Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

AND(x, or(y, z)) -> AND(x, z)
AND(x, or(y, z)) -> AND(x, y)

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(AND(x_1, x_2)) = 1 + x_1 + x_2
POL(or(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   AND(x, or(y, z)) -> AND(x, z)
   AND(x, or(y, z)) -> AND(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.516 seconds.