Term Rewriting System R:
[x, y]
and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(not(x), not(y)) -> implies(y, and(x, y))

where the Polynomial interpretation:
  POL(and(x1, x2))=  1 + x1 + x2  
  POL(false)=  0  
  POL(implies(x1, x2))=  1 + 2·x1 + 2·x2  
  POL(not(x1))=  1 + 2·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, false) -> not(x)
implies(false, y) -> not(false)

where the Polynomial interpretation:
  POL(false)=  0  
  POL(implies(x1, x2))=  1 + x1 + x2  
  POL(not(x1))=  x1  
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