Term Rewriting System R:
[x, y, z]
app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(f, app(s, x))
APP(app(app(f, 0), 1), x) -> APP(s, x)
APP(app(app(f, x), y), app(s, z)) -> APP(s, app(app(app(f, 0), 1), z))
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(f, 0)

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)

Rules:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Strategy:

innermost

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

APP(app(app(f, 0), 1), x) -> APP(app(f, app(s, x)), x)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)

Rules:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Strategy:

innermost

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

APP(app(app(f, x), y), app(s, z)) -> APP(app(f, 0), 1)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pairs:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)

Rules:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Narrowing Transformation`

Dependency Pair:

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)

Rules:

app(app(app(f, 0), 1), x) -> app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) -> app(s, app(app(app(f, 0), 1), z))

Strategy:

innermost

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

APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), x)
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

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