Term Rewriting System R:

   [x, y, z]
   and(x, or(y, z)) -> or(and(x, y), and(x, z))
   and(x, and(y, y)) -> and(x, y)
   and(x, true) -> x
   and(false, y) -> false
   and(x, x) -> x
   or(or(x, y), and(y, z)) -> or(x, y)
   or(x, and(x, y)) -> x
   or(true, y) -> true
   or(x, false) -> x
   or(x, x) -> x
   or(x, or(y, y)) -> or(x, y)

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

OR(x, or(y, y)) -> OR(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(OR(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:

   OR(x, or(y, y)) -> OR(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.586 seconds.