 ### Gave a proper name and place to theorem toto.

parent 416d16fe
 ... ... @@ -55,39 +55,6 @@ Definition Rnd_ZR_pt (F : R -> Prop) (x f : R) := Definition Rnd_ZR (F : R -> Prop) (rnd : R -> R) := forall x : R, Rnd_ZR_pt F x (rnd x). Theorem toto : forall (F : R -> Prop) (rnd: R-> R), Rnd_ZR F rnd -> forall x : R, (Rabs (rnd x) <= Rabs x)%R. Proof. intros F rnd H x. assert (F 0%R). replace 0%R with (rnd 0%R). eapply H. apply Rle_refl. destruct (H 0%R) as (H1, H2). apply Rle_antisym. apply H1. apply Rle_refl. apply H2. apply Rle_refl. (* . *) destruct (Rle_or_lt 0 x). (* positive *) rewrite Rabs_right. rewrite Rabs_right; auto with real. now apply (proj1 (H x)). apply Rle_ge. now apply (proj1 (H x)). (* negative *) rewrite Rabs_left1. rewrite Rabs_left1 ; auto with real. apply Ropp_le_contravar. apply (proj2 (H x)). auto with real. apply (proj2 (H x)) ; auto with real. Qed. (* property of being a rounding to nearest *) Definition Rnd_N_pt (F : R -> Prop) (x f : R) := F f /\ ... ...
 ... ... @@ -325,6 +325,39 @@ apply Hd. apply Hu. Qed. Theorem Rnd_ZR_abs : forall (F : R -> Prop) (rnd: R-> R), Rnd_ZR F rnd -> forall x : R, (Rabs (rnd x) <= Rabs x)%R. Proof. intros F rnd H x. assert (F 0%R). replace 0%R with (rnd 0%R). eapply H. apply Rle_refl. destruct (H 0%R) as (H1, H2). apply Rle_antisym. apply H1. apply Rle_refl. apply H2. apply Rle_refl. (* . *) destruct (Rle_or_lt 0 x). (* positive *) rewrite Rabs_right. rewrite Rabs_right; auto with real. now apply (proj1 (H x)). apply Rle_ge. now apply (proj1 (H x)). (* negative *) rewrite Rabs_left1. rewrite Rabs_left1 ; auto with real. apply Ropp_le_contravar. apply (proj2 (H x)). auto with real. apply (proj2 (H x)) ; auto with real. Qed. Theorem Rnd_N_pt_DN_or_UP : forall F : R -> Prop, forall x f : R, ... ...
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