Term Rewriting System R:
[x]
ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AP(ap(ff, x), x) -> AP(ap(x, ap(ff, x)), ap(ap(cons, x), nil))
AP(ap(ff, x), x) -> AP(x, ap(ff, x))
AP(ap(ff, x), x) -> AP(ap(cons, x), nil)
AP(ap(ff, x), x) -> AP(cons, x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

AP(ap(ff, x), x) -> AP(ap(cons, x), nil)
AP(ap(ff, x), x) -> AP(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

Rule:

ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

Strategy:

innermost

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

AP(ap(ff, x), x) -> AP(ap(x, ap(ff, x)), ap(ap(cons, x), nil))
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 Pair:

AP(ap(ff, x), x) -> AP(ap(cons, x), nil)

Rule:

ap(ap(ff, x), x) -> ap(ap(x, ap(ff, x)), ap(ap(cons, x), nil))

Strategy:

innermost

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

AP(ap(ff, x), x) -> AP(ap(cons, x), nil)
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