Module mathcomp.analysis_stdlib.showcase.uniform_bigO
From Stdlib Require Import Reals.From Corelib Require Import ssreflect ssrfun ssrbool.
From mathcomp Require Import ssrnat eqtype choice fintype bigop order ssralg ssrnum.
From mathcomp Require Import boolp reals Rstruct_topology ereal classical_sets.
From mathcomp Require Import interval_inference topology normedtype landau.
Unset SsrOldRewriteGoalsOrder.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Local Open Scope ring_scope.
Section UniformBigO.
This section shows how we can formalize the uniform bigO from:
Boldo, Clément, Filliâtre, Mayero, Melquiond, Weis,
Wave Equation Numerical Resolution: A Comprehensive Mechanized Proof of a
C Program, Journal of Automated Reasoning 2013. The corresponding source
code is here: http://fost.saclay.inria.fr/coq_total/BigO.html.
Context ( : Type) ( : set (R * R)).
Definition
OuP
Source code
( : A -> R * R -> R) ( : R * R -> R) :=Source code
{
alp
Source code
: R & { : R |Source code
0 < alp /\ 0 < C /\
forall : A, forall : R * R,
sqrt (Rsqr (fst dX) + Rsqr (snd dX)) < alp -> P dX ->
Rabs (f X dX) <= C * Rabs (g dX)}}.
Let
normedR2
Source code
: normedModType _ := (R^o * R^o)%type.Source code
Definition
OuPex
Source code
( : A -> R * R -> R^o) ( : R * R -> R^o) :=Source code
exists2
alp
Source code
, 0 < alp & exists2 , 0 < C &Source code
forall , forall : normedR2,
`|dX| < alp -> P dX -> `|f X dX| <= C * `|g dX|.
Lemma
ler_norm2
Source code
( : normedR2) :Source code
`|x| <= sqrt (Rsqr (fst x) + Rsqr (snd x)) <= Num.sqrt 2 * `|x|.
Proof.
rewrite RsqrtE !Rsqr_pow2 !RpowE; apply/andP; split.
by rewrite ge_max; apply/andP; split;
rewrite -[`|_|]sqrtr_sqr ler_wsqrtr // (lerDl, lerDr) sqr_ge0.
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 lerD2r.
rewrite -[leLHS]ger0_norm ?sqr_ge0 // -[leRHS]ger0_norm ?sqr_ge0 //.
by rewrite !normrX lerXn2r // nnegrE normr_ge0.
Qed.
by rewrite ge_max; apply/andP; split;
rewrite -[`|_|]sqrtr_sqr ler_wsqrtr // (lerDl, lerDr) sqr_ge0.
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 lerD2r.
rewrite -[leLHS]ger0_norm ?sqr_ge0 // -[leRHS]ger0_norm ?sqr_ge0 //.
by rewrite !normrX lerXn2r // nnegrE normr_ge0.
Qed.
Lemma
OuP_to_ex
Source code
: OuP f g -> OuPex f g.Source code
Proof.
Lemma
Ouex_to_P
Source code
: OuPex f g -> OuP f g.Source code
Proof.
move=> /exists2P /getPex; set Q := fun => _ /\ _ => - [lt0getQ].
move=> /exists2P /getPex; set R := fun => _ /\ _ => - [lt0getR fOg].
apply: existT (get Q) _; apply: exist (get R) _; split=> //; split => //.
move=> x dx ltdxgetQ; apply: fOg; apply: le_lt_trans ltdxgetQ.
by have /andP [] := ler_norm2 dx.
Qed.
move=> /exists2P /getPex; set R := fun => _ /\ _ => - [lt0getR fOg].
apply: existT (get Q) _; apply: exist (get R) _; split=> //; split => //.
move=> x dx ltdxgetQ; apply: fOg; apply: le_lt_trans ltdxgetQ.
by have /andP [] := ler_norm2 dx.
Qed.
Definition
OuO
Source code
( : A -> R * R -> R^o) ( : R * R -> R^o) :=Source code
(fun => f x.1 x.2) =O_ (filter_prod [set setT]%classic
(within P (nbhs (0%R:R^o, 0%R:R^o)))) (fun => g x.2).
Lemma
OuP_to_O
Source code
: OuP f g -> OuO f g.Source code
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=> ? []].
- by rewrite ler_pM.
- 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 gt_max; apply/andP.
Unshelve. all: by end_near. Qed.
apply/eqOP; near=> k; near=> x; apply: le_trans (fOg _ _ _ _) _; last 2 first.
- by near: x; exists (setT, P); [split=> //=; apply: withinT|move=> ? []].
- by rewrite ler_pM.
- 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 gt_max; apply/andP.
Unshelve. all: by end_near. Qed.
Lemma
OuO_to_P
Source code
: OuO f g -> OuP f g.Source code
Proof.
move=> fOg; apply/Ouex_to_P; move: fOg => /eqOP [k [kreal hk]].
have /hk [Q [->]] : k < maxr 1 (k + 1) by rewrite lt_max ltrDl orbC ltr01.
move=> [R [[_/posnumP[e1] Re1] [_/posnumP[e2] Re2]] sRQ] fOg.
exists (minr e1%:num e2%:num) => //.
exists (maxr 1 (k + 1)); first by rewrite lt_max ltr01.
move=> x dx dxe Pdx; apply: (fOg (x, dx)); split=> //=.
move: dxe; rewrite gt_max !lt_min => /andP[/andP [dxe11 _] /andP [_ dxe22]].
by apply/sRQ => //; split; [apply/Re1|apply/Re2]; rewrite /= distrC subr0.
Qed.
have /hk [Q [->]] : k < maxr 1 (k + 1) by rewrite lt_max ltrDl orbC ltr01.
move=> [R [[_/posnumP[e1] Re1] [_/posnumP[e2] Re2]] sRQ] fOg.
exists (minr e1%:num e2%:num) => //.
exists (maxr 1 (k + 1)); first by rewrite lt_max ltr01.
move=> x dx dxe Pdx; apply: (fOg (x, dx)); split=> //=.
move: dxe; rewrite gt_max !lt_min => /andP[/andP [dxe11 _] /andP [_ dxe22]].
by apply/sRQ => //; split; [apply/Re1|apply/Re2]; rewrite /= distrC subr0.
Qed.
End UniformBigO.