Termination w.r.t. Q proof of /tmp/tmpDRcEzM/PALINDROME_nokinds

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 DependencyGraphProof (⇔)

    ↳14 TRUE

(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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → ISNELIST(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → ISQID(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ISQID(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, __(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → ISNEPAL(activate(V))
ISPAL(V) → ACTIVATE(V)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __¹(X1, X2)
ACTIVATE(n__isList(X)) → ISLIST(X)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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 2 SCCs with 10 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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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__isList(X)) → ISLIST(X)
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(V) → ACTIVATE(V)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(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
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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__isList(X)) → ISLIST(X)
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(V) → ACTIVATE(V)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(n____(I, __(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, __(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_nil}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_o}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > u > n_{_u}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > [n_{_e}, e]
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > ISNELIST₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > a > n_{_a}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_nil}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_o}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > u > n_{_u}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > [n_{_e}, e]
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isList₁}, ISLIST₁, AND₂, isList₁] > [n_{_isNeList₁}, isNeList₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > a > n_{_a}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_nil}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_o}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > u > n_{_u}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > [n_{_e}, e]
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > ISPAL₁ > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > a > n_{_a}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_nil}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > n_{_o}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > u > n_{_u}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > [n_{_e}, e]
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > i > n_{_i}
[n_{_{_{_₂}}}, _{_₂}] > [n_{_isPal₁}, isPal₁] > [isNePal₁, and₂] > isQid₁ > tt > [ACTIVATE₁, ISNEPAL₁] > [activate₁, nil, o] > a > n_{_a}

Status:

i: []
n_{_u}: []
__₂: [1,2]
n_{_i}: []
activate₁: [1]
n_{_nil}: []
and₂: [1,2]
n_{_a}: []
tt: []
ISPAL₁: [1]
AND₂: [2,1]
n_{_isList₁}: [1]
nil: []
ACTIVATE₁: [1]
a: []
isList₁: [1]
ISNELIST₁: [1]
n_{_isPal₁}: [1]
e: []
n_{_e}: []
isNePal₁: [1]
o: []
n_{_{_{_₂}}}: [1,2]
isQid₁: [1]
n_{_isNeList₁}: [1]
isPal₁: [1]
n_{_o}: []
u: []
isNeList₁: [1]
ISLIST₁: [1]
ISNEPAL₁: [1]

The following usable rules [FROCOS05] were oriented:

isNeList(V) → isQid(activate(V))
isNePal(V) → isQid(activate(V))
isList(n__nil) → tt
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
activate(n__o) → o
activate(n__u) → u
activate(n__e) → e
activate(n__i) → i
activate(X) → X
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
o → n__o
u → n__u
activate(n__a) → a
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isPal(V) → isNePal(activate(V))
activate(n__isPal(X)) → isPal(X)
isNeList(X) → n__isNeList(X)
isList(X) → n__isList(X)
__(X1, X2) → n____(X1, X2)
nil → n__nil
i → n__i
e → n__e
a → n__a
isPal(X) → n__isPal(X)
isQid(n__a) → tt
isPal(n__nil) → tt
isQid(n__u) → tt
isQid(n__o) → tt
isQid(n__i) → tt
isQid(n__e) → tt

(12) Obligation:

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

ISNEPAL(V) → ACTIVATE(V)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, __(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
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)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
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)) → __(X1, X2)
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
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 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) TRUE