Term Rewriting System R:
[t, u, v, x, f]
app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(rec, t)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(app(app(rectuv, t), u), v), n) -> APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) -> APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(f, n)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) -> APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) -> APP(app(u, x), app(app(app(app(rec, t), u), v), x))


Rules:


app(app(app(app(rec, t), u), v), 0) -> t
app(app(app(app(rec, t), u), v), app(s, x)) -> app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) -> app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) -> app(app(app(app(rec, t), u), v), n)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:12 minutes