Term Rewriting System R:
[f, x, y]
app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

Rules:

app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))

Strategy:

innermost

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

APP(app(app(rec, f), x), app(s, y)) -> APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
three new Dependency Pairs are created:

APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

APP(app(app(rec, f''), x''), app(s, app(s, y''))) -> APP(app(f'', app(s, app(s, y''))), app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')))
APP(app(app(rec, f''), x''), app(s, 0)) -> APP(app(f'', app(s, 0)), x'')
APP(app(app(rec, app(app(rec, f''), x'')), x), app(s, y'')) -> APP(app(app(f'', app(s, y'')), app(app(app(rec, f''), x''), y'')), app(app(app(rec, app(app(rec, f''), x'')), x), y''))
APP(app(app(rec, f), x), app(s, y)) -> APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) -> APP(app(app(rec, f), x), y)

Rules:

app(app(app(rec, f), x), 0) -> x
app(app(app(rec, f), x), app(s, y)) -> app(app(f, app(s, y)), app(app(app(rec, f), x), y))

Strategy:

innermost

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