Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.


↳ QTRS
  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ISPAL → ISNEPAL
ISLIST → ISLIST
ISNELIST → ISLIST
U711(tt) → ISPAL
ISNEPAL → U611(isQid)
__1(__(X, Y), Z) → __1(X, __(Y, Z))
U511(tt) → U521(isList)
ISNELIST → U411(isList)
U211(tt) → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
U411(tt) → ISNELIST
ISNEPAL → ISQID
U711(tt) → U721(isPal)
__1(__(X, Y), Z) → __1(Y, Z)
ISNELIST → ISNELIST
ISLIST → ISNELIST
U211(tt) → U221(isList)
U411(tt) → U421(isNeList)
ISPAL → U811(isNePal)
ISNEPAL → U711(isQid)
ISLIST → U111(isNeList)
ISNELIST → U311(isQid)
ISNELIST → ISQID
ISLIST → U211(isList)

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.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

ISPAL → ISNEPAL
ISLIST → ISLIST
ISNELIST → ISLIST
U711(tt) → ISPAL
ISNEPAL → U611(isQid)
__1(__(X, Y), Z) → __1(X, __(Y, Z))
U511(tt) → U521(isList)
ISNELIST → U411(isList)
U211(tt) → ISLIST
ISNELIST → U511(isNeList)
U511(tt) → ISLIST
U411(tt) → ISNELIST
ISNEPAL → ISQID
U711(tt) → U721(isPal)
__1(__(X, Y), Z) → __1(Y, Z)
ISNELIST → ISNELIST
ISLIST → ISNELIST
U211(tt) → U221(isList)
U411(tt) → U421(isNeList)
ISPAL → U811(isNePal)
ISNEPAL → U711(isQid)
ISLIST → U111(isNeList)
ISNELIST → U311(isQid)
ISNELIST → ISQID
ISLIST → U211(isList)

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.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 10 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

ISPAL → ISNEPAL
U711(tt) → ISPAL
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.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

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.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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.
We use the reduction pair processor [15].


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.
none
Used ordering: Polynomial interpretation [25,35]:

POL(__(x1, x2)) = 13/4 + (2)x_1 + (4)x_2   
POL(__1(x1, x2)) = (4)x_1   
POL(nil) = 0   
The value of delta used in the strict ordering is 13.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof

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