Term Rewriting System R:

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

Innermost Termination of R to be shown.


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

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

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

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

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

   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: 0.386 seconds.