Termination w.r.t. Q proof of /tmp/tmpdV62zl/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)
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 remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2) = A____(x0)
MARK(x0, x1) = MARK(x0)
A__AND(x0, x1, x2) = A__AND(x0, x2)
A__ISLIST(x0, x1) = A__ISLIST(x1)
A__ISNELIST(x0, x1) = A__ISNELIST(x0, x1)
A__ISNEPAL(x0, x1) = A__ISNEPAL(x1)
A__ISPAL(x0, x1) = A__ISPAL(x1)

Tags:
A____ has argument tags [10,0,6] and root tag 0
MARK has argument tags [10,8] and root tag 0
A__AND has argument tags [6,5,13] and root tag 4
A__ISLIST has argument tags [1,4] and root tag 3
A__ISNELIST has argument tags [10,4] and root tag 2
A__ISNEPAL has argument tags [11,4] and root tag 2
A__ISPAL has argument tags [0,4] and root tag 7

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
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(x1, 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) = x1
a__isQid(x1) = x1
isPal(x1) = isPal(x1)
A__ISPAL(x1) = A__ISPAL
nil = nil
a__and(x1, x2) = a__and(x1, x2)
a__isNePal(x1) = a__isNePal(x1)
a__isPal(x1) = a__isPal(x1)
isQid(x1) = x1
a = a
e = e
i = i
o = o
u = u

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

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

Status:

A_{_{_{_₂}}}: [1,2]
__₂: [1,2]
a_{_{_{_₂}}}: [1,2]
and₂: multiset
A_{_AND₂}: multiset
tt: multiset
isList₁: multiset
A_{_ISLIST₁}: multiset
A_{_ISNELIST₁}: [1]
a_{_isList₁}: multiset
isNeList₁: multiset
a_{_isNeList₁}: multiset
isNePal₁: [1]
isPal₁: multiset
A_{_ISPAL}: []
nil: multiset
a_{_and₂}: multiset
a_{_isNePal₁}: [1]
a_{_isPal₁}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

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))

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 1 less node.

(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