Term Rewriting System R:
[x, y]
f(g(x, y), f(y, y)) -> f(g(y, x), y)

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  1 + x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Termination of R successfully shown.
Duration:
0:00 minutes