Definition SF_sup_r (f : R -> R) (a b : R) (n : nat) : StepFun a b.

(* Specifications: *)
Lemma SF_sup_subdiv (f : R -> R) (a b : R) (n : nat) : subdivision (SF_sup_r f a b n) = match (Rle_dec a b) with | left _ => seq2Rlist (unif_part a b n) | right _ => seq2Rlist (unif_part b a n) end.
Lemma SF_sup_subdiv_val (f : R -> R) (a b : R) (n : nat) : subdivision_val (SF_sup_r f a b n) = match (Rle_dec a b) with | left _ => (seq2Rlist (behead (pairmap (fun x y => real (Sup_fct f x y)) 0 (unif_part a b n)))) | right _ => (seq2Rlist (behead (pairmap (fun x y => real (Sup_fct f x y)) 0 (unif_part b a n)))) end.
Lemma SF_sup_r_bound (f : R -> R) (a b : R) (n : nat) : forall x, SF_sup_r f a b n x = SF_sup_r f b a n x.
Lemma ad_SF_inf_r (f : R -> R) (a b : R) (n : nat) : ((a <= b) -> adapted_couple (fun x => real (SF_inf_fun f a b n x)) a b (seq2Rlist (unif_part a b n)) (seq2Rlist (behead (pairmap (fun x y => real (Inf_fct f x y)) 0 (unif_part a b n))))) /\ (~(a <= b) -> adapted_couple (fun x => real (SF_inf_fun f a b n x)) a b (seq2Rlist (unif_part b a n)) (seq2Rlist (behead (pairmap (fun x y => real (Inf_fct f x y)) 0 (unif_part b a n))))).