Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/SK90SLASH2.36.trs

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:

implies₂(not₁(x), y) -> or₂(x, y)
implies₂(not₁(x), or₂(y, z)) -> implies₂(y, or₂(x, z))
implies₂(x, or₂(y, z)) -> or₂(y, implies₂(x, z))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

implies₂(not₁(x), y) -> or₂(x, y)
implies₂(not₁(x), or₂(y, z)) -> implies₂(y, or₂(x, z))
implies₂(x, or₂(y, z)) -> or₂(y, implies₂(x, z))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IMPLIES₂(x, or₂(y, z)) -> IMPLIES₂(x, z)
IMPLIES₂(not₁(x), or₂(y, z)) -> IMPLIES₂(y, or₂(x, z))

The TRS R consists of the following rules:

implies₂(not₁(x), y) -> or₂(x, y)
implies₂(not₁(x), or₂(y, z)) -> implies₂(y, or₂(x, z))
implies₂(x, or₂(y, z)) -> or₂(y, implies₂(x, z))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

IMPLIES₂(x, or₂(y, z)) -> IMPLIES₂(x, z)
IMPLIES₂(not₁(x), or₂(y, z)) -> IMPLIES₂(y, or₂(x, z))

The TRS R consists of the following rules:

implies₂(not₁(x), y) -> or₂(x, y)
implies₂(not₁(x), or₂(y, z)) -> implies₂(y, or₂(x, z))
implies₂(x, or₂(y, z)) -> or₂(y, implies₂(x, z))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

IMPLIES₂(x, or₂(y, z)) -> IMPLIES₂(x, z)

The remaining pairs can at least by weakly be oriented.

IMPLIES₂(not₁(x), or₂(y, z)) -> IMPLIES₂(y, or₂(x, z))

Used ordering: Combined order from the following AFS and order.
IMPLIES₂(x1, x2) = x2
or₂(x1, x2) = or₁(x2)
not₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

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

IMPLIES₂(not₁(x), or₂(y, z)) -> IMPLIES₂(y, or₂(x, z))

The TRS R consists of the following rules:

implies₂(not₁(x), y) -> or₂(x, y)
implies₂(not₁(x), or₂(y, z)) -> implies₂(y, or₂(x, z))
implies₂(x, or₂(y, z)) -> or₂(y, implies₂(x, z))

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