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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

Strategy:

innermost

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

APP(app(app(curry, f), x), y) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)

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

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(plus) =  0 POL(0) =  1 POL(curry) =  1 POL(s) =  0 POL(app(x1, x2)) =  x1 + 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`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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