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:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(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(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), 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(x, y)
F(f(0, x), 1) → F(x, x)
F(f(0, x), 1) → F(g(f(x, x)), x)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

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

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

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

F(g(x), y) → F(x, y)
F(f(0, x), 1) → F(x, x)
F(f(0, x), 1) → F(g(f(x, x)), x)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

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

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

F(g(x), y) → F(x, y)
F(f(0, x), 1) → F(x, x)
F(f(0, x), 1) → F(g(f(x, x)), x)

The TRS R consists of the following rules:

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))

The set Q consists of the following terms:

f(f(0, x0), 1)
f(g(x0), x1)

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