Term Rewriting System R:

   [x, y, z]
   f(0, 1, x) -> f(x, x, x)
   f(x, y, z) -> 2
   0 -> 2
   1 -> 2
   g(x, x, y) -> y
   g(x, y, y) -> x

Innermost Termination of R to be shown.


Removing the following rules from R which left hand sides contain non normal subterms

   f(0, 1, x) -> f(x, x, x)

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

   f(x, y, z) -> 2
   g(x, x, y) -> y
   g(x, y, y) -> x

where the Polynomial interpretation:
POL(g(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3
POL(1) = 0
POL(f(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
POL(2) = 0
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 -> 2

where the Polynomial interpretation:
POL(1) = 0
POL(2) = 0
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: 

   1 -> 2

where the Polynomial interpretation:
POL(1) = 1
POL(2) = 0
was used. 



All Rules of R can be deleted.



Innermost Termination of R successfully shown.


Duration: 0.376 seconds.