Term Rewriting System R:
[x, y]
app(app(, x), x) -> e
app(app(, e), x) -> x
app(app(, x), app(app(., x), y)) -> y
app(app(, app(app(/, x), y)), x) -> y
app(app(/, x), x) -> e
app(app(/, x), e) -> x
app(app(/, app(app(., y), x)), x) -> y
app(app(/, x), app(app(, y), x)) -> y
app(app(., e), x) -> x
app(app(., x), e) -> x
app(app(., x), app(app(, x), y)) -> y
app(app(., app(app(/, y), x)), x) -> y

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

app(app(, x), x) -> e
app(app(, e), x) -> x
app(app(, x), app(app(., x), y)) -> y
app(app(, app(app(/, x), y)), x) -> y
app(app(/, x), app(app(, y), x)) -> y
app(app(., x), app(app(, x), y)) -> y

where the Polynomial interpretation:
  POL(e)=  0  
  POL(.)=  0  
  POL(/)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL()=  1  
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:

app(app(., app(app(/, y), x)), x) -> y
app(app(., x), e) -> x
app(app(/, app(app(., y), x)), x) -> y
app(app(., e), x) -> x

where the Polynomial interpretation:
  POL(e)=  0  
  POL(.)=  1  
  POL(/)=  0  
  POL(app(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:

app(app(/, x), x) -> e
app(app(/, x), e) -> x

where the Polynomial interpretation:
  POL(e)=  0  
  POL(/)=  1  
  POL(app(x1, x2))=  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