Termination Proof using AProVE

Term Rewriting System R:
[z, x, y]
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z), app(e, x)) -> APP(h, app(c, z))
APP(app(h, z), app(e, x)) -> APP(c, z)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(d, z)
APP(app(d, z), app(app(g, 0), 0)) -> APP(e, 0)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(d, z), app(app(g, x), y)) -> APP(g, app(e, x))
APP(app(d, z), app(app(g, x), y)) -> APP(e, x)
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(g, app(app(d, app(c, z)), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(d, z)
APP(app(g, app(e, x)), app(e, y)) -> APP(e, app(app(g, x), y))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(g, app(e, x)), app(e, y)) -> APP(g, x)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

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

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))

three new Dependency Pairs are created:

APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

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

APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))

three new Dependency Pairs are created:

APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))

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(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))
APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

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

APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))

eight new Dependency Pairs are created:

APP(app(d, app(c, z'')), app(app(g, app(app(g, 0), 0)), 0)) -> APP(app(g, app(e, 0)), app(app(d, z''), app(app(g, 0), 0)))
APP(app(d, app(c, z'')), app(app(g, app(app(g, x''), y'')), 0)) -> APP(app(g, app(app(g, app(e, x'')), app(app(d, app(c, z'')), y''))), app(app(d, z''), app(app(g, x''), y'')))
APP(app(d, app(c, z'')), app(app(g, app(app(g, app(app(g, x''), y'')), 0)), 0)) -> APP(app(g, app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y'')))), app(app(d, z''), app(app(g, app(app(g, x''), y'')), 0)))
APP(app(d, app(c, z)), app(app(g, app(app(g, app(e, x'')), app(e, y''))), 0)) -> APP(app(g, app(app(d, app(c, z)), app(e, app(app(g, x''), y'')))), app(app(d, z), app(app(g, app(e, x'')), app(e, y''))))
APP(app(d, app(c, z'')), app(app(g, app(app(g, 0), 0)), 0)) -> APP(app(g, app(app(d, app(c, z'')), app(app(g, 0), 0))), app(e, 0))
APP(app(d, app(c, z'')), app(app(g, app(app(g, x''), y'')), 0)) -> APP(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(d, app(c, app(c, z''))), app(app(g, app(app(g, app(app(g, x''), y'')), 0)), 0)) -> APP(app(g, app(app(d, app(c, app(c, z''))), app(app(g, app(app(g, x''), y'')), 0))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))
APP(app(d, app(c, z)), app(app(g, app(app(g, app(e, x'')), app(e, y''))), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, app(e, x'')), app(e, y'')))), app(app(d, z), app(e, app(app(g, x''), y''))))

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(d, app(c, app(c, z''))), app(app(g, app(app(g, app(app(g, x''), y'')), 0)), 0)) -> APP(app(g, app(app(d, app(c, app(c, z''))), app(app(g, app(app(g, x''), y'')), 0))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))
APP(app(d, app(c, z'')), app(app(g, app(app(g, x''), y'')), 0)) -> APP(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(d, app(c, z'')), app(app(g, app(app(g, 0), 0)), 0)) -> APP(app(g, app(app(d, app(c, z'')), app(app(g, 0), 0))), app(e, 0))
APP(app(d, app(c, z'')), app(app(g, app(app(g, app(app(g, x''), y'')), 0)), 0)) -> APP(app(g, app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y'')))), app(app(d, z''), app(app(g, app(app(g, x''), y'')), 0)))
APP(app(d, app(c, z'')), app(app(g, app(app(g, x''), y'')), 0)) -> APP(app(g, app(app(g, app(e, x'')), app(app(d, app(c, z'')), y''))), app(app(d, z''), app(app(g, x''), y'')))
APP(app(d, app(c, z'')), app(app(g, app(app(g, 0), 0)), 0)) -> APP(app(g, app(e, 0)), app(app(d, z''), app(app(g, 0), 0)))
APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(h, app(c, z'')), app(e, app(app(g, app(app(g, x''), y')), 0))) -> APP(app(h, app(c, app(c, z''))), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y'))), app(app(d, z''), app(app(g, x''), y'))))
APP(app(h, z''), app(e, app(app(g, x''), y'))) -> APP(app(h, app(c, z'')), app(app(g, app(e, x'')), app(app(d, z''), y')))
APP(app(h, z''), app(e, app(app(g, 0), 0))) -> APP(app(h, app(c, z'')), app(e, 0))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))

Rules:

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:22 minutes