Term Rewriting System R:
[f, x, xs]
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




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

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
four new Dependency Pairs are created:

APP(app(map, app(map, f'')), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, app(map, f'')), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, 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(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




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

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

APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, nil), xs'')), xs)) -> APP(app(cons, app(app(cons, nil), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, app(app(cons, x'), xs''')), xs'')), xs)) -> APP(app(cons, app(app(cons, app(app(cons, app(f', x')), app(app(map, f'), xs'''))), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), nil)), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(cons, app(f''', x')), app(app(map, f'''), xs''')))), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), nil)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), nil)
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), app(app(cons, app(app(map, f'''), x')), app(app(map, app(map, 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(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), app(app(cons, app(app(map, f'''), x')), app(app(map, app(map, f''')), xs''')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(cons, app(f''', x')), app(app(map, f'''), xs''')))), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), nil)), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, app(app(cons, x'), xs''')), xs'')), xs)) -> APP(app(cons, app(app(cons, app(app(cons, app(f', x')), app(app(map, f'), xs'''))), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, nil), xs'')), xs)) -> APP(app(cons, app(app(cons, nil), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




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

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

APP(app(map, app(map, f')), app(app(cons, nil), app(app(cons, x''), xs''))) -> APP(app(cons, nil), app(app(cons, app(app(map, f'), x'')), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, app(app(cons, x'''), xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(cons, app(app(map, f'), x'')), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, nil), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, nil), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, app(app(cons, x'''), xs')), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(map, app(map, f')), xs'')))
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), nil))
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), app(app(cons, app(f''', x''')), app(app(map, f'''), xs'))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), app(app(cons, app(f''', x''')), app(app(map, f'''), xs'))))
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), nil))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, app(app(cons, x'''), xs')), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, nil), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, nil), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, app(app(cons, x'''), xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(cons, app(app(map, f'), x'')), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(cons, app(f''', x')), app(app(map, f'''), xs''')))), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), nil)), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, app(app(cons, x'), xs''')), xs'')), xs)) -> APP(app(cons, app(app(cons, app(app(cons, app(f', x')), app(app(map, f'), xs'''))), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, nil), xs'')), xs)) -> APP(app(cons, app(app(cons, nil), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), app(app(cons, app(app(map, f'''), x')), app(app(map, app(map, f''')), xs''')))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, 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(map, f), app(app(cons, x), xs)) -> APP(f, x)
11 new Dependency Pairs are created:

APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, nil), xs'''')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, nil), xs'''')), xs''))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), nil)), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs''''''))), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs'''''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs'''''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, nil), xs''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, nil), xs'''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs'''''')))
APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), nil))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs''')))), xs)) -> APP(app(map, 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 5
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs''')))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), nil))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs'''''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs''''''))), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), nil)), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs''))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, nil), xs''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, nil), xs'''')))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, nil), xs'''')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, nil), xs'''')), xs''))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), nil))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, app(app(cons, x'''), xs')), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, nil), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, nil), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, app(app(cons, x'''), xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(cons, app(app(map, f'), x'')), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), app(app(cons, app(app(map, f'''), x')), app(app(map, app(map, f''')), xs''')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(cons, app(f''', x')), app(app(map, f'''), xs''')))), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), nil)), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, app(app(cons, x'), xs''')), xs'')), xs)) -> APP(app(cons, app(app(cons, app(app(cons, app(f', x')), app(app(map, f'), xs'''))), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, nil), xs'')), xs)) -> APP(app(cons, app(app(cons, nil), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), app(app(cons, app(f''', x''')), app(app(map, f'''), xs'))))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), nil))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), nil))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, app(app(cons, x'''), xs')), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, x), app(app(cons, nil), xs''))) -> APP(app(cons, app(app(map, f'), x)), app(app(cons, nil), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f')), app(app(cons, app(app(cons, x'''), xs')), app(app(cons, x''), xs''))) -> APP(app(cons, app(app(cons, app(f', x''')), app(app(map, f'), xs'))), app(app(cons, app(app(map, f'), x'')), app(app(map, app(map, f')), xs'')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), xs'')), app(app(cons, x'), xs'''))) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(map, f'''), xs''))), app(app(cons, app(app(map, f'''), x')), app(app(map, app(map, f''')), xs''')))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), app(app(cons, x'), xs'''))), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), app(app(cons, app(f''', x')), app(app(map, f'''), xs''')))), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(cons, app(app(cons, app(f''', x'')), nil)), app(app(map, app(map, f''')), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, app(app(cons, x'), xs''')), xs'')), xs)) -> APP(app(cons, app(app(cons, app(app(cons, app(f', x')), app(app(map, f'), xs'''))), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, app(map, app(map, f'))), app(app(cons, app(app(cons, nil), xs'')), xs)) -> APP(app(cons, app(app(cons, nil), app(app(map, app(map, f')), xs''))), app(app(map, app(map, app(map, f'))), xs))
APP(app(map, f'''), app(app(cons, x), app(app(cons, x''), app(app(cons, x'''), xs')))) -> APP(app(cons, app(f''', x)), app(app(cons, app(f''', x'')), app(app(cons, app(f''', x''')), app(app(map, f'''), xs'))))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(map)=  1  
  POL(nil)=  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 6
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs''')))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), nil))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs'''''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs''''''))), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), nil)), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs''))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, nil), xs''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, nil), xs'''')))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, nil), xs'''')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, nil), xs'''')), xs''))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs''')))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''0), app(app(cons, x''''''), xs'''))))
APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), nil))), xs)) -> APP(app(map, f'''''), app(app(cons, x''), app(app(cons, x''''), nil)))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, app(app(cons, x'''''), xs''''')), xs'''''')))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs''''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, app(app(cons, x'''''), xs''''')), app(app(cons, x''0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs''''''))), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), xs''''0)), app(app(cons, x'0'), xs'''''')))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), app(app(cons, x'0'), xs'''''))), xs''))
APP(app(map, app(map, app(map, f'''''))), app(app(cons, app(app(cons, app(app(cons, x''''), nil)), xs'')), xs)) -> APP(app(map, app(map, f''''')), app(app(cons, app(app(cons, x''''), nil)), xs''))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, app(app(cons, x'''), xs''''')), xs''0')), xs''))
APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, x''), app(app(cons, nil), xs''''))), xs)) -> APP(app(map, app(map, f''')), app(app(cons, x''), app(app(cons, nil), xs'''')))
APP(app(map, app(map, app(map, app(map, f''')))), app(app(cons, app(app(cons, app(app(cons, nil), xs'''')), xs'')), xs)) -> APP(app(map, app(map, app(map, f'''))), app(app(cons, app(app(cons, nil), xs'''')), xs''))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(map, f''), app(app(cons, x''), xs''))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(map)=  0  
  POL(nil)=  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 7
                 ↳Polynomial Ordering


Dependency Pair:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons)=  0  
  POL(map)=  0  
  POL(nil)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(APP(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Dependency Graph


Dependency Pair:


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:21 minutes