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

Termination of R to be shown.



   R
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(s(x1))=  x1  
  POL(f(x1, x2, x3))=  2·x1 + x2 + x3  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.


   R
RRRPolo
       →TRS2
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(f(x1, x2, x3))=  2·x1 + x2 + x3  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
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
RRRPolo
           →TRS3
OC
             ...
               →TRS4
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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