Term Rewriting System R:
[f, n, x, xs]
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(hd, app(app(map, f), app(app(cons, 0), nil)))
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(map, f)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(cons, 0)
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(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)


Rules:


app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))





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

APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
three new Dependency Pairs are created:

APP(app(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)

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


Rules:


app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))




Termination of R could not be shown.
Duration:
0:00 minutes