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

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(g2(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  
  POL(h(x1, x2))=  x1 + x2  
  POL(a)=  0  
  POL(f(x1, x2, x3, x4, x5))=  1 + 2·x1 + 2·x2 + x3 + x4 + x5  
  POL(g1(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  
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:

g2(y, y, x, a) -> h(x, y)
g2(x, y, y, a) -> h(x, y)
g1(x, x, y, a) -> h(x, y)
g1(y, x, x, a) -> h(x, y)

where the Polynomial interpretation:
  POL(g2(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  
  POL(h(x1, x2))=  x1 + x2  
  POL(a)=  1  
  POL(g1(x1, x2, x3, x4))=  x1 + x2 + x3 + x4  
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:

h(x, x) -> x

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

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS4
                 ↳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
             ↳RRRPolo
             ...
               →TRS5
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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