analysis.convex.krein_milman | 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

27929793

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

0b7c740e

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -80,7 +80,7 @@ theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.None
     rw [←
       hBmin ({z ∈ t | ∀ w ∈ t, l w ≤ l z})
         ⟨⟨z, hzt, hz⟩, h.is_closed htclos, hst.trans h.is_extreme⟩ (t.sep_subset _)] at
-      hyB 
+      hyB
     exact hl.not_le (hyB.2 x hxt)
   refine' zorn_superset _ fun F hFS hF => _
   obtain rfl | hFnemp := F.eq_empty_or_nonempty

0b7c740e

bump to nightly-2024-03-01-08

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

f62598c4

Diff

@@ -93,7 +93,7 @@ theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.None
   haveI : Nonempty ↥F := hFnemp.to_subtype
   rw [sInter_eq_Inter]
   refine'
-    IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun t u => _)
+    IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ (fun t u => _)
       (fun t => (hFS t.Mem).1)
       (fun t => IsCompact.of_isClosed_subset hscomp (hFS t.Mem).2.1 (hFS t.Mem).2.2.1) fun t =>
       (hFS t.Mem).2.1

0b7c740e

bump to nightly-2023-12-02-06

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

e637da5d

Diff

@@ -62,14 +62,14 @@ open scoped Classical
 variable {E : Type _} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
   [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E}
 
-#print IsCompact.has_extreme_point /-
+#print IsCompact.extremePoints_nonempty /-
 /-- **Krein-Milman lemma**: In a LCTVS, any nonempty compact set has an extreme point. -/
-theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
+theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
     (s.extremePoints ℝ).Nonempty :=
   by
   let S : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t}
   rsuffices ⟨t, ⟨⟨x, hxt⟩, htclos, hst⟩, hBmin⟩ : ∃ t ∈ S, ∀ u ∈ S, u ⊆ t → u = t
-  · refine' ⟨x, mem_extremePoints_iff_extreme_singleton.2 _⟩
+  · refine' ⟨x, isExtreme_singleton.2 _⟩
     rwa [← eq_singleton_iff_unique_mem.2 ⟨hxt, fun y hyB => _⟩]
     by_contra hyx
     obtain ⟨l, hl⟩ := geometric_hahn_banach_point_point hyx
@@ -99,7 +99,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
       (hFS t.Mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
   exacts [⟨t, subset.rfl, htu⟩, ⟨u, hut, subset.rfl⟩]
-#align is_compact.has_extreme_point IsCompact.has_extreme_point
+#align is_compact.has_extreme_point IsCompact.extremePoints_nonempty
 -/
 
 #print closure_convexHull_extremePoints /-
@@ -115,7 +115,7 @@ theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex
     geometric_hahn_banach_closed_point (convex_convexHull _ _).closure isClosed_closure hxt
   have h : IsExposed ℝ s ({y ∈ s | ∀ z ∈ s, l z ≤ l y}) := fun _ => ⟨l, rfl⟩
   obtain ⟨z, hzA, hz⟩ := hscomp.exists_forall_ge ⟨x, hxA⟩ l.continuous.continuous_on
-  obtain ⟨y, hy⟩ := (h.is_compact hscomp).has_extreme_point ⟨z, hzA, hz⟩
+  obtain ⟨y, hy⟩ := (h.is_compact hscomp).extremePoints_nonempty ⟨z, hzA, hz⟩
   linarith [hlr _
       (subset_closure <|
         subset_convexHull _ _ <| h.is_extreme.extreme_points_subset_extreme_points hy),

0b7c740e

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) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathbin.Analysis.Convex.Exposed
-import Mathbin.Analysis.NormedSpace.HahnBanach.Separation
+import Analysis.Convex.Exposed
+import Analysis.NormedSpace.HahnBanach.Separation
 
 #align_import analysis.convex.krein_milman from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
 
@@ -74,7 +74,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
     by_contra hyx
     obtain ⟨l, hl⟩ := geometric_hahn_banach_point_point hyx
     obtain ⟨z, hzt, hz⟩ :=
-      (isCompact_of_isClosed_subset hscomp htclos hst.1).exists_forall_ge ⟨x, hxt⟩
+      (IsCompact.of_isClosed_subset hscomp htclos hst.1).exists_forall_ge ⟨x, hxt⟩
         l.continuous.continuous_on
     have h : IsExposed ℝ t ({z ∈ t | ∀ w ∈ t, l w ≤ l z}) := fun h => ⟨l, rfl⟩
     rw [←
@@ -95,7 +95,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
   refine'
     IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun t u => _)
       (fun t => (hFS t.Mem).1)
-      (fun t => isCompact_of_isClosed_subset hscomp (hFS t.Mem).2.1 (hFS t.Mem).2.2.1) fun t =>
+      (fun t => IsCompact.of_isClosed_subset hscomp (hFS t.Mem).2.1 (hFS t.Mem).2.2.1) fun t =>
       (hFS t.Mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
   exacts [⟨t, subset.rfl, htu⟩, ⟨u, hut, subset.rfl⟩]

0b7c740e

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) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
-
-! This file was ported from Lean 3 source module analysis.convex.krein_milman
-! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Convex.Exposed
 import Mathbin.Analysis.NormedSpace.HahnBanach.Separation
 
+#align_import analysis.convex.krein_milman from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-!
 # The Krein-Milman theorem

0b7c740e

bump to nightly-2023-06-04-18

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

3fb4268b

Diff

@@ -70,7 +70,7 @@ variable {E : Type _} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Sp
 theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
     (s.extremePoints ℝ).Nonempty :=
   by
-  let S : Set (Set E) := { t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t }
+  let S : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t}
   rsuffices ⟨t, ⟨⟨x, hxt⟩, htclos, hst⟩, hBmin⟩ : ∃ t ∈ S, ∀ u ∈ S, u ⊆ t → u = t
   · refine' ⟨x, mem_extremePoints_iff_extreme_singleton.2 _⟩
     rwa [← eq_singleton_iff_unique_mem.2 ⟨hxt, fun y hyB => _⟩]
@@ -79,9 +79,9 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
     obtain ⟨z, hzt, hz⟩ :=
       (isCompact_of_isClosed_subset hscomp htclos hst.1).exists_forall_ge ⟨x, hxt⟩
         l.continuous.continuous_on
-    have h : IsExposed ℝ t ({ z ∈ t | ∀ w ∈ t, l w ≤ l z }) := fun h => ⟨l, rfl⟩
+    have h : IsExposed ℝ t ({z ∈ t | ∀ w ∈ t, l w ≤ l z}) := fun h => ⟨l, rfl⟩
     rw [←
-      hBmin ({ z ∈ t | ∀ w ∈ t, l w ≤ l z })
+      hBmin ({z ∈ t | ∀ w ∈ t, l w ≤ l z})
         ⟨⟨z, hzt, hz⟩, h.is_closed htclos, hst.trans h.is_extreme⟩ (t.sep_subset _)] at
       hyB 
     exact hl.not_le (hyB.2 x hxt)
@@ -116,7 +116,7 @@ theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex
   obtain ⟨x, hxA, hxt⟩ := not_subset.1 hs
   obtain ⟨l, r, hlr, hrx⟩ :=
     geometric_hahn_banach_closed_point (convex_convexHull _ _).closure isClosed_closure hxt
-  have h : IsExposed ℝ s ({ y ∈ s | ∀ z ∈ s, l z ≤ l y }) := fun _ => ⟨l, rfl⟩
+  have h : IsExposed ℝ s ({y ∈ s | ∀ z ∈ s, l z ≤ l y}) := fun _ => ⟨l, rfl⟩
   obtain ⟨z, hzA, hz⟩ := hscomp.exists_forall_ge ⟨x, hxA⟩ l.continuous.continuous_on
   obtain ⟨y, hy⟩ := (h.is_compact hscomp).has_extreme_point ⟨z, hzA, hz⟩
   linarith [hlr _

0b7c740e

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -83,7 +83,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
     rw [←
       hBmin ({ z ∈ t | ∀ w ∈ t, l w ≤ l z })
         ⟨⟨z, hzt, hz⟩, h.is_closed htclos, hst.trans h.is_extreme⟩ (t.sep_subset _)] at
-      hyB
+      hyB 
     exact hl.not_le (hyB.2 x hxt)
   refine' zorn_superset _ fun F hFS hF => _
   obtain rfl | hFnemp := F.eq_empty_or_nonempty
@@ -101,7 +101,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
       (fun t => isCompact_of_isClosed_subset hscomp (hFS t.Mem).2.1 (hFS t.Mem).2.2.1) fun t =>
       (hFS t.Mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
-  exacts[⟨t, subset.rfl, htu⟩, ⟨u, hut, subset.rfl⟩]
+  exacts [⟨t, subset.rfl, htu⟩, ⟨u, hut, subset.rfl⟩]
 #align is_compact.has_extreme_point IsCompact.has_extreme_point
 -/

0b7c740e

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -60,7 +60,7 @@ See chapter 8 of [Barry Simon, *Convexity*][simon2011]
 
 open Set
 
-open Classical
+open scoped Classical
 
 variable {E : Type _} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
   [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E}

0b7c740e

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 
 ! This file was ported from Lean 3 source module analysis.convex.krein_milman
-! leanprover-community/mathlib commit 279297937dede7b1b3451b7b0f1786352ad011fa
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Analysis.NormedSpace.HahnBanach.Separation
 /-!
 # The Krein-Milman theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file proves the Krein-Milman lemma and the Krein-Milman theorem.
 
 ## The lemma

27929793

bump to nightly-2023-05-25-20

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

e9c63ad2

Diff

@@ -62,6 +62,7 @@ open Classical
 variable {E : Type _} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
   [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E}
 
+#print IsCompact.has_extreme_point /-
 /-- **Krein-Milman lemma**: In a LCTVS, any nonempty compact set has an extreme point. -/
 theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
     (s.extremePoints ℝ).Nonempty :=
@@ -99,7 +100,9 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
   obtain htu | hut := hF.total t.mem u.mem
   exacts[⟨t, subset.rfl, htu⟩, ⟨u, hut, subset.rfl⟩]
 #align is_compact.has_extreme_point IsCompact.has_extreme_point
+-/
 
+#print closure_convexHull_extremePoints /-
 /-- **Krein-Milman theorem**: In a LCTVS, any compact convex set is the closure of the convex hull
     of its extreme points. -/
 theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex ℝ s) :
@@ -118,4 +121,5 @@ theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex
         subset_convexHull _ _ <| h.is_extreme.extreme_points_subset_extreme_points hy),
     hy.1.2 x hxA]
 #align closure_convex_hull_extreme_points closure_convexHull_extremePoints
+-/

27929793

bump to nightly-2023-05-14-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

d31da313

Diff

@@ -86,13 +86,13 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
   · exact ⟨s, ⟨hsnemp, hscomp.is_closed, IsExtreme.rfl⟩, fun _ => False.elim⟩
   refine'
     ⟨⋂₀ F,
-      ⟨_, isClosed_interₛ fun t ht => (hFS ht).2.1,
-        isExtreme_interₛ hFnemp fun t ht => (hFS ht).2.2⟩,
+      ⟨_, isClosed_sInter fun t ht => (hFS ht).2.1,
+        isExtreme_sInter hFnemp fun t ht => (hFS ht).2.2⟩,
       fun t ht => sInter_subset_of_mem ht⟩
   haveI : Nonempty ↥F := hFnemp.to_subtype
   rw [sInter_eq_Inter]
   refine'
-    IsCompact.nonempty_interᵢ_of_directed_nonempty_compact_closed _ (fun t u => _)
+    IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun t u => _)
       (fun t => (hFS t.Mem).1)
       (fun t => isCompact_of_isClosed_subset hscomp (hFS t.Mem).2.1 (hFS t.Mem).2.2.1) fun t =>
       (hFS t.Mem).2.1

27929793

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

27929793

feat(Mathlib.Topology.Compactness.Compact): add sInter version of Cantor's intersection theorem (#10956)

The iInter version of Cantor's intersection theorem is already present: IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed. The proof of the added sInter version takes advantage of the iInter version.

Much of the addition due to conversation with Sebastien Gouezel on Zulip.

0bbdc654

Diff

@@ -82,7 +82,7 @@ theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.None
     isExtreme_sInter hFnemp fun t ht => (hFS ht).2.2⟩, fun t ht => sInter_subset_of_mem ht⟩
   haveI : Nonempty (↥F) := hFnemp.to_subtype
   rw [sInter_eq_iInter]
-  refine' IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun t u => _)
+  refine' IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ (fun t u => _)
     (fun t => (hFS t.mem).1)
     (fun t => hscomp.of_isClosed_subset (hFS t.mem).2.1 (hFS t.mem).2.2.1) fun t =>
       (hFS t.mem).2.1

27929793

chore(*): replace $ with <| (#9319)

See Zulip thread for the discussion.

9f6d3388

Diff

@@ -112,13 +112,14 @@ lemma surjOn_extremePoints_image (f : E →A[ℝ] F) (hs : IsCompact s) :
     SurjOn f (extremePoints ℝ s) (extremePoints ℝ (f '' s)) := by
   rintro w hw
   -- The fiber of `w` is nonempty and compact
-  have ht : IsCompact {x ∈ s | f x = w} := hs.inter_right $ isClosed_singleton.preimage f.continuous
+  have ht : IsCompact {x ∈ s | f x = w} :=
+    hs.inter_right <| isClosed_singleton.preimage f.continuous
   have ht₀ : {x ∈ s | f x = w}.Nonempty := by simpa using extremePoints_subset hw
   -- Hence by the Krein-Milman lemma it has an extreme point `x`
   obtain ⟨x, ⟨hx, rfl⟩, hyt⟩ := ht.extremePoints_nonempty ht₀
   -- `f x = w` and `x` is an extreme point of `s`, so we're done
   refine mem_image_of_mem _ ⟨hx, fun y hy z hz hxyz ↦ ?_⟩
   have := by simpa using image_openSegment _ f.toAffineMap y z
-  have := hw.2 (mem_image_of_mem _ hy) (mem_image_of_mem _ hz) $ by
+  have := hw.2 (mem_image_of_mem _ hy) (mem_image_of_mem _ hz) <| by
     rw [← this]; exact mem_image_of_mem _ hxyz
   exact hyt ⟨hy, this.1⟩ ⟨hz, this.2⟩ hxyz

27929793

chore: space after ← (#8178)

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

5fb9beab

Diff

@@ -120,5 +120,5 @@ lemma surjOn_extremePoints_image (f : E →A[ℝ] F) (hs : IsCompact s) :
   refine mem_image_of_mem _ ⟨hx, fun y hy z hz hxyz ↦ ?_⟩
   have := by simpa using image_openSegment _ f.toAffineMap y z
   have := hw.2 (mem_image_of_mem _ hy) (mem_image_of_mem _ hz) $ by
-    rw [←this]; exact mem_image_of_mem _ hxyz
+    rw [← this]; exact mem_image_of_mem _ hxyz
   exact hyt ⟨hy, this.1⟩ ⟨hz, this.2⟩ hxyz

27929793

feat: Mapping extreme points under continuous maps (#8574)

Prove that extreme points are preserved under affine equivalences, and the less trivial statement that a continuous affine map sends extreme points of a compact set to a superset of the extreme points of the image of that set.

Also fix a few name and tweak the API a bit.

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

b9b9fdbc

Diff

@@ -5,6 +5,7 @@ Authors: Yaël Dillies
 -/
 import Mathlib.Analysis.Convex.Exposed
 import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
+import Mathlib.Topology.Algebra.ContinuousAffineMap
 
 #align_import analysis.convex.krein_milman from "leanprover-community/mathlib"@"279297937dede7b1b3451b7b0f1786352ad011fa"
 
@@ -51,20 +52,19 @@ See chapter 8 of [Barry Simon, *Convexity*][simon2011]
 
 -/
 
-
 open Set
+open scoped Classical
 
-open Classical
-
-variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
+variable {E F : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
   [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E}
+  [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [T1Space F]
 
 /-- **Krein-Milman lemma**: In a LCTVS, any nonempty compact set has an extreme point. -/
-theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
+theorem IsCompact.extremePoints_nonempty (hscomp : IsCompact s) (hsnemp : s.Nonempty) :
     (s.extremePoints ℝ).Nonempty := by
   let S : Set (Set E) := { t | t.Nonempty ∧ IsClosed t ∧ IsExtreme ℝ s t }
   rsuffices ⟨t, ⟨⟨x, hxt⟩, htclos, hst⟩, hBmin⟩ : ∃ t ∈ S, ∀ u ∈ S, u ⊆ t → u = t
-  · refine' ⟨x, mem_extremePoints_iff_extreme_singleton.2 _⟩
+  · refine ⟨x, IsExtreme.mem_extremePoints ?_⟩
     rwa [← eq_singleton_iff_unique_mem.2 ⟨hxt, fun y hyB => ?_⟩]
     by_contra hyx
     obtain ⟨l, hl⟩ := geometric_hahn_banach_point_point hyx
@@ -88,7 +88,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
       (hFS t.mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
   exacts [⟨t, Subset.rfl, htu⟩, ⟨u, hut, Subset.rfl⟩]
-#align is_compact.has_extreme_point IsCompact.has_extreme_point
+#align is_compact.has_extreme_point IsCompact.extremePoints_nonempty
 
 /-- **Krein-Milman theorem**: In a LCTVS, any compact convex set is the closure of the convex hull
     of its extreme points. -/
@@ -101,7 +101,24 @@ theorem closure_convexHull_extremePoints (hscomp : IsCompact s) (hAconv : Convex
     geometric_hahn_banach_closed_point (convex_convexHull _ _).closure isClosed_closure hxt
   have h : IsExposed ℝ s ({ y ∈ s | ∀ z ∈ s, l z ≤ l y }) := fun _ => ⟨l, rfl⟩
   obtain ⟨z, hzA, hz⟩ := hscomp.exists_forall_ge ⟨x, hxA⟩ l.continuous.continuousOn
-  obtain ⟨y, hy⟩ := (h.isCompact hscomp).has_extreme_point ⟨z, hzA, hz⟩
+  obtain ⟨y, hy⟩ := (h.isCompact hscomp).extremePoints_nonempty ⟨z, hzA, hz⟩
   linarith [hlr _ (subset_closure <| subset_convexHull _ _ <|
     h.isExtreme.extremePoints_subset_extremePoints hy), hy.1.2 x hxA]
 #align closure_convex_hull_extreme_points closure_convexHull_extremePoints
+
+/-- A continuous affine map is surjective from the extreme points of a compact set to the extreme
+points of the image of that set. This inclusion is in general strict. -/
+lemma surjOn_extremePoints_image (f : E →A[ℝ] F) (hs : IsCompact s) :
+    SurjOn f (extremePoints ℝ s) (extremePoints ℝ (f '' s)) := by
+  rintro w hw
+  -- The fiber of `w` is nonempty and compact
+  have ht : IsCompact {x ∈ s | f x = w} := hs.inter_right $ isClosed_singleton.preimage f.continuous
+  have ht₀ : {x ∈ s | f x = w}.Nonempty := by simpa using extremePoints_subset hw
+  -- Hence by the Krein-Milman lemma it has an extreme point `x`
+  obtain ⟨x, ⟨hx, rfl⟩, hyt⟩ := ht.extremePoints_nonempty ht₀
+  -- `f x = w` and `x` is an extreme point of `s`, so we're done
+  refine mem_image_of_mem _ ⟨hx, fun y hy z hz hxyz ↦ ?_⟩
+  have := by simpa using image_openSegment _ f.toAffineMap y z
+  have := hw.2 (mem_image_of_mem _ hy) (mem_image_of_mem _ hz) $ by
+    rw [←this]; exact mem_image_of_mem _ hxyz
+  exact hyt ⟨hy, this.1⟩ ⟨hz, this.2⟩ hxyz

27929793

chore(Topology/SubsetProperties): rename isCompact_of_isClosed_subset (#7298)

As discussed on Zulip.

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

b68cba43

Diff

@@ -69,7 +69,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
     by_contra hyx
     obtain ⟨l, hl⟩ := geometric_hahn_banach_point_point hyx
     obtain ⟨z, hzt, hz⟩ :=
-      (isCompact_of_isClosed_subset hscomp htclos hst.1).exists_forall_ge ⟨x, hxt⟩
+      (hscomp.of_isClosed_subset htclos hst.1).exists_forall_ge ⟨x, hxt⟩
         l.continuous.continuousOn
     have h : IsExposed ℝ t ({ z ∈ t | ∀ w ∈ t, l w ≤ l z }) := fun _ => ⟨l, rfl⟩
     rw [← hBmin ({ z ∈ t | ∀ w ∈ t, l w ≤ l z })
@@ -84,7 +84,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
   rw [sInter_eq_iInter]
   refine' IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed _ (fun t u => _)
     (fun t => (hFS t.mem).1)
-    (fun t => isCompact_of_isClosed_subset hscomp (hFS t.mem).2.1 (hFS t.mem).2.2.1) fun t =>
+    (fun t => hscomp.of_isClosed_subset (hFS t.mem).2.1 (hFS t.mem).2.2.1) fun t =>
       (hFS t.mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
   exacts [⟨t, Subset.rfl, htu⟩, ⟨u, hut, Subset.rfl⟩]

27929793

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -56,7 +56,7 @@ open Set
 
 open Classical
 
-variable {E : Type _} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
+variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E]
   [TopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E}
 
 /-- **Krein-Milman lemma**: In a LCTVS, any nonempty compact set has an extreme point. -/

27929793

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) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
-
-! This file was ported from Lean 3 source module analysis.convex.krein_milman
-! leanprover-community/mathlib commit 279297937dede7b1b3451b7b0f1786352ad011fa
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Convex.Exposed
 import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
 
+#align_import analysis.convex.krein_milman from "leanprover-community/mathlib"@"279297937dede7b1b3451b7b0f1786352ad011fa"
+
 /-!
 # The Krein-Milman theorem

27929793

chore: fix grammar 1/3 (#5001)

All of these are doc fixes

d8caa37d

Diff

@@ -22,7 +22,7 @@ The lemma states that a nonempty compact set `s` has an extreme point. The proof
 1. Using Zorn's lemma, find a minimal nonempty closed `t` that is an extreme subset of `s`. We will
   show that `t` is a singleton, thus corresponding to an extreme point.
 2. By contradiction, `t` contains two distinct points `x` and `y`.
-3. With the (geometric) Hahn-Banach theorem, find an hyperplane that separates `x` and `y`.
+3. With the (geometric) Hahn-Banach theorem, find a hyperplane that separates `x` and `y`.
 4. Look at the extreme (actually exposed) subset of `t` obtained by going the furthest away from
   the separating hyperplane in the direction of `x`. It is nonempty, closed and an extreme subset
   of `s`.
@@ -33,7 +33,7 @@ The lemma states that a nonempty compact set `s` has an extreme point. The proof
 The theorem states that a compact convex set `s` is the closure of the convex hull of its extreme
 points. It is an almost immediate strengthening of the lemma. The proof goes:
 1. By contradiction, `s \ closure (convexHull ℝ (extremePoints ℝ s))` is nonempty, say with `x`.
-2. With the (geometric) Hahn-Banach theorem, find an hyperplane that separates `x` from
+2. With the (geometric) Hahn-Banach theorem, find a hyperplane that separates `x` from
   `closure (convexHull ℝ (extremePoints ℝ s))`.
 3. Look at the extreme (actually exposed) subset of
   `s \ closure (convexHull ℝ (extremePoints ℝ s))` obtained by going the furthest away from the

27929793

chore: add space after exacts (#4945)

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

bf799bb9

Diff

@@ -90,7 +90,7 @@ theorem IsCompact.has_extreme_point (hscomp : IsCompact s) (hsnemp : s.Nonempty)
     (fun t => isCompact_of_isClosed_subset hscomp (hFS t.mem).2.1 (hFS t.mem).2.2.1) fun t =>
       (hFS t.mem).2.1
   obtain htu | hut := hF.total t.mem u.mem
-  exacts[⟨t, Subset.rfl, htu⟩, ⟨u, hut, Subset.rfl⟩]
+  exacts [⟨t, Subset.rfl, htu⟩, ⟨u, hut, Subset.rfl⟩]
 #align is_compact.has_extreme_point IsCompact.has_extreme_point
 
 /-- **Krein-Milman theorem**: In a LCTVS, any compact convex set is the closure of the convex hull

27929793

feat: port Analysis.Convex.KreinMilman (#4347)

868ddb6b

`analysis.convex.krein_milman` ⟷ `Mathlib.Analysis.Convex.KreinMilman`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 814

analysis.convex.krein_milman ⟷ Mathlib.Analysis.Convex.KreinMilman

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 12 + 814

`analysis.convex.krein_milman` ⟷ `Mathlib.Analysis.Convex.KreinMilman`