Term Rewriting System R:
[x, y]
app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(f, app(s, x)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, x), y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(if, app(f, x))
APP(app(g, x), app(c, y)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, app(s, x)), y))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(g, app(s, x))
APP(app(g, x), app(c, y)) -> APP(s, x)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:

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