Definition SF_fun (s : SF_seq) (y0 : T) (x : R) :=
  SF_fun_aux (SF_h s,y0) (SF_t s) y0 x.

(* Specifications: *)
Lemma SF_fun_incr (s : SF_seq) (y0 : T) (x : R) Hs Hx : SF_fun s y0 x = match (sorted_dec (SF_lx s) 0 x Hs Hx) with | inleft H => nth y0 (SF_ly s) (proj1_sig H) | inright _ => nth y0 (SF_ly s) (SF_size s -1)%nat end.
Lemma SF_fun_map {T T0 : Type} (f : T -> T0) (s : SF_seq) y0 : forall x, SF_fun (SF_map f s) (f y0) x = f (SF_fun s y0 x).