Term Rewriting System R:

   [a, x, k, d]
   f(a, empty) -> g(a, empty)
   f(a, cons(x, k)) -> f(cons(x, a), k)
   g(empty, d) -> d
   g(cons(x, k), d) -> g(k, cons(x, d))

Termination of R to be shown.


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

   f(a, empty) -> g(a, empty)

where the Polynomial interpretation:
POL(g(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(empty) = 0
POL(cons(x_1, x_2)) = x_1 + x_2
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: 

   f(a, cons(x, k)) -> f(cons(x, a), k)
   g(empty, d) -> d

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 2 + x_1 + x_2
POL(f(x_1, x_2)) = 2 + x_1 + 2*x_2
POL(empty) = 2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
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: 

   g(cons(x, k), d) -> g(k, cons(x, d))

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 2 + 2*x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.400 seconds.