Term Rewriting System R:
[x, y]
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(quot, 0), app(s, y)) -> 0
app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y)))
app(log, app(s, 0)) -> 0
app(log, app(s, app(s, x))) -> app(s, app(log, app(s, app(app(quot, x), app(s, app(s, 0))))))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, x)
APP(app(quot, app(s, x)), app(s, y)) -> APP(s, app(app(quot, app(app(minus, x), y)), app(s, y)))
APP(app(quot, app(s, x)), app(s, y)) -> APP(app(quot, app(app(minus, x), y)), app(s, y))
APP(app(quot, app(s, x)), app(s, y)) -> APP(quot, app(app(minus, x), y))
APP(app(quot, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(quot, app(s, x)), app(s, y)) -> APP(minus, x)
APP(log, app(s, app(s, x))) -> APP(s, app(log, app(s, app(app(quot, x), app(s, app(s, 0))))))
APP(log, app(s, app(s, x))) -> APP(log, app(s, app(app(quot, x), app(s, app(s, 0)))))
APP(log, app(s, app(s, x))) -> APP(s, app(app(quot, x), app(s, app(s, 0))))
APP(log, app(s, app(s, x))) -> APP(app(quot, x), app(s, app(s, 0)))
APP(log, app(s, app(s, x))) -> APP(quot, x)
APP(log, app(s, app(s, x))) -> APP(s, app(s, 0))
APP(log, app(s, app(s, x))) -> APP(s, 0)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(log, app(s, app(s, x))) -> APP(app(quot, x), app(s, app(s, 0)))
APP(log, app(s, app(s, x))) -> APP(log, app(s, app(app(quot, x), app(s, app(s, 0)))))
APP(app(quot, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(quot, app(s, x)), app(s, y)) -> APP(app(quot, app(app(minus, x), y)), app(s, y))
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)


Rules:


app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, x), y)
app(app(quot, 0), app(s, y)) -> 0
app(app(quot, app(s, x)), app(s, y)) -> app(s, app(app(quot, app(app(minus, x), y)), app(s, y)))
app(log, app(s, 0)) -> 0
app(log, app(s, app(s, x))) -> app(s, app(log, app(s, app(app(quot, x), app(s, app(s, 0))))))


Strategy:

innermost



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