Term Rewriting System R:

   [x, y, z]
   +(*(x, y), *(x, z)) -> *(x, +(y, z))
   +(+(x, y), z) -> +(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

Termination of R to be shown.


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

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

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.391 seconds.