Term Rewriting System R:
[x, y, z]
.(1, x) -> x
.(x, 1) -> x
.(i(x), x) -> 1
.(x, i(x)) -> 1
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(1) -> 1
i(i(x)) -> x
i(.(x, y)) -> .(i(y), i(x))

Termination of R to be shown.



   R
Removing Redundant Rules



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

.(1, x) -> x
.(x, 1) -> x
.(i(x), x) -> 1
.(x, i(x)) -> 1
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z

where the Polynomial interpretation:
  POL(i(x1))=  x1  
  POL(1)=  0  
  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:

i(i(x)) -> x
i(1) -> 1

where the Polynomial interpretation:
  POL(i(x1))=  1 + 2·x1  
  POL(1)=  0  
  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:

i(.(x, y)) -> .(i(y), i(x))

where the Polynomial interpretation:
  POL(i(x1))=  2·x1  
  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
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

.(.(x, y), z) -> .(x, .(y, z))

where the Polynomial interpretation:
  POL(.(x1, x2))=  1 + 2·x1 + x2  
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