Term Rewriting System R:

   [x, y]
   \(x, x) -> e
   \(e, x) -> x
   \(x, .(x, y)) -> y
   \(/(x, y), x) -> y
   /(x, x) -> e
   /(x, e) -> x
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   .(e, x) -> x
   .(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y

Termination of R to be shown.


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

   \(x, x) -> e
   \(e, x) -> x
   \(x, .(x, y)) -> y
   \(/(x, y), x) -> y
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   .(e, x) -> x
   .(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y

where the Polynomial interpretation:
POL(e) = 0
POL(/(x_1, x_2)) = x_1 + x_2
POL(\(x_1, x_2)) = 1 + x_1 + x_2
POL(.(x_1, x_2)) = 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, x) -> e
   /(x, e) -> x

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.396 seconds.