Termination w.r.t. Q proof of /tmp/tmpf2cTOZ/PALINDROME_nokinds

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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:

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISNELIST(V)
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(V) → A__ISQID(V)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNEPAL(V) → A__ISQID(V)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISPAL(V) → A__ISNEPAL(V)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isQid(X)) → A__ISQID(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(isPal(X)) → A__ISPAL(X)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(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 1 SCC with 4 less nodes.

(4) Obligation:

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

A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__ISNEPAL(V)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isList(V1), isNeList(V2))
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(__(V1, V2)) → A__AND(a__isList(V1), isList(V2))
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__AND(a__isNeList(V1), isList(V2))
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPal(P))
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A____(x1, x2) = A____(x1, x2)
__(x1, x2) = __(x1, x2)
MARK(x1) = x1
mark(x1) = x1
a____(x1, x2) = a____(x1, x2)
and(x1, x2) = and(x1, x2)
A__AND(x1, x2) = A__AND(x2)
tt = tt
isList(x1) = isList(x1)
A__ISLIST(x1) = A__ISLIST(x1)
A__ISNELIST(x1) = A__ISNELIST(x1)
a__isList(x1) = a__isList(x1)
isNeList(x1) = isNeList(x1)
a__isNeList(x1) = a__isNeList(x1)
isNePal(x1) = isNePal(x1)
A__ISNEPAL(x1) = A__ISNEPAL(x1)
a__isQid(x1) = a__isQid
isPal(x1) = isPal(x1)
A__ISPAL(x1) = A__ISPAL(x1)
nil = nil
a__and(x1, x2) = a__and(x1, x2)
a__isNePal(x1) = a__isNePal(x1)
a__isPal(x1) = a__isPal(x1)
isQid(x1) = isQid
a = a
e = e
i = i
o = o
u = u

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

[isList₁, A_{_ISLIST₁}, A_{_ISNELIST₁}, a_{_isList₁}, isNeList₁, a_{_isNeList₁}] > [and₂, a_{_and₂}] > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, A_{_AND₁}, tt, a_{_isQid}, nil, isQid, a, e, i, o, u]
[isNePal₁, A_{_ISNEPAL₁}, isPal₁, A_{_ISPAL₁}, a_{_isNePal₁}, a_{_isPal₁}] > [and₂, a_{_and₂}] > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, A_{_AND₁}, tt, a_{_isQid}, nil, isQid, a, e, i, o, u]

Status:

A_{_{_{_₂}}}: [1,2]
__₂: [1,2]
a_{_{_{_₂}}}: [1,2]
and₂: [2,1]
A_{_AND₁}: [1]
tt: []
isList₁: [1]
A_{_ISLIST₁}: [1]
A_{_ISNELIST₁}: [1]
a_{_isList₁}: [1]
isNeList₁: [1]
a_{_isNeList₁}: [1]
isNePal₁: [1]
A_{_ISNEPAL₁}: [1]
a_{_isQid}: []
isPal₁: [1]
A_{_ISPAL₁}: [1]
nil: []
a_{_and₂}: [2,1]
a_{_isNePal₁}: [1]
a_{_isPal₁}: [1]
isQid: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented:

a____(X, nil) → mark(X)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(nil, X) → mark(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__and(tt, X) → mark(X)
mark(isList(X)) → a__isList(X)
a__isList(V) → a__isNeList(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
mark(isNeList(X)) → a__isNeList(X)
mark(isNePal(X)) → a__isNePal(X)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
mark(isPal(X)) → a__isPal(X)
a__isPal(V) → a__isNePal(V)
mark(isQid(X)) → a__isQid(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__isList(nil) → tt
a__isList(X) → isList(X)
a__isNeList(V) → a__isQid(V)
a__isNeList(X) → isNeList(X)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
a__isQid(X) → isQid(X)
a__and(X1, X2) → and(X1, X2)
a__isNePal(V) → a__isQid(V)
a__isNePal(X) → isNePal(X)
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(6) Obligation:

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

MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(isList(X)) → A__ISLIST(X)
A__ISLIST(V) → A__ISNELIST(V)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__ISNEPAL(V)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) TRUE