Term Rewriting System R:
[X]
f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

f(0) -> cons(0, nf(s(0)))
f(X) -> nf(X)
activate(X) -> X

where the Polynomial interpretation:
  POL(n__f(x1))=  x1  
  POL(activate(x1))=  1 + x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(p(x1))=  x1  
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:

p(s(0)) -> 0

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(n__f(x1))=  x1  
  POL(0)=  1  
  POL(s(x1))=  2·x1  
  POL(f(x1))=  x1  
  POL(p(x1))=  x1  
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:

activate(nf(X)) -> f(X)

where the Polynomial interpretation:
  POL(activate(x1))=  1 + x1  
  POL(n__f(x1))=  x1  
  POL(0)=  0  
  POL(s(x1))=  x1  
  POL(f(x1))=  x1  
  POL(p(x1))=  x1  
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
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))

R contains no SCCs.

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