Section RInt_val.
Context {V : ModuleSpace R_Ring}.
Definition RInt_val (f : R -> V) (a b : R) (n : nat) :=
Riemann_sum f (SF_seq_f2 (fun x y => (x + y) / 2) (unif_part a b n)).
End RInt_val.
(** ** From SF_seq to StepFun *)
(** Alternative Chasles relation *)
Fixpoint seq_cut_down' (s : seq (R*R)) (x x0 : R) : seq (R*R) :=
match s with
| [::] => [:: (x,x0)]
| h :: t =>
match Rle_dec (fst h) x with
| right _ => [:: (x,snd h)]
| left _ => h :: (seq_cut_down' t x (snd h))
end
end.
Fixpoint seq_cut_up' (s : seq (R*R)) (x x0 : R) : seq (R*R) :=
match s with
| [::] => [:: (x,x0)]
| h :: t =>
match Rle_dec (fst h) x with
| right _ => (x,x0)::h::t
| left _ => seq_cut_up' t x (snd h)
end
end.
Definition SF_cut_down' (sf : @SF_seq R) (x : R) x0 :=
let s := seq_cut_down' ((SF_h sf,x0) :: (SF_t sf)) x x0 in
mkSF_seq (fst (head (SF_h sf,x0) s)) (behead s).
Definition SF_cut_up' (sf : @SF_seq R) (x : R) x0 :=
let s := seq_cut_up' ((SF_h sf,x0) :: (SF_t sf)) x x0 in
mkSF_seq (fst (head (SF_h sf,x0) s)) (behead s).
(** Build StepFun using SF_seq *)
Definition SF_compat_le (s : @SF_seq R) (pr : SF_sorted Rle s) :
StepFun (head 0 (SF_lx s)) (last 0 (SF_lx s)).
(** Build StepFun using uniform partition *)
Definition sf_SF_val_fun (f : R -> R) (a b : R) (n : nat) : StepFun a b.
(** ** Upper and lower step functions *)

Lemma ex_Im_fct (f : R -> R) (a b : R) : a <> b ->
exists x, (fun y => exists x, y = f x /\ Rmin a b < x < Rmax a b) x.