Term Rewriting System R:
[f, x, xs, y]
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(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))

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(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, x)
APP(app(div, app(s, x)), app(s, y)) -> APP(s, app(app(div, app(app(minus, x), y)), app(s, y)))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(div, app(s, x)), app(s, y)) -> APP(div, app(app(minus, x), y))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(minus, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
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(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


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))
eight 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(minus, x'')), app(app(cons, 0), xs)) -> APP(app(cons, x''), app(app(map, app(minus, x'')), xs))
APP(app(map, app(minus, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(app(minus, x''), y')), app(app(map, app(minus, app(s, x''))), xs))
APP(app(map, app(div, 0)), app(app(cons, app(s, y')), xs)) -> APP(app(cons, 0), app(app(map, app(div, 0)), xs))
APP(app(map, app(div, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(s, app(app(div, app(app(minus, x''), y')), app(s, y')))), app(app(map, app(div, app(s, x''))), 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(div, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(s, app(app(div, app(app(minus, x''), y')), app(s, y')))), app(app(map, app(div, app(s, x''))), xs))
APP(app(map, app(div, 0)), app(app(cons, app(s, y')), xs)) -> APP(app(cons, 0), app(app(map, app(div, 0)), xs))
APP(app(map, app(minus, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(app(minus, x''), y')), app(app(map, app(minus, app(s, x''))), xs))
APP(app(map, app(minus, x'')), app(app(cons, 0), xs)) -> APP(app(cons, x''), app(app(map, app(minus, x'')), xs))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
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(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)


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(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




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

APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
two new Dependency Pairs are created:

APP(app(div, app(s, x'')), app(s, 0)) -> APP(app(div, x''), app(s, 0))
APP(app(div, app(s, app(s, x''))), app(s, app(s, y''))) -> APP(app(div, app(app(minus, x''), y'')), app(s, app(s, y'')))

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(div, app(s, app(s, x''))), app(s, app(s, y''))) -> APP(app(div, app(app(minus, x''), y'')), app(s, app(s, y'')))
APP(app(div, app(s, x'')), app(s, 0)) -> APP(app(div, x''), app(s, 0))
APP(app(map, app(div, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(s, app(app(div, app(app(minus, x''), y')), app(s, y')))), app(app(map, app(div, app(s, x''))), xs))
APP(app(map, app(div, 0)), app(app(cons, app(s, y')), xs)) -> APP(app(cons, 0), app(app(map, app(div, 0)), xs))
APP(app(map, app(minus, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(app(minus, x''), y')), app(app(map, app(minus, app(s, x''))), xs))
APP(app(map, app(minus, x'')), app(app(cons, 0), xs)) -> APP(app(cons, x''), app(app(map, app(minus, x'')), xs))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
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(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(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


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(minus, app(s, x''))), app(app(cons, app(s, y'')), xs)) -> APP(app(minus, app(s, x'')), app(s, y''))
APP(app(map, app(div, app(s, x''))), app(app(cons, app(s, y'')), xs)) -> APP(app(div, app(s, x'')), app(s, y''))
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(minus, x''''))), app(app(cons, app(app(cons, 0), xs'')), xs)) -> APP(app(map, app(minus, x'''')), app(app(cons, 0), xs''))
APP(app(map, app(map, app(minus, app(s, x'''')))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(minus, app(s, x''''))), app(app(cons, app(s, y''')), xs''))
APP(app(map, app(map, app(div, 0))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(div, 0)), app(app(cons, app(s, y''')), xs''))
APP(app(map, app(map, app(div, app(s, x'''')))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(div, app(s, x''''))), app(app(cons, app(s, y''')), 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'''')))
APP(app(map, app(div, app(s, x''''))), app(app(cons, app(s, 0)), xs)) -> APP(app(div, app(s, x'''')), app(s, 0))
APP(app(map, app(div, app(s, app(s, x'''')))), app(app(cons, app(s, app(s, y''''))), xs)) -> APP(app(div, app(s, app(s, x''''))), app(s, app(s, y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
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(div, app(s, app(s, x'''')))), app(app(cons, app(s, app(s, y''''))), xs)) -> APP(app(div, app(s, app(s, x''''))), app(s, app(s, y'''')))
APP(app(map, app(div, app(s, x''''))), app(app(cons, app(s, 0)), xs)) -> APP(app(div, app(s, x'''')), app(s, 0))
APP(app(map, app(map, app(div, app(s, x'''')))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(div, app(s, x''''))), app(app(cons, app(s, y''')), xs''))
APP(app(map, app(map, app(div, 0))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(div, 0)), app(app(cons, app(s, y''')), xs''))
APP(app(map, app(map, app(minus, app(s, x'''')))), app(app(cons, app(app(cons, app(s, y''')), xs'')), xs)) -> APP(app(map, app(minus, app(s, x''''))), app(app(cons, app(s, y''')), xs''))
APP(app(map, app(map, app(minus, x''''))), app(app(cons, app(app(cons, 0), xs'')), xs)) -> APP(app(map, app(minus, x'''')), app(app(cons, 0), 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(div, app(s, x''))), app(app(cons, app(s, y'')), xs)) -> APP(app(div, app(s, x'')), app(s, y''))
APP(app(map, app(minus, app(s, x''))), app(app(cons, app(s, y'')), xs)) -> APP(app(minus, app(s, x'')), app(s, y''))
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(div, app(s, x'')), app(s, 0)) -> APP(app(div, x''), app(s, 0))
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(div, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(s, app(app(div, app(app(minus, x''), y')), app(s, y')))), app(app(map, app(div, app(s, x''))), xs))
APP(app(map, app(div, 0)), app(app(cons, app(s, y')), xs)) -> APP(app(cons, 0), app(app(map, app(div, 0)), xs))
APP(app(map, app(minus, app(s, x''))), app(app(cons, app(s, y')), xs)) -> APP(app(cons, app(app(minus, x''), y')), app(app(map, app(minus, app(s, x''))), xs))
APP(app(map, app(minus, x'')), app(app(cons, 0), xs)) -> APP(app(cons, x''), app(app(map, app(minus, x'')), xs))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
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(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, app(s, x''))), app(s, app(s, y''))) -> APP(app(div, app(app(minus, x''), y'')), app(s, app(s, y'')))


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(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:19 minutes