Term Rewriting System R:
[f, x, xs, g]
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))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)

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)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(twice, f) -> APP(app(comp, f), f)
APP(twice, f) -> APP(comp, f)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(twice, f) -> APP(app(comp, f), f)
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
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))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




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

APP(twice, f) -> APP(app(comp, f), f)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
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))
APP(app(app(comp, f), g), x) -> APP(g, x)


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))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


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))
six 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, app(app(comp, f''), g')), app(app(cons, x''), xs)) -> APP(app(cons, app(f'', app(g', x''))), app(app(map, app(app(comp, f''), g')), xs))
APP(app(map, twice), app(app(cons, x'), xs)) -> APP(app(cons, app(app(comp, x'), x')), app(app(map, twice), 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
             ↳Nar
             ...
               →DP Problem 3
                 ↳Forward Instantiation 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, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, twice), app(app(cons, x'), xs)) -> APP(app(cons, app(app(comp, x'), x')), app(app(map, twice), xs))
APP(app(map, app(app(comp, f''), g')), app(app(cons, x''), xs)) -> APP(app(cons, app(f'', app(g', x''))), app(app(map, app(app(comp, f''), g')), xs))
APP(app(app(comp, f), g), x) -> APP(g, x)
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(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))


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))
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


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)
seven 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(app(comp, f''), g'')), app(app(cons, x''), xs)) -> APP(app(app(comp, f''), g''), x'')
APP(app(map, app(map, app(map, f''''))), app(app(cons, app(app(cons, app(app(cons, x''''), xs'''')), xs'')), xs)) -> APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''''), xs'''')), xs''))
APP(app(map, app(map, app(app(comp, f''''), g'''))), app(app(cons, app(app(cons, x''''), xs'')), xs)) -> APP(app(map, app(app(comp, f''''), g''')), app(app(cons, x''''), xs''))
APP(app(map, app(map, twice)), app(app(cons, app(app(cons, x'''), xs'')), xs)) -> APP(app(map, twice), app(app(cons, x'''), xs''))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), nil)), xs)) -> APP(app(map, f''''), app(app(cons, x''), nil))
APP(app(map, app(map, f'''')), app(app(cons, app(app(cons, x''), app(app(cons, x''''), xs''''))), xs)) -> APP(app(map, 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
             ↳Nar
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Strategy:

innermost




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

APP(app(app(comp, f), g), x) -> APP(g, x)
13 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:17 minutes