alternative proof of gauss integral #1584

IshiguroYoshihiro · 2025-04-22T21:52:21Z

Motivation for this change

I added an alternative proof of gauss integral and placed it in theories/showcase.
This is an experimental approach to evaluating improper integral.
I have already separate reusable lemmas and made PRs ( #1579 is under review now), so this file intended to showcases the difference of proof strategy.
In contrast to the existing proof, this version mainly relies on lemmas related to integration over unbounded intervals.

Checklist

added corresponding entries in CHANGELOG_UNRELEASED.md

added corresponding documentation in the headers

Reference: How to document

Merge policy

As a rule of thumb:

PRs with several commits that make sense individually and that
all compile are preferentially merged into master.
PRs with disorganized commits are very likely to be squash-rebased.

Reminder to reviewers

Read this Checklist
Put a milestone if possible
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hoheinzollern

Please consider the comments above.

theories/showcase/gauss_integral_alternative.v

hoheinzollern · 2025-05-12T05:03:17Z

CHANGELOG_UNRELEASED.md

 ### Added

+- in `theories/showcase`
+  + add new file `gauss_integral_alternative.v`


The new lemmas and definitions should also be added in the changelog

Indeed. I had a misunderstanding when I wrote it.
I will correct it.

I rewrited changelog as describe what lemmas are included in gauss_integral_unbounded.v.
I'm glad if you check. (PR is reopened #1646 )

theories/showcase/gauss_integral_alternative.v

hoheinzollern · 2025-05-12T07:05:04Z

theories/showcase/gauss_integral_alternative.v

+
+Lemma max_y_ge0 : (0 <= max_y)%R. Proof. by rewrite mulr_ge0. Qed.
+
+Definition helper (x : R) := (2 * x * expR (- x ^+ 2))%R.


This and the next definitions and lemmas are non-descriptive. I recommend finding a better name.

I renamed this function ndgauss meaning derivative of - gauss_fun.

theories/showcase/gauss_integral_alternative.v

hoheinzollern · 2025-05-12T07:38:34Z

theories/showcase/gauss_integral_alternative.v

+
+Let dudx x0 y0 : R := (u^~ y0)^`() x0.
+
+Lemma int0yu_fin_num x : \int[mu]_(x0 in `[0%R, +oo[) (u x x0)%:E \is a fin_num.


Should this be using int0yu?

I'm not sure how to resolve this.
When I use int0yu in int0yu_fin_num as

Lemma int0yu_fin_num x : (int0yu x)%:E \is a fin_num.

Is this the expected change?
I should unfold the notation Rintegral in the proof of above but I feel I should avoid to unfold because Rintegral is hidden by the notation.

hoheinzollern

Looking good to me, please consider the alternate file names and merge.

theories/showcase/gauss_integral_alternative.v

affeldt-aist · 2025-06-17T08:41:51Z

theories/showcase/gauss_integral_alternative.v

+- by rewrite {}/ballx {}/c; near: e; exact: u_dominates.
+Unshelve. all: end_near. Qed.
+
+Lemma rc_int0yu0 : int0yu x @[x --> 0^'+] --> int0yu 0.


Suggested change

Lemma rc_int0yu0 : int0yu x @[x --> 0^'+] --> int0yu 0.

Lemma cvg_int0yu : int0yu x @[x --> 0^'+] --> int0yu 0.

I renamed it to rcvg0_int0yu.
rcvg0 means "converge at 0 from the right".

theories/showcase/gauss_integral_alternative.v

affeldt-aist · 2025-06-17T08:45:54Z

theories/showcase/gauss_integral_alternative.v

+- by rewrite {}/G; near: e; exact: u_dominates.
+Unshelve. end_near. Qed.
+
+Let dint0yuE_substep2 x (x0 : (0 < x)%R) :


Suggested change

Let dint0yuE_substep2 x (x0 : (0 < x)%R) :

Let integral_pderive_u x : (0 < x)%R ->

I renamed to derive1_int0yuE_subproof2.
Indeed more descriptive name is better.
However, I haven't found any other application of this lemma so I'm putting renaming to integral_pderive_u off now.
Should I rename to it?

affeldt-aist · 2025-06-17T08:47:40Z

theories/showcase/gauss_integral_alternative.v

+by rewrite mulrDr mulr1 mulNr -exprMn expRD.
+Qed.
+
+Let der_mulrE (x : R) : (fun z => (x * z)%R)^`() = cst x.


try to use derive1Mr in this proof

Since derive1Mr can't apply directly at where der_mulrE is used, I rewrited the proof of der_mulrE with derive1Mr.

* generic lemmas from the prob_lang PR Co-authored-by: Cyril Cohen <cyril.cohen@inria.fr>

…1592)

In order to not forget it when adding a new release to opam repo.

Analysis is not yet compatible with rocq (without the coq-core compat shim) - Calling rocq didn't work with Coq 8.19 and 8.20 - Compile without coq shim on Rocq 9.0

* lemmas about measurability of comparison * measurability of poweR Co-authored-by: Alessandro Bruni <alessandro.bruni@gmail.com>

* add is_derive1_powR

--------- Co-authored-by: Cyril Cohen <CohenCyril@users.noreply.github.com> Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

* theory of essential supremum Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp> Co-authored-by: Cyril Cohen <cohen@crans.org> Co-authored-by: Pierre Roux <pierre.roux@onera.fr> Co-authored-by: Takafumi Saikawa <tscompor@gmail.com>

@yosakaon

* fixes math-comp#1632 Co-authored-by: @yosakaon

* change the convention for the convex combination operation --------- Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

* repair structure names in tvs.v * repair uniform algebraic structures * fix Zmodules

- this commit adds an implementation of the inverse for extended reals, `inve` a total involutive inversion function on `\bar R`, denoted `^-1` in the `ereal_scope` coinciding with `x^-1%R` when `x != 0` but such that `0^-1 = +oo` and `-oo^-1 = -oo`, - notation `x / y` in `ereal_scope` for `x / y = x * y^-1`/, - lemmas `inver`, `inveP`, `fine_invr`, `inve0`, `inve1`, `invey`, `invey`, `inveNy`, `inveK`, `invr_inj`, `inveN`, `inve_eq0`, `inve_ge0`, `inve_gt0`, - a predicate `inveM_def` with notation `x *^-1? y` defining a sufficient condition for the inverse and product to commute, - it changes `mule` to have special cases optimizing computation for +oo and -oo adn `mule_def` has been rewritten to optimize computation in several cases. - it adds a compatibility lemma `mule_defE` to bridge the former definition of `mule_def` with the new one. Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp> Co-authored-by: Pierre Roux <pierre.roux@onera.fr>

* Chore: 📝 🔧 `make html` to make the html documentation using Rocqnavi This change has been inspire in the repository: https://github.com/CoqHott/logrel-coq

* alias for pseudometric and factory for normed module --------- Co-authored-by: Quentin Vermande <quentin.vermande@ens.fr>

* is_deriveX, etc. --------- Co-authored-by: Quentin VERMANDE <quentin.vermande@orange.fr>

Co-authored-by: IshiguroYoshihiro <jb.15r.1213@s.thers.ac.jp>

* expR_ge1Dxn

- the set of tuples is measurable - tnth, cons, behead are measurable functions Co-authored-by: Alessandro Bruni <brun@itu.dk> Co-authored-by: Cyril Cohen <cohen@crans.org> Co-authored-by: Pierre Roux <pierre.roux@onera.fr> Co-authored-by: Takafumi Saikawa <tscompor@gmail.com>

--------- Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

IshiguroYoshihiro · 2025-06-20T04:59:11Z

I'm sorry, I accidentally pushed an unsuccessfully rebased commit.
I will fix it immediately.

IshiguroYoshihiro · 2025-06-20T16:14:07Z

@hoheinzollern and @affeldt-aist , I've opened a clearer PR as #1646 .
Could you review that one instead? Thanks.

IshiguroYoshihiro added 3 commits April 22, 2025 19:33

add gauss_integral_alternative.v

2faee91

CHANGELOG

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cleaning

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IshiguroYoshihiro changed the title ~~Gauss integral alternative 20250422~~ alternative proof of gauss integral Apr 22, 2025

hoheinzollern self-requested a review April 23, 2025 04:06

IshiguroYoshihiro added 2 commits April 24, 2025 12:16

refine dependencies

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minor cleaning

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affeldt-aist modified the milestones: 1.11.0, 1.12.0 May 1, 2025

hoheinzollern suggested changes May 12, 2025

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IshiguroYoshihiro added 3 commits June 3, 2025 18:53

some Let -> Lemma in gauss_integral_proof

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remove deplicated definition

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add changelog

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hoheinzollern self-requested a review June 17, 2025 06:31

hoheinzollern approved these changes Jun 17, 2025

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affeldt-aist reviewed Jun 17, 2025

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affeldt-aist and others added 9 commits June 20, 2025 13:42

fixes math-comp#1586 (math-comp#1587)

af1417d

typos (math-comp#1588)

47b2359

Adapt to math-comp/math-comp#1354

0e57c04

generic lemmas from the prob_lang PR (math-comp#1583)

130e0a9

* generic lemmas from the prob_lang PR Co-authored-by: Cyril Cohen <cyril.cohen@inria.fr>

lemma extracted from the sampling branch to help rebasing (math-comp#…

4c50e90

…1592)

[opam] Add upper bound on MC

abc05b1

In order to not forget it when adding a new release to opam repo.

Fix coqc in Makefile (math-comp#1596)

54f107c

Analysis is not yet compatible with rocq (without the coq-core compat shim) - Calling rocq didn't work with Coq 8.19 and 8.20 - Compile without coq shim on Rocq 9.0

Measurable realfun 20250430 (math-comp#1597)

a2cd995

* lemmas about measurability of comparison * measurability of poweR Co-authored-by: Alessandro Bruni <alessandro.bruni@gmail.com>

is_derive1_powR (math-comp#1566)

c258eeb

* add is_derive1_powR

affeldt-aist and others added 26 commits June 20, 2025 13:43

changelog for version 1.11.0

eb95ed6

almost everywhere equality (math-comp#1600)

fc0d863

--------- Co-authored-by: Cyril Cohen <CohenCyril@users.noreply.github.com> Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

cleanup README / INSTALL instructions (math-comp#1619)

bc41f7b

[CI] Update

06a90c7

[CI] Disactivate currently broken generate_doc

357bf60

Essential supremum (math-comp#1606)

f950195

* theory of essential supremum Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp> Co-authored-by: Cyril Cohen <cohen@crans.org> Co-authored-by: Pierre Roux <pierre.roux@onera.fr> Co-authored-by: Takafumi Saikawa <tscompor@gmail.com>

CI: Rocqnavi released version (math-comp#1614)

5c0041e

Minor cleanup (math-comp#1625)

4eb694a

Fix the affiliation of Kazuhiko Sakaguchi in AUTHORS.md

fc59bb8

adapt to rocq#19987 (math-comp#1532)

b152c2e

fixes math-comp#1632 (math-comp#1633)

7abc6e0

* fixes math-comp#1632 Co-authored-by: @yosakaon

Conv convention (math-comp#1630)

2ccacda

* change the convention for the convex combination operation --------- Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

Repair structure names in tvs.v (math-comp#1631)

eb1a01d

* repair structure names in tvs.v * repair uniform algebraic structures * fix Zmodules

📝 🔧 make html to make the html documentation (math-comp#1622)

07b94c9

* Chore: 📝 🔧 `make html` to make the html documentation using Rocqnavi This change has been inspire in the repository: https://github.com/CoqHott/logrel-coq

pseudometric alias (math-comp#1628)

9408d90

* alias for pseudometric and factory for normed module --------- Co-authored-by: Quentin Vermande <quentin.vermande@ens.fr>

is_deriveX: is that a generalization? (math-comp#1637)

2f6ebac

* is_deriveX, etc. --------- Co-authored-by: Quentin VERMANDE <quentin.vermande@orange.fr>

exponential distribution (math-comp#1635)

3c941f0

Co-authored-by: IshiguroYoshihiro <jb.15r.1213@s.thers.ac.jp>

expR_ge1Dxn (math-comp#1638)

7473552

* expR_ge1Dxn

lemmas for integrals on increasing set sequences (math-comp#1579)

b7b5a3d

--------- Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>

CHANGELOG

66709f2

add changelog

d2ea14b

rename

713f548

fix

58a16fc

cleaning

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IshiguroYoshihiro closed this Jun 20, 2025

affeldt-aist mentioned this pull request Jul 1, 2025

Alternative Proof of Gauss Integral with improper integral #1646

Open

2 tasks


		Lemma max_y_ge0 : (0 <= max_y)%R. Proof. by rewrite mulr_ge0. Qed.

		Definition helper (x : R) := (2 * x * expR (- x ^+ 2))%R.


		Let dudx x0 y0 : R := (u^~ y0)^`() x0.

		Lemma int0yu_fin_num x : \int[mu]_(x0 in `[0%R, +oo[) (u x x0)%:E \is a fin_num.

	Lemma rc_int0yu0 : int0yu x @[x --> 0^'+] --> int0yu 0.
	Lemma cvg_int0yu : int0yu x @[x --> 0^'+] --> int0yu 0.

	Let dint0yuE_substep2 x (x0 : (0 < x)%R) :
	Let integral_pderive_u x : (0 < x)%R ->

alternative proof of gauss integral #1584

alternative proof of gauss integral #1584

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IshiguroYoshihiro commented Apr 22, 2025

Motivation for this change

Checklist

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IshiguroYoshihiro commented Jun 20, 2025 •

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