Term Rewriting System R:

   [y, x]
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

Termination of R to be shown.


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

   +(0, y) -> y
   -(x, 0) -> x

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0) = 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: 

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

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = 2*x_1 + x_2
POL(0) = 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: 

   -(0, y) -> 0

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.388 seconds.