End Rcompare.
Section Rle_bool.
Definition Rle_bool x y :=
match Rcompare x y with
| Gt => false
| _ => true
end.
Inductive Rle_bool_prop (x y : R) : bool -> Prop :=
| Rle_bool_true_ : (x <= y)%R -> Rle_bool_prop x y true
| Rle_bool_false_ : (y < x)%R -> Rle_bool_prop x y false.

Theorem Rle_bool_false :
forall x y,
(y < x)%R -> Rle_bool x y = false.