Term Rewriting System R:
[X, Y, Z, X1, X2]
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X

Termination of R to be shown.



   R
Removing Redundant Rules



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

first(0, X) -> nil

where the Polynomial interpretation:
  POL(from(x1))=  2·x1  
  POL(n__from(x1))=  x1  
  POL(activate(x1))=  2·x1  
  POL(first(x1, x2))=  x1 + 2·x2  
  POL(0)=  1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(n__first(x1, x2))=  x1 + x2  
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:

first(X1, X2) -> nfirst(X1, X2)
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))

where the Polynomial interpretation:
  POL(from(x1))=  2·x1  
  POL(n__from(x1))=  x1  
  POL(activate(x1))=  2·x1  
  POL(first(x1, x2))=  2 + x1 + 2·x2  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(n__first(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
RRRPolo
       →TRS2
RRRPolo
           →TRS3
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(nfirst(X1, X2)) -> first(X1, X2)
activate(X) -> X

where the Polynomial interpretation:
  POL(from(x1))=  1 + 2·x1  
  POL(n__from(x1))=  x1  
  POL(activate(x1))=  1 + 2·x1  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(n__first(x1, x2))=  x1 + 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:

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
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS6
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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