Term Rewriting System R:

   [x, y, z]
   app(app(., 1), x) -> x
   app(app(., x), 1) -> x
   app(app(., app(i, x)), x) -> 1
   app(app(., x), app(i, x)) -> 1
   app(app(., app(i, y)), app(app(., y), z)) -> z
   app(app(., y), app(app(., app(i, y)), z)) -> z
   app(i, 1) -> 1
   app(i, app(i, x)) -> x

Termination of R to be shown.


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

   app(app(., 1), x) -> x
   app(app(., x), 1) -> x
   app(app(., app(i, x)), x) -> 1
   app(app(., x), app(i, x)) -> 1
   app(app(., app(i, y)), app(app(., y), z)) -> z
   app(app(., y), app(app(., app(i, y)), z)) -> z

where the Polynomial interpretation:
POL(1) = 0
POL(i) = 0
POL(app(x_1, x_2)) = x_1 + x_2
POL(.) = 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: 

   app(i, 1) -> 1
   app(i, app(i, x)) -> x

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.406 seconds.