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)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



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. 


   R
     ↳DPs
       →DP Problem 1
         ↳Non Termination


Dependency Pairs:

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))


Rules:


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)


Strategy:

innermost




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, fix), app(fix, app(fix, fix)))
evaluates to t =APP(app(fix, fix), app(fix, app(fix, fix)))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:01 minutes