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upd misc/uniform_bigO.v #1142

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Jan 9, 2024
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upd misc/uniform_bigO.v #1142

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11 changes: 5 additions & 6 deletions theories/misc/uniform_bigO.v
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Original file line number Diff line number Diff line change
Expand Up @@ -2,8 +2,8 @@
Require Import Reals.
From Coq Require Import ssreflect ssrfun ssrbool.
From mathcomp Require Import ssrnat eqtype choice fintype bigop order ssralg ssrnum.
Require Import boolp reals Rstruct Rbar.
Require Import classical_sets posnum topology normedtype landau.
From mathcomp Require Import boolp reals Rstruct ereal.
From mathcomp Require Import classical_sets signed topology normedtype landau.

Set Implicit Arguments.
Unset Strict Implicit.
Expand Down Expand Up @@ -55,6 +55,7 @@ rewrite !Rsqr_pow2 !RpowE; apply/andP; split.
wlog lex12 : x / (`|x.1| <= `|x.2|).
move=> ler_norm; case: (lerP `|x.1| `|x.2|) => [/ler_norm|] //.
rewrite lt_leAnge => /andP [lex21 _].
rewrite RplusE.
by rewrite addrC [`|_|]maxC (ler_norm (x.2, x.1)).
rewrite [`|_|]max_r // -[X in X * _]ger0_norm // -normrM.
rewrite -sqrtr_sqr ler_wsqrtr // exprMn sqr_sqrtr // mulr_natl mulr2n ler_add2r.
Expand Down Expand Up @@ -90,14 +91,12 @@ Proof.
move=> /OuP_to_ex [_/posnumP[a] [_/posnumP[C] fOg]].
apply/eqOP; near=> k; near=> x; apply: le_trans (fOg _ _ _ _) _; last 2 first.
- by near: x; exists (setT, P); [split=> //=; apply: withinT|move=> ? []].
- rewrite ler_pmul => //; near: k; exists C%:num; split.
exact: posnum_real.
by move=> ?; rewrite lt_leAnge => /andP[].
- by rewrite ler_pmul.
- near: x; exists (setT, ball (0 : R^o * R^o) a%:num).
by split=> //=; rewrite /within; near=> x =>_; near: x; apply: nbhsx_ballx.
move=> x [_ [/=]]; rewrite -ball_normE /= distrC subr0 distrC subr0.
by move=> ??; rewrite lt_maxl; apply/andP.
Grab Existential Variables. all: end_near. Qed.
Unshelve. all: by end_near. Qed.

Lemma OuO_to_P f g : OuO f g -> OuP f g.
Proof.
Expand Down
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