Term Rewriting System R:
[x, a, k, y]
f(x, empty) -> x
f(empty, cons(a, k)) -> f(cons(a, k), k)
f(cons(a, k), y) -> f(y, k)

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

f(x, empty) -> x

where the Polynomial interpretation:
  POL(cons(x1, x2))=  x1 + 2·x2  
  POL(f(x1, x2))=  1 + x1 + 2·x2  
  POL(empty)=  0  
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:

f(empty, cons(a, k)) -> f(cons(a, k), k)

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

f(cons(a, k), y) -> f(y, k)

where the Polynomial interpretation:
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  x1 + x2  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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