Term Rewriting System R:
[Z, X, Y, X1, X2]
fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
activate(nfst(X1, X2)) -> fst(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nadd(X1, X2)) -> add(X1, X2)
activate(nlen(X)) -> len(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:

fst(0, Z) -> nil
add(0, X) -> X
len(nil) -> 0

where the Polynomial interpretation:
  POL(from(x1))=  2·x1  
  POL(activate(x1))=  2·x1  
  POL(len(x1))=  2·x1  
  POL(n__fst(x1, x2))=  x1 + x2  
  POL(add(x1, x2))=  2·x1 + x2  
  POL(n__from(x1))=  x1  
  POL(0)=  1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nil)=  1  
  POL(fst(x1, x2))=  2·x1 + 2·x2  
  POL(s(x1))=  x1  
  POL(n__len(x1))=  x1  
  POL(n__add(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:

add(X1, X2) -> nadd(X1, X2)
add(s(X), Y) -> s(nadd(activate(X), Y))

where the Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(from(x1))=  2·x1  
  POL(activate(x1))=  2·x1  
  POL(len(x1))=  2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(n__fst(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(fst(x1, x2))=  2·x1 + 2·x2  
  POL(n__len(x1))=  x1  
  POL(n__add(x1, x2))=  1 + x1 + x2  
  POL(add(x1, x2))=  2 + 2·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:

activate(nlen(X)) -> len(X)

where the Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(from(x1))=  2·x1  
  POL(activate(x1))=  2·x1  
  POL(len(x1))=  1 + 2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(n__fst(x1, x2))=  x1 + x2  
  POL(fst(x1, x2))=  2·x1 + 2·x2  
  POL(s(x1))=  x1  
  POL(n__len(x1))=  1 + x1  
  POL(n__add(x1, x2))=  x1 + x2  
  POL(add(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(nadd(X1, X2)) -> add(X1, X2)

where the Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(from(x1))=  2·x1  
  POL(activate(x1))=  2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(len(x1))=  2·x1  
  POL(n__fst(x1, x2))=  x1 + x2  
  POL(fst(x1, x2))=  2·x1 + 2·x2  
  POL(s(x1))=  x1  
  POL(n__len(x1))=  x1  
  POL(n__add(x1, x2))=  1 + x1 + x2  
  POL(add(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
             ...
               →TRS5
Removing Redundant Rules



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

fst(X1, X2) -> nfst(X1, X2)
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))

where the Polynomial interpretation:
  POL(from(x1))=  2·x1  
  POL(n__from(x1))=  x1  
  POL(activate(x1))=  2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(len(x1))=  2·x1  
  POL(n__fst(x1, x2))=  1 + x1 + x2  
  POL(fst(x1, x2))=  2 + 2·x1 + 2·x2  
  POL(s(x1))=  x1  
  POL(n__len(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
             ...
               →TRS6
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(X) -> X
activate(nfst(X1, X2)) -> fst(X1, X2)
len(X) -> nlen(X)

where the Polynomial interpretation:
  POL(from(x1))=  1 + 2·x1  
  POL(n__from(x1))=  x1  
  POL(activate(x1))=  1 + 2·x1  
  POL(len(x1))=  1 + 2·x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(n__fst(x1, x2))=  x1 + x2  
  POL(fst(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(n__len(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
             ...
               →TRS7
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))=  1 + x1  
  POL(from(x1))=  x1  
  POL(activate(x1))=  x1  
  POL(len(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(n__len(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
             ...
               →TRS8
Removing Redundant Rules



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

len(cons(X, Z)) -> s(nlen(activate(Z)))

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(len(x1))=  1 + x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(n__len(x1))=  x1  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS9
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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