Term Rewriting System R:

   [x, y, z]
   i(0) -> 0
   i(i(x)) -> x
   i(+(x, y)) -> +(i(x), i(y))
   +(0, y) -> y
   +(x, 0) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(y)) -> x

Termination of R to be shown.


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

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

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

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

where the Polynomial interpretation:
POL(i(x_1)) = 1 + x_1
POL(+(x_1, x_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: 

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

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.409 seconds.