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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(add, app(s, x)), y) -> APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
APP(app(add, app(s, x)), y) -> APP(add, x)
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))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)


Rules:


app(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), 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))


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))
seven new Dependency Pairs are created:

APP(app(map, id), app(app(cons, x''), xs)) -> APP(app(cons, x''), app(app(map, id), xs))
APP(app(map, add), app(app(cons, 0), xs)) -> APP(app(cons, id), app(app(map, add), xs))
APP(app(map, app(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(add, x''), x0))), app(app(map, app(add, app(s, x''))), xs))
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
             ↳Remaining Obligation(s)




The following remains to be proven:
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, 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(add, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(cons, app(s, app(app(add, x''), x0))), app(app(map, app(add, app(s, x''))), xs))
APP(app(map, add), app(app(cons, 0), xs)) -> APP(app(cons, id), app(app(map, add), xs))
APP(app(add, app(s, x)), y) -> APP(app(add, x), y)
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(id, x) -> x
app(add, 0) -> id
app(app(add, app(s, x)), y) -> app(s, app(app(add, x), 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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:04 minutes