topology.metric_space.kuratowski | Mathlib porting status

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

95d4f658

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

2ebc1d6c

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Analysis.NormedSpace.LpSpace
+import Analysis.NormedSpace.lpSpace
 import Topology.Sets.Compacts
 
 #align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"

2ebc1d6c

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,8 +3,8 @@ Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Analysis.NormedSpace.LpSpace
-import Mathbin.Topology.Sets.Compacts
+import Analysis.NormedSpace.LpSpace
+import Topology.Sets.Compacts
 
 #align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"

2ebc1d6c

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.kuratowski
-! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.NormedSpace.LpSpace
 import Mathbin.Topology.Sets.Compacts
 
+#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"2ebc1d6c2fed9f54c95bbc3998eaa5570527129a"
+
 /-!
 # The Kuratowski embedding

2ebc1d6c

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -27,7 +27,6 @@ open Set Metric TopologicalSpace
 
 open scoped ENNReal
 
--- mathport name: exprℓ_infty_ℝ
 local notation "ℓ_infty_ℝ" => lp (fun n : ℕ => ℝ) ∞
 
 universe u v w
@@ -41,6 +40,7 @@ namespace kuratowskiEmbedding
 
 variable {f g : ℓ_infty_ℝ} {n : ℕ} {C : ℝ} [MetricSpace α] (x : ℕ → α) (a b : α)
 
+#print KuratowskiEmbedding.embeddingOfSubset /-
 /-- A metric space can be embedded in `l^∞(ℝ)` via the distances to points in
 a fixed countable set, if this set is dense. This map is given in `Kuratowski_embedding`,
 without density assumptions. -/
@@ -52,11 +52,15 @@ def embeddingOfSubset : ℓ_infty_ℝ :=
     rintro - ⟨n, rfl⟩
     exact abs_dist_sub_le _ _ _⟩
 #align Kuratowski_embedding.embedding_of_subset KuratowskiEmbedding.embeddingOfSubset
+-/
 
+#print KuratowskiEmbedding.embeddingOfSubset_coe /-
 theorem embeddingOfSubset_coe : embeddingOfSubset x a n = dist a (x n) - dist (x 0) (x n) :=
   rfl
 #align Kuratowski_embedding.embedding_of_subset_coe KuratowskiEmbedding.embeddingOfSubset_coe
+-/
 
+#print KuratowskiEmbedding.embeddingOfSubset_dist_le /-
 /-- The embedding map is always a semi-contraction. -/
 theorem embeddingOfSubset_dist_le (a b : α) :
     dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b :=
@@ -66,7 +70,9 @@ theorem embeddingOfSubset_dist_le (a b : α) :
   convert abs_dist_sub_le a b (x n) using 2
   ring
 #align Kuratowski_embedding.embedding_of_subset_dist_le KuratowskiEmbedding.embeddingOfSubset_dist_le
+-/
 
+#print KuratowskiEmbedding.embeddingOfSubset_isometry /-
 /-- When the reference set is dense, the embedding map is an isometry on its image. -/
 theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) :=
   by
@@ -98,7 +104,9 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
       _ = dist (embedding_of_subset x b) (embedding_of_subset x a) + e := by ring
   simpa [dist_comm] using this
 #align Kuratowski_embedding.embedding_of_subset_isometry KuratowskiEmbedding.embeddingOfSubset_isometry
+-/
 
+#print KuratowskiEmbedding.exists_isometric_embedding /-
 /-- Every separable metric space embeds isometrically in `ℓ_infty_ℝ`. -/
 theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
     ∃ f : α → ℓ_infty_ℝ, Isometry f :=
@@ -114,22 +122,28 @@ theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpac
     -- Use embedding_of_subset to construct the desired isometry
     exact ⟨embedding_of_subset x, embedding_of_subset_isometry x (S_dense.mono x_range)⟩
 #align Kuratowski_embedding.exists_isometric_embedding KuratowskiEmbedding.exists_isometric_embedding
+-/
 
 end kuratowskiEmbedding
 
 open TopologicalSpace kuratowskiEmbedding
 
+#print kuratowskiEmbedding /-
 /-- The Kuratowski embedding is an isometric embedding of a separable metric space in `ℓ^∞(ℝ)`. -/
 def kuratowskiEmbedding (α : Type u) [MetricSpace α] [SeparableSpace α] : α → ℓ_infty_ℝ :=
   Classical.choose (KuratowskiEmbedding.exists_isometric_embedding α)
 #align Kuratowski_embedding kuratowskiEmbedding
+-/
 
+#print kuratowskiEmbedding.isometry /-
 /-- The Kuratowski embedding is an isometry. -/
 protected theorem kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [SeparableSpace α] :
     Isometry (kuratowskiEmbedding α) :=
   Classical.choose_spec (exists_isometric_embedding α)
 #align Kuratowski_embedding.isometry kuratowskiEmbedding.isometry
+-/
 
+#print NonemptyCompacts.kuratowskiEmbedding /-
 /-- Version of the Kuratowski embedding for nonempty compacts -/
 def NonemptyCompacts.kuratowskiEmbedding (α : Type u) [MetricSpace α] [CompactSpace α]
     [Nonempty α] : NonemptyCompacts ℓ_infty_ℝ
@@ -138,4 +152,5 @@ def NonemptyCompacts.kuratowskiEmbedding (α : Type u) [MetricSpace α] [Compact
   is_compact' := isCompact_range (kuratowskiEmbedding.isometry α).Continuous
   nonempty' := range_nonempty _
 #align nonempty_compacts.Kuratowski_embedding NonemptyCompacts.kuratowskiEmbedding
+-/

2ebc1d6c

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -96,7 +96,6 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
               (embedding_of_subset x b - embedding_of_subset x a) n
         nlinarith
       _ = dist (embedding_of_subset x b) (embedding_of_subset x a) + e := by ring
-      
   simpa [dist_comm] using this
 #align Kuratowski_embedding.embedding_of_subset_isometry KuratowskiEmbedding.embeddingOfSubset_isometry

2ebc1d6c

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.kuratowski
-! leanprover-community/mathlib commit 95d4f6586d313c8c28e00f36621d2a6a66893aa6
+! leanprover-community/mathlib commit 2ebc1d6c2fed9f54c95bbc3998eaa5570527129a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Topology.Sets.Compacts
 /-!
 # The Kuratowski embedding
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Any separable metric space can be embedded isometrically in `ℓ^∞(ℝ)`.
 -/
 
@@ -81,7 +84,7 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
       _ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by
         apply_rules [add_le_add_left, le_abs_self]
       _ ≤ 2 * (e / 2) + |embedding_of_subset x b n - embedding_of_subset x a n| := by rw [C];
-        apply_rules [add_le_add, mul_le_mul_of_nonneg_left, hn.le, le_refl] ; norm_num
+        apply_rules [add_le_add, mul_le_mul_of_nonneg_left, hn.le, le_refl]; norm_num
       _ ≤ 2 * (e / 2) + dist (embedding_of_subset x b) (embedding_of_subset x a) :=
         by
         have :

95d4f658

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -22,7 +22,7 @@ noncomputable section
 
 open Set Metric TopologicalSpace
 
-open ENNReal
+open scoped ENNReal
 
 -- mathport name: exprℓ_infty_ℝ
 local notation "ℓ_infty_ℝ" => lp (fun n : ℕ => ℝ) ∞

95d4f658

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -77,10 +77,7 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
   have :=
     calc
       dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
-      _ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) :=
-        by
-        simp [dist_comm]
-        ring
+      _ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [dist_comm]; ring
       _ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by
         apply_rules [add_le_add_left, le_abs_self]
       _ ≤ 2 * (e / 2) + |embedding_of_subset x b n - embedding_of_subset x a n| := by rw [C];
@@ -105,9 +102,7 @@ theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpac
     ∃ f : α → ℓ_infty_ℝ, Isometry f :=
   by
   cases' (univ : Set α).eq_empty_or_nonempty with h h
-  · use fun _ => 0
-    intro x
-    exact absurd h (nonempty.ne_empty ⟨x, mem_univ x⟩)
+  · use fun _ => 0; intro x; exact absurd h (nonempty.ne_empty ⟨x, mem_univ x⟩)
   · -- We construct a map x : ℕ → α with dense image
     rcases h with ⟨basepoint⟩
     haveI : Inhabited α := ⟨basepoint⟩

95d4f658

bump to nightly-2023-02-27-10

mathlib commit https://github.com/leanprover-community/mathlib/commit/eb0cb4511aaef0da2462207b67358a0e1fe1e2ee

49ff026a

Diff

@@ -22,7 +22,7 @@ noncomputable section
 
 open Set Metric TopologicalSpace
 
-open Ennreal
+open ENNReal
 
 -- mathport name: exprℓ_infty_ℝ
 local notation "ℓ_infty_ℝ" => lp (fun n : ℕ => ℝ) ∞
@@ -92,7 +92,7 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
             dist (embedding_of_subset x b) (embedding_of_subset x a) :=
           by
           simpa [dist_eq_norm] using
-            lp.norm_apply_le_norm Ennreal.top_ne_zero
+            lp.norm_apply_le_norm ENNReal.top_ne_zero
               (embedding_of_subset x b - embedding_of_subset x a) n
         nlinarith
       _ = dist (embedding_of_subset x b) (embedding_of_subset x a) + e := by ring

95d4f658

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

95d4f658

chore: tidy various files (#10311)

890dba74

Diff

@@ -142,7 +142,7 @@ theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {ι
     ∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by
   -- Construct the coordinate-wise extensions
   rw [LipschitzOnWith.coordinate] at hfl
-  have (i: ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=
+  have (i : ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=
     LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
   choose g hgl hgeq using this
   rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩

95d4f658

chore(Topology): remove autoImplicit in some files (#9689)

... where this is easy to do.

Co-authored-by: grunweg <grunweg@posteo.de>

9410c2d8

Diff

@@ -16,9 +16,6 @@ Any partially defined Lipschitz map into `ℓ^∞` can be extended to the whole
 
 -/
 
-set_option autoImplicit true
-
-
 noncomputable section
 
 set_option linter.uppercaseLean3 false
@@ -140,19 +137,19 @@ Theorem 2.2 of [Assaf Naor, *Metric Embeddings and Lipschitz Extensions*][Naor-2
 The same result for the case of a finite type `ι` is implemented in
 `LipschitzOnWith.extend_pi`.
 -/
-theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {f : α → ℓ^∞(ι)}
-    {K : ℝ≥0} (hfl : LipschitzOnWith K f s): ∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by
+theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {ι : Type*}
+    {f : α → ℓ^∞(ι)} {K : ℝ≥0} (hfl : LipschitzOnWith K f s) :
+    ∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by
   -- Construct the coordinate-wise extensions
   rw [LipschitzOnWith.coordinate] at hfl
-  have : ∀ i : ι, ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s
-  · intro i
-    exact LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
+  have (i: ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=
+    LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
   choose g hgl hgeq using this
   rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩
   · exact ⟨0, LipschitzWith.const' 0, by simp⟩
   · -- Show that the extensions are uniformly bounded
-    have hf_extb : ∀ a : α, Memℓp (swap g a) ∞
-    · apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
+    have hf_extb : ∀ a : α, Memℓp (swap g a) ∞ := by
+      apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
       use ‖f a₀‖
       rintro - ⟨i, rfl⟩
       simp_rw [← hgeq i ha₀_in_s]

95d4f658

chore: remove uses of cases' (#9171)

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

3fca282c

Diff

@@ -93,7 +93,7 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
 /-- Every separable metric space embeds isometrically in `ℓ^∞(ℕ)`. -/
 theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
     ∃ f : α → ℓ^∞(ℕ), Isometry f := by
-  cases' (univ : Set α).eq_empty_or_nonempty with h h
+  rcases (univ : Set α).eq_empty_or_nonempty with h | h
   · use fun _ => 0; intro x; exact absurd h (Nonempty.ne_empty ⟨x, mem_univ x⟩)
   · -- We construct a map x : ℕ → α with dense image
     rcases h with ⟨basepoint⟩

95d4f658

chore: space after ← (#8178)

Co-authored-by: Moritz Firsching <firsching@google.com>

5fb9beab

Diff

@@ -155,7 +155,7 @@ theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {f :
     · apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
       use ‖f a₀‖
       rintro - ⟨i, rfl⟩
-      simp_rw [←hgeq i ha₀_in_s]
+      simp_rw [← hgeq i ha₀_in_s]
       exact lp.norm_apply_le_norm top_ne_zero (f a₀) i
     -- Construct witness by bundling the function with its certificate of membership in ℓ^∞
     let f_ext' : α → ℓ^∞(ι) := fun i ↦ ⟨swap g i, hf_extb i⟩

95d4f658

fix: disable autoImplicit globally (#6528)

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

Assuming variables are in scope, but pasting the lemma in the wrong section
Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

0c24f831

Diff

@@ -16,6 +16,8 @@ Any partially defined Lipschitz map into `ℓ^∞` can be extended to the whole
 
 -/
 
+set_option autoImplicit true
+
 
 noncomputable section

95d4f658

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,15 +2,12 @@
 Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.kuratowski
-! leanprover-community/mathlib commit 95d4f6586d313c8c28e00f36621d2a6a66893aa6
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.NormedSpace.lpSpace
 import Mathlib.Topology.Sets.Compacts
 
+#align_import topology.metric_space.kuratowski from "leanprover-community/mathlib"@"95d4f6586d313c8c28e00f36621d2a6a66893aa6"
+
 /-!
 # The Kuratowski embedding

95d4f658

chore: fix focusing dots (#5708)

This PR is the result of running

find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;

which firstly replaces . focusing dots with · and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.

cb02d09e

Diff

@@ -150,10 +150,10 @@ theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {f :
     exact LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
   choose g hgl hgeq using this
   rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩
-  . exact ⟨0, LipschitzWith.const' 0, by simp⟩
+  · exact ⟨0, LipschitzWith.const' 0, by simp⟩
   · -- Show that the extensions are uniformly bounded
     have hf_extb : ∀ a : α, Memℓp (swap g a) ∞
-    . apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
+    · apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
       use ‖f a₀‖
       rintro - ⟨i, rfl⟩
       simp_rw [←hgeq i ha₀_in_s]

95d4f658

feat: Lipschitz extensions of maps into l^infty (#5107)

A function f : α → ℓ^∞(ι, ℝ) which is K-Lipschitz on a subset s admits a K-Lipschitz extension to the whole space.

Co-authored-by: Ian Bunner <31334766+ian-bunner@users.noreply.github.com>

Co-authored-by: Chris Camano <53490775+chriscamano@users.noreply.github.com>

2372d46e

Diff

@@ -14,7 +14,9 @@ import Mathlib.Topology.Sets.Compacts
 /-!
 # The Kuratowski embedding
 
-Any separable metric space can be embedded isometrically in `ℓ^∞(ℝ)`.
+Any separable metric space can be embedded isometrically in `ℓ^∞(ℕ, ℝ)`.
+Any partially defined Lipschitz map into `ℓ^∞` can be extended to the whole space.
+
 -/
 
 
@@ -22,11 +24,7 @@ noncomputable section
 
 set_option linter.uppercaseLean3 false
 
-open Set Metric TopologicalSpace
-
-open scoped ENNReal
-
-local notation "ℓ_infty_ℝ" => lp (fun n : ℕ => ℝ) ∞
+open Set Metric TopologicalSpace NNReal ENNReal lp Function
 
 universe u v w
 
@@ -34,15 +32,15 @@ variable {α : Type u} {β : Type v} {γ : Type w}
 
 namespace KuratowskiEmbedding
 
-/-! ### Any separable metric space can be embedded isometrically in ℓ^∞(ℝ) -/
+/-! ### Any separable metric space can be embedded isometrically in ℓ^∞(ℕ, ℝ) -/
 
 
-variable {f g : ℓ_infty_ℝ} {n : ℕ} {C : ℝ} [MetricSpace α] (x : ℕ → α) (a b : α)
+variable {f g : ℓ^∞(ℕ)} {n : ℕ} {C : ℝ} [MetricSpace α] (x : ℕ → α) (a b : α)
 
 /-- A metric space can be embedded in `l^∞(ℝ)` via the distances to points in
 a fixed countable set, if this set is dense. This map is given in `kuratowskiEmbedding`,
 without density assumptions. -/
-def embeddingOfSubset : ℓ_infty_ℝ :=
+def embeddingOfSubset : ℓ^∞(ℕ) :=
   ⟨fun n => dist a (x n) - dist (x 0) (x n), by
     apply memℓp_infty
     use dist a (x 0)
@@ -93,9 +91,9 @@ theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSub
   simpa [dist_comm] using this
 #align Kuratowski_embedding.embedding_of_subset_isometry KuratowskiEmbedding.embeddingOfSubset_isometry
 
-/-- Every separable metric space embeds isometrically in `ℓ_infty_ℝ`. -/
+/-- Every separable metric space embeds isometrically in `ℓ^∞(ℕ)`. -/
 theorem exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
-    ∃ f : α → ℓ_infty_ℝ, Isometry f := by
+    ∃ f : α → ℓ^∞(ℕ), Isometry f := by
   cases' (univ : Set α).eq_empty_or_nonempty with h h
   · use fun _ => 0; intro x; exact absurd h (Nonempty.ne_empty ⟨x, mem_univ x⟩)
   · -- We construct a map x : ℕ → α with dense image
@@ -112,12 +110,15 @@ end KuratowskiEmbedding
 
 open TopologicalSpace KuratowskiEmbedding
 
-/-- The Kuratowski embedding is an isometric embedding of a separable metric space in `ℓ^∞(ℝ)`. -/
-def kuratowskiEmbedding (α : Type u) [MetricSpace α] [SeparableSpace α] : α → ℓ_infty_ℝ :=
+/-- The Kuratowski embedding is an isometric embedding of a separable metric space in `ℓ^∞(ℕ, ℝ)`.
+-/
+def kuratowskiEmbedding (α : Type u) [MetricSpace α] [SeparableSpace α] : α → ℓ^∞(ℕ) :=
   Classical.choose (KuratowskiEmbedding.exists_isometric_embedding α)
 #align Kuratowski_embedding kuratowskiEmbedding
 
-/-- The Kuratowski embedding is an isometry. -/
+/--
+The Kuratowski embedding is an isometry.
+Theorem 2.1 of [Assaf Naor, *Metric Embeddings and Lipschitz Extensions*][Naor-2015]. -/
 protected theorem kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [SeparableSpace α] :
     Isometry (kuratowskiEmbedding α) :=
   Classical.choose_spec (exists_isometric_embedding α)
@@ -125,8 +126,43 @@ protected theorem kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [S
 
 /-- Version of the Kuratowski embedding for nonempty compacts -/
 nonrec def NonemptyCompacts.kuratowskiEmbedding (α : Type u) [MetricSpace α] [CompactSpace α]
-    [Nonempty α] : NonemptyCompacts ℓ_infty_ℝ where
+    [Nonempty α] : NonemptyCompacts ℓ^∞(ℕ) where
   carrier := range (kuratowskiEmbedding α)
   isCompact' := isCompact_range (kuratowskiEmbedding.isometry α).continuous
   nonempty' := range_nonempty _
 #align nonempty_compacts.Kuratowski_embedding NonemptyCompacts.kuratowskiEmbedding
+
+/--
+A function `f : α → ℓ^∞(ι, ℝ)` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz
+extension to the whole space.
+
+Theorem 2.2 of [Assaf Naor, *Metric Embeddings and Lipschitz Extensions*][Naor-2015]
+
+The same result for the case of a finite type `ι` is implemented in
+`LipschitzOnWith.extend_pi`.
+-/
+theorem LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {f : α → ℓ^∞(ι)}
+    {K : ℝ≥0} (hfl : LipschitzOnWith K f s): ∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by
+  -- Construct the coordinate-wise extensions
+  rw [LipschitzOnWith.coordinate] at hfl
+  have : ∀ i : ι, ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s
+  · intro i
+    exact LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
+  choose g hgl hgeq using this
+  rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩
+  . exact ⟨0, LipschitzWith.const' 0, by simp⟩
+  · -- Show that the extensions are uniformly bounded
+    have hf_extb : ∀ a : α, Memℓp (swap g a) ∞
+    . apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
+      use ‖f a₀‖
+      rintro - ⟨i, rfl⟩
+      simp_rw [←hgeq i ha₀_in_s]
+      exact lp.norm_apply_le_norm top_ne_zero (f a₀) i
+    -- Construct witness by bundling the function with its certificate of membership in ℓ^∞
+    let f_ext' : α → ℓ^∞(ι) := fun i ↦ ⟨swap g i, hf_extb i⟩
+    refine ⟨f_ext', ?_, ?_⟩
+    · rw [LipschitzWith.coordinate]
+      exact hgl
+    · intro a hyp
+      ext i
+      exact (hgeq i) hyp

95d4f658

feat: port Analysis.NormedSpace.LpEquiv (#4554)

This PR also renames Analysis.NormedSpace.LpSpace to Analysis.NormedSpace.lpSpace per this Zulip poll

9b438dac

Diff

@@ -8,7 +8,7 @@ Authors: Sébastien Gouëzel
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathlib.Analysis.NormedSpace.LpSpace
+import Mathlib.Analysis.NormedSpace.lpSpace
 import Mathlib.Topology.Sets.Compacts
 
 /-!

95d4f658

feat: port Topology.MetricSpace.Kuratowski (#4544)

a8e87892

`topology.metric_space.kuratowski` ⟷ `Mathlib.Topology.MetricSpace.Kuratowski`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 779

topology.metric_space.kuratowski ⟷ Mathlib.Topology.MetricSpace.Kuratowski

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 779

`topology.metric_space.kuratowski` ⟷ `Mathlib.Topology.MetricSpace.Kuratowski`