Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/D33SLASH20.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:

not₁(not₁(x)) -> x
not₁(or₂(x, y)) -> and₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))
not₁(and₂(x, y)) -> or₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

not₁(not₁(x)) -> x
not₁(or₂(x, y)) -> and₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))
not₁(and₂(x, y)) -> or₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))

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:

NOT₁(and₂(x, y)) -> NOT₁(x)
NOT₁(and₂(x, y)) -> NOT₁(not₁(x))
NOT₁(and₂(x, y)) -> NOT₁(y)
NOT₁(or₂(x, y)) -> NOT₁(x)
NOT₁(or₂(x, y)) -> NOT₁(not₁(not₁(x)))
NOT₁(or₂(x, y)) -> NOT₁(y)
NOT₁(and₂(x, y)) -> NOT₁(not₁(not₁(y)))
NOT₁(or₂(x, y)) -> NOT₁(not₁(x))
NOT₁(and₂(x, y)) -> NOT₁(not₁(y))
NOT₁(or₂(x, y)) -> NOT₁(not₁(y))
NOT₁(and₂(x, y)) -> NOT₁(not₁(not₁(x)))
NOT₁(or₂(x, y)) -> NOT₁(not₁(not₁(y)))

The TRS R consists of the following rules:

not₁(not₁(x)) -> x
not₁(or₂(x, y)) -> and₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))
not₁(and₂(x, y)) -> or₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

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

NOT₁(and₂(x, y)) -> NOT₁(x)
NOT₁(and₂(x, y)) -> NOT₁(not₁(x))
NOT₁(and₂(x, y)) -> NOT₁(y)
NOT₁(or₂(x, y)) -> NOT₁(x)
NOT₁(or₂(x, y)) -> NOT₁(not₁(not₁(x)))
NOT₁(or₂(x, y)) -> NOT₁(y)
NOT₁(and₂(x, y)) -> NOT₁(not₁(not₁(y)))
NOT₁(or₂(x, y)) -> NOT₁(not₁(x))
NOT₁(and₂(x, y)) -> NOT₁(not₁(y))
NOT₁(or₂(x, y)) -> NOT₁(not₁(y))
NOT₁(and₂(x, y)) -> NOT₁(not₁(not₁(x)))
NOT₁(or₂(x, y)) -> NOT₁(not₁(not₁(y)))

The TRS R consists of the following rules:

not₁(not₁(x)) -> x
not₁(or₂(x, y)) -> and₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))
not₁(and₂(x, y)) -> or₂(not₁(not₁(not₁(x))), not₁(not₁(not₁(y))))

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