Term Rewriting System R:

   [x, y, u, z]
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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: 


   F(s(x), 0, z, u) -> F(x, u, minus(z, s(x)), u)
   F(s(x), s(y), z, u) -> F(s(x), minus(y, x), z, u)
   F(s(x), s(y), z, u) -> F(x, u, z, u)
   PERFECTP(s(x)) -> F(x, s(0), s(x), s(x))

Furthermore, R contains one SCC.


SCC1:

F(s(x), s(y), z, u) -> F(x, u, z, u)
F(s(x), 0, z, u) -> F(x, u, minus(z, s(x)), u)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(minus(x_1, x_2)) = 0
POL(F(x_1, x_2, x_3, x_4)) = x_1 + x_4
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.705 seconds.