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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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