Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
  ↳6 QDP
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 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) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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)
U211(tt) → U221(isList)
U211(tt) → ISLIST
U411(tt) → U421(isNeList)
U411(tt) → ISNELIST
U511(tt) → U521(isList)
U511(tt) → ISLIST
U711(tt) → U721(isPal)
U711(tt) → ISPAL
ISLIST → U111(isNeList)
ISLIST → ISNELIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U311(isQid)
ISNELIST → ISQID
ISNELIST → U411(isList)
ISNELIST → ISLIST
ISNELIST → U511(isNeList)
ISNELIST → ISNELIST
ISNEPAL → U611(isQid)
ISNEPAL → ISQID
ISNEPAL → U711(isQid)
ISPAL → U811(isNePal)
ISPAL → ISNEPAL

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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 10 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

U711(tt) → ISPAL
ISPAL → ISNEPAL
ISNEPAL → U711(isQid)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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

(6) Obligation:

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

U211(tt) → ISLIST
ISLIST → ISNELIST
ISNELIST → U411(isList)
U411(tt) → ISNELIST
ISNELIST → ISLIST
ISLIST → U211(isList)
ISLIST → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
ISNELIST → ISNELIST

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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

(7) 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) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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

(8) 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: Combined order from the following AFS and order.
__1(x1, x2)  =  __1(x1, x2)
__(x1, x2)  =  __(x1, x2)
nil  =  nil
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U22(x1)  =  U22
isList  =  isList
U31(x1)  =  x1
U41(x1)  =  U41(x1)
U42(x1)  =  U42
isNeList  =  isNeList
U51(x1)  =  U51
U52(x1)  =  U52(x1)
U61(x1)  =  x1
U71(x1)  =  U71(x1)
U72(x1)  =  x1
isPal  =  isPal
U81(x1)  =  U81
isQid  =  isQid
isNePal  =  isNePal

Lexicographic path order with status [LPO].
Precedence:
_^12 > _2
isNeList > U411 > U42 > tt
isNeList > U51 > isList > U11 > tt
isNeList > U51 > isList > U21 > U22 > tt
isNeList > U51 > U521 > tt
isNeList > isQid > tt
isNePal > U711 > isPal > U81 > tt
isNePal > isQid > tt

Status:
isPal: []
U21: []
U51: []
_2: [1,2]
U22: []
U411: [1]
U42: []
isNeList: []
U11: []
isList: []
tt: []
U521: [1]
_^12: [1,2]
U81: []
isNePal: []
isQid: []
U711: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

(9) 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) → tt
U21(tt) → U22(isList)
U22(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNeList)
U42(tt) → tt
U51(tt) → U52(isList)
U52(tt) → tt
U61(tt) → tt
U71(tt) → U72(isPal)
U72(tt) → tt
U81(tt) → tt
isList → U11(isNeList)
isList → tt
isList → U21(isList)
isNeList → U31(isQid)
isNeList → U41(isList)
isNeList → U51(isNeList)
isNePal → U61(isQid)
isNePal → U71(isQid)
isPal → U81(isNePal)
isPal → tt
isQid → tt

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE