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
             ↳Narrowing 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 Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(maptlist, f''), app(app(cons, app(leaf, x'')), xs)) -> APP(app(cons, app(leaf, app(f'', x''))), app(app(maptlist, f''), xs))
six new Dependency Pairs are created:

APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, nil)), xs)) -> APP(app(cons, app(leaf, nil)), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), nil)) -> APP(app(cons, app(leaf, app(f''', x''))), nil)
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(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
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

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




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, app(node, xs'')), xs)) -> APP(app(cons, app(node, app(app(maptlist, f''), xs''))), app(app(maptlist, f''), xs))
four new Dependency Pairs are created:

APP(app(maptlist, f'''), app(app(cons, app(node, nil)), xs)) -> APP(app(cons, app(node, nil)), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), nil)) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), nil)
APP(app(maptlist, f'''), app(app(cons, app(node, xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), 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
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

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




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), 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'')))
six new Dependency Pairs are created:

APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(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')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), 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(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
             ↳Nar
             ...
               →DP Problem 5
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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




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)
15 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, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(leaf, x'''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, 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), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(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'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, 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''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(app(mapt, f'''), x)), 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, x), app(app(cons, x''), nil))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(app(mapt, f'''), x'')), nil))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(node, xs')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, x), app(app(cons, app(leaf, x''')), xs''))) -> APP(app(cons, app(app(mapt, f'''), x)), app(app(cons, app(leaf, app(f''', x'''))), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(node, xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs'))), app(app(cons, app(app(mapt, f'''), x'')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x''')), app(app(cons, x''), xs''))) -> APP(app(cons, app(leaf, app(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'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(node, app(app(maptlist, f'''), xs''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))
APP(app(maptlist, f'''), app(app(cons, app(node, app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(node, app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs''')))), app(app(maptlist, f'''), xs))
APP(app(maptlist, f'''), app(app(cons, app(leaf, x'')), app(app(cons, x'), xs''))) -> APP(app(cons, app(leaf, app(f''', x''))), app(app(cons, app(app(mapt, f'''), x')), app(app(maptlist, f'''), xs'')))
APP(app(maptlist, app(maptlist, f')), app(app(cons, app(leaf, app(app(cons, x'), xs''))), xs)) -> APP(app(cons, app(leaf, app(app(cons, app(app(mapt, f'), x')), app(app(maptlist, f'), xs'')))), app(app(maptlist, app(maptlist, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(node, xs''))), xs)) -> APP(app(cons, app(leaf, app(node, app(app(maptlist, f'), xs'')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, app(mapt, f')), app(app(cons, app(leaf, app(leaf, x'))), xs)) -> APP(app(cons, app(leaf, app(leaf, app(f', x')))), app(app(maptlist, app(mapt, f')), xs))
APP(app(maptlist, f'''), app(app(cons, x), 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(cons, app(app(mapt, f'''), x''')), app(app(maptlist, f'''), xs'))))


Additionally, 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: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(mapt)=  1  
  POL(maptlist)=  1  
  POL(nil)=  0  
  POL(node)=  0  
  POL(leaf)=  0  
  POL(app(x1, x2))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, 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''''0), app(app(cons, x''''''), xs'''))))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, x''''), nil)))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(node, xs''''')), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, x''), app(app(cons, app(leaf, x''''')), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''')), app(app(cons, x''''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x''''')), app(app(cons, x''0'), xs'''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, xs''''0)), app(app(cons, x'''), xs'''''')))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))) -> APP(app(maptlist, f'''''), app(app(cons, app(node, app(app(cons, x'''), xs'''''))), xs''))
APP(app(mapt, app(maptlist, f''''')), app(leaf, app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))) -> APP(app(maptlist, f'''''), app(app(cons, app(leaf, x'''')), app(app(cons, x'0'), xs'''')))
APP(app(mapt, app(maptlist, app(maptlist, f'''))), app(leaf, app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))) -> APP(app(maptlist, app(maptlist, f''')), app(app(cons, app(leaf, app(app(cons, x'''), xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(node, xs''''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, app(node, xs''''))), xs''))
APP(app(mapt, app(maptlist, app(mapt, f'''))), app(leaf, app(app(cons, app(leaf, app(leaf, x'''))), xs''))) -> APP(app(maptlist, app(mapt, f''')), app(app(cons, app(leaf, 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''))


Additionally, 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: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(mapt)=  0  
  POL(maptlist)=  0  
  POL(nil)=  0  
  POL(node)=  0  
  POL(leaf)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳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:
0:33 minutes