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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

The following usable rule for innermost can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
APP > app

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:

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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