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. 



   R
     ↳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. 


   R
     ↳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: 



   R
     ↳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: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


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




The following dependency pairs can be strictly oriented:

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), 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 following usable rules for innermost can be oriented:

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))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{nil, maptlist, node, mapt} > cons
{nil, maptlist, node, mapt} > leaf

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> x1
app(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Argument Filtering and Ordering


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, 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)


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




The following dependency pairs can be strictly oriented:

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''))


The following usable rules for innermost can be oriented:

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))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> x1
app(x1, x2) -> app(x2)


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
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(mapt, f), app(node, 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



Innermost Termination of R could not be shown.
Duration:
0:20 minutes