Termination w.r.t. Q proof of /tmp/tmpqxtfM3/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 DependencyGraphProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 DependencyGraphProof (⇔)

    ↳23 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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(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(activate(X1), activate(X2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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 3 SCCs with 41 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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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: Lexicographic path order with status [LPO].
Quasi-Precedence:

__^1₂ > [_{_₂}, n_{_{_{_₂}}}]

Status:

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

The following usable rules [FROCOS05] were oriented:

__(X1, X2) → n____(X1, X2)
__(X, nil) → X
__(__(X, Y), Z) → __(X, __(Y, Z))
__(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(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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(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(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
U71¹(tt, V) → ACTIVATE(V)
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isPalListKind(X)) → ISPALLISTKIND(X)
ISPALLISTKIND(n____(V1, V2)) → AND(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISPALLISTKIND(n____(V1, V2)) → ISPALLISTKIND(activate(V1))
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V1)
ISPALLISTKIND(n____(V1, V2)) → ACTIVATE(V2)
ISNEPAL(n____(I, n____(P, I))) → AND(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPalListKind(activate(I)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE(x1)
n____(x1, x2) = n____(x1, x2)
n__isPalListKind(x1) = n__isPalListKind(x1)
ISPALLISTKIND(x1) = ISPALLISTKIND(x1)
AND(x1, x2) = AND(x2)
isPalListKind(x1) = isPalListKind(x1)
activate(x1) = x1
tt = tt
n__and(x1, x2) = n__and(x1, x2)
n__isPal(x1) = x1
ISPAL(x1) = ISPAL(x1)
U71¹(x1, x2) = U71¹(x2)
ISNEPAL(x1) = ISNEPAL(x1)
U61¹(x1, x2) = U61¹(x2)
and(x1, x2) = and(x1, x2)
isQid(x1) = x1
o = o
n__o = n__o
i = i
n__i = n__i
e = e
n__e = n__e
a = a
n__a = n__a
__(x1, x2) = __(x1, x2)
n__nil = n__nil
nil = nil
u = u
n__u = n__u
isPal(x1) = x1
isNePal(x1) = isNePal(x1)
U71(x1, x2) = x2
U61(x1, x2) = U61(x1)
U72(x1) = U72
U62(x1) = U62

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

[ACTIVATE₁, n_{_{_{_₂}}}, ISPALLISTKIND₁, AND₁, ISPAL₁, U71^1₁, ISNEPAL₁, U61^1₁, _{_₂}] > [n_{_{isPalListKind₁}}, isPalListKind₁] > [n_{_and₂}, and₂] > [tt, i, n_{_i}, U72]
[o, n_{_o}] > [tt, i, n_{_i}, U72]
[e, n_{_e}] > [tt, i, n_{_i}, U72]
[a, n_{_a}] > [tt, i, n_{_i}, U72]
[n_{_nil}, nil] > [tt, i, n_{_i}, U72]
[u, n_{_u}] > [tt, i, n_{_i}, U72]
isNePal₁ > [n_{_{isPalListKind₁}}, isPalListKind₁] > [n_{_and₂}, and₂] > [tt, i, n_{_i}, U72]
isNePal₁ > [U61₁, U62] > [tt, i, n_{_i}, U72]

Status:

i: []
__₂: [1,2]
n_{_u}: []
n_{_i}: []
n_{_nil}: []
and₂: [1,2]
n_{_a}: []
tt: []
ISPAL₁: [1]
isPalListKind₁: [1]
U72: []
nil: []
ACTIVATE₁: [1]
a: []
U61₁: [1]
U62: []
AND₁: [1]
e: []
n_{_e}: []
isNePal₁: [1]
o: []
n_{_{_{_₂}}}: [1,2]
n_{_and₂}: [1,2]
n_{_o}: []
u: []
ISPALLISTKIND₁: [1]
U61^1₁: [1]
U71^1₁: [1]
n_{_{isPalListKind₁}}: [1]
ISNEPAL₁: [1]

The following usable rules [FROCOS05] were oriented:

o → n__o
i → n__i
e → n__e
a → n__a
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__nil) → nil
u → n__u
activate(n__e) → e
activate(n__a) → a
activate(n__isPal(X)) → isPal(X)
__(X, nil) → X
activate(X) → X
__(__(X, Y), Z) → __(X, __(Y, Z))
activate(n__u) → u
activate(n__o) → o
__(nil, X) → X
activate(n__i) → i
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
isQid(n__u) → tt
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt

(12) Obligation:

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

AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISNEPAL(V) → ISPALLISTKIND(activate(V))
ISNEPAL(V) → ACTIVATE(V)
U71¹(tt, V) → ACTIVATE(V)
ISPAL(V) → ISPALLISTKIND(activate(V))
ISPAL(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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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 1 SCC with 3 less nodes.

(14) Obligation:

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

ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ACTIVATE(V)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ACTIVATE(V)
ISNEPAL(V) → ACTIVATE(V)
U71¹(tt, 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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__isPal(X)) → ISPAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISPAL(x1) = ISPAL(x1)
U71¹(x1, x2) = U71¹(x2)
isPalListKind(x1) = x1
activate(x1) = x1
tt = tt
ISNEPAL(x1) = ISNEPAL(x1)
U61¹(x1, x2) = U61¹(x2)
ACTIVATE(x1) = ACTIVATE(x1)
n__isPal(x1) = n__isPal(x1)
o = o
n__o = n__o
i = i
n__i = n__i
e = e
n__e = n__e
a = a
n__a = n__a
n__isPalListKind(x1) = x1
n__and(x1, x2) = n__and(x1, x2)
and(x1, x2) = and(x1, x2)
n____(x1, x2) = n____(x1, x2)
__(x1, x2) = __(x1, x2)
n__nil = n__nil
nil = nil
u = u
n__u = n__u
isPal(x1) = isPal(x1)
isNePal(x1) = x1
isQid(x1) = isQid(x1)
U71(x1, x2) = x1
U61(x1, x2) = x1
U72(x1) = U72
U62(x1) = U62

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

[tt, o, n_{_o}, i, n_{_i}, e, n_{_e}, a, n_{_a}, n_{_nil}, nil, u, n_{_u}, U72, U62] > [ISPAL₁, U71^1₁, ISNEPAL₁, U61^1₁, ACTIVATE₁]
[tt, o, n_{_o}, i, n_{_i}, e, n_{_e}, a, n_{_a}, n_{_nil}, nil, u, n_{_u}, U72, U62] > [n_{_isPal₁}, n_{_{_{_₂}}}, _{_₂}, isPal₁, isQid₁] > [n_{_and₂}, and₂]

Status:

i: []
n_{_u}: []
__₂: [1,2]
n_{_i}: []
n_{_nil}: []
and₂: [2,1]
n_{_a}: []
tt: []
ISPAL₁: [1]
U72: []
nil: []
ACTIVATE₁: [1]
a: []
U62: []
n_{_isPal₁}: [1]
e: []
n_{_e}: []
n_{_{_{_₂}}}: [1,2]
o: []
isQid₁: [1]
n_{_and₂}: [2,1]
isPal₁: [1]
n_{_o}: []
u: []
U61^1₁: [1]
U71^1₁: [1]
ISNEPAL₁: [1]

The following usable rules [FROCOS05] were oriented:

o → n__o
i → n__i
e → n__e
a → n__a
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__nil) → nil
u → n__u
activate(n__e) → e
activate(n__a) → a
activate(n__isPal(X)) → isPal(X)
activate(X) → X
__(X, nil) → X
activate(n__u) → u
__(__(X, Y), Z) → __(X, __(Y, Z))
activate(n__o) → o
activate(n__i) → i
__(nil, X) → X
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
isNePal(V) → U61(isPalListKind(activate(V)), activate(V))
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
isQid(n__u) → tt
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt

(16) Obligation:

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

ISPAL(V) → U71¹(isPalListKind(activate(V)), activate(V))
U71¹(tt, V) → ISNEPAL(activate(V))
ISNEPAL(V) → U61¹(isPalListKind(activate(V)), activate(V))
U61¹(tt, V) → ACTIVATE(V)
ISPAL(V) → ACTIVATE(V)
ISNEPAL(V) → ACTIVATE(V)
U71¹(tt, 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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes.

(18) TRUE

(19) 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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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) 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))
U51¹(tt, V1, V2) → U52¹(isNeList(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))
U21¹(tt, V1, V2) → ISLIST(activate(V1))
U51¹(tt, V1, V2) → ISNELIST(activate(V1))
U41¹(tt, V1, V2) → ISLIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2) = U11¹(x2)
tt = tt
ISNELIST(x1) = ISNELIST(x1)
activate(x1) = x1
n____(x1, x2) = n____(x1, x2)
U41¹(x1, x2, x3) = U41¹(x1, x2, x3)
and(x1, x2) = x2
isPalListKind(x1) = isPalListKind
n__isPalListKind(x1) = n__isPalListKind
U42¹(x1, x2) = U42¹(x1, x2)
isList(x1) = isList
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2) = U52¹(x2)
isNeList(x1) = isNeList(x1)
ISLIST(x1) = ISLIST(x1)
U21¹(x1, x2, x3) = U21¹(x2, x3)
U22¹(x1, x2) = U22¹(x2)
U51(x1, x2, x3) = U51(x2)
U52(x1, x2) = x1
U43(x1) = x1
U53(x1) = U53
U32(x1) = x1
U31(x1, x2) = x2
isQid(x1) = x1
U42(x1, x2) = U42(x1, x2)
U41(x1, x2, x3) = U41(x2, x3)
U21(x1, x2, x3) = x1
U22(x1, x2) = x1
U12(x1) = U12
U23(x1) = U23
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1, x2) = U11
isPal(x1) = isPal(x1)
n__nil = n__nil
n__a = n__a
isNePal(x1) = isNePal
n__and(x1, x2) = x2
n__isPal(x1) = n__isPal(x1)
U71(x1, x2) = x1
U61(x1, x2) = U61
U72(x1) = U72
U62(x1) = U62
n__u = n__u
n__o = n__o
n__i = n__i
n__e = n__e
o = o
i = i
e = e
a = a
u = u

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

[n_{_{_{_₂}}}, _{_₂}] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[n_{_{_{_₂}}}, _{_₂}] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_{_{_₂}}}, _{_₂}] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]
[n_{_{_{_₂}}}, _{_₂}] > U41^1₃ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_{_{_₂}}}, _{_₂}] > U41^1₃ > U42^1₂ > ISNELIST₁
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_{_{_₂}}}, _{_₂}] > U51^1₂ > [isNeList₁, U51₁]
[n_{_{_{_₂}}}, _{_₂}] > U21^1₂ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[nil, n_{_nil}]
[n_{_a}, a] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[n_{_a}, a] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_a}, a] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]
[isNePal, U61] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[isNePal, U61] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[isNePal, U61] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]
[n_{_u}, u] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[n_{_u}, u] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_u}, u] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]
[n_{_o}, o] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[n_{_o}, o] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_o}, o] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]
[n_{_e}, e] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42^1₂ > ISNELIST₁
[n_{_e}, e] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U52^1₁ > [U11^1₁, ISLIST₁, U22^1₁] > ISNELIST₁
[n_{_e}, e] > [tt, isPalListKind, n_{_{isPalListKind}}, isList, U53, U41₂, U12, U23, U11, isPal₁, n_{_isPal₁}, U72, U62, n_{_i}, i] > U42₂ > [isNeList₁, U51₁]

Status:

i: []
U51^1₂: [1,2]
U11^1₁: [1]
__₂: [1,2]
n_{_u}: []
isPalListKind: []
n_{_i}: []
U21^1₂: [1,2]
U11: []
n_{_nil}: []
n_{_a}: []
isList: []
U61: []
tt: []
U22^1₁: [1]
U72: []
isNePal: []
U23: []
nil: []
a: []
ISNELIST₁: [1]
U62: []
U42₂: [1,2]
n_{_isPal₁}: [1]
e: []
U53: []
U12: []
n_{_e}: []
n_{_{_{_₂}}}: [1,2]
o: []
U41^1₃: [1,2,3]
U52^1₁: [1]
isPal₁: [1]
n_{_o}: []
U42^1₂: [1,2]
n_{_{isPalListKind}}: []
U41₂: [1,2]
u: []
isNeList₁: [1]
ISLIST₁: [1]
U51₁: [1]

The following usable rules [FROCOS05] were oriented:

U51(tt, V1, V2) → U52(isNeList(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isQid(activate(V)))
U42(tt, V2) → U43(isNeList(activate(V2)))
U41(tt, V1, V2) → U42(isList(activate(V1)), activate(V2))
U21(tt, V1, V2) → U22(isList(activate(V1)), activate(V2))
U12(tt) → tt
U23(tt) → tt
U22(tt, V2) → U23(isList(activate(V2)))
__(X, nil) → X
__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt, V) → U12(isNeList(activate(V)))
__(nil, X) → X
isPal(n__nil) → tt
isPalListKind(n__a) → tt
isNePal(n____(I, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P))))
isPal(V) → U71(isPalListKind(activate(V)), activate(V))
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))
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))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2))
isList(V) → U11(isPalListKind(activate(V)), activate(V))
U71(tt, V) → U72(isNePal(activate(V)))
U72(tt) → tt
U61(tt, V) → U62(isQid(activate(V)))
U62(tt) → tt
isPal(X) → n__isPal(X)
and(X1, X2) → n__and(X1, X2)
isPalListKind(X) → n__isPalListKind(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
isQid(n__u) → tt
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt
isQid(n__a) → tt
isPalListKind(n__u) → tt
isPalListKind(n__o) → tt
isPalListKind(n__nil) → tt
isPalListKind(n__i) → tt
isPalListKind(n__e) → tt
o → n__o
i → n__i
e → n__e
a → n__a
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
and(tt, X) → activate(X)
isPalListKind(n____(V1, V2)) → and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2)))
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__nil) → nil
u → n__u
activate(n__e) → e
activate(n__a) → a
activate(n__isPal(X)) → isPal(X)
activate(X) → X
activate(n__u) → u
activate(n__o) → o
activate(n__i) → i

(21) Obligation:

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

ISLIST(V) → U11¹(isPalListKind(activate(V)), activate(V))
U22¹(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(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, n____(P, I))) → and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__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)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isPalListKind(X)) → isPalListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isPal(X)) → isPal(X)
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.

(22) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) DependencyGraphProof (EQUIVALENT transformation)

(23) TRUE