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

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))))).