Commit a85b73a8 by Guillaume Melquiond

Rnd_NA_monotone

parent 221e165b
 ... ... @@ -10,6 +10,24 @@ apply Rge_minus. now apply Rle_ge. Qed. Lemma Rabs_eq_Rabs : forall x y : R, Rabs x = Rabs y -> x = y \/ x = Ropp y. Proof. intros x y H. unfold Rabs in H. destruct (Rcase_abs x) as [_|_]. assert (H' := f_equal Ropp H). rewrite Ropp_involutive in H'. rewrite H'. destruct (Rcase_abs y) as [_|_]. left. apply Ropp_involutive. now right. rewrite H. now destruct (Rcase_abs y) as [_|_] ; [right|left]. Qed. Section Z2R. Fixpoint P2R (p : positive) := ... ...
 ... ... @@ -448,6 +448,68 @@ intros x Hx. now apply Rnd_N_pt_involutive with F. Qed. Theorem Rnd_NA_pt_monotone : forall F : R -> Prop, F 0 -> forall x y f g : R, Rnd_NA_pt F x f -> Rnd_NA_pt F y g -> x <= y -> f <= g. Proof. intros F HF x y f g (Hf,Hx) (Hg,Hy) [Hxy|Hxy]. now apply Rnd_N_pt_monotone with F x y. apply Req_le. rewrite <- Hxy in Hg, Hy. clear y Hxy. assert (K: f = g \/ f = -g). apply Rabs_eq_Rabs. apply Rle_antisym. now apply Hy. now apply Hx. destruct K as [K|K]. exact K. rewrite K. rewrite K in Hf. clear f Hx Hy K. unfold Rnd_N_pt in Hf, Hg. assert (L: g + x = g - x \/ g + x = x - g). rewrite <- (Ropp_minus_distr g x). apply Rabs_eq_Rabs. rewrite <- Rabs_Ropp. rewrite Ropp_plus_distr. fold (-g - x). apply Rle_antisym. now apply Hf. now apply Hg. destruct L as [L|L]. assert (g = 0). apply Rnd_N_pt_involutive with F. replace 0 with x. exact Hg. apply Rmult_eq_reg_l with 2. rewrite Rmult_0_r. rewrite <- (Rminus_diag_eq _ _ L). ring. now apply (Z2R_neq 2 0). exact HF. rewrite H. apply Ropp_0. apply Rplus_eq_reg_l with x. fold (x - g). rewrite <- L. apply Rplus_comm. Qed. Theorem Rnd_NA_monotone : forall F : R -> Prop, F 0 -> forall rnd : R -> R, Rnd_NA F rnd -> MonotoneP rnd. Proof. intros F rnd Hr x y Hxy. now apply Rnd_NA_pt_monotone with F. Qed. Theorem Rnd_NA_pt_involutive : forall F : R -> Prop, forall x f : R, ... ...
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