Term Rewriting System R:

   [x, y]
   not(not(x)) -> x
   not(or(x, y)) -> and(not(not(not(x))), not(not(not(y))))
   not(and(x, y)) -> or(not(not(not(x))), not(not(not(y))))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   NOT(and(x, y)) -> NOT(not(not(x)))
   NOT(and(x, y)) -> NOT(not(x))
   NOT(and(x, y)) -> NOT(x)
   NOT(and(x, y)) -> NOT(not(not(y)))
   NOT(and(x, y)) -> NOT(not(y))
   NOT(and(x, y)) -> NOT(y)
   NOT(or(x, y)) -> NOT(not(not(x)))
   NOT(or(x, y)) -> NOT(not(x))
   NOT(or(x, y)) -> NOT(x)
   NOT(or(x, y)) -> NOT(not(not(y)))
   NOT(or(x, y)) -> NOT(not(y))
   NOT(or(x, y)) -> NOT(y)

Furthermore, R contains one SCC.


SCC1:

NOT(or(x, y)) -> NOT(y)
NOT(or(x, y)) -> NOT(not(y))
NOT(or(x, y)) -> NOT(not(not(y)))
NOT(or(x, y)) -> NOT(x)
NOT(or(x, y)) -> NOT(not(x))
NOT(or(x, y)) -> NOT(not(not(x)))
NOT(and(x, y)) -> NOT(y)
NOT(and(x, y)) -> NOT(not(y))
NOT(and(x, y)) -> NOT(not(not(y)))
NOT(and(x, y)) -> NOT(x)
NOT(and(x, y)) -> NOT(not(x))
NOT(and(x, y)) -> NOT(not(not(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(NOT(x_1)) = x_1
POL(not(x_1)) = x_1
POL(or(x_1, x_2)) = 1 + x_1 + x_2
POL(and(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   NOT(or(x, y)) -> NOT(y)
   NOT(or(x, y)) -> NOT(not(y))
   NOT(or(x, y)) -> NOT(not(not(y)))
   NOT(or(x, y)) -> NOT(x)
   NOT(or(x, y)) -> NOT(not(x))
   NOT(or(x, y)) -> NOT(not(not(x)))
   NOT(and(x, y)) -> NOT(y)
   NOT(and(x, y)) -> NOT(not(y))
   NOT(and(x, y)) -> NOT(not(not(y)))
   NOT(and(x, y)) -> NOT(x)
   NOT(and(x, y)) -> NOT(not(x))
   NOT(and(x, y)) -> NOT(not(not(x)))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 2.36 seconds.