Termination Proof using AProVE

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.

     ↳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.

     ↳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:

     ↳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:

     ↳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:

     ↳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 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.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Remaining Obligation(s)

The following remains to be proven:
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

Innermost Termination of R could not be shown.
Duration:
0:02 minutes

POL(*)	= 1
POL(app(x₁, x₂))	= x₁
POL(+)	= 0
POL(APP(x₁, x₂))	= x₁