fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
(#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
TotallyBounded t := by
/- Let `δ>0`, and `ε = δ/... | theorem | Topology | [
"Mathlib.Logic.Encodable.Pi",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.Topology.MetricSpace.Closeds",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Topology.MetricSpace.GromovHausdorffRealized",
"Mathlib.Topology.MetricSpace.Kuratowski"
] | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | totallyBounded | The Gromov-Hausdorff space is second countable. -/
instance : SecondCountableTopology GHSpace := by
refine secondCountable_of_countable_discretization fun δ δpos => ?_
let ε := 2 / 5 * δ
have εpos : 0 < ε := mul_pos (by simp) δpos
have : ∀ p : GHSpace, ∃ s : Set p.Rep, s.Finite ∧ univ ⊆ ⋃ x ∈ s, ball x ε := fun... |
AuxGluingStruct (A : Type) [MetricSpace A] : Type 1 where
Space : Type
metric : MetricSpace Space
embed : A → Space
isom : Isometry embed
attribute [local instance] AuxGluingStruct.metric | structure | Topology | [
"Mathlib.Logic.Encodable.Pi",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.Topology.MetricSpace.Closeds",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Topology.MetricSpace.GromovHausdorffRealized",
"Mathlib.Topology.MetricSpace.Kuratowski"
] | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | AuxGluingStruct | Auxiliary structure used to glue metric spaces below, recording an isometric embedding
of a type `A` in another metric space. |
auxGluing (n : ℕ) : AuxGluingStruct (X n) :=
Nat.recOn n default fun n Y =>
{ Space := GlueSpace Y.isom (isometry_optimalGHInjl (X n) (X (n + 1)))
metric := by infer_instance
embed :=
toGlueR Y.isom (isometry_optimalGHInjl (X n) (X (n + 1))) ∘ optimalGHInjr (X n) (X (n + 1))
isom := (toG... | def | Topology | [
"Mathlib.Logic.Encodable.Pi",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.Topology.MetricSpace.Closeds",
"Mathlib.Topology.MetricSpace.Completion",
"Mathlib.Topology.MetricSpace.GromovHausdorffRealized",
"Mathlib.Topology.MetricSpace.Kuratowski"
] | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | auxGluing | Auxiliary sequence of metric spaces, containing copies of `X 0`, ..., `X n`, where each
`X i` is glued to `X (i+1)` in an optimal way. The space at step `n+1` is obtained from the space
at step `n` by adding `X (n+1)`, glued in an optimal way to the `X n` already sitting there. |
private ProdSpaceFun : Type _ :=
(X ⊕ Y) × (X ⊕ Y) → ℝ | abbrev | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | ProdSpaceFun | null |
private Cb : Type _ :=
BoundedContinuousFunction ((X ⊕ Y) × (X ⊕ Y)) ℝ | abbrev | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | Cb | null |
private maxVar : ℝ≥0 :=
2 * ⟨diam (univ : Set X), diam_nonneg⟩ + 1 + 2 * ⟨diam (univ : Set Y), diam_nonneg⟩ | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | maxVar | null |
private one_le_maxVar : 1 ≤ maxVar X Y :=
calc
(1 : Real) = 2 * 0 + 1 + 2 * 0 := by simp
_ ≤ 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by gcongr <;> positivity | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | one_le_maxVar | null |
candidates : Set (ProdSpaceFun X Y) :=
{ f | (((((∀ x y : X, f (Sum.inl x, Sum.inl y) = dist x y) ∧
∀ x y : Y, f (Sum.inr x, Sum.inr y) = dist x y) ∧
∀ x y, f (x, y) = f (y, x)) ∧
∀ x y z, f (x, z) ≤ f (x, y) + f (y, z)) ∧
∀ x, f (x, x) = 0) ∧
∀ x y, f (x, y) ≤ maxVar X Y } | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates | The set of functions on `X ⊕ Y` that are candidates distances to realize the
minimum of the Hausdorff distances between `X` and `Y` in a coupling. |
private candidatesB : Set (Cb X Y) :=
{ f : Cb X Y | (f : _ → ℝ) ∈ candidates X Y } | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesB | Version of the set of candidates in bounded_continuous_functions, to apply Arzela-Ascoli. |
private maxVar_bound [CompactSpace X] [Nonempty X] [CompactSpace Y] [Nonempty Y] :
dist x y ≤ maxVar X Y :=
calc
dist x y ≤ diam (univ : Set (X ⊕ Y)) :=
dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
_ = diam (range inl ∪ range inr : Set (X ⊕ Y)) := by rw [range_inl_union_ra... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | maxVar_bound | null |
private candidates_symm (fA : f ∈ candidates X Y) : f (x, y) = f (y, x) :=
fA.1.1.1.2 x y | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_symm | null |
private candidates_triangle (fA : f ∈ candidates X Y) : f (x, z) ≤ f (x, y) + f (y, z) :=
fA.1.1.2 x y z | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_triangle | null |
private candidates_refl (fA : f ∈ candidates X Y) : f (x, x) = 0 :=
fA.1.2 x | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_refl | null |
private candidates_nonneg (fA : f ∈ candidates X Y) : 0 ≤ f (x, y) := by
have : 0 ≤ 2 * f (x, y) :=
calc
0 = f (x, x) := (candidates_refl fA).symm
_ ≤ f (x, y) + f (y, x) := candidates_triangle fA
_ = f (x, y) + f (x, y) := by rw [candidates_symm fA]
_ = 2 * f (x, y) := by ring
linarith | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_nonneg | null |
private candidates_dist_inl (fA : f ∈ candidates X Y) (x y : X) :
f (inl x, inl y) = dist x y :=
fA.1.1.1.1.1 x y | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_dist_inl | null |
private candidates_dist_inr (fA : f ∈ candidates X Y) (x y : Y) :
f (inr x, inr y) = dist x y :=
fA.1.1.1.1.2 x y | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_dist_inr | null |
private candidates_le_maxVar (fA : f ∈ candidates X Y) : f (x, y) ≤ maxVar X Y :=
fA.2 x y | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_le_maxVar | null |
private candidates_dist_bound (fA : f ∈ candidates X Y) :
∀ {x y : X ⊕ Y}, f (x, y) ≤ maxVar X Y * dist x y
| inl x, inl y =>
calc
f (inl x, inl y) = dist x y := candidates_dist_inl fA x y
_ = dist (α := X ⊕ Y) (inl x) (inl y) := by
rw [@Sum.dist_eq X Y]
rfl
_ = 1 * dist (α :... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_dist_bound | candidates are bounded by `maxVar X Y` |
private candidates_lipschitz_aux (fA : f ∈ candidates X Y) :
f (x, y) - f (z, t) ≤ 2 * maxVar X Y * dist (x, y) (z, t) :=
calc
f (x, y) - f (z, t) ≤ f (x, t) + f (t, y) - f (z, t) := by gcongr; exact candidates_triangle fA
_ ≤ f (x, z) + f (z, t) + f (t, y) - f (z, t) := by gcongr; exact candidates_triang... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_lipschitz_aux | Technical lemma to prove that candidates are Lipschitz |
private candidates_lipschitz (fA : f ∈ candidates X Y) :
LipschitzWith (2 * maxVar X Y) f := by
apply LipschitzWith.of_dist_le_mul
rintro ⟨x, y⟩ ⟨z, t⟩
rw [Real.dist_eq, abs_sub_le_iff]
use candidates_lipschitz_aux fA
rw [dist_comm]
exact candidates_lipschitz_aux fA | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidates_lipschitz | Candidates are Lipschitz |
private closed_candidatesB : IsClosed (candidatesB X Y) := by
have I1 : ∀ x y, IsClosed { f : Cb X Y | f (inl x, inl y) = dist x y } := fun x y =>
isClosed_eq continuous_eval_const continuous_const
have I2 : ∀ x y, IsClosed { f : Cb X Y | f (inr x, inr y) = dist x y } := fun x y =>
isClosed_eq continuous_ev... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | closed_candidatesB | To apply Arzela-Ascoli, we need to check that the set of candidates is closed and
equicontinuous. Equicontinuity follows from the Lipschitz control, we check closedness. |
HD (f : Cb X Y) :=
max (⨆ x, ⨅ y, f (inl x, inr y)) (⨆ y, ⨅ x, f (inl x, inr y))
/- We will show that `HD` is continuous on `BoundedContinuousFunction`s, to deduce that its
minimum on the compact set `candidatesB` is attained. Since it is defined in terms of
infimum and supremum on `ℝ`, which is only conditionally co... | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD | We will then choose the candidate minimizing the Hausdorff distance. Except that we are not
in a metric space setting, so we need to define our custom version of Hausdorff distance,
called `HD`, and prove its basic properties. |
HD_below_aux1 {f : Cb X Y} (C : ℝ) {x : X} :
BddBelow (range fun y : Y => f (inl x, inr y) + C) :=
let ⟨cf, hcf⟩ := f.isBounded_range.bddBelow
⟨cf + C, forall_mem_range.2 fun _ => add_le_add_right ((fun x => hcf (mem_range_self x)) _) _⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_below_aux1 | null |
private HD_bound_aux1 [Nonempty Y] (f : Cb X Y) (C : ℝ) :
BddAbove (range fun x : X => ⨅ y, f (inl x, inr y) + C) := by
obtain ⟨Cf, hCf⟩ := f.isBounded_range.bddAbove
refine ⟨Cf + C, forall_mem_range.2 fun x => ?_⟩
calc
⨅ y, f (inl x, inr y) + C ≤ f (inl x, inr default) + C := ciInf_le (HD_below_aux1 C) d... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_bound_aux1 | null |
HD_below_aux2 {f : Cb X Y} (C : ℝ) {y : Y} :
BddBelow (range fun x : X => f (inl x, inr y) + C) :=
let ⟨cf, hcf⟩ := f.isBounded_range.bddBelow
⟨cf + C, forall_mem_range.2 fun _ => add_le_add_right ((fun x => hcf (mem_range_self x)) _) _⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_below_aux2 | null |
private HD_bound_aux2 [Nonempty X] (f : Cb X Y) (C : ℝ) :
BddAbove (range fun y : Y => ⨅ x, f (inl x, inr y) + C) := by
obtain ⟨Cf, hCf⟩ := f.isBounded_range.bddAbove
refine ⟨Cf + C, forall_mem_range.2 fun y => ?_⟩
calc
⨅ x, f (inl x, inr y) + C ≤ f (inl default, inr y) + C := ciInf_le (HD_below_aux2 C) d... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_bound_aux2 | null |
private HD_lipschitz_aux1 (f g : Cb X Y) :
(⨆ x, ⨅ y, f (inl x, inr y)) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g := by
obtain ⟨cg, hcg⟩ := g.isBounded_range.bddBelow
have Hcg : ∀ x, cg ≤ g x := fun x => hcg (mem_range_self x)
obtain ⟨cf, hcf⟩ := f.isBounded_range.bddBelow
have Hcf : ∀ x, cf ≤ f x := fun x ... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_lipschitz_aux1 | null |
private HD_lipschitz_aux2 (f g : Cb X Y) :
(⨆ y, ⨅ x, f (inl x, inr y)) ≤ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g := by
obtain ⟨cg, hcg⟩ := g.isBounded_range.bddBelow
have Hcg : ∀ x, cg ≤ g x := fun x => hcg (mem_range_self x)
obtain ⟨cf, hcf⟩ := f.isBounded_range.bddBelow
have Hcf : ∀ x, cf ≤ f x := fun x ... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_lipschitz_aux2 | null |
private HD_lipschitz_aux3 (f g : Cb X Y) :
HD f ≤ HD g + dist f g :=
max_le (le_trans (HD_lipschitz_aux1 f g) (add_le_add_right (le_max_left _ _) _))
(le_trans (HD_lipschitz_aux2 f g) (add_le_add_right (le_max_right _ _) _)) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_lipschitz_aux3 | null |
private HD_continuous : Continuous (HD : Cb X Y → ℝ) :=
LipschitzWith.continuous (LipschitzWith.of_le_add HD_lipschitz_aux3) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_continuous | Conclude that `HD`, being Lipschitz, is continuous |
private isCompact_candidatesB : IsCompact (candidatesB X Y) := by
refine arzela_ascoli₂
(Icc 0 (maxVar X Y) : Set ℝ) isCompact_Icc (candidatesB X Y) closed_candidatesB ?_ ?_
· rintro f ⟨x1, x2⟩ hf
simp only [Set.mem_Icc]
exact ⟨candidates_nonneg hf, candidates_le_maxVar hf⟩
· refine equicontinuous_o... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | isCompact_candidatesB | Compactness of candidates (in `BoundedContinuousFunction`s) follows. |
candidatesBOfCandidates (f : ProdSpaceFun X Y) (fA : f ∈ candidates X Y) : Cb X Y :=
BoundedContinuousFunction.mkOfCompact ⟨f, (candidates_lipschitz fA).continuous⟩ | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesBOfCandidates | candidates give rise to elements of `BoundedContinuousFunction`s |
candidatesBOfCandidates_mem (f : ProdSpaceFun X Y) (fA : f ∈ candidates X Y) :
candidatesBOfCandidates f fA ∈ candidatesB X Y :=
fA
variable [Nonempty X] [Nonempty Y] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesBOfCandidates_mem | null |
private dist_mem_candidates :
(fun p : (X ⊕ Y) × (X ⊕ Y) => dist p.1 p.2) ∈ candidates X Y := by
simp_rw [candidates, Set.mem_setOf_eq, dist_comm, dist_triangle, dist_self, maxVar_bound,
forall_const, and_true]
exact ⟨fun x y => rfl, fun x y => rfl⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | dist_mem_candidates | The distance on `X ⊕ Y` is a candidate |
candidatesBDist (X : Type u) (Y : Type v) [MetricSpace X] [CompactSpace X] [Nonempty X]
[MetricSpace Y] [CompactSpace Y] [Nonempty Y] : Cb X Y :=
candidatesBOfCandidates _ dist_mem_candidates | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesBDist | The distance on `X ⊕ Y` as a candidate |
candidatesBDist_mem_candidatesB :
candidatesBDist X Y ∈ candidatesB X Y :=
candidatesBOfCandidates_mem _ _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesBDist_mem_candidatesB | null |
private candidatesB_nonempty : (candidatesB X Y).Nonempty :=
⟨_, candidatesBDist_mem_candidatesB⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | candidatesB_nonempty | null |
HD_candidatesBDist_le :
HD (candidatesBDist X Y) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := by
refine max_le (ciSup_le fun x => ?_) (ciSup_le fun y => ?_)
· have A : ⨅ y, candidatesBDist X Y (inl x, inr y) ≤ candidatesBDist X Y (inl x, inr default) :=
ciInf_le (by simpa using HD_below_aux1 0) defa... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_candidatesBDist_le | Explicit bound on `HD (dist)`. This means that when looking for minimizers it will
be sufficient to look for functions with `HD(f)` bounded by this bound. |
private exists_minimizer : ∃ f ∈ candidatesB X Y, ∀ g ∈ candidatesB X Y, HD f ≤ HD g :=
isCompact_candidatesB.exists_isMinOn candidatesB_nonempty HD_continuous.continuousOn | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | exists_minimizer | null |
private optimalGHDist : Cb X Y :=
Classical.choose (exists_minimizer X Y) | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | optimalGHDist | null |
private optimalGHDist_mem_candidatesB : optimalGHDist X Y ∈ candidatesB X Y := by
cases Classical.choose_spec (exists_minimizer X Y)
assumption | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | optimalGHDist_mem_candidatesB | null |
private HD_optimalGHDist_le (g : Cb X Y) (hg : g ∈ candidatesB X Y) :
HD (optimalGHDist X Y) ≤ HD g :=
let ⟨_, Z2⟩ := Classical.choose_spec (exists_minimizer X Y)
Z2 g hg | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | HD_optimalGHDist_le | null |
premetricOptimalGHDist : PseudoMetricSpace (X ⊕ Y) where
dist p q := optimalGHDist X Y (p, q)
dist_self _ := candidates_refl (optimalGHDist_mem_candidatesB X Y)
dist_comm _ _ := candidates_symm (optimalGHDist_mem_candidatesB X Y)
dist_triangle _ _ _ := candidates_triangle (optimalGHDist_mem_candidatesB X Y)
att... | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | premetricOptimalGHDist | With the optimal candidate, construct a premetric space structure on `X ⊕ Y`, on which the
predistance is given by the candidate. Then, we will identify points at `0` predistance
to obtain a genuine metric space. |
OptimalGHCoupling : Type _ :=
@SeparationQuotient (X ⊕ Y) (premetricOptimalGHDist X Y).toUniformSpace.toTopologicalSpace | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | OptimalGHCoupling | A metric space which realizes the optimal coupling between `X` and `Y` |
optimalGHInjl (x : X) : OptimalGHCoupling X Y :=
Quotient.mk'' (inl x) | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | optimalGHInjl | Injection of `X` in the optimal coupling between `X` and `Y` |
isometry_optimalGHInjl : Isometry (optimalGHInjl X Y) :=
Isometry.of_dist_eq fun _ _ => candidates_dist_inl (optimalGHDist_mem_candidatesB X Y) _ _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | isometry_optimalGHInjl | The injection of `X` in the optimal coupling between `X` and `Y` is an isometry. |
optimalGHInjr (y : Y) : OptimalGHCoupling X Y :=
Quotient.mk'' (inr y) | def | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | optimalGHInjr | Injection of `Y` in the optimal coupling between `X` and `Y` |
isometry_optimalGHInjr : Isometry (optimalGHInjr X Y) :=
Isometry.of_dist_eq fun _ _ => candidates_dist_inr (optimalGHDist_mem_candidatesB X Y) _ _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | isometry_optimalGHInjr | The injection of `Y` in the optimal coupling between `X` and `Y` is an isometry. |
compactSpace_optimalGHCoupling : CompactSpace (OptimalGHCoupling X Y) := ⟨by
rw [← range_quotient_mk']
exact isCompact_range (continuous_sum_dom.2
⟨(isometry_optimalGHInjl X Y).continuous, (isometry_optimalGHInjr X Y).continuous⟩)⟩ | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | compactSpace_optimalGHCoupling | The optimal coupling between two compact spaces `X` and `Y` is still a compact space |
hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
refine le_trans (le_of_forall_gt_imp_ge_of_dense fun r hr => ?_) (HD_optimalGHDist_le X Y f h)
have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y),... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli",
"Mathlib.Topology.ContinuousMap.Bounded.Normed",
"Mathlib.Topology.MetricSpace.Gluing",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | hausdorffDist_optimal_le_HD | For any candidate `f`, `HD(f)` is larger than or equal to the Hausdorff distance in the
optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. |
@[irreducible] noncomputable dimH (s : Set X) : ℝ≥0∞ := by
borelize X; exact ⨆ (d : ℝ≥0) (_ : @hausdorffMeasure X _ _ ⟨rfl⟩ d s = ∞), d
/-! | def | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH | Hausdorff dimension of a set in an (e)metric space. |
dimH_def (s : Set X) : dimH s = ⨆ (d : ℝ≥0) (_ : μH[d] s = ∞), (d : ℝ≥0∞) := by
borelize X; rw [dimH] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_def | Unfold the definition of `dimH` using `[MeasurableSpace X] [BorelSpace X]` from the
environment. |
hausdorffMeasure_of_lt_dimH {s : Set X} {d : ℝ≥0} (h : ↑d < dimH s) : μH[d] s = ∞ := by
simp only [dimH_def, lt_iSup_iff] at h
rcases h with ⟨d', hsd', hdd'⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at hdd'
exact top_unique (hsd' ▸ hausdorffMeasure_mono hdd'.le _) | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | hausdorffMeasure_of_lt_dimH | null |
dimH_le {s : Set X} {d : ℝ≥0∞} (H : ∀ d' : ℝ≥0, μH[d'] s = ∞ → ↑d' ≤ d) : dimH s ≤ d :=
(dimH_def s).trans_le <| iSup₂_le H | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_le | null |
dimH_le_of_hausdorffMeasure_ne_top {s : Set X} {d : ℝ≥0} (h : μH[d] s ≠ ∞) : dimH s ≤ d :=
le_of_not_gt <| mt hausdorffMeasure_of_lt_dimH h | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_le_of_hausdorffMeasure_ne_top | null |
le_dimH_of_hausdorffMeasure_eq_top {s : Set X} {d : ℝ≥0} (h : μH[d] s = ∞) :
↑d ≤ dimH s := by
rw [dimH_def]; exact le_iSup₂ (α := ℝ≥0∞) d h | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | le_dimH_of_hausdorffMeasure_eq_top | null |
hausdorffMeasure_of_dimH_lt {s : Set X} {d : ℝ≥0} (h : dimH s < d) : μH[d] s = 0 := by
rw [dimH_def] at h
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨d', hsd', hd'd⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at hd'd
exact (hausdorffMeasure_zero_or_top hd'd s).resolve_right fun h₂ => hsd'.not_ge <|
... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | hausdorffMeasure_of_dimH_lt | null |
measure_zero_of_dimH_lt {μ : Measure X} {d : ℝ≥0} (h : μ ≪ μH[d]) {s : Set X}
(hd : dimH s < d) : μ s = 0 :=
h <| hausdorffMeasure_of_dimH_lt hd | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | measure_zero_of_dimH_lt | null |
le_dimH_of_hausdorffMeasure_ne_zero {s : Set X} {d : ℝ≥0} (h : μH[d] s ≠ 0) : ↑d ≤ dimH s :=
le_of_not_gt <| mt hausdorffMeasure_of_dimH_lt h | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | le_dimH_of_hausdorffMeasure_ne_zero | null |
dimH_of_hausdorffMeasure_ne_zero_ne_top {d : ℝ≥0} {s : Set X} (h : μH[d] s ≠ 0)
(h' : μH[d] s ≠ ∞) : dimH s = d :=
le_antisymm (dimH_le_of_hausdorffMeasure_ne_top h') (le_dimH_of_hausdorffMeasure_ne_zero h) | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_of_hausdorffMeasure_ne_zero_ne_top | null |
dimH_eq_iInf (s : Set X) : dimH s = ⨅ (d : ℝ≥0) (_ : μH[d] s = 0), (d : ℝ≥0∞) := by
apply le_antisymm
· rw [dimH_def]
simp only [le_iInf_iff, iSup_le_iff, ENNReal.coe_le_coe]
intro i hi j hj
by_contra! hij
simpa [hi, hj] using hausdorffMeasure_mono hij.le s
· by_contra! h
rcases ENNReal.lt_iff... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_eq_iInf | The Hausdorff dimension of a set `s` is the infimum of all `d : ℝ≥0` such that the
`d`-dimensional Hausdorff measure of `s` is zero. This infimum is taken in `ℝ≥0∞`.
This gives an equivalent definition of the Hausdorff dimension. |
@[mono]
dimH_mono {s t : Set X} (h : s ⊆ t) : dimH s ≤ dimH t := by
borelize X
exact dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top <| top_unique <| hd ▸ measure_mono h | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_mono | null |
dimH_subsingleton {s : Set X} (h : s.Subsingleton) : dimH s = 0 := by
borelize X
apply le_antisymm _ (zero_le _)
refine dimH_le_of_hausdorffMeasure_ne_top ?_
exact ((hausdorffMeasure_le_one_of_subsingleton h le_rfl).trans_lt ENNReal.one_lt_top).ne
alias Set.Subsingleton.dimH_zero := dimH_subsingleton
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_subsingleton | null |
dimH_empty : dimH (∅ : Set X) = 0 :=
subsingleton_empty.dimH_zero
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_empty | null |
dimH_singleton (x : X) : dimH ({x} : Set X) = 0 :=
subsingleton_singleton.dimH_zero
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_singleton | null |
dimH_iUnion {ι : Sort*} [Countable ι] (s : ι → Set X) :
dimH (⋃ i, s i) = ⨆ i, dimH (s i) := by
borelize X
refine le_antisymm (dimH_le fun d hd => ?_) (iSup_le fun i => dimH_mono <| subset_iUnion _ _)
contrapose! hd
have : ∀ i, μH[d] (s i) = 0 := fun i =>
hausdorffMeasure_of_dimH_lt ((le_iSup (fun i => ... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_iUnion | null |
dimH_bUnion {s : Set ι} (hs : s.Countable) (t : ι → Set X) :
dimH (⋃ i ∈ s, t i) = ⨆ i ∈ s, dimH (t i) := by
haveI := hs.toEncodable
rw [biUnion_eq_iUnion, dimH_iUnion, ← iSup_subtype'']
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_bUnion | null |
dimH_sUnion {S : Set (Set X)} (hS : S.Countable) : dimH (⋃₀ S) = ⨆ s ∈ S, dimH s := by
rw [sUnion_eq_biUnion, dimH_bUnion hS]
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_sUnion | null |
dimH_union (s t : Set X) : dimH (s ∪ t) = max (dimH s) (dimH t) := by
rw [union_eq_iUnion, dimH_iUnion, iSup_bool_eq, cond, cond] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_union | null |
dimH_countable {s : Set X} (hs : s.Countable) : dimH s = 0 :=
biUnion_of_singleton s ▸ by simp only [dimH_bUnion hs, dimH_singleton, ENNReal.iSup_zero]
alias Set.Countable.dimH_zero := dimH_countable | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_countable | null |
dimH_finite {s : Set X} (hs : s.Finite) : dimH s = 0 :=
hs.countable.dimH_zero
alias Set.Finite.dimH_zero := dimH_finite
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_finite | null |
dimH_coe_finset (s : Finset X) : dimH (s : Set X) = 0 :=
s.finite_toSet.dimH_zero
alias Finset.dimH_zero := dimH_coe_finset
/-! | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_coe_finset | null |
exists_mem_nhdsWithin_lt_dimH_of_lt_dimH {s : Set X} {r : ℝ≥0∞} (h : r < dimH s) :
∃ x ∈ s, ∀ t ∈ 𝓝[s] x, r < dimH t := by
contrapose! h; choose! t htx htr using h
rcases countable_cover_nhdsWithin htx with ⟨S, hSs, hSc, hSU⟩
calc
dimH s ≤ dimH (⋃ x ∈ S, t x) := dimH_mono hSU
_ = ⨆ x ∈ S, dimH (t x) ... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | exists_mem_nhdsWithin_lt_dimH_of_lt_dimH | If `r` is less than the Hausdorff dimension of a set `s` in an (extended) metric space with
second countable topology, then there exists a point `x ∈ s` such that every neighborhood
`t` of `x` within `s` has Hausdorff dimension greater than `r`. |
bsupr_limsup_dimH (s : Set X) : ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets = dimH s := by
refine le_antisymm (iSup₂_le fun x _ => ?_) ?_
· refine limsup_le_of_le isCobounded_le_of_bot ?_
exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩
· refine le_of_forall_lt_imp_le_of_dense fun r hr =... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | bsupr_limsup_dimH | In an (extended) metric space with second countable topology, the Hausdorff dimension
of a set `s` is the supremum over `x ∈ s` of the limit superiors of `dimH t` along
`(𝓝[s] x).smallSets`. |
iSup_limsup_dimH (s : Set X) : ⨆ x, limsup dimH (𝓝[s] x).smallSets = dimH s := by
refine le_antisymm (iSup_le fun x => ?_) ?_
· refine limsup_le_of_le isCobounded_le_of_bot ?_
exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩
· rw [← bsupr_limsup_dimH]; exact iSup₂_le_iSup _ _ | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | iSup_limsup_dimH | In an (extended) metric space with second countable topology, the Hausdorff dimension
of a set `s` is the supremum over all `x` of the limit superiors of `dimH t` along
`(𝓝[s] x).smallSets`. |
HolderOnWith.dimH_image_le (h : HolderOnWith C r f s) (hr : 0 < r) :
dimH (f '' s) ≤ dimH s / r := by
borelize X Y
refine dimH_le fun d hd => ?_
have := h.hausdorffMeasure_image_le hr d.coe_nonneg
rw [hd, ← ENNReal.coe_rpow_of_nonneg _ d.coe_nonneg, top_le_iff] at this
have Hrd : μH[(r * d : ℝ≥0)] s = ⊤ :... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | HolderOnWith.dimH_image_le | If `f` is a Hölder continuous map with exponent `r > 0`, then `dimH (f '' s) ≤ dimH s / r`. |
dimH_image_le (h : HolderWith C r f) (hr : 0 < r) (s : Set X) :
dimH (f '' s) ≤ dimH s / r :=
(h.holderOnWith s).dimH_image_le hr | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image_le | If `f : X → Y` is Hölder continuous with a positive exponent `r`, then the Hausdorff dimension
of the image of a set `s` is at most `dimH s / r`. |
dimH_range_le (h : HolderWith C r f) (hr : 0 < r) :
dimH (range f) ≤ dimH (univ : Set X) / r :=
@image_univ _ _ f ▸ h.dimH_image_le hr univ | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_range_le | If `f` is a Hölder continuous map with exponent `r > 0`, then the Hausdorff dimension of its
range is at most the Hausdorff dimension of its domain divided by `r`. |
dimH_image_le_of_locally_holder_on [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y}
(hr : 0 < r) {s : Set X} (hf : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, HolderOnWith C r f t) :
dimH (f '' s) ≤ dimH s / r := by
choose! C t htn hC using hf
rcases countable_cover_nhdsWithin htn with ⟨u, hus, huc, huU⟩
replace ... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image_le_of_locally_holder_on | If `s` is a set in a space `X` with second countable topology and `f : X → Y` is Hölder
continuous in a neighborhood within `s` of every point `x ∈ s` with the same positive exponent `r`
but possibly different coefficients, then the Hausdorff dimension of the image `f '' s` is at most
the Hausdorff dimension of `s` div... |
dimH_range_le_of_locally_holder_on [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y}
(hr : 0 < r) (hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, HolderOnWith C r f s) :
dimH (range f) ≤ dimH (univ : Set X) / r := by
rw [← image_univ]
refine dimH_image_le_of_locally_holder_on hr fun x _ => ?_
simpa only [exists_pro... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_range_le_of_locally_holder_on | If `f : X → Y` is Hölder continuous in a neighborhood of every point `x : X` with the same
positive exponent `r` but possibly different coefficients, then the Hausdorff dimension of the range
of `f` is at most the Hausdorff dimension of `X` divided by `r`. |
LipschitzOnWith.dimH_image_le (h : LipschitzOnWith K f s) : dimH (f '' s) ≤ dimH s := by
simpa using h.holderOnWith.dimH_image_le zero_lt_one | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | LipschitzOnWith.dimH_image_le | If `f : X → Y` is Lipschitz continuous on `s`, then `dimH (f '' s) ≤ dimH s`. |
dimH_image_le (h : LipschitzWith K f) (s : Set X) : dimH (f '' s) ≤ dimH s :=
h.lipschitzOnWith.dimH_image_le | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image_le | If `f` is a Lipschitz continuous map, then `dimH (f '' s) ≤ dimH s`. |
dimH_range_le (h : LipschitzWith K f) : dimH (range f) ≤ dimH (univ : Set X) :=
@image_univ _ _ f ▸ h.dimH_image_le univ | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_range_le | If `f` is a Lipschitz continuous map, then the Hausdorff dimension of its range is at most the
Hausdorff dimension of its domain. |
dimH_image_le_of_locally_lipschitzOn [SecondCountableTopology X] {f : X → Y} {s : Set X}
(hf : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t) : dimH (f '' s) ≤ dimH s := by
have : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t := by
simpa only [holderOnWith_one] using hf
simpa only [ENNR... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image_le_of_locally_lipschitzOn | If `s` is a set in an extended metric space `X` with second countable topology and `f : X → Y`
is Lipschitz in a neighborhood within `s` of every point `x ∈ s`, then the Hausdorff dimension of
the image `f '' s` is at most the Hausdorff dimension of `s`. |
dimH_range_le_of_locally_lipschitzOn [SecondCountableTopology X] {f : X → Y}
(hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s) :
dimH (range f) ≤ dimH (univ : Set X) := by
rw [← image_univ]
refine dimH_image_le_of_locally_lipschitzOn fun x _ => ?_
simpa only [exists_prop, nhdsWithin_univ] using... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_range_le_of_locally_lipschitzOn | If `f : X → Y` is Lipschitz in a neighborhood of each point `x : X`, then the Hausdorff
dimension of `range f` is at most the Hausdorff dimension of `X`. |
dimH_preimage_le (hf : AntilipschitzWith K f) (s : Set Y) : dimH (f ⁻¹' s) ≤ dimH s := by
borelize X Y
refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_
have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
rw [hd, top_le_iff] at this
contrapose! this
exact ENNReal.mul_ne_top (by simp) this | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_preimage_le | null |
le_dimH_image (hf : AntilipschitzWith K f) (s : Set X) : dimH s ≤ dimH (f '' s) :=
calc
dimH s ≤ dimH (f ⁻¹' (f '' s)) := dimH_mono (subset_preimage_image _ _)
_ ≤ dimH (f '' s) := hf.dimH_preimage_le _ | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | le_dimH_image | null |
Isometry.dimH_image (hf : Isometry f) (s : Set X) : dimH (f '' s) = dimH s :=
le_antisymm (hf.lipschitz.dimH_image_le _) (hf.antilipschitz.le_dimH_image _) | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | Isometry.dimH_image | null |
@[simp]
dimH_image (e : X ≃ᵢ Y) (s : Set X) : dimH (e '' s) = dimH s :=
e.isometry.dimH_image s
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image | null |
dimH_preimage (e : X ≃ᵢ Y) (s : Set Y) : dimH (e ⁻¹' s) = dimH s := by
rw [← e.image_symm, e.symm.dimH_image] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_preimage | null |
dimH_univ (e : X ≃ᵢ Y) : dimH (univ : Set X) = dimH (univ : Set Y) := by
rw [← e.dimH_preimage univ, preimage_univ] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ | null |
@[simp]
dimH_image (e : E ≃L[𝕜] F) (s : Set E) : dimH (e '' s) = dimH s :=
le_antisymm (e.lipschitz.dimH_image_le s) <| by
simpa only [e.symm_image_image] using e.symm.lipschitz.dimH_image_le (e '' s)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_image | null |
dimH_preimage (e : E ≃L[𝕜] F) (s : Set F) : dimH (e ⁻¹' s) = dimH s := by
rw [← e.image_symm_eq_preimage, e.symm.dimH_image] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_preimage | null |
dimH_univ (e : E ≃L[𝕜] F) : dimH (univ : Set E) = dimH (univ : Set F) := by
rw [← e.dimH_preimage, preimage_univ] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ | null |
dimH_ball_pi (x : ι → ℝ) {r : ℝ} (hr : 0 < r) :
dimH (Metric.ball x r) = Fintype.card ι := by
cases isEmpty_or_nonempty ι
· rwa [dimH_subsingleton, eq_comm, Nat.cast_eq_zero, Fintype.card_eq_zero_iff]
exact fun x _ y _ => Subsingleton.elim x y
· rw [← ENNReal.coe_natCast]
have : μH[Fintype.card ι] (Me... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_ball_pi | null |
dimH_ball_pi_fin {n : ℕ} (x : Fin n → ℝ) {r : ℝ} (hr : 0 < r) :
dimH (Metric.ball x r) = n := by rw [dimH_ball_pi x hr, Fintype.card_fin] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_ball_pi_fin | null |
dimH_univ_pi (ι : Type*) [Fintype ι] : dimH (univ : Set (ι → ℝ)) = Fintype.card ι := by
simp only [← Metric.iUnion_ball_nat_succ (0 : ι → ℝ), dimH_iUnion,
dimH_ball_pi _ (Nat.cast_add_one_pos _), iSup_const] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ_pi | null |
dimH_univ_pi_fin (n : ℕ) : dimH (univ : Set (Fin n → ℝ)) = n := by
rw [dimH_univ_pi, Fintype.card_fin] | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ_pi_fin | null |
dimH_of_mem_nhds {x : E} {s : Set E} (h : s ∈ 𝓝 x) : dimH s = finrank ℝ E := by
have e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ :=
ContinuousLinearEquiv.ofFinrankEq (Module.finrank_fin_fun ℝ).symm
rw [← e.dimH_image]
refine le_antisymm ?_ ?_
· exact (dimH_mono (subset_univ _)).trans_eq (dimH_univ_pi_fin _)
· have... | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_of_mem_nhds | null |
dimH_of_nonempty_interior {s : Set E} (h : (interior s).Nonempty) : dimH s = finrank ℝ E :=
let ⟨_, hx⟩ := h
dimH_of_mem_nhds (mem_interior_iff_mem_nhds.1 hx)
variable (E) | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_of_nonempty_interior | null |
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