Term Rewriting System R:
[z, x, y]
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z), app(e, x)) -> APP(h, app(c, z))
APP(app(h, z), app(e, x)) -> APP(c, z)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(d, z)
APP(app(d, z), app(app(g, 0), 0)) -> APP(e, 0)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(d, z), app(app(g, x), y)) -> APP(g, app(e, x))
APP(app(d, z), app(app(g, x), y)) -> APP(e, x)
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(g, app(app(d, app(c, z)), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(d, z)
APP(app(g, app(e, x)), app(e, y)) -> APP(e, app(app(g, x), y))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(g, app(e, x)), app(e, y)) -> APP(g, x)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes