Term Rewriting System R:

   [x, y]
   not(true) -> false
   not(false) -> true
   odd(0) -> false
   odd(s(x)) -> not(odd(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))

Termination of R to be shown.


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

   not(true) -> false
   not(false) -> true

where the Polynomial interpretation:
POL(not(x_1)) = 1 + x_1
POL(odd(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(true) = 0
POL(+(x_1, x_2)) = x_1 + x_2
POL(0) = 0
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: 

   odd(0) -> false
   +(x, 0) -> x

where the Polynomial interpretation:
POL(odd(x_1)) = x_1
POL(s(x_1)) = x_1
POL(not(x_1)) = x_1
POL(+(x_1, x_2)) = x_1 + x_2
POL(0) = 1
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: 

   odd(s(x)) -> not(odd(x))

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(odd(x_1)) = x_1
POL(not(x_1)) = x_1
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, s(y)) -> s(+(x, y))

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(+(x_1, x_2)) = x_1 + 2*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: 

   +(s(x), y) -> s(+(x, y))

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(+(x_1, x_2)) = 2*x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.442 seconds.