Term Rewriting System R:
[X, Y, Z, X1, X2]
2nd(cons(X, ncons(Y, Z))) -> activate(Y)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
cons(X1, X2) -> ncons(X1, X2)
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Innermost Termination of R to be shown.



   R
Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

2nd(cons(X, ncons(Y, Z))) -> activate(Y)


   R
RRRI
       →TRS2
Removing Redundant Rules



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

from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(ncons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

where the Polynomial interpretation:
  POL(from(x1))=  1 + 2·x1  
  POL(n__from(x1))=  x1  
  POL(n__cons(x1, x2))=  x1 + x2  
  POL(activate(x1))=  1 + 2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.


   R
RRRI
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



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

cons(X1, X2) -> ncons(X1, X2)

where the Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(from(x1))=  x1  
  POL(n__cons(x1, x2))=  x1 + x2  
  POL(activate(x1))=  x1  
  POL(cons(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
RRRI
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

activate(nfrom(X)) -> from(X)

where the Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(from(x1))=  x1  
  POL(activate(x1))=  1 + x1  
was used.

All Rules of R can be deleted.


   R
RRRI
       →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