Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, x), z))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, x)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, app(app(:, x), y)), z))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(:, app(app(:, x), y))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

Rule:

app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

Strategy:

innermost

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

APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

four new Dependency Pairs are created:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''')), y)), z'')), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, C), x''))), y'')), z'')), u'') -> APP(app(:, app(app(:, app(app(:, C), x'')), z'')), app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), u'')))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, C), x'')), y''))), y0)), z'')), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), app(app(:, app(app(:, app(app(:, x''), y0)), app(app(:, app(app(:, app(app(:, x''), y'')), y0)), z''))), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), z)), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rewriting Transformation

Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), z)), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, C), x'')), y''))), y0)), z'')), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), app(app(:, app(app(:, app(app(:, x''), y0)), app(app(:, app(app(:, app(app(:, x''), y'')), y0)), z''))), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, C), x''))), y'')), z'')), u'') -> APP(app(:, app(app(:, app(app(:, C), x'')), z'')), app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), u'')))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''')), y)), z'')), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)

Rule:

app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

Strategy:

innermost

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

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''')), y)), z'')), u))

one new Dependency Pair is created:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), y))), z'')), u))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 3

                 ↳Rewriting Transformation

Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), y))), z'')), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, C), x'')), y''))), y0)), z'')), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), app(app(:, app(app(:, app(app(:, x''), y0)), app(app(:, app(app(:, app(app(:, x''), y'')), y0)), z''))), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, C), x''))), y'')), z'')), u'') -> APP(app(:, app(app(:, app(app(:, C), x'')), z'')), app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), u'')))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), z)), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))

Rule:

app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

Strategy:

innermost

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

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), z)), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))

one new Dependency Pair is created:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), z))), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Rw

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z''))), y0)), z)), u) -> APP(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), z))), app(app(:, app(app(:, app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), y0))), z)), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, C), x'')), y''))), y0)), z'')), u) -> APP(app(:, app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'')), app(app(:, app(app(:, app(app(:, x''), y0)), app(app(:, app(app(:, app(app(:, x''), y'')), y0)), z''))), u))
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, C), x''))), y'')), z'')), u'') -> APP(app(:, app(app(:, app(app(:, C), x'')), z'')), app(app(:, app(app(:, x''), z'')), app(app(:, app(app(:, app(app(:, x''), y'')), z'')), u'')))
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), y)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, x), y)), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, app(app(:, app(app(:, x), y)), z)), u)
APP(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> APP(app(:, x), z)
APP(app(:, app(app(:, app(app(:, app(app(:, C), app(app(:, app(app(:, app(app(:, C), x'')), y'')), z'''))), y)), z'')), u) -> APP(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), z''))), app(app(:, app(app(:, app(app(:, app(app(:, x''), z''')), app(app(:, app(app(:, app(app(:, x''), y'')), z''')), y))), z'')), u))

Rule:

app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) -> app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes