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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(f, app(h, x))

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))

Rules:

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

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

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)

Rules:

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

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

APP(g, app(g, x)) -> APP(h, app(g, x))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

Rules:

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

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

APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

Rules:

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

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

APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
three new Dependency Pairs are created:

APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))

Rules:

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

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

APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Narrowing Transformation`

Dependency Pairs:

APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

Rules:

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

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

APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
three new Dependency Pairs are created:

APP(g, app(g, app(g, x'''))) -> APP(h, app(g, x'''))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(g, app(g, x')))) -> APP(h, app(g, app(h, app(g, app(h, app(g, x'))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 7`
`                 ↳Forward Instantiation Transformation`

Dependency Pairs:

APP(g, app(g, app(g, app(g, x')))) -> APP(h, app(g, app(h, app(g, app(h, app(g, x'))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))
APP(g, app(g, app(g, x'''))) -> APP(h, app(g, x'''))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(h, app(g, x'))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))
APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(h, app(g, app(h, app(g, x'))))

Rules:

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

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

APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
no new Dependency Pairs are created.
The transformation is resulting in two new DP problems:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 8`
`                 ↳Polynomial Ordering`

Dependency Pair:

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

Rules:

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

The following dependency pair can be strictly oriented:

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

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

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 10`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 9`
`                 ↳Polynomial Ordering`

Dependency Pairs:

APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

Rules:

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

The following dependency pairs can be strictly oriented:

APP(g, app(g, app(g, app(h, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(g, x'''))) -> APP(g, app(h, app(g, x''')))
APP(g, app(g, app(h, app(g, app(g, x'))))) -> APP(g, app(h, app(g, app(h, app(g, x')))))
APP(g, app(g, app(h, app(g, app(h, app(g, x')))))) -> APP(g, app(h, app(g, x')))
APP(g, app(g, app(g, app(g, x')))) -> APP(g, app(h, app(g, app(h, app(g, app(h, app(g, x')))))))

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

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

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

resulting in one new DP problem.

Termination of R successfully shown.
Duration:
0:01 minutes