Termination w.r.t. Q proof of /tmp/tmpxB1afB/PALINDROME_complete

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 QDPOrderProof (⇔)

        ↳14 QDP

        ↳15 DependencyGraphProof (⇔)

        ↳16 TRUE

    ↳17 QDP

        ↳18 QDPOrderProof (⇔)

        ↳19 QDP

        ↳20 DependencyGraphProof (⇔)

        ↳21 TRUE

    ↳22 QDP

        ↳23 QDPOrderProof (⇔)

        ↳24 QDP

        ↳25 DependencyGraphProof (⇔)

        ↳26 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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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(isNeList(activate(V)))
U11¹(tt, V) → ISNELIST(activate(V))
U11¹(tt, V) → ACTIVATE(V)
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U21¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V2) → U23¹(isList(activate(V2)))
U22¹(tt, V2) → ISLIST(activate(V2))
U22¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, V) → U32¹(isQid(activate(V)))
U31¹(tt, V) → ISQID(activate(V))
U31¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U41¹(tt, V1, V2) → ISLIST(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V2) → U43¹(isNeList(activate(V2)))
U42¹(tt, V2) → ISNELIST(activate(V2))
U42¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → U53¹(isList(activate(V2)))
U52¹(tt, V2) → ISLIST(activate(V2))
U52¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V) → U62¹(isQid(activate(V)))
U61¹(tt, V) → ISQID(activate(V))
U61¹(tt, V) → ACTIVATE(V)
U71¹(tt, V) → U72¹(isNePal(activate(V)))
U71¹(tt, V) → ISNEPAL(activate(V))
U71¹(tt, V) → ACTIVATE(V)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(V) → ISPALLISTKIND(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISLIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(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(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
ISNELIST(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(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(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), 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))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(V) → ACTIVATE(V)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(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__isPalListKind(X)) → ISPALLISTKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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 52 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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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(x1, x2) = __¹(x1)

Tags:
__¹ has tags [0,1]

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

nil > [_{_₂}, n_{_{_{_₂}}}]

Status:

__₂: multiset
nil: multiset
n_{_{_{_₂}}}: multiset

The following usable rules [FROCOS05] were oriented: none

(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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:

ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → 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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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.

ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x1) = ACTIVATE(x1)
ISPALLISTKIND(x1) = ISPALLISTKIND(x1)
AND(x1, x2) = AND(x2)

Tags:
ACTIVATE has tags [3]
ISPALLISTKIND has tags [0]
AND has tags [0,3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
n__isPalListKind(x1) = n__isPalListKind(x1)
n____(x1, x2) = n____(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
activate(x1) = activate(x1)
tt = tt
n__and(x1, x2) = x2
n__nil = n__nil
nil = nil
__(x1, x2) = __(x1, x2)
and(x1, x2) = and(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

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

[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > [isPalListKind₁, tt] > n_{_{isPalListKind₁}}
[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > nil > n_{_nil}
[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > e > n_{_e}
[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > n_{_o}
[n_{_{_{_₂}}}, _{_₂}] > [activate₁, and₁, o] > u > n_{_u}
[n_{_a}, a] > [isPalListKind₁, tt] > n_{_{isPalListKind₁}}

Status:

n_{_{isPalListKind₁}}: [1]
n_{_{_{_₂}}}: [1,2]
isPalListKind₁: multiset
activate₁: [1]
tt: multiset
n_{_nil}: multiset
nil: multiset
__₂: [1,2]
and₁: [1]
n_{_a}: multiset
a: multiset
n_{_e}: multiset
e: multiset
n_{_i}: multiset
i: multiset
n_{_o}: multiset
o: multiset
n_{_u}: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(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(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
__(__(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:
The TRS P consists of the following rules:

AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
AND(x1, x2) = AND(x2)
ACTIVATE(x1) = ACTIVATE(x1)

Tags:
AND has tags [0,3]
ACTIVATE has tags [3]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
tt = tt
n__and(x1, x2) = n__and(x2)

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

trivial

Status:

tt: multiset
n_{_and₁}: multiset

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt, V) → U12(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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.

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

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

ISNEPAL(n____(I, __(P, I))) → ISPAL(activate(P))
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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) 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))) → ISPAL(activate(P))
U71¹(tt, V) → ISNEPAL(activate(V))

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

Tags:
ISNEPAL has tags [1]
ISPAL has tags [2]
U71¹ has tags [1,2]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
activate(x1) = x1
isPalListKind(x1) = x1
tt = tt
n__nil = n__nil
nil = nil
n__isPalListKind(x1) = x1
and(x1, x2) = and(x1, x2)
n__and(x1, x2) = n__and(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

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

[n_{_nil}, nil] > [tt, n_{_i}, i] > [n_{_{_{_₂}}}, _{_₂}, and₂, n_{_and₂}]
[n_{_a}, a] > [tt, n_{_i}, i] > [n_{_{_{_₂}}}, _{_₂}, and₂, n_{_and₂}]
[n_{_e}, e] > [tt, n_{_i}, i] > [n_{_{_{_₂}}}, _{_₂}, and₂, n_{_and₂}]
[n_{_o}, o] > [tt, n_{_i}, i] > [n_{_{_{_₂}}}, _{_₂}, and₂, n_{_and₂}]
[n_{_u}, u] > [tt, n_{_i}, i] > [n_{_{_{_₂}}}, _{_₂}, and₂, n_{_and₂}]

Status:

n_{_{_{_₂}}}: [1,2]
__₂: [1,2]
tt: multiset
n_{_nil}: multiset
nil: multiset
and₂: [1,2]
n_{_and₂}: [1,2]
n_{_a}: multiset
a: multiset
n_{_e}: multiset
e: multiset
n_{_i}: multiset
i: multiset
n_{_o}: multiset
o: multiset
n_{_u}: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(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(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
__(__(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

(19) Obligation:

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

ISPAL(V) → U71¹(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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.

(20) DependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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

U11¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
U52¹(tt, V2) → ISLIST(activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U22¹(tt, V2) → ISLIST(activate(V2))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U41¹(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(tt, V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → U41¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isList(activate(V1)), activate(V2))
U42¹(tt, V2) → ISNELIST(activate(V2))
ISNELIST(n____(V1, V2)) → U51¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, V2) → ISLIST(activate(V2))
ISLIST(n____(V1, V2)) → U21¹(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U21¹(tt, V1, V2) → U22¹(isList(activate(V1)), activate(V2))
U22¹(tt, V2) → ISLIST(activate(V2))
U41¹(tt, V1, V2) → ISLIST(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(x1, x2) = U11¹(x2)
ISNELIST(x1) = ISNELIST(x1)
U41¹(x1, x2, x3) = U41¹(x2, x3)
U42¹(x1, x2) = U42¹(x2)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2) = U52¹(x2)
ISLIST(x1) = ISLIST(x1)
U21¹(x1, x2, x3) = U21¹(x1, x2, x3)
U22¹(x1, x2) = U22¹(x1, x2)

Tags:
U11¹ has tags [31,1]
ISNELIST has tags [0]
U41¹ has tags [30,24,23]
U42¹ has tags [31,16]
U51¹ has tags [28,0,31]
U52¹ has tags [15,31]
ISLIST has tags [1]
U21¹ has tags [16,1,6]
U22¹ has tags [9,4]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
tt = tt
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
and(x1, x2) = and(x1, x2)
isPalListKind(x1) = isPalListKind(x1)
n__isPalListKind(x1) = n__isPalListKind(x1)
isList(x1) = isList
isNeList(x1) = x1
n__nil = n__nil
nil = nil
__(x1, x2) = __(x1, x2)
n__and(x1, x2) = n__and(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
U11(x1, x2) = U11
U21(x1, x2, x3) = U21
U31(x1, x2) = U31
U41(x1, x2, x3) = U41(x1, x3)
U51(x1, x2, x3) = U51(x1, x2, x3)
U42(x1, x2) = x1
U12(x1) = U12
U52(x1, x2) = U52(x2)
U22(x1, x2) = x1
U43(x1) = U43
U53(x1) = U53(x1)
U23(x1) = U23
U32(x1) = U32(x1)
isQid(x1) = x1

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

[n_{_{_{_₂}}}, _{_₂}, U51₃] > [isPalListKind₁, n_{_{isPalListKind₁}}] > [tt, and₂, isList, n_{_nil}, nil, n_{_and₂}, n_{_e}, e, U11, U21, U41₂, U12, U43, U23] > [U52₁, U53₁]
[n_{_a}, a]
[n_{_i}, i] > [tt, and₂, isList, n_{_nil}, nil, n_{_and₂}, n_{_e}, e, U11, U21, U41₂, U12, U43, U23] > [U52₁, U53₁]
[n_{_o}, o] > [tt, and₂, isList, n_{_nil}, nil, n_{_and₂}, n_{_e}, e, U11, U21, U41₂, U12, U43, U23] > [U52₁, U53₁]
[n_{_u}, u]
U32₁ > [tt, and₂, isList, n_{_nil}, nil, n_{_and₂}, n_{_e}, e, U11, U21, U41₂, U12, U43, U23] > [U52₁, U53₁]

Status:

tt: multiset
n_{_{_{_₂}}}: [1,2]
and₂: [1,2]
isPalListKind₁: [1]
n_{_{isPalListKind₁}}: [1]
isList: []
n_{_nil}: multiset
nil: multiset
__₂: [1,2]
n_{_and₂}: [1,2]
n_{_a}: multiset
a: multiset
n_{_e}: multiset
e: multiset
n_{_i}: multiset
i: multiset
n_{_o}: multiset
o: multiset
n_{_u}: multiset
u: multiset
U11: []
U21: []
U31: []
U41₂: multiset
U51₃: multiset
U12: []
U52₁: multiset
U43: multiset
U53₁: multiset
U23: []
U32₁: multiset

The following usable rules [FROCOS05] were oriented:

activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__isPalListKind(X)) → isPalListKind(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__and(X1, X2)) → and(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(X) → n__isPalListKind(X)
and(X1, X2) → n__and(X1, X2)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
U11(tt, V) → U12(isNeList(activate(V)))
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U12(tt) → tt
U23(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

(24) Obligation:

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

U51¹(tt, V1, V2) → U52¹(isNeList(activate(V1)), activate(V2))
ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U51¹(tt, V1, V2) → ISNELIST(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(isNeList(activate(V)))
U12(tt) → tt
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U22(tt, V2) → U23(isList(activate(V2)))
U23(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNeList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U52(tt, V2) → U53(isList(activate(V2)))
U53(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
and(tt, X) → activate(X)
isList(V) → U11(isPalListKind(activate(V)), activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(V) → U31(isPalListKind(activate(V)), activate(V))
isNeList(n____(V1, V2)) → U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNeList(n____(V1, V2)) → U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
isNePal(n____(I, __(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(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)) → and(isPalListKind(activate(V1)), n__isPalListKind(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)
isPalListKind(X) → n__isPalListKind(X)
and(X1, X2) → n__and(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__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(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.

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 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) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) TRUE

(17) Obligation:

(18) QDPOrderProof (EQUIVALENT transformation)

(19) Obligation:

(20) DependencyGraphProof (EQUIVALENT transformation)

(21) TRUE

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) TRUE