Termination Proof using AProVE

Term Rewriting System R:
[x]
app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(f, app(h, x))

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Strategy:
innermost
Dependency Pair:
APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Strategy:
innermost
Dependency Pair:
APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))

Rules:

app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes