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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)

Furthermore, R contains one SCC.

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

Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)

Rules:

app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

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

Rules:

app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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