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(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

p2(a1(x0), p2(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))

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(x1, p2(x2, x3))) -> P2(a1(x3), x3)
P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x0, p2(a1(x3), x3))
P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x1, p2(x0, p2(a1(x3), x3)))

The TRS R consists of the following rules:

p2(a1(x0), p2(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))

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(x1, p2(x2, x3))) -> P2(a1(x3), x3)
P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x0, p2(a1(x3), x3))
P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x1, p2(x0, p2(a1(x3), x3)))

The TRS R consists of the following rules:

p2(a1(x0), p2(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))

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(x1, p2(x2, x3))) -> P2(a1(x3), x3)
P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x0, p2(a1(x3), x3))
The remaining pairs can at least be oriented weakly.

P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x1, p2(x0, p2(a1(x3), x3)))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( P2(x1, x2) ) = max{0, x2 - 1}


POL( p2(x1, x2) ) = x2 + 1


POL( a1(x1) ) = x1



The following usable rules [14] were oriented:

p2(a1(x0), p2(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP

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

P2(a1(x0), p2(x1, p2(x2, x3))) -> P2(x1, p2(x0, p2(a1(x3), x3)))

The TRS R consists of the following rules:

p2(a1(x0), p2(x1, p2(x2, x3))) -> p2(x1, p2(x0, p2(a1(x3), x3)))

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