Term Rewriting System R:
[x, y]
merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

merge(x, nil) -> x
merge(nil, y) -> y

where the Polynomial interpretation:
  POL(v)=  0  
  POL(merge(x1, x2))=  x1 + x2  
  POL(++(x1, x2))=  x1 + x2  
  POL(nil)=  1  
  POL(u)=  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:

merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))

where the Polynomial interpretation:
  POL(v)=  0  
  POL(merge(x1, x2))=  2·x1 + x2  
  POL(++(x1, x2))=  1 + x1 + x2  
  POL(u)=  0  
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:

merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

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