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

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

f(a, empty) -> g(a, empty)

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

g(empty, d) -> d

where the Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(cons(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  x1 + 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
RRRPolo
           →TRS3
Removing Redundant Rules



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

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

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

g(cons(x, k), d) -> g(k, cons(x, d))

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

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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