Added theorems about reflexivity of standard predicates.

src/Flocq_rnd_prop.v

@@ -201,6 +201,18 @@ rewrite Ropp_involutive.
 apply H1.
 Qed.
 
+Theorem Rnd_DN_pt_refl :
+  forall F : R -> Prop,
+  forall x : R, F x ->
+  Rnd_DN_pt F x x.
+Proof.
+intros F x Hx.
+repeat split.
+exact Hx.
+apply Rle_refl.
+now intros.
+Qed.
+
 Theorem Rnd_DN_pt_idempotent :
   forall F : R -> Prop,
   forall x f : R,
@@ -229,6 +241,18 @@ intros x Hx.
 now apply Rnd_DN_pt_idempotent with (2 := Hx).
 Qed.
 
+Theorem Rnd_UP_pt_refl :
+  forall F : R -> Prop,
+  forall x : R, F x ->
+  Rnd_UP_pt F x x.
+Proof.
+intros F x Hx.
+repeat split.
+exact Hx.
+apply Rle_refl.
+now intros.
+Qed.
+
 Theorem Rnd_UP_pt_idempotent :
   forall F : R -> Prop,
   forall x f : R,
@@ -551,6 +575,20 @@ rewrite Hxy.
 apply Rle_refl.
 Qed.
 
+Theorem Rnd_N_pt_refl :
+  forall F : R -> Prop,
+  forall x : R, F x ->
+  Rnd_N_pt F x x.
+Proof.
+intros F x Hx.
+repeat split.
+exact Hx.
+intros g _.
+unfold Rminus at 1.
+rewrite Rplus_opp_r, Rabs_R0.
+apply Rabs_pos.
+Qed.
+
 Theorem Rnd_N_pt_idempotent :
   forall F : R -> Prop,
   forall x f : R,
@@ -884,6 +922,20 @@ intros F rnd Hr x y Hxy.
 now apply Rnd_NA_pt_monotone with F.
 Qed.
 
+Theorem Rnd_NA_pt_refl :
+  forall F : R -> Prop,
+  forall x : R, F x ->
+  Rnd_NA_pt F x x.
+Proof.
+intros F x Hx.
+split.
+now apply Rnd_N_pt_refl.
+intros f Hxf.
+apply Req_le.
+apply f_equal.
+now apply Rnd_N_pt_idempotent with (1 := Hxf).
+Qed.
+
 Theorem Rnd_NA_pt_idempotent :
   forall F : R -> Prop,
   forall x f : R,