Term Rewriting System R:

   [x]
   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   evenodd(0, s(0)) -> false

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(s(x_1)) = x_1
POL(true) = 0
POL(0) = 0
POL(false) = 0
POL(evenodd(x_1, x_2)) = 1 + x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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


R contains the following Dependency Pairs: 


   EVENODD(s(x), s(0)) -> EVENODD(x, 0)
   EVENODD(x, 0) -> NOT(evenodd(x, s(0)))
   EVENODD(x, 0) -> EVENODD(x, s(0))

Furthermore, R contains one SCC.


SCC1:

EVENODD(x, 0) -> EVENODD(x, s(0))
EVENODD(s(x), s(0)) -> EVENODD(x, 0)

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(EVENODD(x_1, x_2)) = x_1
POL(s(x_1)) = 1 + x_1
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.592 seconds.