(* Some general properties of rounded predicate :
   -Projector A bounded is rounded to something equal to itself 
  - Monotone : the rounded predicate is monotone *)
 
Definition ProjectorP (P : R -> float -> Prop) :=
  forall p q : float, Fbounded b p -> P p q -> p = q :>R.Definition MonotoneP (P : R -> float -> Prop) :=
  forall (p q : R) (p' q' : float),
  (p < q)%R -> P p p' -> P q q' -> (p' <= q')%R.(* What it is to be a minimum*)
 
Definition isMin (r : R) (min : float) :=
  Fbounded b min /\
  (min <= r)%R /\
  (forall f : float, Fbounded b f -> (f <= r)%R -> (f <= min)%R).

(* Specifications: *)
Theorem isMin_inv1 : forall (p : float) (r : R), isMin r p -> (p <= r)%R.
Theorem ProjectMin : ProjectorP isMin.
Theorem MonotoneMin : MonotoneP isMin.