Term Rewriting System R:
[ys, x, xs, f]
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(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) -> nil
app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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(flatwith, f), app(leaf, x)) -> APP(app(cons, app(f, x)), nil)
APP(app(flatwith, f), app(leaf, x)) -> APP(cons, app(f, x))
APP(app(flatwith, f), app(leaf, x)) -> APP(f, x)
APP(app(flatwith, f), app(node, xs)) -> APP(app(flatwithsub, f), xs)
APP(app(flatwith, f), app(node, xs)) -> APP(flatwithsub, f)
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(append, app(app(flatwith, f), x))
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(flatwith, f), x)
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(flatwith, f)
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(flatwithsub, f), xs)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(flatwithsub, f), xs)
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(flatwith, f), x)
APP(app(flatwithsub, f), app(app(cons, x), xs)) -> APP(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
APP(app(flatwith, f), app(node, xs)) -> APP(app(flatwithsub, f), xs)
APP(app(flatwith, f), app(leaf, x)) -> APP(f, x)
APP(app(flatwith, f), app(leaf, x)) -> APP(app(cons, app(f, x)), nil)
APP(app(append, app(app(cons, x), xs)), ys) -> APP(app(append, xs), ys)


Rules:


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(flatwith, f), app(leaf, x)) -> app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) -> app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) -> nil
app(app(flatwithsub, f), app(app(cons, x), xs)) -> app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))


Strategy:

innermost



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