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:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(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:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

The set Q consists of the following terms:

g(f(x0), x1)
h(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:

H(x, y) → G(x, f(y))
G(f(x), y) → H(x, y)

The TRS R consists of the following rules:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

The set Q consists of the following terms:

g(f(x0), x1)
h(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:

H(x, y) → G(x, f(y))
G(f(x), y) → H(x, y)

The TRS R consists of the following rules:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

The set Q consists of the following terms:

g(f(x0), x1)
h(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
              ↳ QDPOrderProof

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

H(x, y) → G(x, f(y))
G(f(x), y) → H(x, y)

The TRS R consists of the following rules:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

The set Q consists of the following terms:

g(f(x0), x1)
h(x0, 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.


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

Lexicographic Path Order [19].
Precedence:
[H1, f1] > G1


The following usable rules [14] were oriented: none



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

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

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

The set Q consists of the following terms:

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

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