Term Rewriting System R:
[f, x, h, t]
app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(twice, f), x) -> APP(f, x)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(map, f), app(app(cons, h), t)) -> APP(cons, app(f, h))
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(cons, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(fmap, tf)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))





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

APP(app(fmap, app(app(cons, f), tf)), x) -> APP(app(fmap, tf), x)
no new Dependency Pairs are created.
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(fmap, app(app(cons, f), tf)), x) -> APP(app(cons, app(f, x)), app(app(fmap, tf), x))
APP(app(map, f), app(app(cons, h), t)) -> APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) -> APP(f, h)
APP(app(map, f), app(app(cons, h), t)) -> APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(twice, f), x) -> APP(f, x)
APP(app(twice, f), x) -> APP(f, app(f, x))
APP(app(fmap, app(app(cons, f), tf)), x) -> APP(f, x)


Rules:


app(app(twice, f), x) -> app(f, app(f, x))
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, h), t)) -> app(app(cons, app(f, h)), app(app(map, f), t))
app(app(fmap, nil), x) -> nil
app(app(fmap, app(app(cons, f), tf)), x) -> app(app(cons, app(f, x)), app(app(fmap, tf), x))




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