feat(analysis/cauchy_equation): Add Cauchy's Functional Equation #12933

MantasBaksys · 2022-03-25T15:46:10Z

Cauchy's functional equation over the real-line for a Lebesgue measurable function and some of the usual variants from which linearity of a given function is deduced.

depends on: [Merged by Bors] - feat(topology/algebra/group): add small topology lemma #12931
depends on: [Merged by Bors] - feat(measure_theory/measure/haar): Add the Steinhaus Theorem #12932
depends on: [Merged by Bors] - feat(order/bounds): Boundedness of monotone/antitone functions #13079
depends on: [Merged by Bors] - feat(analysis/normed_space/pointwise): Addition of balls #13381
depends on: [Merged by Bors] - feat({data/{finset,set},order/filter}/pointwise): More lemmas #14216

…-Equation

…tion

…-Equation

urkud

Thank you for working on this! The proof can be golfed a lot using lemmas I mention in my comments.

urkud · 2022-05-29T13:37:14Z

src/analysis/cauchy_equation.lean

+
+/-- **Cauchy's functional equation**. An additive monoid homomorphism automatically preserves `ℚ`.
+-/
+theorem add_monoid_hom.is_linear_map_rat (f : ℝ →+ ℝ) : is_linear_map ℚ f :=


See also add_monoid_hom.to_rat_linear_map

See #12933 (comment)

src/analysis/cauchy_equation.lean

…sitive measure (#14449) Motivated by #12933 Co-authored-by: Mantas Bakšys <baksysmantas@gmail.com>

…-Equation

…sitive measure (#14449) Motivated by #12933 Co-authored-by: Mantas Bakšys <baksysmantas@gmail.com>

src/analysis/cauchy_equation.lean

urkud · 2022-06-11T14:29:10Z

src/analysis/cauchy_equation.lean

+λ h, is_open_univ.measure_ne_zero μ univ_nonempty $ by rw [h, coe_zero, pi.zero_apply]
+
+lemma exists_zero_nhds_bounded (f : ℝ →+ ℝ) (h : measurable f) :
+  ∃ s, s ∈ 𝓝 (0 : ℝ) ∧ bounded (f '' s) :=


What is the right generality for this lemma? Probably, it's true at least for all finite dimensional real normed spaces.

urkud · 2022-06-11T14:40:11Z

src/analysis/cauchy_equation.lean

+end
+
+lemma additive_continuous_at_zero_of_bounded_nhds_zero (f : ℝ →+ ℝ) (hs : s ∈ 𝓝 (0 : ℝ))
+  (hbounded : bounded (f '' s)) : continuous_at f 0 :=


If you upgrade f to a ℚ-linear map, then you can use linear_map.bound_of_shell or linear_map.bound_of_ball_bound. Again, you can upgrade your lemma to a real TVS.

urkud · 2022-06-11T14:41:38Z

src/analysis/cauchy_equation.lean

+  { rw [smul_eq_mul, smul_eq_mul, h (c * x), h x, ←mul_assoc, mul_comm _ c, mul_assoc] }
+end
+
+lemma is_linear_rat (f : ℝ →+ ℝ) : ∀ (q : ℚ), f q = f 1 * q :=


This lemma and most lemmas below are already in mathlib, aren't they?

…-Equation

eric-wieser · 2023-07-16T15:26:18Z

@YaelDillies, are you interested in finishing this, or are your own PRs more important to you?

YaelDillies · 2023-07-16T18:47:43Z

Yes, I in fact started fixing it earlier today. It needs some more cleanup but the gist is here.

urkud · 2023-09-10T17:45:25Z

src/analysis/cauchy_equation.lean

+
+local notation `ℝⁿ` := ι → ℝ
+
+lemma add_monoid_hom.measurable_of_continuous (f : ℝ →+ ℝ) (h : measurable f) : continuous f :=


Name says measurable_of_continuous but the theorem says measurable f -> continuous f.

urkud · 2023-09-10T17:46:21Z

src/measure_theory/measure/measure_space.lean

@@ -2329,6 +2329,8 @@ by rw [← empty_mem_iff_bot, mem_ae_iff, compl_empty, measure_univ_eq_zero]
 @[simp] lemma ae_ne_bot : μ.ae.ne_bot ↔ μ ≠ 0 :=
 ne_bot_iff.trans (not_congr ae_eq_bot)

+instance ae_ne_bot.to_ne_zero [μ.ae.ne_bot] : ne_zero μ := ⟨ae_ne_bot.1 ‹_›⟩


We have [NeZero μ] : μ.ae.NeBot in Lean 4.

urkud · 2023-09-10T17:57:34Z

src/analysis/cauchy_equation.lean

+open topological_space
+variables {G H : Type*} [seminormed_add_group G] [topological_add_group G] [is_R_or_C H] {s : set G}
+
+lemma add_monoid_hom.continuous_of_bounded_nhds_zero (f : G →+ H) (hs : s ∈ 𝓝 (0 : G))


Isn't [is_R_or_C H] too strong? Is it true for any normed space over rationals?

YaelDillies · 2023-11-18T11:19:22Z

Ported to LeanCamCombi

MantasBaksys and others added 13 commits March 21, 2022 14:56

testing

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update

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reduce imports

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steinhaus to prove

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blueprint

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update

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Merge remote-tracking branch 'origin/master' into Cauchy's-Functional…

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…-Equation

finish steinhaus_theorem

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update and add variants

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update

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fix line length

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remove overhead + add steinhaus_theorem

128c15c

bring back cauchy

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MantasBaksys added the WIP Work in progress label Mar 25, 2022

leanprover-community-bot-assistant added the blocked-by-other-PR This PR depends on another PR which is still in the queue. A bot manages this label via PR comment. label Mar 25, 2022

MantasBaksys changed the title ~~Cauchy's functional equation~~ feat (analysis/cauchy_equation) : Add Cauchy's Functional Equation Mar 25, 2022

MantasBaksys added 14 commits March 25, 2022 19:25

replace occurences

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fix typos

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Merge branch 'mantas-small-lemma' into mantas-steinhaus-theorem

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update proofs and statements

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fix dependency

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Merge remote-tracking branch 'origin/master' into mantas-small-lemma

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segment into lemmas + golf

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Merge branch 'mantas-small-lemma' into mantas-steinhaus-theorem

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fix lint

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Merge branch 'mantas-small-lemma' into mantas-steinhaus-theorem

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truly fix lint

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fix errors

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Merge branch 'mantas-steinhaus-theorem' into Cauchy's-Functional-Equa…

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…tion

make compile

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YaelDillies added 2 commits May 29, 2022 01:30

Merge remote-tracking branch 'origin/master' into Cauchy's-Functional…

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…-Equation

bump, golf

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YaelDillies added RFC Request for comment and removed WIP Work in progress labels May 29, 2022

urkud mentioned this pull request May 29, 2022

[Merged by Bors] - feat(measure_theory/measure/measure_space): there exists a ball of positive measure #14449

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urkud reviewed May 29, 2022

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feat(measure_theory/measure/measure_space): there exists a ball of po…

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…sitive measure (#14449) Motivated by #12933 Co-authored-by: Mantas Bakšys <baksysmantas@gmail.com>

YaelDillies added 2 commits June 5, 2022 14:07

Merge remote-tracking branch 'origin/master' into Cauchy's-Functional…

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…-Equation

bump

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feat(measure_theory/measure/measure_space): there exists a ball of po…

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…sitive measure (#14449) Motivated by #12933 Co-authored-by: Mantas Bakšys <baksysmantas@gmail.com>

more bump

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Merge remote-tracking branch 'origin/master' into Cauchy's-Functional…

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…-Equation

move is_open_pos_measure.to_ae_ne_bot

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YaelDillies requested a review from a team as a code owner July 16, 2023 18:47

github-actions bot added the modifies-synchronized-file This PR touches a files that has already been ported to mathlib4, and may need a synchronization PR. label Jul 16, 2023

YaelDillies added the awaiting-CI The author would like to see what CI has to say before doing more work. label Jul 16, 2023

kim-em added the not-too-late This PR was ready at the point mathlib3 was frozen: we will try to merge it and port it to mathlib4 label Jul 16, 2023

YaelDillies added 2 commits August 3, 2023 08:15

fix build, generalise more

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move ae_ne_bot.to_ne_zero

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YaelDillies deleted the Cauchy's-Functional-Equation branch November 18, 2023 11:19

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feat(analysis/cauchy_equation): Add Cauchy's Functional Equation #12933

feat(analysis/cauchy_equation): Add Cauchy's Functional Equation #12933

MantasBaksys commented Mar 25, 2022 •

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YaelDillies commented Nov 18, 2023


		local notation `ℝⁿ` := ι → ℝ

		lemma add_monoid_hom.measurable_of_continuous (f : ℝ →+ ℝ) (h : measurable f) : continuous f :=

feat(analysis/cauchy_equation): Add Cauchy's Functional Equation #12933

feat(analysis/cauchy_equation): Add Cauchy's Functional Equation #12933

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