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/5`. For each `p`, we construct a finite subset `s p` of `p`, which
is `ε`-dense and has cardinality at most `K n`. Encoding the mutual distances of points
in `s p`, up to `ε`, we will get a map `F` associating to `p` finitely many data, and making
it possible to reconstruct `p` up to `ε`. This is enough to prove total boundedness. -/
refine Metric.totallyBounded_of_finite_discretization fun δ δpos => ?_
let ε := 1 / 5 * δ
have εpos : 0 < ε := mul_pos (by simp) δpos
rcases Metric.tendsto_atTop.1 ulim ε εpos with ⟨n, hn⟩
have u_le_ε : u n ≤ ε := by
have := hn n le_rfl
simp only [Real.dist_eq, add_zero, sub_eq_add_neg, neg_zero] at this
exact le_of_lt (lt_of_le_of_lt (le_abs_self _) this)
have :
∀ p : GHSpace,
∃ s : Set p.Rep, ∃ N ≤ K n, ∃ _ : Equiv s (Fin N), p ∈ t → univ ⊆ ⋃ x ∈ s, ball x (u n) := by
intro p
by_cases hp : p ∉ t
· have : Nonempty (Equiv (∅ : Set p.Rep) (Fin 0)) := by
rw [← Fintype.card_eq, card_empty, Fintype.card_fin]
use ∅, 0, bot_le, this.some
exact fun hp' => (hp hp').elim
· rcases hcov _ (Set.not_notMem.1 hp) n with ⟨s, ⟨scard, scover⟩⟩
rcases Cardinal.lt_aleph0.1 (lt_of_le_of_lt scard (Cardinal.nat_lt_aleph0 _)) with ⟨N, hN⟩
rw [hN, Nat.cast_le] at scard
have : #s = #(Fin N) := by rw [hN, Cardinal.mk_fin]
obtain ⟨E⟩ := Quotient.exact this
use s, N, scard, E
simp only [scover, imp_true_iff]
choose s N hN E hs using this
let M := ⌊ε⁻¹ * max C 0⌋₊
let F : GHSpace → Σ k : Fin (K n).succ, Fin k → Fin k → Fin M.succ := fun p =>
⟨⟨N p, lt_of_le_of_lt (hN p) (Nat.lt_succ_self _)⟩, fun a b =>
⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊,
(min_le_left _ _).trans_lt (Nat.lt_succ_self _)⟩⟩
refine ⟨_, ?_, fun p => F p, ?_⟩
· infer_instance
rintro ⟨p, pt⟩ ⟨q, qt⟩ hpq
have Npq : N p = N q := Fin.ext_iff.1 (Sigma.mk.inj_iff.1 hpq).1
let Ψ : s p → s q := fun x => (E q).symm (Fin.cast Npq ((E p) x))
let Φ : s p → q.Rep := fun x => Ψ x
have main : ghDist p.Rep q.Rep ≤ ε + ε / 2 + ε := by
refine ghDist_le_of_approx_subsets Φ ?_ ?_ ?_
· show ∀ x : p.Rep, ∃ y ∈ s p, dist x y ≤ ε
intro x
have : x ∈ ⋃ y ∈ s p, ball y (u n) := (hs p pt) (mem_univ _)
rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
... | 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 p => by
simpa only [subset_univ, true_and] using
finite_cover_balls_of_compact (X := p.Rep) isCompact_univ εpos
-- for each `p`, `s p` is a finite `ε`-dense subset of `p` (or rather the metric space
-- `p.rep` representing `p`)
choose s hs using this
have : ∀ p : GHSpace, ∀ t : Set p.Rep, t.Finite → ∃ n : ℕ, ∃ _ : Equiv t (Fin n), True := by
intro p t ht
let _ : Fintype t := Finite.fintype ht
exact ⟨Fintype.card t, Fintype.equivFin t, trivial⟩
choose N e _ using this
-- cardinality of the nice finite subset `s p` of `p.rep`, called `N p`
let N := fun p : GHSpace => N p (s p) (hs p).1
-- equiv from `s p`, a nice finite subset of `p.rep`, to `Fin (N p)`, called `E p`
let E := fun p : GHSpace => e p (s p) (hs p).1
-- A function `F` associating to `p : GHSpace` the data of all distances between points
-- in the `ε`-dense set `s p`.
let F : GHSpace → Σ n : ℕ, Fin n → Fin n → ℤ := fun p =>
⟨N p, fun a b => ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
refine ⟨Σ n, Fin n → Fin n → ℤ, by infer_instance, F, fun p q hpq => ?_⟩
/- As the target space of F is countable, it suffices to show that two points
`p` and `q` with `F p = F q` are at distance `≤ δ`.
For this, we construct a map `Φ` from `s p ⊆ p.rep` (representing `p`)
to `q.rep` (representing `q`) which is almost an isometry on `s p`, and
with image `s q`. For this, we compose the identification of `s p` with `Fin (N p)`
and the inverse of the identification of `s q` with `Fin (N q)`. Together with
the fact that `N p = N q`, this constructs `Ψ` between `s p` and `s q`, and then
composing with the canonical inclusion we get `Φ`. -/
have Npq : N p = N q := (Sigma.mk.inj_iff.1 hpq).1
let Ψ : s p → s q := fun x => (E q).symm (Fin.cast Npq ((E p) x))
let Φ : s p → q.Rep := fun x => Ψ x
-- Use the almost isometry `Φ` to show that `p.rep` and `q.rep`
-- are within controlled Gromov-Hausdorff distance.
have main : ghDist p.Rep q.Rep ≤ ε + ε / 2 + ε := by
refine ghDist_le_of_approx_subsets Φ ?_ ?_ ?_
· show ∀ x : p.Rep, ∃ y ∈ s p, dist x y ≤ ε
-- by construction, `s p` is `ε`-dense
intro x
have : x ∈ ⋃ y ∈ s p, ball y ε := (hs p).2 (mem_univ _)
rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
exact ⟨y, ys, le_of_lt hy⟩
· show ∀ x : q.Rep, ∃ z : s p, dist x (Φ z) ≤ ε
-- by construction, `s q` is `ε`-dense, and it is the range of `Φ`
intro x
have : x ∈ ⋃ y ∈ s q, ball y ε := (hs q).2 (mem_univ _)
rcases mem_iUnion₂.1 this with ⟨y, ys, hy⟩
let i : ℕ := E q ⟨y, ys⟩
let hi := ((E q) ⟨y, ys⟩).is_lt
have ihi_eq : (⟨i, hi⟩ : Fin (N q)) = (E q) ⟨y, ys⟩ := by rw [Fin.ext_iff, Fin.val_mk]
have hiq : i < N q := hi
have hip : i < N p := by rwa [Npq.symm] at hiq
let z := (E p).symm ⟨i, hip⟩
use z
have C1 : (E p) z = ⟨i, hip⟩ := (E p).apply_symm_apply ⟨i, hip⟩
have C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩ := rfl
have C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩ := by
rw [ihi_eq]; exact (E q).symm_apply_apply ⟨y, ys⟩
have : Φ z = y := by simp only [Φ, Ψ]; rw [C1, C2, C3]
rw [this]
exact le_of_lt hy
· show ∀ x y : s p, |dist x y - dist (Φ x) (Φ y)| ≤ ε
/- the distance between `x` and `y` is encoded in `F p`, and the distance between
`Φ x` and `Φ y` (two points of `s q`) is encoded in `F q`, all this up to `ε`.
As `F p = F q`, the distances are almost equal. -/
intro x y
-- introduce `i`, that codes both `x` and `Φ x` in `Fin (N p) = Fin (N q)`
let i : ℕ := E p x
have hip : i < N p := ((E p) x).2
have hiq : i < N q := by rwa [Npq] at hip
have i' : i = (E q) (Ψ x) := by simp only [i, Ψ, Equiv.apply_symm_apply, Fin.coe_cast]
-- introduce `j`, that codes both `y` and `Φ y` in `Fin (N p) = Fin (N q)`
let j : ℕ := E p y
have hjp : j < N p := ((E p) y).2
have hjq : j < N q := by rwa [Npq] at hjp
have j' : j = ((E q) (Ψ y)).1 := by
simp only [j, Ψ, Equiv.apply_symm_apply, Fin.coe_cast]
-- Express `dist x y` in terms of `F p`
have : (F p).2 ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋ := by
simp only [F, (E p).symm_apply_apply]
have Ap : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = ⌊ε⁻¹ * dist x y⌋ := by rw [← this]
-- Express `dist (Φ x) (Φ y)` in terms of `F q`
have : (F q).2 ((E q) (Ψ x)) ((E q) (Ψ y)) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ := by
simp only [F, (E q).symm_apply_apply]
have Aq : (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋ := by
rw [← this]
congr!
-- use the equality between `F p` and `F q` to deduce that the distances have equal
-- integer parts
have : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = (F q).2 ⟨i, hiq⟩ ⟨j, hjq⟩ := by
have hpq' : (F p).snd ≍ (F q).snd := (Sigma.mk.inj_iff.1 hpq).2
rw [Fin.heq_fun₂_iff Npq Npq] at hpq'
rw [← hpq']
rw [Ap, Aq] at this
-- deduce that the distances coincide up to `ε`, by a straightforward computation
-- that should be automated
have I :=
calc
|ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)| = |ε⁻¹ * (dist x y - dist (Ψ x) (Ψ y))| :=
(abs_mul _ _).symm
_ = |ε⁻¹ * dist x y - ε⁻¹ * dist (Ψ x) (Ψ y)| := by congr; ring
_ ≤ 1 := le_of_lt (abs_sub_lt_one_of_floor_eq_floor this)
calc
|dist x y - dist (Ψ x) (Ψ y)| = ε * ε⁻¹ * |dist x y - dist (Ψ x) (Ψ y)| := by
rw [mul_inv_cancel₀ (ne_of_gt εpos), one_mul]
_ = ε * (|ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)|) := by
rw [abs_of_nonneg (le_of_lt (inv_pos.2 εpos)), mul_assoc]
_ ≤ ε * 1 := mul_le_mul_of_nonneg_left I (le_of_lt εpos)
_ = ε := mul_one _
calc
dist p q = ghDist p.Rep q.Rep := dist_ghDist p q
_ ≤ ε + ε / 2 + ε := main
_ = δ := by ring
/-- Compactness criterion: a closed set of compact metric spaces is compact if the spaces have
a uniformly bounded diameter, and for all `ε` the number of balls of radius `ε` required
to cover the spaces is uniformly bounded. This is an equivalence, but we only prove the
interesting direction that these conditions imply compactness. |
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 := (toGlueR_isometry _ _).comp (isometry_optimalGHInjr (X n) (X (n + 1))) } | 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_range_inr]
_ ≤ diam (range inl : Set (X ⊕ Y)) + dist (inl default) (inr default) +
diam (range inr : Set (X ⊕ Y)) :=
(diam_union (mem_range_self _) (mem_range_self _))
_ = diam (univ : Set X) + (dist (α := X) default default + 1 + dist (α := Y) default default) +
diam (univ : Set Y) := by
rw [isometry_inl.diam_range, isometry_inr.diam_range]
rfl
_ = 1 * diam (univ : Set X) + 1 + 1 * diam (univ : Set Y) := by simp
_ ≤ 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by gcongr <;> norm_num | 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 := X ⊕ Y) (inl x) (inl y) := by ring
_ ≤ maxVar X Y * dist (inl x) (inl y) := by gcongr; exact one_le_maxVar X Y
| inl x, inr y =>
calc
f (inl x, inr y) ≤ maxVar X Y := candidates_le_maxVar fA
_ = maxVar X Y * 1 := by simp
_ ≤ maxVar X Y * dist (inl x) (inr y) := by gcongr; apply Sum.one_le_dist_inl_inr
| inr x, inl y =>
calc
f (inr x, inl y) ≤ maxVar X Y := candidates_le_maxVar fA
_ = maxVar X Y * 1 := by simp
_ ≤ maxVar X Y * dist (inl x) (inr y) := by gcongr; apply Sum.one_le_dist_inl_inr
| inr x, inr y =>
calc
f (inr x, inr y) = dist x y := candidates_dist_inr fA x y
_ = dist (α := X ⊕ Y) (inr x) (inr y) := by
rw [@Sum.dist_eq X Y]
rfl
_ = 1 * dist (α := X ⊕ Y) (inr x) (inr y) := by ring
_ ≤ maxVar X Y * dist (inr x) (inr y) := by gcongr; exact one_le_maxVar 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_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_triangle fA
_ = f (x, z) + f (t, y) := by simp [sub_eq_add_neg, add_assoc]
_ ≤ maxVar X Y * dist x z + maxVar X Y * dist t y := by
gcongr <;> apply candidates_dist_bound fA
_ ≤ maxVar X Y * max (dist x z) (dist t y) + maxVar X Y * max (dist x z) (dist t y) := by
gcongr
· apply le_max_left
· apply le_max_right
_ = 2 * maxVar X Y * max (dist x z) (dist y t) := by
rw [dist_comm t y]
ring
_ = 2 * maxVar X Y * dist (x, y) (z, t) := 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 | 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_eval_const continuous_const
have I3 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) = f (y, x) } := fun x y =>
isClosed_eq continuous_eval_const continuous_eval_const
have I4 : ∀ x y z, IsClosed { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) } := fun x y z =>
isClosed_le continuous_eval_const (continuous_eval_const.add continuous_eval_const)
have I5 : ∀ x, IsClosed { f : Cb X Y | f (x, x) = 0 } := fun x =>
isClosed_eq continuous_eval_const continuous_const
have I6 : ∀ x y, IsClosed { f : Cb X Y | f (x, y) ≤ maxVar X Y } := fun x y =>
isClosed_le continuous_eval_const continuous_const
have : candidatesB X Y = (((((⋂ (x) (y), { f : Cb X Y | f (@inl X Y x, @inl X Y y) = dist x y }) ∩
⋂ (x) (y), { f : Cb X Y | f (@inr X Y x, @inr X Y y) = dist x y }) ∩
⋂ (x) (y), { f : Cb X Y | f (x, y) = f (y, x) }) ∩
⋂ (x) (y) (z), { f : Cb X Y | f (x, z) ≤ f (x, y) + f (y, z) }) ∩
⋂ x, { f : Cb X Y | f (x, x) = 0 }) ∩
⋂ (x) (y), { f : Cb X Y | f (x, y) ≤ maxVar X Y } := by
ext
simp only [candidatesB, candidates, mem_inter_iff, mem_iInter, mem_setOf_eq]
rw [this]
repeat'
first
| apply IsClosed.inter _ _
| apply isClosed_iInter _
| apply I1 _ _ | apply I2 _ _ | apply I3 _ _ | apply I4 _ _ _ | apply I5 _ | apply I6 _ _
| intro 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 | 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 complete, we will need all the time
to check that the defining sets are bounded below or above. This is done in the next few
technical lemmas. -/ | 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) default
_ ≤ Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_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 | 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) default
_ ≤ Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_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 | 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 => hcf (mem_range_self x)
have Z : (⨆ x, ⨅ y, f (inl x, inr y)) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g :=
ciSup_mono (HD_bound_aux1 _ (dist f g)) fun x =>
ciInf_mono ⟨cf, forall_mem_range.2 fun i => Hcf _⟩ fun y => coe_le_coe_add_dist
have E1 : ∀ x, (⨅ y, g (inl x, inr y)) + dist f g = ⨅ y, g (inl x, inr y) + dist f g := by
intro x
refine Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) ?_ ?_
· intro x y hx
simpa
· change BddBelow (range fun y : Y => g (inl x, inr y))
exact ⟨cg, forall_mem_range.2 fun i => Hcg _⟩
have E2 : (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g = ⨆ x, (⨅ y, g (inl x, inr y)) + dist f g := by
refine Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) ?_ ?_
· intro x y hx
simpa
· simpa using HD_bound_aux1 _ 0
simpa [E2, E1, Function.comp] | 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 => hcf (mem_range_self x)
have Z : (⨆ y, ⨅ x, f (inl x, inr y)) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g :=
ciSup_mono (HD_bound_aux2 _ (dist f g)) fun y =>
ciInf_mono ⟨cf, forall_mem_range.2 fun i => Hcf _⟩ fun y => coe_le_coe_add_dist
have E1 : ∀ y, (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g := by
intro y
refine Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) ?_ ?_
· intro x y hx
simpa
· change BddBelow (range fun x : X => g (inl x, inr y))
exact ⟨cg, forall_mem_range.2 fun i => Hcg _⟩
have E2 : (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g = ⨆ y, (⨅ x, g (inl x, inr y)) + dist f g := by
refine Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) ?_ ?_
· intro x y hx
simpa
· simpa using HD_bound_aux2 _ 0
simpa [E2, E1] | 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_of_continuity_modulus (fun t => 2 * maxVar X Y * t) ?_ _ ?_
· have : Tendsto (fun t : ℝ => 2 * (maxVar X Y : ℝ) * t) (𝓝 0) (𝓝 (2 * maxVar X Y * 0)) :=
tendsto_const_nhds.mul tendsto_id
simpa using this
· rintro x y ⟨f, hf⟩
exact (candidates_lipschitz hf).dist_le_mul _ _ | 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) default
have B : dist (inl x) (inr default) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) :=
calc
dist (inl x) (inr (default : Y)) = dist x (default : X) + 1 + dist default default := rfl
_ ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := by
gcongr <;>
exact dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
exact le_trans A B
· have A : ⨅ x, candidatesBDist X Y (inl x, inr y) ≤ candidatesBDist X Y (inl default, inr y) :=
ciInf_le (by simpa using HD_below_aux2 0) default
have B : dist (inl default) (inr y) ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) :=
calc
dist (inl (default : X)) (inr y) = dist default default + 1 + dist default y := rfl
_ ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := by
gcongr <;>
exact dist_le_diam_of_mem isBounded_of_compactSpace (mem_univ _) (mem_univ _)
exact le_trans A B | 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)
attribute [local instance] premetricOptimalGHDist | 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), dist x y ≤ r := by
rintro _ ⟨z, rfl⟩
have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
lt_of_le_of_lt (le_max_left _ _) hr
have I2 :
⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y) :=
le_csSup (by simpa using HD_bound_aux1 _ 0) (mem_range_self _)
have I : ⨅ y, optimalGHDist X Y (inl z, inr y) < r := lt_of_le_of_lt I2 I1
rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
exact ⟨optimalGHInjr X Y z', mem_range_self _, le_of_lt hr'⟩
refine hausdorffDist_le_of_mem_dist ?_ A ?_
· inhabit X
rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
exact le_trans dist_nonneg hy
· rintro _ ⟨z, rfl⟩
have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
lt_of_le_of_lt (le_max_right _ _) hr
have I2 :
⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
have I : ⨅ x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1
rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
refine ⟨optimalGHInjl X Y z', mem_range_self _, le_of_lt ?_⟩
rwa [dist_comm] | 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 <|
le_iSup₂ (α := ℝ≥0∞) d' h₂ | 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_exists_nnreal_btwn.1 h with ⟨d', hdim_lt, hlt⟩
have h0 : μH[d'] s = 0 := hausdorffMeasure_of_dimH_lt hdim_lt
exact hlt.not_ge (iInf₂_le d' h0) | 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 => dimH (s i)) i).trans_lt hd)
rw [measure_iUnion_null this]
exact ENNReal.zero_ne_top
@[simp] | 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) := dimH_bUnion hSc _
_ ≤ r := iSup₂_le fun x hx => htr x <| hSs hx | 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 => ?_
rcases exists_mem_nhdsWithin_lt_dimH_of_lt_dimH hr with ⟨x, hxs, hxr⟩
refine le_iSup₂_of_le x hxs ?_; rw [limsup_eq]; refine le_sInf fun b hb => ?_
rcases eventually_smallSets.1 hb with ⟨t, htx, ht⟩
exact (hxr t htx).le.trans (ht t Subset.rfl) | 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 = ⊤ := by
contrapose this
finiteness
rw [ENNReal.le_div_iff_mul_le, mul_comm, ← ENNReal.coe_mul]
exacts [le_dimH_of_hausdorffMeasure_eq_top Hrd, Or.inl (mt ENNReal.coe_eq_zero.1 hr.ne'),
Or.inl ENNReal.coe_ne_top] | 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 huU := inter_eq_self_of_subset_left huU; rw [inter_iUnion₂] at huU
rw [← huU, image_iUnion₂, dimH_bUnion huc, dimH_bUnion huc]; simp only [ENNReal.iSup_div]
exact iSup₂_mono fun x hx => ((hC x (hus hx)).mono inter_subset_right).dimH_image_le hr | 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` divided by `r`. |
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_prop, nhdsWithin_univ] using hf x
/-! | 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 [ENNReal.coe_one, div_one] using dimH_image_le_of_locally_holder_on zero_lt_one this | 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 hf x | 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] (Metric.ball x r) = ENNReal.ofReal ((2 * r) ^ Fintype.card ι) := by
rw [hausdorffMeasure_pi_real, Real.volume_pi_ball _ hr]
refine dimH_of_hausdorffMeasure_ne_zero_ne_top ?_ ?_ <;> rw [NNReal.coe_natCast, this]
· simp [pow_pos (mul_pos (zero_lt_two' ℝ) hr)]
· exact ENNReal.ofReal_ne_top | 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 : e '' s ∈ 𝓝 (e x) := by rw [← e.map_nhds_eq]; exact image_mem_map h
rcases Metric.nhds_basis_ball.mem_iff.1 this with ⟨r, hr0, hr⟩
simpa only [dimH_ball_pi_fin (e x) hr0] using dimH_mono hr | 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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