Termination w.r.t. Q proof of TRS/Endrullispair2simple2.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:

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

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:

P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(a₁(b₁(x0))), x2)
P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

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:

P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(a₁(b₁(x0))), x2)
P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

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 oriented strictly and are deleted.

P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(a₁(b₁(x0))), x2)

The remaining pairs can at least be oriented weakly.

P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

Used ordering: Polynomial interpretation [21]:

POL(P₂(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a₁(x₁)) = 1
POL(b₁(x₁)) = 1 + 2·x₁
POL(p₂(x₁, x₂)) = 2 + 2·x₂

The following usable rules [14] were oriented:

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

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

P₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> P₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

The TRS R consists of the following rules:

p₂(a₁(x0), p₂(a₁(a₁(a₁(x1))), x2)) -> p₂(a₁(x2), p₂(a₁(a₁(b₁(x0))), x2))

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