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Tietze's theorem #971

Merged
merged 16 commits into from
Jul 28, 2023
Merged

Tietze's theorem #971

merged 16 commits into from
Jul 28, 2023

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zstone1
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@zstone1 zstone1 commented Jul 7, 2023

Motivation for this change

In the apparently never-ending results needed for fundamental theorem, here's Tietze's Extension theorem. We're continuing along #965 to show that continuous functions are dense in L1.

A few things to note:

  1. Normally, Tietze's theorem assumes X is normal. Here, we assume it has urysohn extensions. Urysohn's lemma (Urysohn's Lemma #900) is the proof that these are equivalent. We could directly show that pseudometric spaces have urysohn extensions, once we get that far if we had to. Or we can merge in the urysohns' lemma PR, move to HB, and just do it that way.
  2. Urysohn's lemma has several variants, including a form unbounded functions. The proof is straightforward, by composing f with x/(1+|x|) and applying the bounded version. We don't need it right now, so we can deal with that later.
  3. The proof itself is a rather straightforward duplication of the textbook argument. It's about 80% analysis with geometric series. Big thanks to all the posnum stuff automating away several days of work!
Things done/to do
  • added corresponding entries in CHANGELOG_UNRELEASED.md
  • added corresponding documentation in the headers
Compatibility with MathComp 2.0
  • I added the label TODO: HB port to make sure someone ports this PR to
    the hierarchy-builder branch or I already opened an issue or PR (please cross reference).
Automatic note to reviewers

Read this Checklist and put a milestone if possible.

@zstone1 zstone1 added the TODO: MC2 port This PR must be ported to mathcomp 2 now that the. Remove this label when the port is done. label Jul 7, 2023
@zstone1 zstone1 requested a review from affeldt-aist July 7, 2023 21:18
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zstone1 and others added 7 commits July 10, 2023 13:43
Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
@affeldt-aist affeldt-aist self-requested a review July 27, 2023 07:44
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I re-read the scripts and could find a number of shortcuts that I have committed.
What about using notations such as:
Local Notation "'2/3" := (PosNum twothirds_pos).
in continuous_bounded_extension?
That could make intermediate goals a bit shorter and more readable.
(Yet, inside this proof the problem is not here ^_^.)

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zstone1 commented Jul 28, 2023

Turns out Local Notation "3" := 3%:R. is good enough to clean up the annoying notational issues. Just using 2/3 infers positivity as expected.

@zstone1 zstone1 merged commit 505f9ea into math-comp:master Jul 28, 2023
23 of 34 checks passed
affeldt-aist added a commit to affeldt-aist/analysis that referenced this pull request Jul 29, 2023
* tietze lemma done

* nearly done with tietze

* cauchy with admits

* cauchy part done

* tietze continuity done

* tietze bounded

* proof of tietze is done

* linting and changelog

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* handling interval shifting inside tietze

* simplifying a bit calculations

* shorten proofs, nitpicking

* positive 3 notation

---------

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>
proux01 pushed a commit that referenced this pull request Jul 31, 2023
* tietze lemma done

* nearly done with tietze

* cauchy with admits

* cauchy part done

* tietze continuity done

* tietze bounded

* proof of tietze is done

* linting and changelog

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* handling interval shifting inside tietze

* simplifying a bit calculations

* shorten proofs, nitpicking

* positive 3 notation

---------

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>
@proux01 proux01 removed the TODO: MC2 port This PR must be ported to mathcomp 2 now that the. Remove this label when the port is done. label Jul 31, 2023
IshiguroYoshihiro pushed a commit to IshiguroYoshihiro/analysis that referenced this pull request Sep 7, 2023
* tietze lemma done

* nearly done with tietze

* cauchy with admits

* cauchy part done

* tietze continuity done

* tietze bounded

* proof of tietze is done

* linting and changelog

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* Update theories/numfun.v

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>

* handling interval shifting inside tietze

* simplifying a bit calculations

* shorten proofs, nitpicking

* positive 3 notation

---------

Co-authored-by: affeldt-aist <33154536+affeldt-aist@users.noreply.github.com>
Co-authored-by: Reynald Affeldt <reynald.affeldt@aist.go.jp>
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3 participants