Commit d8afcec3 by Guillaume Melquiond

### s/involutive/idempotent/

parent a85b73a8
 ... ... @@ -23,12 +23,12 @@ Definition MonotoneP (rnd : R -> R) := (x <= y)%R -> (rnd x <= rnd y)%R. Definition InvolutiveP (F : R -> Prop) (rnd : R -> R) := Definition IdempotentP (F : R -> Prop) (rnd : R -> R) := (forall x : R, F (rnd x)) /\ (forall x : R, F x -> rnd x = x). Definition Rounding_for_Format (F : R -> Prop) (rnd : R -> R) := MonotoneP rnd /\ InvolutiveP F rnd. MonotoneP rnd /\ IdempotentP F rnd. (* unbounded floating-point format *) Definition FLX_format (prec : Z) (x : R) := ... ...
 ... ... @@ -119,7 +119,7 @@ intros F rnd Hr x y Hxy. now eapply Rnd_DN_pt_monotone. Qed. Theorem Rnd_DN_pt_involutive : Theorem Rnd_DN_pt_idempotent : forall F : R -> Prop, forall x f : R, Rnd_DN_pt F x f -> F x -> ... ... @@ -133,18 +133,18 @@ exact Hx. apply Rle_refl. Qed. Theorem Rnd_DN_involutive : Theorem Rnd_DN_idempotent : forall F : R -> Prop, forall rnd : R -> R, Rnd_DN F rnd -> InvolutiveP F rnd. IdempotentP F rnd. Proof. intros F rnd Hr. split. intros. eapply Hr. intros x Hx. now apply Rnd_DN_pt_involutive with (2 := Hx). now apply Rnd_DN_pt_idempotent with (2 := Hx). Qed. Theorem Rnd_UP_pt_monotone : ... ... @@ -169,7 +169,7 @@ intros F rnd Hr x y Hxy. now eapply Rnd_UP_pt_monotone. Qed. Theorem Rnd_UP_pt_involutive : Theorem Rnd_UP_pt_idempotent : forall F : R -> Prop, forall x f : R, Rnd_UP_pt F x f -> F x -> ... ... @@ -183,18 +183,18 @@ apply Rle_refl. exact Hx1. Qed. Theorem Rnd_UP_involutive : Theorem Rnd_UP_idempotent : forall F : R -> Prop, forall rnd : R -> R, Rnd_UP F rnd -> InvolutiveP F rnd. IdempotentP F rnd. Proof. intros F rnd Hr. split. intros. eapply Hr. intros x Hx. now apply Rnd_UP_pt_involutive with (2 := Hx). now apply Rnd_UP_pt_idempotent with (2 := Hx). Qed. Theorem Rnd_DN_pt_le_rnd : ... ... @@ -414,7 +414,7 @@ rewrite Hxy. apply Rle_refl. Qed. Theorem Rnd_N_pt_involutive : Theorem Rnd_N_pt_idempotent : forall F : R -> Prop, forall x f : R, Rnd_N_pt F x f -> F x -> ... ... @@ -434,18 +434,18 @@ apply Rabs_R0. apply Rabs_pos. Qed. Theorem Rnd_N_involutive : Theorem Rnd_N_idempotent : forall F : R -> Prop, forall rnd : R -> R, Rnd_N F rnd -> InvolutiveP F rnd. IdempotentP F rnd. Proof. intros F rnd Hr. split. intros x. eapply Hr. intros x Hx. now apply Rnd_N_pt_involutive with F. now apply Rnd_N_pt_idempotent with F. Qed. Theorem Rnd_NA_pt_monotone : ... ... @@ -482,7 +482,7 @@ now apply Hf. now apply Hg. destruct L as [L|L]. assert (g = 0). apply Rnd_N_pt_involutive with F. apply Rnd_N_pt_idempotent with F. replace 0 with x. exact Hg. apply Rmult_eq_reg_l with 2. ... ... @@ -510,28 +510,28 @@ intros F rnd Hr x y Hxy. now apply Rnd_NA_pt_monotone with F. Qed. Theorem Rnd_NA_pt_involutive : Theorem Rnd_NA_pt_idempotent : forall F : R -> Prop, forall x f : R, Rnd_NA_pt F x f -> F x -> f = x. Proof. intros F x f (Hf,_) Hx. now apply Rnd_N_pt_involutive with F. now apply Rnd_N_pt_idempotent with F. Qed. Theorem Rnd_NA_involutive : Theorem Rnd_NA_idempotent : forall F : R -> Prop, forall rnd : R -> R, Rnd_NA F rnd -> InvolutiveP F rnd. IdempotentP F rnd. Proof. intros F rnd Hr. split. intros x. eapply Hr. intros x Hx. now apply Rnd_NA_pt_involutive with F. now apply Rnd_NA_pt_idempotent with F. Qed. Theorem Rnd_0 : ... ... @@ -603,7 +603,7 @@ exists (fun x => match Rle_dec 0 x with assert (K : Rounding_for_Format F rnd). split. now apply Rnd_DN_monotone with F. now apply Rnd_DN_involutive. now apply Rnd_DN_idempotent. intros x. destruct (Rle_dec 0 x) as [Hx|Hx] ; split. (* positive or zero *) ... ... @@ -784,10 +784,10 @@ destruct (Hdn (-x)%R) as (yn,(H1,(H2,H3))). exists (-yn)%R. repeat split. now apply Hneg. rewrite <- (Ropp_involutive x). rewrite <- (Ropp_idempotent x). now apply Ropp_le_contravar. intros z Hz Hxz. rewrite <- (Ropp_involutive z). rewrite <- (Ropp_idempotent z). apply Ropp_le_contravar. apply H3. now apply Hneg. ... ...
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