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

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Forward Instantiation Transformation

Dependency Pair:

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

Rules:

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

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

APP(f, app(g, x)) -> APP(f, x)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

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

Rules:

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

Strategy:

innermost

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