Term Rewriting System R:
[x, y]
g(0, f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0)
g(s(x), y) -> g(f(x, y), 0)
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  x1 + x2  
  POL(s(x1))=  1 + x1  
  POL(f(x1, x2))=  1 + x1 + x2  
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:

g(s(x), y) -> g(f(x, y), 0)
g(x, s(y)) -> g(f(x, y), 0)

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  x1 + x2  
  POL(s(x1))=  1 + x1  
  POL(f(x1, x2))=  x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



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

g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  2·x1 + x2  
  POL(f(x1, x2))=  1 + x1 + x2  
was used.

All Rules of R can be deleted.


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



R contains no Dependency Pairs and therefore no SCCs.

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