End Rle_bool.
Section Rlt_bool.
Definition Rlt_bool x y :=
match Rcompare x y with
| Lt => true
| _ => false
end.
Inductive Rlt_bool_prop (x y : R) : bool -> Prop :=
| Rlt_bool_true_ : (x < y)%R -> Rlt_bool_prop x y true
| Rlt_bool_false_ : (y <= x)%R -> Rlt_bool_prop x y false.

(* Specifications: *)
Theorem Rlt_bool_spec :
forall x y, Rlt_bool_prop x y (Rlt_bool x y).
Theorem negb_Rlt_bool :
forall x y,
negb (Rle_bool x y) = Rlt_bool y x.
Theorem negb_Rle_bool :
forall x y,
negb (Rlt_bool x y) = Rle_bool y x.
Theorem Rlt_bool_true :
forall x y,
(x < y)%R -> Rlt_bool x y = true.
Theorem Rlt_bool_false :
forall x y,
(y <= x)%R -> Rlt_bool x y = false.
Theorem Rlt_bool_spec :
forall x y, Rlt_bool_prop x y (Rlt_bool x y).
Theorem negb_Rlt_bool :
forall x y,
negb (Rle_bool x y) = Rlt_bool y x.
Theorem negb_Rle_bool :
forall x y,
negb (Rlt_bool x y) = Rle_bool y x.
Theorem Rlt_bool_true :
forall x y,
(x < y)%R -> Rlt_bool x y = true.
Theorem Rlt_bool_false :
forall x y,
(y <= x)%R -> Rlt_bool x y = false.