Term Rewriting System R:
[x, y, p, f]
app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(app(app(until, p), f), x) -> app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(until, p), f), x) -> APP(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))
APP(app(app(until, p), f), x) -> APP(app(if, app(p, x)), x)
APP(app(app(until, p), f), x) -> APP(if, app(p, x))
APP(app(app(until, p), f), x) -> APP(p, x)
APP(app(app(until, p), f), x) -> APP(app(app(until, p), f), app(f, x))
APP(app(app(until, p), f), x) -> APP(f, x)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(app(until, p), f), x) -> APP(f, x)
APP(app(app(until, p), f), x) -> APP(app(app(until, p), f), app(f, x))
APP(app(app(until, p), f), x) -> APP(p, x)
APP(app(app(until, p), f), x) -> APP(app(if, app(p, x)), x)
APP(app(app(until, p), f), x) -> APP(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))


Rules:


app(app(app(if, true), x), y) -> x
app(app(app(if, false), x), y) -> y
app(app(app(until, p), f), x) -> app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
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