Term Rewriting System R:

   [x, y, f, g]
   app(app(const, x), y) -> x
   app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
   app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
   APP(app(app(subst, f), g), x) -> APP(f, x)
   APP(app(app(subst, f), g), x) -> APP(g, x)
   APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
   APP(app(fix, f), x) -> APP(f, app(fix, f))

Furthermore, R contains one SCC.


SCC1:

APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))

Found an infinite P-chain over R:
P = APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
R = [app(app(const, x), y) -> x, app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x)), app(app(fix, f), x) -> app(app(f, app(fix, f)), x)]
s = APP(app(fix, app(fix, fix)), app(fix, app(fix, fix))) evaluates to t = APP(app(fix, app(fix, fix)), app(fix, app(fix, fix)))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.737 seconds.