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

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

where the Polynomial interpretation:
  POL(e)=  0  
  POL(.(x1, x2))=  x1 + x2  
  POL(/(x1, x2))=  x1 + x2  
  POL((x1, x2))=  1 + 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
         ↳Removing Redundant Rules



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

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

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

.(e, x) -> x
.(x, e) -> x

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