Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar4/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.

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

      ↳ QDPAfsSolverProof

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.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

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

Used argument filtering: IMPLIES₂(x1, x2) = x2
or₂(x1, x2) = or₁(x2)
Used ordering: Quasi Precedence: trivial

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPAfsSolverProof

        ↳ 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.