Term Rewriting System R:

   [Z, Y, X]
   fst(0, Z) -> nil
   fst(s, cons(Y)) -> cons(Y)
   from(X) -> cons(X)
   add(0, X) -> X
   add(s, Y) -> s
   len(nil) -> 0
   len(cons(X)) -> s

Termination of R to be shown.


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

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

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s) = 0
POL(from(x_1)) = x_1
POL(fst(x_1, x_2)) = x_1 + x_2
POL(len(x_1)) = 1 + x_1
POL(0) = 1
POL(cons(x_1)) = x_1
was used. 



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


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

   fst(s, cons(Y)) -> cons(Y)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s) = 0
POL(from(x_1)) = x_1
POL(len(x_1)) = x_1
POL(fst(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(cons(x_1)) = x_1
was used. 



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


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

   from(X) -> cons(X)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s) = 0
POL(from(x_1)) = 1 + x_1
POL(len(x_1)) = x_1
POL(0) = 0
POL(cons(x_1)) = x_1
was used. 



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


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

   add(s, Y) -> s

where the Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(s) = 0
POL(len(x_1)) = x_1
POL(0) = 0
was used. 



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


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

   len(nil) -> 0

where the Polynomial interpretation:
POL(nil) = 0
POL(len(x_1)) = 1 + x_1
POL(0) = 0
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.407 seconds.