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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

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

five new Dependency Pairs are created:

APP(app(*, app(app(+, y''), z'')), app(app(+, y0), z)) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(*, app(app(*, x''), y'')), app(app(+, y0), z)) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), 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'')))
APP(app(*, app(app(+, y''), z''')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(*, x''), y'')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(*, app(app(*, x''), y'')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))
APP(app(*, app(app(+, y''), z''')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(*, x''), y'')), app(app(+, y0), z)) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y''), z'')), app(app(+, y0), z)) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), 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'')))
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)

Rules:

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

Strategy:

innermost

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

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

five new Dependency Pairs are created:

APP(app(*, app(app(+, y0), z)), app(app(+, y''), z'')) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(*, app(app(+, y0), z)), app(app(*, x''), y'')) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y), app(app(+, y''), z''))), x'') -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
APP(app(*, app(app(+, y), z'')), app(app(+, y''), z''')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(+, y), z'')), app(app(*, x''), y'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), 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(*, app(app(+, y), z'')), app(app(*, x''), y'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))
APP(app(*, app(app(+, y), z'')), app(app(+, y''), z''')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(+, y), app(app(+, y''), z''))), x'') -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
APP(app(*, app(app(+, y0), z)), app(app(*, x''), y'')) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y0), z)), app(app(+, y''), z'')) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(*, app(app(+, y''), z''')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(*, x''), y'')), app(app(+, y0), z)) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y''), z'')), app(app(+, y0), z)) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), 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'')))
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, x), app(app(*, y), z))
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, app(app(*, x''), y'')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))

Rules:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

APP(app(*, app(app(*, x), y0)), app(app(+, y''), z'')) -> APP(app(*, x), app(app(+, app(app(*, y0), y'')), app(app(*, y0), z'')))
APP(app(*, app(app(*, x), app(app(*, x''), y''))), z'') -> APP(app(*, x), app(app(*, x''), app(app(*, y''), z'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(*, app(app(*, x), app(app(*, x''), y''))), z'') -> APP(app(*, x), app(app(*, x''), app(app(*, y''), z'')))
APP(app(*, app(app(*, x), y0)), app(app(+, y''), z'')) -> APP(app(*, x), app(app(+, app(app(*, y0), y'')), app(app(*, y0), z'')))
APP(app(*, app(app(+, y), z'')), app(app(+, y''), z''')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), z''')))
APP(app(*, app(app(+, y), app(app(+, y''), z''))), x'') -> APP(app(+, app(app(*, x''), y)), app(app(+, app(app(*, x''), y'')), app(app(*, x''), z'')))
APP(app(*, app(app(+, y0), z)), app(app(*, x''), y'')) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y0), z)), app(app(+, y''), z'')) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(*, app(app(*, x''), y'')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))
APP(app(*, app(app(+, y''), z''')), app(app(+, y), z'')) -> APP(app(+, app(app(*, app(app(+, y''), z''')), y)), app(app(+, app(app(*, z''), y'')), app(app(*, z''), 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'')))
APP(app(*, app(app(*, x''), y'')), app(app(+, y0), z)) -> APP(app(+, app(app(*, x''), app(app(*, y''), y0))), app(app(*, app(app(*, x''), y'')), z))
APP(app(*, app(app(+, y''), z'')), app(app(+, y0), z)) -> APP(app(+, app(app(+, app(app(*, y0), y'')), app(app(*, y0), z''))), app(app(*, app(app(+, y''), z'')), z))
APP(app(+, app(app(+, x), y)), z) -> APP(app(+, y), z)
APP(app(*, app(app(*, x), y)), z) -> APP(app(*, y), z)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) -> APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) -> APP(app(*, x), y)
APP(app(*, app(app(+, y), z'')), app(app(*, x''), y'')) -> APP(app(+, app(app(*, app(app(*, x''), y'')), y)), app(app(*, x''), app(app(*, y''), z'')))

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:14 minutes