Term Rewriting System R:
[f, x, xs, ys, yss, xss]
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(append, xs), nil) -> xs
app(app(append, nil), ys) -> ys
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) -> yss
app(app(zip, xss), nil) -> xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) -> xs
app(app(combine, xs), app(app(cons, ys), yss)) -> app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) -> app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))

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(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), app(app(append, xs), ys))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(append, xs)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(cons, app(app(append, xs), ys))
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(append, xs), ys)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(append, xs)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(zip, xss), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(zip, xss)
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(combine, app(app(zip, xs), ys)), yss)
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(combine, app(app(zip, xs), ys))
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(zip, xs), ys)
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(zip, xs)
APP(levels, app(app(node, x), xs)) -> APP(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))
APP(levels, app(app(node, x), xs)) -> APP(cons, app(app(cons, x), nil))
APP(levels, app(app(node, x), xs)) -> APP(app(cons, x), nil)
APP(levels, app(app(node, x), xs)) -> APP(cons, x)
APP(levels, app(app(node, x), xs)) -> APP(app(combine, nil), app(app(map, levels), xs))
APP(levels, app(app(node, x), xs)) -> APP(combine, nil)
APP(levels, app(app(node, x), xs)) -> APP(app(map, levels), xs)
APP(levels, app(app(node, x), xs)) -> APP(map, levels)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(levels, app(app(node, x), xs)) -> APP(app(map, levels), xs)
APP(levels, app(app(node, x), xs)) -> APP(app(combine, nil), app(app(map, levels), xs))
APP(levels, app(app(node, x), xs)) -> APP(app(cons, x), nil)
APP(levels, app(app(node, x), xs)) -> APP(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(zip, xs), ys)
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(combine, app(app(zip, xs), ys)), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(zip, xss), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(append, xs), ys)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), app(app(append, xs), ys))
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(append, xs), nil) -> xs
app(app(append, nil), ys) -> ys
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) -> yss
app(app(zip, xss), nil) -> xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) -> xs
app(app(combine, xs), app(app(cons, ys), yss)) -> app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) -> app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))





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

APP(levels, app(app(node, x), xs)) -> APP(app(cons, x), nil)
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(levels, app(app(node, x), xs)) -> APP(app(combine, nil), app(app(map, levels), xs))
APP(levels, app(app(node, x), xs)) -> APP(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(zip, xs), ys)
APP(app(combine, xs), app(app(cons, ys), yss)) -> APP(app(combine, app(app(zip, xs), ys)), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(zip, xss), yss)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(append, xs), ys)
APP(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> APP(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(cons, x), app(app(append, xs), ys))
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(levels, app(app(node, x), xs)) -> APP(app(map, levels), 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(append, xs), nil) -> xs
app(app(append, nil), ys) -> ys
app(app(append, app(app(cons, x), xs)), ys) -> app(app(cons, x), app(app(append, xs), ys))
app(app(zip, nil), yss) -> yss
app(app(zip, xss), nil) -> xss
app(app(zip, app(app(cons, xs), xss)), app(app(cons, ys), yss)) -> app(app(cons, app(app(append, xs), ys)), app(app(zip, xss), yss))
app(app(combine, xs), nil) -> xs
app(app(combine, xs), app(app(cons, ys), yss)) -> app(app(combine, app(app(zip, xs), ys)), yss)
app(levels, app(app(node, x), xs)) -> app(app(cons, app(app(cons, x), nil)), app(app(combine, nil), app(app(map, levels), xs)))




Termination of R could not be shown.
Duration:
0:01 minutes