Term Rewriting System R:

   [x, y, z]
   xor(x, F) -> x
   xor(x, neg(x)) -> F
   xor(x, x) -> F
   and(x, T) -> x
   and(x, F) -> F
   and(x, x) -> x
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))
   impl(x, y) -> xor(and(x, y), xor(x, T))
   or(x, y) -> xor(and(x, y), xor(x, y))
   equiv(x, y) -> xor(x, xor(y, T))
   neg(x) -> xor(x, T)

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)
   IMPL(x, y) -> XOR(and(x, y), xor(x, T))
   IMPL(x, y) -> AND(x, y)
   IMPL(x, y) -> XOR(x, T)
   NEG(x) -> XOR(x, T)
   OR(x, y) -> XOR(and(x, y), xor(x, y))
   OR(x, y) -> AND(x, y)
   OR(x, y) -> XOR(x, y)
   EQUIV(x, y) -> XOR(x, xor(y, T))
   EQUIV(x, y) -> XOR(y, T)

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: 0.631 seconds.