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))

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(-, 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(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)

Furthermore, R contains one SCC.


SCC1:

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(-, x), y)) -> APP(D, y)
APP(D, app(app(-, 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(+) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.861 seconds.