Term Rewriting System R:
[x, y]
app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(D, app(app(+, x), y)) -> APP(app(+, app(D, x)), app(D, y))
APP(D, app(app(+, x), y)) -> APP(+, app(D, x))
APP(D, app(app(+, x), y)) -> APP(D, x)
APP(D, app(app(+, x), y)) -> APP(D, y)
APP(D, app(app(*, x), y)) -> APP(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
APP(D, app(app(*, x), y)) -> APP(+, app(app(*, y), app(D, x)))
APP(D, app(app(*, x), y)) -> APP(app(*, y), app(D, x))
APP(D, app(app(*, x), y)) -> APP(*, y)
APP(D, app(app(*, x), y)) -> APP(D, x)
APP(D, app(app(*, x), y)) -> APP(app(*, x), app(D, y))
APP(D, app(app(*, x), y)) -> APP(D, y)
APP(D, app(app(-, x), y)) -> APP(app(-, app(D, x)), app(D, y))
APP(D, app(app(-, x), y)) -> APP(-, app(D, x))
APP(D, app(app(-, x), y)) -> APP(D, x)
APP(D, app(app(-, x), y)) -> APP(D, y)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(D, app(app(-, x), y)) -> APP(D, y)
APP(D, app(app(-, x), y)) -> APP(D, x)
APP(D, app(app(-, x), y)) -> APP(app(-, app(D, x)), app(D, y))
APP(D, app(app(*, x), y)) -> APP(D, y)
APP(D, app(app(*, x), y)) -> APP(D, x)
APP(D, app(app(*, x), y)) -> APP(app(*, y), app(D, x))
APP(D, app(app(*, x), y)) -> APP(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
APP(D, app(app(+, x), y)) -> APP(D, y)
APP(D, app(app(+, x), y)) -> APP(D, x)
APP(D, app(app(+, x), y)) -> APP(app(+, app(D, x)), app(D, y))


Rules:


app(D, t) -> 1
app(D, constant) -> 0
app(D, app(app(+, x), y)) -> app(app(+, app(D, x)), app(D, y))
app(D, app(app(*, x), y)) -> app(app(+, app(app(*, y), app(D, x))), app(app(*, x), app(D, y)))
app(D, app(app(-, x), y)) -> app(app(-, app(D, x)), app(D, y))


Strategy:

innermost



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