Term Rewriting System R:
[x, y]
not(x) -> xor(x, true)
implies(x, y) -> xor(and(x, y), xor(x, true))
or(x, y) -> xor(and(x, y), xor(x, y))
=(x, y) -> xor(x, xor(y, true))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

not(x) -> xor(x, true)

where the Polynomial interpretation:
  POL(and(x1, x2))=  x1 + x2  
  POL(xor(x1, x2))=  x1 + x2  
  POL(=(x1, x2))=  x1 + x2  
  POL(implies(x1, x2))=  2·x1 + x2  
  POL(true)=  0  
  POL(or(x1, x2))=  2·x1 + 2·x2  
  POL(not(x1))=  1 + x1  
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:

implies(x, y) -> xor(and(x, y), xor(x, true))

where the Polynomial interpretation:
  POL(and(x1, x2))=  x1 + x2  
  POL(xor(x1, x2))=  x1 + x2  
  POL(=(x1, x2))=  x1 + x2  
  POL(or(x1, x2))=  2·x1 + 2·x2  
  POL(true)=  0  
  POL(implies(x1, x2))=  1 + 2·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:

or(x, y) -> xor(and(x, y), xor(x, y))

where the Polynomial interpretation:
  POL(and(x1, x2))=  x1 + x2  
  POL(xor(x1, x2))=  x1 + x2  
  POL(=(x1, x2))=  x1 + x2  
  POL(true)=  0  
  POL(or(x1, x2))=  1 + 2·x1 + 2·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
             ↳RRRPolo
             ...
               →TRS4
                 ↳Removing Redundant Rules



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

=(x, y) -> xor(x, xor(y, true))

where the Polynomial interpretation:
  POL(xor(x1, x2))=  x1 + x2  
  POL(=(x1, x2))=  1 + x1 + x2  
  POL(true)=  0  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS5
                 ↳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
             ...
               →TRS6
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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