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)))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   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)
   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(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(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)

Furthermore, R contains one SCC.


SCC1:

APP(D, app(ln, 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(D, y)
APP(D, app(app(+, x), y)) -> APP(D, x)
APP(D, app(app(div, x), y)) -> APP(D, y)
APP(D, app(app(div, x), y)) -> APP(D, x)
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(pow, x), y)) -> APP(D, y)
APP(D, app(app(pow, x), y)) -> APP(D, x)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(*) = 1
POL(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(-) = 1
POL(D) = 1
POL(ln) = 1
POL(div) = 1
POL(minus) = 1
POL(+) = 1
POL(pow) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   APP(D, app(ln, 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(D, y)
   APP(D, app(app(+, x), y)) -> APP(D, x)
   APP(D, app(app(div, x), y)) -> APP(D, y)
   APP(D, app(app(div, x), y)) -> APP(D, x)
   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(pow, x), y)) -> APP(D, y)
   APP(D, app(app(pow, x), y)) -> APP(D, x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 3.392 seconds.