how can we determine function k if we know kl-1fl holds for all functions f of type int-int and l of type int?

Do you want mathematical proof?

(1) k(i - 1) = f(i)

(2) j = i - 1

from (1) & (2):

(3) k(j) = f(i)

from (2):

(4) i = j + 1

from (3) & (4)

(5) k(j) = f(j + 1)

q.e.d.

Of course (fun v -> k(v+1)) is your k and the inside k is f