Termination Proof using AProVE

Term Rewriting System R:
[f, x, xs]
app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(mapt, f), app(leaf, x)) -> APP(leaf, app(f, x))
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(mapt, f), app(node, xs)) -> APP(node, app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(node, xs)) -> APP(maptlist, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(cons, app(app(mapt, f), x))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(mapt, f)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

four new Dependency Pairs are created:

APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, x), nil)) -> APP(app(cons, app(app(mapt, f''), x)), nil)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(mapt, f), app(leaf, x)) -> APP(f, x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

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

APP(app(mapt, f), app(leaf, x)) -> APP(f, x)

six new Dependency Pairs are created:

APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), xs'''')))) -> APP(app(maptlist, f''''), app(app(cons, x''), app(app(cons, x''''), xs'''')))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(node, xs'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(node, xs'''')), xs''))
APP(app(mapt, app(maptlist, f'''')), app(leaf, app(app(cons, app(leaf, x'''')), xs''))) -> APP(app(maptlist, f''''), app(app(cons, app(leaf, x'''')), xs''))
APP(app(mapt, app(maptlist, f'')), app(leaf, app(app(cons, x''), xs''))) -> APP(app(maptlist, f''), app(app(cons, x''), xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(node, xs''))) -> APP(app(mapt, f''), app(node, xs''))
APP(app(mapt, app(mapt, f'')), app(leaf, app(leaf, x''))) -> APP(app(mapt, f''), app(leaf, x''))
APP(app(maptlist, f''), app(app(cons, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) -> APP(app(mapt, f), x)
APP(app(mapt, f), app(node, xs)) -> APP(app(maptlist, f), xs)
APP(app(maptlist, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(mapt, f''), x)), app(app(cons, app(app(mapt, f''), x'')), app(app(maptlist, f''), xs'')))

Rules:

app(app(mapt, f), app(leaf, x)) -> app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) -> app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) -> nil
app(app(maptlist, f), app(app(cons, x), xs)) -> app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:03 minutes