Term Rewriting System R:

   [x, y, z]
   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))
   xor(x, false) -> x
   xor(x, x) -> false

Termination of R to be shown.


R contains the following Dependency Pairs: 


   AND(xor(x, y), z) -> XOR(and(x, z), and(y, z))
   AND(xor(x, y), z) -> AND(x, z)
   AND(xor(x, y), z) -> AND(y, z)
   IMPLIES(x, y) -> XOR(and(x, y), xor(x, true))
   IMPLIES(x, y) -> AND(x, y)
   IMPLIES(x, y) -> XOR(x, true)
   OR(x, y) -> XOR(and(x, y), xor(x, y))
   OR(x, y) -> AND(x, y)
   OR(x, y) -> XOR(x, y)
   NOT(x) -> XOR(x, true)

Furthermore, R contains one SCC.


SCC1:

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

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(xor(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.579 seconds.