Term Rewriting System R:

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

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   not(x) -> xor(x, true)

where the Polynomial interpretation:
POL(not(x_1)) = 1 + x_1
POL(implies(x_1, x_2)) = 2*x_1 + x_2
POL(and(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(or(x_1, x_2)) = 2*x_1 + 2*x_2
POL(xor(x_1, x_2)) = x_1 + x_2
POL(=(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   implies(x, y) -> xor(and(x, y), xor(x, true))

where the Polynomial interpretation:
POL(implies(x_1, x_2)) = 1 + 2*x_1 + x_2
POL(or(x_1, x_2)) = 2*x_1 + 2*x_2
POL(and(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(xor(x_1, x_2)) = x_1 + x_2
POL(=(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   or(x, y) -> xor(and(x, y), xor(x, y))

where the Polynomial interpretation:
POL(true) = 0
POL(or(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
POL(and(x_1, x_2)) = x_1 + x_2
POL(xor(x_1, x_2)) = x_1 + x_2
POL(=(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   =(x, y) -> xor(x, xor(y, true))

where the Polynomial interpretation:
POL(true) = 0
POL(xor(x_1, x_2)) = x_1 + x_2
POL(=(x_1, x_2)) = 1 + x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.433 seconds.