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))
app(D, app(minus, x)) -> app(minus, app(D, x))
app(D, app(app(div, x), y)) -> app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) -> app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) -> app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, 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)
APP(D, app(minus, x)) -> APP(minus, app(D, x))
APP(D, app(minus, x)) -> APP(D, x)
APP(D, app(app(div, x), y)) -> APP(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
APP(D, app(app(div, x), y)) -> APP(-, app(app(div, app(D, x)), y))
APP(D, app(app(div, x), y)) -> APP(app(div, app(D, x)), y)
APP(D, app(app(div, x), y)) -> APP(div, app(D, x))
APP(D, app(app(div, x), y)) -> APP(D, x)
APP(D, app(app(div, x), y)) -> APP(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2))
APP(D, app(app(div, x), y)) -> APP(div, app(app(*, x), app(D, y)))
APP(D, app(app(div, x), y)) -> APP(app(*, x), app(D, y))
APP(D, app(app(div, x), y)) -> APP(*, x)
APP(D, app(app(div, x), y)) -> APP(D, y)
APP(D, app(app(div, x), y)) -> APP(app(pow, y), 2)
APP(D, app(app(div, x), y)) -> APP(pow, y)
APP(D, app(ln, x)) -> APP(app(div, app(D, x)), x)
APP(D, app(ln, x)) -> APP(div, app(D, x))
APP(D, app(ln, x)) -> APP(D, x)
APP(D, app(app(pow, x), y)) -> APP(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
APP(D, app(app(pow, x), y)) -> APP(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x)))
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))
APP(D, app(app(pow, x), y)) -> APP(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1))))
APP(D, app(app(pow, x), y)) -> APP(app(*, y), app(app(pow, x), app(app(-, y), 1)))
APP(D, app(app(pow, x), y)) -> APP(*, y)
APP(D, app(app(pow, x), y)) -> APP(app(pow, x), app(app(-, y), 1))
APP(D, app(app(pow, x), y)) -> APP(app(-, y), 1)
APP(D, app(app(pow, x), y)) -> APP(-, y)
APP(D, app(app(pow, x), y)) -> APP(D, x)
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y))
APP(D, app(app(pow, x), y)) -> APP(*, app(app(*, app(app(pow, x), y)), app(ln, x)))
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(pow, x), y)), app(ln, x))
APP(D, app(app(pow, x), y)) -> APP(*, app(app(pow, x), y))
APP(D, app(app(pow, x), y)) -> APP(ln, x)
APP(D, app(app(pow, 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(pow, x), y)) -> APP(D, y)
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(pow, x), y)), app(ln, x))
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y))
APP(D, app(app(pow, x), y)) -> APP(D, x)
APP(D, app(app(pow, x), y)) -> APP(app(*, y), app(app(pow, x), app(app(-, y), 1)))
APP(D, app(app(pow, x), y)) -> APP(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))
APP(D, app(app(pow, x), y)) -> APP(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))
APP(D, app(ln, x)) -> APP(D, x)
APP(D, app(ln, x)) -> APP(app(div, app(D, x)), x)
APP(D, app(app(div, x), y)) -> APP(D, y)
APP(D, app(app(div, x), y)) -> APP(app(*, x), app(D, y))
APP(D, app(app(div, x), y)) -> APP(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2))
APP(D, app(app(div, x), y)) -> APP(D, x)
APP(D, app(app(div, x), y)) -> APP(app(div, app(D, x)), y)
APP(D, app(app(div, x), y)) -> APP(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
APP(D, app(minus, x)) -> APP(D, x)
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))
app(D, app(minus, x)) -> app(minus, app(D, x))
app(D, app(app(div, x), y)) -> app(app(-, app(app(div, app(D, x)), y)), app(app(div, app(app(*, x), app(D, y))), app(app(pow, y), 2)))
app(D, app(ln, x)) -> app(app(div, app(D, x)), x)
app(D, app(app(pow, x), y)) -> app(app(+, app(app(*, app(app(*, y), app(app(pow, x), app(app(-, y), 1)))), app(D, x))), app(app(*, app(app(*, app(app(pow, x), y)), app(ln, x))), app(D, y)))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes