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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g) =  0 POL(APP(x1, x2)) =  x1 + x2 POL(app(x1, x2)) =  1 + x1 + x2 POL(f) =  0

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> app(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pair:

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

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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