Term Rewriting System R:
[x, y]
ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AP(f, x) -> AP(f, app(g, x))

Furthermore, R contains one SCC.

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

Dependency Pair:

AP(f, x) -> AP(f, app(g, x))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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

AP(f, x) -> AP(f, app(g, x))
one new Dependency Pair is created:

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))
one new Dependency Pair is created:

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, x''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 3`
`                 ↳Instantiation Transformation`

Dependency Pair:

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, x''''))))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, x''''))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, x'''''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 4`
`                 ↳Instantiation Transformation`

Dependency Pair:

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, x'''''')))))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, x'''''')))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, x''''''''))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`

Dependency Pair:

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, x''''''''))))))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, x''''''''))))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Inst`
`             ...`
`               →DP Problem 6`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pair:

AP(f, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))))

Rules:

ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

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