Term Rewriting System R:
[x, y]
f(x, y) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)
g1(x, x, y) -> h(x, y)
g1(y, x, x) -> h(x, y)
g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)
h(x, x) -> x

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

f(x, y) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)

where the Polynomial interpretation:
  POL(g2(x1, x2, x3))=  x1 + x2 + x3  
  POL(h(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  1 + 2·x1 + 2·x2  
  POL(g1(x1, x2, x3))=  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:

h(x, x) -> x

where the Polynomial interpretation:
  POL(g2(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(h(x1, x2))=  1 + x1 + x2  
  POL(g1(x1, x2, x3))=  1 + 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
         ↳RRRPolo
           →TRS3
             ↳Removing Redundant Rules



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

g1(y, x, x) -> h(x, y)
g1(x, x, y) -> h(x, y)

where the Polynomial interpretation:
  POL(g2(x1, x2, x3))=  x1 + x2 + x3  
  POL(h(x1, x2))=  x1 + x2  
  POL(g1(x1, x2, x3))=  1 + 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
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS4
                 ↳Removing Redundant Rules



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

g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)

where the Polynomial interpretation:
  POL(g2(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(h(x1, x2))=  x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS5
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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