = a
n__a = n__a
n__e = n__e
n__i = n__i
n__o = n__o
e = e
i = i
o = o
u = u
U21(x1, x2, x3) = x1
U12(x1, x2) = U12(x1)
U13(x1) = U13(x1)
U22(x1, x2, x3) = U22(x2)

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

[tt, isNeList, U12₁] > U42^1₂ > U43^1₂ > [U44^1₂, U45^1₁] > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U52^1₂ > U53^1₂ > U54^1₂ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U24^1₂ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U54₁ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U92₁ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U41₁ > U42₂ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > isQid > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U31₂ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U25₂ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U23₁ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U51₂ > U52₃ > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > [U46, U45] > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U44 > [isPalListKind, U91, U32, U13₁]
[tt, isNeList, U12₁] > U43₂ > [isPalListKind, U91, U32, U13₁]
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U41^1₂ > U42^1₂ > U43^1₂ > [U44^1₂, U45^1₁] > [isPalListKind, U91, U32, U13₁]
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51^1₂ > U52^1₂ > U53^1₂ > U54^1₂ > [isPalListKind, U91, U32, U13₁]
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U24^1₂ > [isPalListKind, U91, U32, U13₁]
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U41₁ > U42₂ > [isPalListKind, U91, U32, U13₁]
[n_{_{_{_₂}}}, U21^1₂, U22^1₂, U23^1₂, _{_₂}] > U51₂ > U52₃ > [isPalListKind, U91, U32, U13₁]
U11₁ > [isPalListKind, U91, U32, U13₁]
U53₂ > U54₁ > [isPalListKind, U91, U32, U13₁]
[n_{_u}, u] > [isPalListKind, U91, U32, U13₁]
[nil, n_{_nil}] > [isPalListKind, U91, U32, U13₁]
[a, n_{_a}] > [isPalListKind, U91, U32, U13₁]
[n_{_e}, e] > [isPalListKind, U91, U32, U13₁]
[n_{_i}, i] > [isPalListKind, U91, U32, U13₁]
[n_{_o}, o] > [isPalListKind, U91, U32, U13₁]
U22₁ > U23₁ > [isPalListKind, U91, U32, U13₁]

Status:

i: multiset
U51^1₂: multiset
U45^1₁: [1]
U53₂: multiset
U25₂: [2,1]
n_{_i}: multiset
n_{_nil}: multiset
U51₂: [1,2]
U23₁: [1]
tt: multiset
nil: multiset
U31₂: multiset
e: multiset
o: multiset
n_{_{_{_₂}}}: [1,2]
n_{_o}: multiset
U22₁: [1]
U41^1₂: [2,1]
U45: multiset
U54^1₂: [1,2]
U52^1₂: multiset
__₂: [1,2]
n_{_u}: multiset
U41₁: multiset
U22^1₂: [1,2]
isNeList: []
isPalListKind: []
U44: []
U21^1₂: [1,2]
n_{_a}: multiset
U92₁: [1]
U53^1₂: [2,1]
U12₁: multiset
U11₁: multiset
isQid: multiset
U44^1₂: [2,1]
U46: []
a: multiset
U32: []
U42₂: multiset
U91: multiset
n_{_e}: multiset
U52₃: multiset
U24^1₂: multiset
U43^1₂: multiset
U42^1₂: [2,1]
u: multiset
U13₁: [1]
U23^1₂: [1,2]
U43₂: multiset
U54₁: multiset

The following usable rules [FROCOS05] were oriented:

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

(22) 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))
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))
U25¹(tt, V2) → ISLIST(activate(V2))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isPalListKind(activate(V)), activate(V))
U12(tt, V) → U13(isNeList(activate(V)))
U13(tt) → tt
U21(tt, V1, V2) → U22(isPalListKind(activate(V1)), activate(V1), activate(V2))
U22(tt, V1, V2) → U23(isPalListKind(activate(V2)), activate(V1), activate(V2))
U23(tt, V1, V2) → U24(isPalListKind(activate(V2)), activate(V1), activate(V2))
U24(tt, V1, V2) → U25(isList(activate(V1)), activate(V2))
U25(tt, V2) → U26(isList(activate(V2)))
U26(tt) → tt
U31(tt, V) → U32(isPalListKind(activate(V)), activate(V))
U32(tt, V) → U33(isQid(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isPalListKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isPalListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isPalListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isList(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNeList(activate(V2)))
U46(tt) → tt
U51(tt, V1, V2) → U52(isPalListKind(activate(V1)), activate(V1), activate(V2))
U52(tt, V1, V2) → U53(isPalListKind(activate(V2)), activate(V1), activate(V2))
U53(tt, V1, V2) → U54(isPalListKind(activate(V2)), activate(V1), activate(V2))
U54(tt, V1, V2) → U55(isNeList(activate(V1)), activate(V2))
U55(tt, V2) → U56(isList(activate(V2)))
U56(tt) → tt
U61(tt, V) → U62(isPalListKind(activate(V)), activate(V))
U62(tt, V) → U63(isQid(activate(V)))
U63(tt) → tt
U71(tt, I, P) → U72(isPalListKind(activate(I)), activate(P))
U72(tt, P) → U73(isPal(activate(P)), activate(P))
U73(tt, P) → U74(isPalListKind(activate(P)))
U74(tt) → tt
U81(tt, V) → U82(isPalListKind(activate(V)), activate(V))
U82(tt, V) → U83(isNePal(activate(V)))
U83(tt) → tt
U91(tt, V2) → U92(isPalListKind(activate(V2)))
U92(tt) → tt
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(isPalListKind(activate(V1)), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(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 8 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) DependencyGraphProof (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