Termination Proof using AProVE

Term Rewriting System R:
[f, x, l, r]
app(app(mapbt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) -> app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(mapbt, f), app(leaf, x)) -> APP(leaf, app(f, x))
APP(app(mapbt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(branch, app(f, x)), app(app(mapbt, f), l))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(branch, app(f, x))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), r)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), r)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(branch, app(f, x)), app(app(mapbt, f), l))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))
APP(app(mapbt, f), app(leaf, x)) -> APP(f, x)

Rules:

app(app(mapbt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) -> app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

Strategy:

innermost

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

APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

six new Dependency Pairs are created:

APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(leaf, x'')), l), r)) -> APP(app(app(branch, app(leaf, app(f'', x''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(app(app(branch, x''), l''), r'')), l), r)) -> APP(app(app(branch, app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, f''), app(app(app(branch, x), app(leaf, x'')), r)) -> APP(app(app(branch, app(f'', x)), app(leaf, app(f'', x''))), app(app(mapbt, f''), r))
APP(app(mapbt, f''), app(app(app(branch, x), app(app(app(branch, x''), l''), r'')), r)) -> APP(app(app(branch, app(f'', x)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, f''), r))
APP(app(mapbt, f''), app(app(app(branch, x), l), app(leaf, x''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(leaf, app(f'', x'')))
APP(app(mapbt, f''), app(app(app(branch, x), l), app(app(app(branch, x''), l''), r''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

APP(app(mapbt, f''), app(app(app(branch, x), l), app(app(app(branch, x''), l''), r''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r'')))
APP(app(mapbt, f''), app(app(app(branch, x), l), app(leaf, x''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(leaf, app(f'', x'')))
APP(app(mapbt, f''), app(app(app(branch, x), app(app(app(branch, x''), l''), r'')), r)) -> APP(app(app(branch, app(f'', x)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, f''), r))
APP(app(mapbt, f''), app(app(app(branch, x), app(leaf, x'')), r)) -> APP(app(app(branch, app(f'', x)), app(leaf, app(f'', x''))), app(app(mapbt, f''), r))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(app(app(branch, x''), l''), r'')), l), r)) -> APP(app(app(branch, app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(leaf, x'')), l), r)) -> APP(app(app(branch, app(leaf, app(f'', x''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(branch, app(f, x)), app(app(mapbt, f), l))
APP(app(mapbt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), r)

Rules:

app(app(mapbt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) -> app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

Strategy:

innermost

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

APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(branch, app(f, x)), app(app(mapbt, f), l))

four new Dependency Pairs are created:

APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(leaf, x'')), l), r)) -> APP(app(branch, app(leaf, app(f'', x''))), app(app(mapbt, app(mapbt, f'')), l))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(app(app(branch, x''), l''), r'')), l), r)) -> APP(app(branch, app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, app(mapbt, f'')), l))
APP(app(mapbt, f''), app(app(app(branch, x), app(leaf, x'')), r)) -> APP(app(branch, app(f'', x)), app(leaf, app(f'', x'')))
APP(app(mapbt, f''), app(app(app(branch, x), app(app(app(branch, x''), l''), r'')), r)) -> APP(app(branch, app(f'', x)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(mapbt, f''), app(app(app(branch, x), app(app(app(branch, x''), l''), r'')), r)) -> APP(app(branch, app(f'', x)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r'')))
APP(app(mapbt, f''), app(app(app(branch, x), app(leaf, x'')), r)) -> APP(app(branch, app(f'', x)), app(leaf, app(f'', x'')))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(app(app(branch, x''), l''), r'')), l), r)) -> APP(app(branch, app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, app(mapbt, f'')), l))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(leaf, x'')), l), r)) -> APP(app(branch, app(leaf, app(f'', x''))), app(app(mapbt, app(mapbt, f'')), l))
APP(app(mapbt, f''), app(app(app(branch, x), l), app(leaf, x''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(leaf, app(f'', x'')))
APP(app(mapbt, f''), app(app(app(branch, x), app(app(app(branch, x''), l''), r'')), r)) -> APP(app(app(branch, app(f'', x)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, f''), r))
APP(app(mapbt, f''), app(app(app(branch, x), app(leaf, x'')), r)) -> APP(app(app(branch, app(f'', x)), app(leaf, app(f'', x''))), app(app(mapbt, f''), r))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(app(app(branch, x''), l''), r'')), l), r)) -> APP(app(app(branch, app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, app(mapbt, f'')), app(app(app(branch, app(leaf, x'')), l), r)) -> APP(app(app(branch, app(leaf, app(f'', x''))), app(app(mapbt, app(mapbt, f'')), l)), app(app(mapbt, app(mapbt, f'')), r))
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), r)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(app(mapbt, f), l)
APP(app(mapbt, f), app(app(app(branch, x), l), r)) -> APP(f, x)
APP(app(mapbt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapbt, f''), app(app(app(branch, x), l), app(app(app(branch, x''), l''), r''))) -> APP(app(app(branch, app(f'', x)), app(app(mapbt, f''), l)), app(app(app(branch, app(f'', x'')), app(app(mapbt, f''), l'')), app(app(mapbt, f''), r'')))

Rules:

app(app(mapbt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapbt, f), app(app(app(branch, x), l), r)) -> app(app(app(branch, app(f, x)), app(app(mapbt, f), l)), app(app(mapbt, f), r))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:15 minutes