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    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.

    Set 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 (A : Type) (P : set (R * R)).

    Definition
    OuP

    OuP : forall [A : Type], set (R * R) -> (A -> R * R -> R) -> (R * R -> R) -> Set OuP is not universe polymorphic Arguments OuP [A]%_type_scope P%_classical_set_scope (f g)%_function_scope OuP is transparent Expands to: Constant mathcomp.analysis_stdlib.showcase.uniform_bigO.OuP Declared in library mathcomp.analysis_stdlib.showcase.uniform_bigO, line 28, characters 11-14

    (f : A -> R * R -> R) (g : R * R -> R) :=
      { alp : R & { C : R |
      0 < alp /\ 0 < C /\
      forall X : A, forall dX : R * R,
      sqrt (Rsqr (fst dX) + Rsqr (snd dX)) < alp -> P dX ->
      Rabs (f X dX) <= C * Rabs (g dX)}}.


    Let normedR2 : normedModType _ := (R^o * R^o)%type.

    Definition
    OuPex

    OuPex : forall [A : Type], set (R * R) -> (A -> R * R -> R^o) -> (R * R -> R^o) -> Prop OuPex is not universe polymorphic Arguments OuPex [A]%_type_scope P%_classical_set_scope (f g)%_function_scope OuPex is transparent Expands to: Constant mathcomp.analysis_stdlib.showcase.uniform_bigO.OuPex Declared in library mathcomp.analysis_stdlib.showcase.uniform_bigO, line 39, characters 11-16

    (f : A -> R * R -> R^o) (g : R * R -> R^o) :=
      exists2 alp, 0 < alp & exists2 C, 0 < C &
        forall X, forall dX : normedR2,
        `|dX| < alp -> P dX -> `|f X dX| <= C * `|g dX|.

    Lemma ler_norm2 (x : normedR2) :
      `|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.

    Lemma OuP_to_ex f g : OuP f g -> OuPex f g.
    Proof.
    move=> [_ [_ [/posnumP[a] [/posnumP[C] fOg]]]].
    exists (a%:num / Num.sqrt 2) => //; exists C%:num => // x dx ltdxa Pdx.
    apply: fOg; move: ltdxa; rewrite ltr_pdivlMr //; apply: le_lt_trans.
    by rewrite mulrC; have /andP[] := ler_norm2 dx.
    Qed.

    Lemma Ouex_to_P f g : OuPex f g -> OuP f g.
    Proof.
    move=> /exists2P /getPex; set Q := fun a => _ /\ _ => - [lt0getQ].
    move=> /exists2P /getPex; set R := fun C => _ /\ _ => - [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

    OuO : forall [A : Type], set (R * R) -> (A -> R * R -> R^o) -> (R * R -> R^o) -> Prop OuO is not universe polymorphic Arguments OuO [A]%_type_scope P%_classical_set_scope (f g)%_function_scope OuO is transparent Expands to: Constant mathcomp.analysis_stdlib.showcase.uniform_bigO.OuO Declared in library mathcomp.analysis_stdlib.showcase.uniform_bigO, line 80, characters 11-14

    (f : A -> R * R -> R^o) (g : R * R -> R^o) :=
      (fun x => f x.1 x.2) =O_ (filter_prod [set setT]%classic
      (within P (nbhs (0%R:R^o, 0%R:R^o)))) (fun x => g x.2).

    Lemma OuP_to_O f g : OuP f g -> OuO f g.
    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.

    Lemma OuO_to_P f g : OuO f g -> OuP f g.
    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.

    End UniformBigO.
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