Term Rewriting System R:

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

Termination of R to be shown.


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

   not(x) -> if(x, false, true)
   =(x, x) -> true
   =(x, y) -> if(x, y, if(y, false, true))

where the Polynomial interpretation:
POL(not(x_1)) = 1 + x_1
POL(implies(x_1, x_2)) = x_1 + x_2
POL(and(x_1, x_2)) = x_1 + x_2
POL(or(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(=(x_1, x_2)) = 1 + x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 0
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: 

   and(x, y) -> if(x, y, false)

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(implies(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(or(x_1, x_2)) = x_1 + x_2
POL(and(x_1, x_2)) = 1 + x_1 + x_2
POL(=(x_1, x_2)) = x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 0
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: 

   if(true, x, y) -> x

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(implies(x_1, x_2)) = 1 + x_1 + x_2
POL(true) = 1
POL(or(x_1, x_2)) = 1 + x_1 + x_2
POL(=(x_1, x_2)) = x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 0
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: 

   if(false, x, y) -> y
   if(x, x, if(x, false, true)) -> true

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(implies(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(or(x_1, x_2)) = x_1 + x_2
POL(=(x_1, x_2)) = x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 1
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) -> if(x, true, y)

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(implies(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(or(x_1, x_2)) = 1 + x_1 + x_2
POL(=(x_1, x_2)) = x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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) -> if(x, y, true)

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(implies(x_1, x_2)) = 1 + x_1 + x_2
POL(true) = 0
POL(=(x_1, x_2)) = x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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) -> if(x, y, not(y))

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(=(x_1, x_2)) = 1 + x_1 + 2*x_2
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.496 seconds.