Term Rewriting System R:
[f, g, x, l, xs]
app(app(app(compose, f), g), x) -> app(g, app(f, x))
app(reverse, l) -> app(app(reverse2, l), nil)
app(app(reverse2, nil), l) -> l
app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) -> x
app(tl, app(app(cons, x), xs)) -> xs
last -> app(app(compose, hd), reverse)
init -> app(app(compose, reverse), app(app(compose, tl), reverse))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(compose, f), g), x) -> APP(g, app(f, x))
APP(app(app(compose, f), g), x) -> APP(f, x)
APP(reverse, l) -> APP(app(reverse2, l), nil)
APP(reverse, l) -> APP(reverse2, l)
APP(app(reverse2, app(app(cons, x), xs)), l) -> APP(app(reverse2, xs), app(app(cons, x), l))
APP(app(reverse2, app(app(cons, x), xs)), l) -> APP(reverse2, xs)
APP(app(reverse2, app(app(cons, x), xs)), l) -> APP(app(cons, x), l)
LAST -> APP(app(compose, hd), reverse)
LAST -> APP(compose, hd)
INIT -> APP(app(compose, reverse), app(app(compose, tl), reverse))
INIT -> APP(compose, reverse)
INIT -> APP(app(compose, tl), reverse)
INIT -> APP(compose, tl)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(reverse2, app(app(cons, x), xs)), l) -> APP(app(cons, x), l)
APP(app(reverse2, app(app(cons, x), xs)), l) -> APP(app(reverse2, xs), app(app(cons, x), l))
APP(reverse, l) -> APP(app(reverse2, l), nil)
APP(app(app(compose, f), g), x) -> APP(f, x)
APP(app(app(compose, f), g), x) -> APP(g, app(f, x))


Rules:


app(app(app(compose, f), g), x) -> app(g, app(f, x))
app(reverse, l) -> app(app(reverse2, l), nil)
app(app(reverse2, nil), l) -> l
app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) -> x
app(tl, app(app(cons, x), xs)) -> xs
last -> app(app(compose, hd), reverse)
init -> app(app(compose, reverse), app(app(compose, tl), reverse))




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