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.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Nar`

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

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

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`       →DP Problem 2`
`         ↳Narrowing Transformation`

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

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

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes