Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/SK90SLASH2.32.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

not₁(x) -> if₃(x, false, true)
and₂(x, y) -> if₃(x, y, false)
or₂(x, y) -> if₃(x, true, y)
implies₂(x, y) -> if₃(x, y, true)
=₂(x, x) -> true
=₂(x, y) -> if₃(x, y, not₁(y))
if₃(true, x, y) -> x
if₃(false, x, y) -> y
if₃(x, x, if₃(x, false, true)) -> true
=₂(x, y) -> if₃(x, y, if₃(y, false, true))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

not₁(x) -> if₃(x, false, true)
and₂(x, y) -> if₃(x, y, false)
or₂(x, y) -> if₃(x, true, y)
implies₂(x, y) -> if₃(x, y, true)
=₂(x, x) -> true
=₂(x, y) -> if₃(x, y, not₁(y))
if₃(true, x, y) -> x
if₃(false, x, y) -> y
if₃(x, x, if₃(x, false, true)) -> true
=₂(x, y) -> if₃(x, y, if₃(y, false, true))

Q is empty.

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

AND₂(x, y) -> IF₃(x, y, false)
NOT₁(x) -> IF₃(x, false, true)
OR₂(x, y) -> IF₃(x, true, y)
=¹₂(x, y) -> NOT₁(y)
=¹₂(x, y) -> IF₃(x, y, if₃(y, false, true))
=¹₂(x, y) -> IF₃(x, y, not₁(y))
IMPLIES₂(x, y) -> IF₃(x, y, true)
=¹₂(x, y) -> IF₃(y, false, true)

The TRS R consists of the following rules:

not₁(x) -> if₃(x, false, true)
and₂(x, y) -> if₃(x, y, false)
or₂(x, y) -> if₃(x, true, y)
implies₂(x, y) -> if₃(x, y, true)
=₂(x, x) -> true
=₂(x, y) -> if₃(x, y, not₁(y))
if₃(true, x, y) -> x
if₃(false, x, y) -> y
if₃(x, x, if₃(x, false, true)) -> true
=₂(x, y) -> if₃(x, y, if₃(y, false, true))

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:

AND₂(x, y) -> IF₃(x, y, false)
NOT₁(x) -> IF₃(x, false, true)
OR₂(x, y) -> IF₃(x, true, y)
=¹₂(x, y) -> NOT₁(y)
=¹₂(x, y) -> IF₃(x, y, if₃(y, false, true))
=¹₂(x, y) -> IF₃(x, y, not₁(y))
IMPLIES₂(x, y) -> IF₃(x, y, true)
=¹₂(x, y) -> IF₃(y, false, true)

The TRS R consists of the following rules:

not₁(x) -> if₃(x, false, true)
and₂(x, y) -> if₃(x, y, false)
or₂(x, y) -> if₃(x, true, y)
implies₂(x, y) -> if₃(x, y, true)
=₂(x, x) -> true
=₂(x, y) -> if₃(x, y, not₁(y))
if₃(true, x, y) -> x
if₃(false, x, y) -> y
if₃(x, x, if₃(x, false, true)) -> true
=₂(x, y) -> if₃(x, y, if₃(y, false, true))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 0 SCCs with 8 less nodes.