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:

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

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:

P3(m, n, s1(r)) -> P3(m, r, n)
P3(m, s1(n), 0) -> P3(0, n, m)

The TRS R consists of the following rules:

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

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:

P3(m, n, s1(r)) -> P3(m, r, n)
P3(m, s1(n), 0) -> P3(0, n, m)

The TRS R consists of the following rules:

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

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.


P3(m, s1(n), 0) -> P3(0, n, m)
The remaining pairs can at least be oriented weakly.

P3(m, n, s1(r)) -> P3(m, r, n)
Used ordering: Polynomial interpretation [21]:

POL(0) = 2   
POL(P3(x1, x2, x3)) = 2·x1·x2·x3 + 2·x2·x3   
POL(s1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof

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

P3(m, n, s1(r)) -> P3(m, r, n)

The TRS R consists of the following rules:

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

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.


P3(m, n, s1(r)) -> P3(m, r, n)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(P3(x1, x2, x3)) = 2·x2·x3 + 2·x3   
POL(s1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

p3(m, n, s1(r)) -> p3(m, r, n)
p3(m, s1(n), 0) -> p3(0, n, m)
p3(m, 0, 0) -> m

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.