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:

f(g(x, y), f(y, y)) → f(g(y, x), y)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(g(x, y), f(y, y)) → f(g(y, x), y)

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(g(x, y), f(y, y)) → f(g(y, x), y)

The set Q consists of the following terms:

f(g(x0, x1), f(x1, x1))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F(g(x, y), f(y, y)) → F(g(y, x), y)

The TRS R consists of the following rules:

f(g(x, y), f(y, y)) → f(g(y, x), y)

The set Q consists of the following terms:

f(g(x0, x1), f(x1, x1))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof

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

F(g(x, y), f(y, y)) → F(g(y, x), y)

The TRS R consists of the following rules:

f(g(x, y), f(y, y)) → f(g(y, x), y)

The set Q consists of the following terms:

f(g(x0, x1), f(x1, x1))

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.


F(g(x, y), f(y, y)) → F(g(y, x), y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
F(x1, x2)  =  F(x2)
g(x1, x2)  =  g(x2)
f(x1, x2)  =  f(x2)

Lexicographic path order with status [19].
Quasi-Precedence:
[g1, f1] > F1

Status:
f1: [1]
F1: [1]
g1: [1]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

f(g(x, y), f(y, y)) → f(g(y, x), y)

The set Q consists of the following terms:

f(g(x0, x1), f(x1, x1))

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