Term Rewriting System R:
[x, y, z]
app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

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

APP(app(*, x), app(app(+, y), z)) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
two new Dependency Pairs are created:

APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

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

APP(app(*, x''), app(app(+, app(app(+, y''), z'')), z)) -> APP(app(+, app(app(+, app(app(*, x''), y'')), app(app(*, x''), z''))), app(app(*, x''), z))
three new Dependency Pairs are created:

APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))

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(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

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

APP(app(*, x''), app(app(+, y), app(app(+, y''), z''))) -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
three new Dependency Pairs are created:

APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`

Dependency Pairs:

APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(*, x'''), app(app(+, y), app(app(+, y''), app(app(+, y'''), z')))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))))
APP(app(*, x'''), app(app(+, y), app(app(+, app(app(+, y'''), z')), z''))) -> APP(app(+, app(app(*, x'''), y)), app(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y'''), z')), app(app(+, y''), z''))) -> APP(app(+, app(app(+, app(app(*, x'''), y''')), app(app(*, x'''), z'))), app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z'')))
APP(app(*, x'''), app(app(+, app(app(+, y''), app(app(+, y'), z'''))), z)) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, app(app(+, y'), z''')), z'')), z)) -> APP(app(+, app(app(+, app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z'''))), app(app(*, x'''), z''))), app(app(*, x'''), z))
APP(app(*, x'''), app(app(+, app(app(+, y''), z'')), app(app(+, y'), z'''))) -> APP(app(+, app(app(+, app(app(*, x'''), y'')), app(app(*, x'''), z''))), app(app(+, app(app(*, x'''), y')), app(app(*, x'''), z''')))

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

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pairs:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)

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 POL(app(x1, x2)) =  x1 + x2 POL(+) =  1 POL(APP(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes