Commit d8afcec3 authored by Guillaume Melquiond's avatar Guillaume Melquiond
Browse files

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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