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 ⌀ |
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ratio_inv (e : X ≃ᵈ X) : ratio (e⁻¹) = (ratio e)⁻¹ := ratio_symm e
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_inv | null |
ratio_pow (e : X ≃ᵈ X) (n : ℕ) : ratio (e ^ n) = ratio e ^ n :=
ratioHom.map_pow _ _
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_pow | null |
ratio_zpow (e : X ≃ᵈ X) (n : ℤ) : ratio (e ^ n) = ratio e ^ n :=
ratioHom.map_zpow _ _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | ratio_zpow | null |
@[simps]
toPerm : (X ≃ᵈ X) →* Equiv.Perm X where
toFun e := e.1
map_mul' _ _ := rfl
map_one' := rfl
@[norm_cast] | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | toPerm | `DilationEquiv.toEquiv` as a monoid homomorphism. |
coe_pow (e : X ≃ᵈ X) (n : ℕ) : ⇑(e ^ n) = e^[n] := by
rw [← coe_toEquiv, ← toPerm_apply, map_pow, Equiv.Perm.coe_pow]; rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | coe_pow | null |
_root_.IsometryEquiv.toDilationEquiv (e : X ≃ᵢ Y) : X ≃ᵈ Y where
edist_eq' := ⟨1, one_ne_zero, by simpa using e.isometry⟩
__ := e.toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.toDilationEquiv | Every isometry equivalence is a dilation equivalence of ratio `1`. |
_root_.IsometryEquiv.toDilationEquiv_apply (e : X ≃ᵢ Y) (x : X) :
e.toDilationEquiv x = e x :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.toDilationEquiv_apply | null |
_root_.IsometryEquiv.toDilationEquiv_symm (e : X ≃ᵢ Y) :
e.symm.toDilationEquiv = e.toDilationEquiv.symm :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.toDilationEquiv_symm | null |
_root_.IsometryEquiv.coe_toDilationEquiv (e : X ≃ᵢ Y) : ⇑e.toDilationEquiv = e :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.coe_toDilationEquiv | null |
_root_.IsometryEquiv.coe_symm_toDilationEquiv (e : X ≃ᵢ Y) :
⇑e.toDilationEquiv.symm = e.symm :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.coe_symm_toDilationEquiv | null |
_root_.IsometryEquiv.toDilationEquiv_toDilation (e : X ≃ᵢ Y) :
(e.toDilationEquiv.toDilation : X →ᵈ Y) = e.isometry.toDilation :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.toDilationEquiv_toDilation | null |
_root_.IsometryEquiv.toDilationEquiv_ratio (e : X ≃ᵢ Y) : ratio e.toDilationEquiv = 1 := by
rw [← ratio_toDilation, IsometryEquiv.toDilationEquiv_toDilation, Isometry.toDilation_ratio] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | _root_.IsometryEquiv.toDilationEquiv_ratio | null |
toHomeomorph (e : X ≃ᵈ Y) : X ≃ₜ Y where
continuous_toFun := Dilation.toContinuous e
continuous_invFun := Dilation.toContinuous e.symm
__ := e.toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | toHomeomorph | Reinterpret a `DilationEquiv` as a homeomorphism. |
toHomeomorph_symm (e : X ≃ᵈ Y) : e.symm.toHomeomorph = e.toHomeomorph.symm :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | toHomeomorph_symm | null |
coe_toHomeomorph (e : X ≃ᵈ Y) : ⇑e.toHomeomorph = e :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | coe_toHomeomorph | null |
coe_symm_toHomeomorph (e : X ≃ᵈ Y) : ⇑e.toHomeomorph.symm = e.symm :=
rfl | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | coe_symm_toHomeomorph | null |
@[simp]
map_cobounded (e : F) : map e (cobounded X) = cobounded Y := by
rw [← Dilation.comap_cobounded e, map_comap_of_surjective (EquivLike.surjective e)] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Dilation"
] | Mathlib/Topology/MetricSpace/DilationEquiv.lean | map_cobounded | null |
equicontinuousAt_iff_right {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {x₀ : β} :
EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) < ε :=
uniformity_basis_dist.equicontinuousAt_iff_right | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | equicontinuousAt_iff_right | Characterization of equicontinuity for families of functions taking values in a (pseudo) metric
space. |
equicontinuousAt_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} {x₀ : β} :
EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ x, dist x x₀ < δ → ∀ i, dist (F i x₀) (F i x) < ε :=
nhds_basis_ball.equicontinuousAt_iff uniformity_basis_dist | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | equicontinuousAt_iff | Characterization of equicontinuity for families of functions between (pseudo) metric spaces. |
protected equicontinuousAt_iff_pair {ι : Type*} [TopologicalSpace β] {F : ι → β → α}
{x₀ : β} :
EquicontinuousAt F x₀ ↔
∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ i, dist (F i x) (F i x') < ε := by
rw [equicontinuousAt_iff_pair]
constructor <;> intro H
· intro ε hε
exact H _ (dist_mem_uniformity hε)
· intro U hU
rcases mem_uniformity_dist.mp hU with ⟨ε, hε, hεU⟩
refine Exists.imp (fun V => And.imp_right fun h => ?_) (H _ hε)
exact fun x hx x' hx' i => hεU (h _ hx _ hx' i) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | equicontinuousAt_iff_pair | Reformulation of `equicontinuousAt_iff_pair` for families of functions taking values in a
(pseudo) metric space. |
uniformEquicontinuous_iff_right {ι : Type*} [UniformSpace β] {F : ι → β → α} :
UniformEquicontinuous F ↔ ∀ ε > 0, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, dist (F i xy.1) (F i xy.2) < ε :=
uniformity_basis_dist.uniformEquicontinuous_iff_right | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | uniformEquicontinuous_iff_right | Characterization of uniform equicontinuity for families of functions taking values in a
(pseudo) metric space. |
uniformEquicontinuous_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} :
UniformEquicontinuous F ↔
∀ ε > 0, ∃ δ > 0, ∀ x y, dist x y < δ → ∀ i, dist (F i x) (F i y) < ε :=
uniformity_basis_dist.uniformEquicontinuous_iff uniformity_basis_dist | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | uniformEquicontinuous_iff | Characterization of uniform equicontinuity for families of functions between
(pseudo) metric spaces. |
equicontinuousAt_of_continuity_modulus {ι : Type*} [TopologicalSpace β] {x₀ : β}
(b : β → ℝ) (b_lim : Tendsto b (𝓝 x₀) (𝓝 0)) (F : ι → β → α)
(H : ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) ≤ b x) : EquicontinuousAt F x₀ := by
rw [Metric.equicontinuousAt_iff_right]
intro ε ε0
filter_upwards [b_lim (Iio_mem_nhds ε0), H] using fun x hx₁ hx₂ i => (hx₂ i).trans_lt hx₁ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | equicontinuousAt_of_continuity_modulus | For a family of functions to a (pseudo) metric spaces, a convenient way to prove
equicontinuity at a point is to show that all of the functions share a common *local* continuity
modulus. |
uniformEquicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
(b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
(H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : UniformEquicontinuous F := by
rw [Metric.uniformEquicontinuous_iff]
intro ε ε0
rcases tendsto_nhds_nhds.1 b_lim ε ε0 with ⟨δ, δ0, hδ⟩
refine ⟨δ, δ0, fun x y hxy i => ?_⟩
calc
dist (F i x) (F i y) ≤ b (dist x y) := H x y i
_ ≤ |b (dist x y)| := le_abs_self _
_ = dist (b (dist x y)) 0 := by simp [Real.dist_eq]
_ < ε := hδ (by simpa only [Real.dist_eq, tsub_zero, abs_dist] using hxy) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | uniformEquicontinuous_of_continuity_modulus | For a family of functions between (pseudo) metric spaces, a convenient way to prove
uniform equicontinuity is to show that all of the functions share a common *global* continuity
modulus. |
equicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
(b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
(H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : Equicontinuous F :=
(uniformEquicontinuous_of_continuity_modulus b b_lim F H).equicontinuous | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] | Mathlib/Topology/MetricSpace/Equicontinuity.lean | equicontinuous_of_continuity_modulus | For a family of functions between (pseudo) metric spaces, a convenient way to prove
equicontinuity is to show that all of the functions share a common *global* continuity modulus. |
glueDist (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : X ⊕ Y → X ⊕ Y → ℝ
| .inl x, .inl y => dist x y
| .inr x, .inr y => dist x y
| .inl x, .inr y => (⨅ p, dist x (Φ p) + dist y (Ψ p)) + ε
| .inr x, .inl y => (⨅ p, dist y (Φ p) + dist x (Ψ p)) + ε | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist | Define a predistance on `X ⊕ Y`, for which `Φ p` and `Ψ p` are at distance `ε` |
private glueDist_self (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : ∀ x, glueDist Φ Ψ ε x x = 0
| .inl _ => dist_self _
| .inr _ => dist_self _ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_self | null |
glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :
glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε := by
have : ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0 := by
have A : ∀ q, 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) := fun _ => by positivity
refine le_antisymm ?_ (le_ciInf A)
have : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p) := by simp
rw [this]
exact ciInf_le ⟨0, forall_mem_range.2 A⟩ p
simp only [glueDist, this, zero_add] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_glued_points | null |
private glueDist_comm (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :
∀ x y, glueDist Φ Ψ ε x y = glueDist Φ Ψ ε y x
| .inl _, .inl _ => dist_comm _ _
| .inr _, .inr _ => dist_comm _ _
| .inl _, .inr _ => rfl
| .inr _, .inl _ => rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_comm | null |
glueDist_swap (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :
∀ x y, glueDist Ψ Φ ε x.swap y.swap = glueDist Φ Ψ ε x y
| .inl _, .inl _ => rfl
| .inr _, .inr _ => rfl
| .inl _, .inr _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm]
| .inr _, .inl _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_swap | null |
le_glueDist_inl_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
ε ≤ glueDist Φ Ψ ε (.inl x) (.inr y) :=
le_add_of_nonneg_left <| Real.iInf_nonneg fun _ => by positivity | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | le_glueDist_inl_inr | null |
le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) := by
rw [glueDist_comm]; apply le_glueDist_inl_inr | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | le_glueDist_inr_inl | null |
private glueDist_triangle_inl_inr_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y z : Y) :
glueDist Φ Ψ ε (.inl x) (.inr z) ≤
glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inr z) := by
simp only [glueDist]
rw [add_right_comm, add_le_add_iff_right]
refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_
· exact forall_mem_range.2 fun _ => by positivity
· linarith [dist_triangle_left z (Ψ p) y] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_triangle_inl_inr_inr | null |
private glueDist_triangle_inl_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)
(H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x : X) (y : Y) (z : X) :
glueDist Φ Ψ ε (.inl x) (.inl z) ≤
glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inl z) := by
simp_rw [glueDist, add_add_add_comm _ ε, add_assoc]
refine le_ciInf_add fun p => ?_
rw [add_left_comm, add_assoc, ← two_mul]
refine le_ciInf_add fun q => ?_
rw [dist_comm z]
linarith [dist_triangle4 x (Φ p) (Φ q) z, dist_triangle_left (Ψ p) (Ψ q) y, (abs_le.1 (H p q)).2] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_triangle_inl_inr_inl | null |
private glueDist_triangle (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)
(H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) :
∀ x y z, glueDist Φ Ψ ε x z ≤ glueDist Φ Ψ ε x y + glueDist Φ Ψ ε y z
| .inl _, .inl _, .inl _ => dist_triangle _ _ _
| .inr _, .inr _, .inr _ => dist_triangle _ _ _
| .inr x, .inl y, .inl z => by
simp only [← glueDist_swap Φ]
apply glueDist_triangle_inl_inr_inr
| .inr x, .inr y, .inl z => by
simpa only [glueDist_comm, add_comm] using glueDist_triangle_inl_inr_inr _ _ _ z y x
| .inl x, .inl y, .inr z => by
simpa only [← glueDist_swap Φ, glueDist_comm, add_comm, Sum.swap_inl, Sum.swap_inr]
using glueDist_triangle_inl_inr_inr Ψ Φ ε z y x
| .inl _, .inr _, .inr _ => glueDist_triangle_inl_inr_inr ..
| .inl x, .inr y, .inl z => glueDist_triangle_inl_inr_inl Φ Ψ ε H x y z
| .inr x, .inl y, .inr z => by
simp only [← glueDist_swap Φ]
apply glueDist_triangle_inl_inr_inl
simpa only [abs_sub_comm] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueDist_triangle | null |
private eq_of_glueDist_eq_zero (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) :
∀ p q : X ⊕ Y, glueDist Φ Ψ ε p q = 0 → p = q
| .inl x, .inl y, h => by rw [eq_of_dist_eq_zero h]
| .inl x, .inr y, h => by exfalso; linarith [le_glueDist_inl_inr Φ Ψ ε x y]
| .inr x, .inl y, h => by exfalso; linarith [le_glueDist_inr_inl Φ Ψ ε x y]
| .inr x, .inr y, h => by rw [eq_of_dist_eq_zero h] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | eq_of_glueDist_eq_zero | null |
Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s := by
simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage]
constructor
· rintro ⟨⟨δX, δX0, hX⟩, δY, δY0, hY⟩
refine ⟨min (min δX δY) ε, lt_min (lt_min δX0 δY0) hε, ?_⟩
rintro (a | a) (b | b) h <;> simp only [lt_min_iff] at h
· exact hX h.1.1
· exact absurd h.2 (le_glueDist_inl_inr _ _ _ _ _).not_gt
· exact absurd h.2 (le_glueDist_inr_inl _ _ _ _ _).not_gt
· exact hY h.1.2
· rintro ⟨ε, ε0, H⟩
constructor <;> exact ⟨ε, ε0, fun _ _ h => H _ _ h⟩ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.mem_uniformity_iff_glueDist | null |
glueMetricApprox [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε)
(H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) : MetricSpace (X ⊕ Y) where
dist := glueDist Φ Ψ ε
dist_self := glueDist_self Φ Ψ ε
dist_comm := glueDist_comm Φ Ψ ε
dist_triangle := glueDist_triangle Φ Ψ ε H
eq_of_dist_eq_zero := eq_of_glueDist_eq_zero Φ Ψ ε ε0 _ _
toUniformSpace := Sum.instUniformSpace
uniformity_dist := uniformity_dist_of_mem_uniformity _ _ <| Sum.mem_uniformity_iff_glueDist ε0 | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | glueMetricApprox | Given two maps `Φ` and `Ψ` intro metric spaces `X` and `Y` such that the distances between
`Φ p` and `Φ q`, and between `Ψ p` and `Ψ q`, coincide up to `2 ε` where `ε > 0`, one can almost
glue the two spaces `X` and `Y` along the images of `Φ` and `Ψ`, so that `Φ p` and `Ψ p` are
at distance `ε`. |
protected Sum.dist : X ⊕ Y → X ⊕ Y → ℝ
| .inl a, .inl a' => dist a a'
| .inr b, .inr b' => dist b b'
| .inl a, .inr b => dist a (Nonempty.some ⟨a⟩) + 1 + dist (Nonempty.some ⟨b⟩) b
| .inr b, .inl a => dist b (Nonempty.some ⟨b⟩) + 1 + dist (Nonempty.some ⟨a⟩) a | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.dist | Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
If the two spaces are bounded, one can say for instance that each point in the first is at distance
`diam X + diam Y + 1` of each point in the second.
Instead, we choose a construction that works for unbounded spaces, but requires basepoints,
chosen arbitrarily.
We embed isometrically each factor, set the basepoints at distance 1,
arbitrarily, and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default. |
Sum.dist_eq_glueDist {p q : X ⊕ Y} (x : X) (y : Y) :
Sum.dist p q =
glueDist (fun _ : Unit => Nonempty.some ⟨x⟩) (fun _ : Unit => Nonempty.some ⟨y⟩) 1 p q := by
cases p <;> cases q <;> first |rfl|simp [Sum.dist, glueDist, dist_comm, add_comm,
add_left_comm, add_assoc] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.dist_eq_glueDist | null |
private Sum.dist_comm (x y : X ⊕ Y) : Sum.dist x y = Sum.dist y x := by
cases x <;> cases y <;> simp [Sum.dist, _root_.dist_comm, add_comm, add_left_comm] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.dist_comm | null |
Sum.one_le_dist_inl_inr {x : X} {y : Y} : 1 ≤ Sum.dist (.inl x) (.inr y) :=
le_trans (le_add_of_nonneg_right dist_nonneg) <|
add_le_add_right (le_add_of_nonneg_left dist_nonneg) _ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.one_le_dist_inl_inr | null |
Sum.one_le_dist_inr_inl {x : X} {y : Y} : 1 ≤ Sum.dist (.inr y) (.inl x) := by
rw [Sum.dist_comm]; exact Sum.one_le_dist_inl_inr | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.one_le_dist_inr_inl | null |
private Sum.mem_uniformity (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ ε > 0, ∀ a b, Sum.dist a b < ε → (a, b) ∈ s := by
constructor
· rintro ⟨hsX, hsY⟩
rcases mem_uniformity_dist.1 hsX with ⟨εX, εX0, hX⟩
rcases mem_uniformity_dist.1 hsY with ⟨εY, εY0, hY⟩
refine ⟨min (min εX εY) 1, lt_min (lt_min εX0 εY0) zero_lt_one, ?_⟩
rintro (a | a) (b | b) h
· exact hX (lt_of_lt_of_le h (le_trans (min_le_left _ _) (min_le_left _ _)))
· cases not_le_of_gt (lt_of_lt_of_le h (min_le_right _ _)) Sum.one_le_dist_inl_inr
· cases not_le_of_gt (lt_of_lt_of_le h (min_le_right _ _)) Sum.one_le_dist_inr_inl
· exact hY (lt_of_lt_of_le h (le_trans (min_le_left _ _) (min_le_right _ _)))
· rintro ⟨ε, ε0, H⟩
constructor <;> rw [Filter.mem_map, mem_uniformity_dist] <;> exact ⟨ε, ε0, fun _ _ h => H _ _ h⟩ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.mem_uniformity | null |
metricSpaceSum : MetricSpace (X ⊕ Y) where
dist := Sum.dist
dist_self x := by cases x <;> simp only [Sum.dist, dist_self]
dist_comm := Sum.dist_comm
dist_triangle
| .inl p, .inl q, .inl r => dist_triangle p q r
| .inl p, .inr q, _ => by
simp only [Sum.dist_eq_glueDist p q]
exact glueDist_triangle _ _ _ (by simp) _ _ _
| _, .inl q, .inr r => by
simp only [Sum.dist_eq_glueDist q r]
exact glueDist_triangle _ _ _ (by simp) _ _ _
| .inr p, _, .inl r => by
simp only [Sum.dist_eq_glueDist r p]
exact glueDist_triangle _ _ _ (by simp) _ _ _
| .inr p, .inr q, .inr r => dist_triangle p q r
eq_of_dist_eq_zero {p q} h := by
rcases p with p | p <;> rcases q with q | q
· rw [eq_of_dist_eq_zero h]
· exact eq_of_glueDist_eq_zero _ _ _ one_pos _ _ ((Sum.dist_eq_glueDist p q).symm.trans h)
· exact eq_of_glueDist_eq_zero _ _ _ one_pos _ _ ((Sum.dist_eq_glueDist q p).symm.trans h)
· rw [eq_of_dist_eq_zero h]
toUniformSpace := Sum.instUniformSpace
uniformity_dist := uniformity_dist_of_mem_uniformity _ _ Sum.mem_uniformity
attribute [local instance] metricSpaceSum | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | metricSpaceSum | The distance on the disjoint union indeed defines a metric space. All the distance properties
follow from our choice of the distance. The harder work is to show that the uniform structure
defined by the distance coincides with the disjoint union uniform structure. |
Sum.dist_eq {x y : X ⊕ Y} : dist x y = Sum.dist x y := rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | Sum.dist_eq | null |
isometry_inl : Isometry (Sum.inl : X → X ⊕ Y) :=
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | isometry_inl | The left injection of a space in a disjoint union is an isometry |
isometry_inr : Isometry (Sum.inr : Y → X ⊕ Y) :=
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | isometry_inr | The right injection of a space in a disjoint union is an isometry |
protected dist : (Σ i, E i) → (Σ i, E i) → ℝ
| ⟨i, x⟩, ⟨j, y⟩ =>
if h : i = j then
haveI : E j = E i := by rw [h]
Dist.dist x (cast this y)
else Dist.dist x (Nonempty.some ⟨x⟩) + 1 + Dist.dist (Nonempty.some ⟨y⟩) y | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | dist | Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
We choose a construction that works for unbounded spaces, but requires basepoints,
chosen arbitrarily.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default. |
instDist : Dist (Σ i, E i) :=
⟨Sigma.dist⟩
attribute [local instance] Sigma.instDist
@[simp] | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | instDist | A `Dist` instance on the disjoint union `Σ i, E i`.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default. |
dist_same (i : ι) (x y : E i) : dist (Sigma.mk i x) ⟨i, y⟩ = dist x y := by
simp [Dist.dist, Sigma.dist]
@[simp] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | dist_same | null |
dist_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ = dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨y⟩) y :=
dif_neg h | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | dist_ne | null |
one_le_dist_of_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
1 ≤ dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ := by
rw [Sigma.dist_ne h x y]
linarith [@dist_nonneg _ _ x (Nonempty.some ⟨x⟩), @dist_nonneg _ _ (Nonempty.some ⟨y⟩) y] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | one_le_dist_of_ne | null |
fst_eq_of_dist_lt_one (x y : Σ i, E i) (h : dist x y < 1) : x.1 = y.1 := by
cases x; cases y
contrapose! h
apply one_le_dist_of_ne h | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | fst_eq_of_dist_lt_one | null |
protected dist_triangle (x y z : Σ i, E i) : dist x z ≤ dist x y + dist y z := by
rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
rcases eq_or_ne i k with (rfl | hik)
· rcases eq_or_ne i j with (rfl | hij)
· simpa using dist_triangle x y z
· simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
calc
dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
simpa only [zero_add, add_zero] using dist_triangle _ _ _
_ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
· rcases eq_or_ne i j with (rfl | hij)
· simp only [Sigma.dist_ne hik, Sigma.dist_same]
calc
dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
apply_rules [add_le_add, le_rfl, dist_triangle]
_ = _ := by abel
· rcases eq_or_ne j k with (rfl | hjk)
· simp only [Sigma.dist_ne hij, Sigma.dist_same]
calc
dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
apply_rules [add_le_add, le_rfl, dist_triangle]
_ = _ := by abel
· simp only [hik, hij, hjk, Sigma.dist_ne, Ne, not_false_iff]
calc
dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z =
dist x (Nonempty.some ⟨x⟩) + 1 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
simp only [add_zero, zero_add]
_ ≤ _ := by apply_rules [add_le_add, zero_le_one, dist_nonneg, le_rfl] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | dist_triangle | null |
protected isOpen_iff (s : Set (Σ i, E i)) :
IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s := by
constructor
· rintro hs ⟨i, x⟩ hx
obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, ball x ε ⊆ Sigma.mk i ⁻¹' s :=
Metric.isOpen_iff.1 (isOpen_sigma_iff.1 hs i) x hx
refine ⟨min ε 1, lt_min εpos zero_lt_one, ?_⟩
rintro ⟨j, y⟩ hy
rcases eq_or_ne i j with (rfl | hij)
· simp only [Sigma.dist_same, lt_min_iff] at hy
exact hε (mem_ball'.2 hy.1)
· apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := hy.trans_le (min_le_right _ _)
· refine fun H => isOpen_sigma_iff.2 fun i => Metric.isOpen_iff.2 fun x hx => ?_
obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, ∀ y, dist (⟨i, x⟩ : Σ j, E j) y < ε → y ∈ s :=
H ⟨i, x⟩ hx
refine ⟨ε, εpos, fun y hy => ?_⟩
apply hε ⟨i, y⟩
rw [Sigma.dist_same]
exact mem_ball'.1 hy | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | isOpen_iff | null |
protected metricSpace : MetricSpace (Σ i, E i) := by
refine MetricSpace.ofDistTopology Sigma.dist ?_ ?_ Sigma.dist_triangle Sigma.isOpen_iff ?_
· rintro ⟨i, x⟩
simp [Sigma.dist]
· rintro ⟨i, x⟩ ⟨j, y⟩
rcases eq_or_ne i j with (rfl | h)
· simp [Sigma.dist, dist_comm]
· simp only [Sigma.dist, dist_comm, h, h.symm, not_false_iff, dif_neg]
abel
· rintro ⟨i, x⟩ ⟨j, y⟩
rcases eq_or_ne i j with (rfl | hij)
· simp [Sigma.dist]
· intro h
apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist (⟨i, x⟩ : Σ k, E k) ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := by rw [h]; exact zero_lt_one
attribute [local instance] Sigma.metricSpace
open Topology
open Filter | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | metricSpace | A metric space structure on the disjoint union `Σ i, E i`.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default. |
isometry_mk (i : ι) : Isometry (Sigma.mk i : E i → Σ k, E k) :=
Isometry.of_dist_eq fun x y => by simp | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | isometry_mk | The injection of a space in a disjoint union is an isometry |
protected completeSpace [∀ i, CompleteSpace (E i)] : CompleteSpace (Σ i, E i) := by
set s : ι → Set (Σ i, E i) := fun i => Sigma.fst ⁻¹' {i}
set U := { p : (Σ k, E k) × Σ k, E k | dist p.1 p.2 < 1 }
have hc : ∀ i, IsComplete (s i) := fun i => by
simp only [s, ← range_sigmaMk]
exact (isometry_mk i).isUniformInducing.isComplete_range
have hd : ∀ (i j), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j := fun i j x hx y hy hxy =>
(Eq.symm hx).trans ((fst_eq_of_dist_lt_one _ _ hxy).trans hy)
refine completeSpace_of_isComplete_univ ?_
convert isComplete_iUnion_separated hc (dist_mem_uniformity zero_lt_one) hd
simp only [s, ← preimage_iUnion, iUnion_of_singleton, preimage_univ] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | completeSpace | A disjoint union of complete metric spaces is complete. |
gluePremetric (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : PseudoMetricSpace (X ⊕ Y) where
dist := glueDist Φ Ψ 0
dist_self := glueDist_self Φ Ψ 0
dist_comm := glueDist_comm Φ Ψ 0
dist_triangle := glueDist_triangle Φ Ψ 0 fun p q => by rw [hΦ.dist_eq, hΨ.dist_eq]; simp | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | gluePremetric | Given two isometric embeddings `Φ : Z → X` and `Ψ : Z → Y`, we define a pseudo metric space
structure on `X ⊕ Y` by declaring that `Φ x` and `Ψ x` are at distance `0`. |
GlueSpace (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Type _ :=
@SeparationQuotient _ (gluePremetric hΦ hΨ).toUniformSpace.toTopologicalSpace | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | GlueSpace | Given two isometric embeddings `Φ : Z → X` and `Ψ : Z → Y`, we define a
space `GlueSpace hΦ hΨ` by identifying in `X ⊕ Y` the points `Φ x` and `Ψ x`. |
toGlueL (hΦ : Isometry Φ) (hΨ : Isometry Ψ) (x : X) : GlueSpace hΦ hΨ :=
Quotient.mk'' (.inl x) | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toGlueL | The canonical map from `X` to the space obtained by gluing isometric subsets in `X` and `Y`. |
toGlueR (hΦ : Isometry Φ) (hΨ : Isometry Ψ) (y : Y) : GlueSpace hΦ hΨ :=
Quotient.mk'' (.inr y) | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toGlueR | The canonical map from `Y` to the space obtained by gluing isometric subsets in `X` and `Y`. |
inhabitedLeft (hΦ : Isometry Φ) (hΨ : Isometry Ψ) [Inhabited X] :
Inhabited (GlueSpace hΦ hΨ) :=
⟨toGlueL _ _ default⟩ | instance | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | inhabitedLeft | null |
inhabitedRight (hΦ : Isometry Φ) (hΨ : Isometry Ψ) [Inhabited Y] :
Inhabited (GlueSpace hΦ hΨ) :=
⟨toGlueR _ _ default⟩ | instance | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | inhabitedRight | null |
toGlue_commute (hΦ : Isometry Φ) (hΨ : Isometry Ψ) :
toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ := by
let i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
let _ := i.toUniformSpace.toTopologicalSpace
funext
simp only [comp, toGlueL, toGlueR]
refine SeparationQuotient.mk_eq_mk.2 (Metric.inseparable_iff.2 ?_)
exact glueDist_glued_points Φ Ψ 0 _ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toGlue_commute | null |
toGlueL_isometry (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (toGlueL hΦ hΨ) :=
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toGlueL_isometry | null |
toGlueR_isometry (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (toGlueR hΦ hΨ) :=
Isometry.of_dist_eq fun _ _ => rfl | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toGlueR_isometry | null |
inductiveLimitDist (f : ∀ n, X n → X (n + 1)) (x y : Σ n, X n) : ℝ :=
dist (leRecOn (le_max_left x.1 y.1) (f _) x.2 : X (max x.1 y.1))
(leRecOn (le_max_right x.1 y.1) (f _) y.2 : X (max x.1 y.1)) | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | inductiveLimitDist | Predistance on the disjoint union `Σ n, X n`. |
inductiveLimitDist_eq_dist (I : ∀ n, Isometry (f n)) (x y : Σ n, X n) :
∀ m (hx : x.1 ≤ m) (hy : y.1 ≤ m), inductiveLimitDist f x y =
dist (leRecOn hx (f _) x.2 : X m) (leRecOn hy (f _) y.2 : X m)
| 0, hx, hy => by
obtain ⟨i, x⟩ := x; obtain ⟨j, y⟩ := y
obtain rfl : i = 0 := nonpos_iff_eq_zero.1 hx
obtain rfl : j = 0 := nonpos_iff_eq_zero.1 hy
rfl
| (m + 1), hx, hy => by
by_cases h : max x.1 y.1 = (m + 1)
· generalize m + 1 = m' at *
subst m'
rfl
· have : max x.1 y.1 ≤ succ m := by simp [hx, hy]
have : max x.1 y.1 ≤ m := by simpa [h] using of_le_succ this
have xm : x.1 ≤ m := le_trans (le_max_left _ _) this
have ym : y.1 ≤ m := le_trans (le_max_right _ _) this
rw [leRecOn_succ xm, leRecOn_succ ym, (I m).dist_eq]
exact inductiveLimitDist_eq_dist I x y m xm ym | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | inductiveLimitDist_eq_dist | The predistance on the disjoint union `Σ n, X n` can be computed in any `X k` for large
enough `k`. |
inductivePremetric (I : ∀ n, Isometry (f n)) : PseudoMetricSpace (Σ n, X n) where
dist := inductiveLimitDist f
dist_self x := by simp [inductiveLimitDist]
dist_comm x y := by
let m := max x.1 y.1
have hx : x.1 ≤ m := le_max_left _ _
have hy : y.1 ≤ m := le_max_right _ _
rw [inductiveLimitDist_eq_dist I x y m hx hy, inductiveLimitDist_eq_dist I y x m hy hx,
dist_comm]
dist_triangle x y z := by
let m := max (max x.1 y.1) z.1
have hx : x.1 ≤ m := le_trans (le_max_left _ _) (le_max_left _ _)
have hy : y.1 ≤ m := le_trans (le_max_right _ _) (le_max_left _ _)
have hz : z.1 ≤ m := le_max_right _ _
calc
inductiveLimitDist f x z = dist (leRecOn hx (f _) x.2 : X m) (leRecOn hz (f _) z.2 : X m) :=
inductiveLimitDist_eq_dist I x z m hx hz
_ ≤ dist (leRecOn hx (f _) x.2 : X m) (leRecOn hy (f _) y.2 : X m) +
dist (leRecOn hy (f _) y.2 : X m) (leRecOn hz (f _) z.2 : X m) :=
(dist_triangle _ _ _)
_ = inductiveLimitDist f x y + inductiveLimitDist f y z := by
rw [inductiveLimitDist_eq_dist I x y m hx hy, inductiveLimitDist_eq_dist I y z m hy hz]
attribute [local instance] inductivePremetric | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | inductivePremetric | Premetric space structure on `Σ n, X n`. |
InductiveLimit (I : ∀ n, Isometry (f n)) : Type _ :=
@SeparationQuotient _ (inductivePremetric I).toUniformSpace.toTopologicalSpace | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | InductiveLimit | The type giving the inductive limit in a metric space context. |
toInductiveLimit (I : ∀ n, Isometry (f n)) (n : ℕ) (x : X n) : Metric.InductiveLimit I :=
Quotient.mk'' (Sigma.mk n x) | def | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toInductiveLimit | Mapping each `X n` to the inductive limit. |
toInductiveLimit_isometry (I : ∀ n, Isometry (f n)) (n : ℕ) :
Isometry (toInductiveLimit I n) :=
Isometry.of_dist_eq fun x y => by
change inductiveLimitDist f ⟨n, x⟩ ⟨n, y⟩ = dist x y
rw [inductiveLimitDist_eq_dist I ⟨n, x⟩ ⟨n, y⟩ n (le_refl n) (le_refl n), leRecOn_self,
leRecOn_self] | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toInductiveLimit_isometry | The map `toInductiveLimit n` mapping `X n` to the inductive limit is an isometry. |
toInductiveLimit_commute (I : ∀ n, Isometry (f n)) (n : ℕ) :
toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n := by
let h := inductivePremetric I
let _ := h.toUniformSpace.toTopologicalSpace
funext x
simp only [comp, toInductiveLimit]
refine SeparationQuotient.mk_eq_mk.2 (Metric.inseparable_iff.2 ?_)
change inductiveLimitDist f ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0
rw [inductiveLimitDist_eq_dist I ⟨n.succ, f n x⟩ ⟨n, x⟩ n.succ, leRecOn_self,
leRecOn_succ, leRecOn_self, dist_self]
· rfl
· rfl
· exact le_succ _ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | toInductiveLimit_commute | The maps `toInductiveLimit n` are compatible with the maps `f n`. |
dense_iUnion_range_toInductiveLimit
{X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)]
{f : (n : ℕ) → X n → X (n + 1)}
(I : ∀ (n : ℕ), Isometry (f n)) :
Dense (⋃ i, range (toInductiveLimit I i)) := by
refine dense_univ.mono ?_
rintro ⟨n, x⟩ _
refine mem_iUnion.2 ⟨n, mem_range.2 ⟨x, rfl⟩⟩ | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | dense_iUnion_range_toInductiveLimit | null |
separableSpaceInductiveLimit_of_separableSpace
{X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)]
[hs : (n : ℕ) → TopologicalSpace.SeparableSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)}
(I : ∀ (n : ℕ), Isometry (f n)) :
TopologicalSpace.SeparableSpace (Metric.InductiveLimit I) := by
choose hsX hcX hdX using (fun n ↦ TopologicalSpace.exists_countable_dense (X n))
let s := ⋃ (i : ℕ), (toInductiveLimit I i '' (hsX i))
refine ⟨s, countable_iUnion (fun n => (hcX n).image _), ?_⟩
refine .of_closure <| (dense_iUnion_range_toInductiveLimit I).mono <| iUnion_subset fun i ↦ ?_
calc
range (toInductiveLimit I i) ⊆ closure (toInductiveLimit I i '' (hsX i)) :=
(toInductiveLimit_isometry I i |>.continuous).range_subset_closure_image_dense (hdX i)
_ ⊆ closure s := closure_mono <| subset_iUnion (fun j ↦ toInductiveLimit I j '' hsX j) i | theorem | Topology | [
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Topology.MetricSpace.Isometry"
] | Mathlib/Topology/MetricSpace/Gluing.lean | separableSpaceInductiveLimit_of_separableSpace | null |
private IsometryRel (x : NonemptyCompacts ℓ_infty_ℝ) (y : NonemptyCompacts ℓ_infty_ℝ) : Prop :=
Nonempty (x ≃ᵢ y) | 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 | IsometryRel | Equivalence relation identifying two nonempty compact sets which are isometric |
private equivalence_isometryRel : Equivalence IsometryRel :=
⟨fun _ => Nonempty.intro (IsometryEquiv.refl _), fun ⟨e⟩ => ⟨e.symm⟩, fun ⟨e⟩ ⟨f⟩ => ⟨e.trans f⟩⟩ | 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 | equivalence_isometryRel | This is indeed an equivalence relation |
IsometryRel.setoid : Setoid (NonemptyCompacts ℓ_infty_ℝ) :=
Setoid.mk IsometryRel equivalence_isometryRel | instance | 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 | IsometryRel.setoid | setoid instance identifying two isometric nonempty compact subspaces of ℓ^∞(ℝ) |
GHSpace : Type :=
Quotient IsometryRel.setoid | 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 | GHSpace | The Gromov-Hausdorff space |
toGHSpace (X : Type u) [MetricSpace X] [CompactSpace X] [Nonempty X] : GHSpace :=
⟦NonemptyCompacts.kuratowskiEmbedding X⟧ | 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 | toGHSpace | Map any nonempty compact type to `GHSpace` |
GHSpace.Rep (p : GHSpace) : Type :=
(Quotient.out p : NonemptyCompacts ℓ_infty_ℝ) | 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 | GHSpace.Rep | A metric space representative of any abstract point in `GHSpace` |
eq_toGHSpace_iff {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{p : NonemptyCompacts ℓ_infty_ℝ} :
⟦p⟧ = toGHSpace X ↔ ∃ Ψ : X → ℓ_infty_ℝ, Isometry Ψ ∧ range Ψ = p := by
simp only [toGHSpace, Quotient.eq]
refine ⟨fun h => ?_, ?_⟩
· rcases Setoid.symm h with ⟨e⟩
have f := (kuratowskiEmbedding.isometry X).isometryEquivOnRange.trans e
use fun x => f x, isometry_subtype_coe.comp f.isometry
rw [range_comp', f.range_eq_univ, Set.image_univ, Subtype.range_coe]
· rintro ⟨Ψ, ⟨isomΨ, rangeΨ⟩⟩
have f :=
((kuratowskiEmbedding.isometry X).isometryEquivOnRange.symm.trans
isomΨ.isometryEquivOnRange).symm
have E : (range Ψ ≃ᵢ NonemptyCompacts.kuratowskiEmbedding X)
= (p ≃ᵢ range (kuratowskiEmbedding X)) := by
dsimp only [NonemptyCompacts.kuratowskiEmbedding]; rw [rangeΨ]; rfl
exact ⟨cast E f⟩ | 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 | eq_toGHSpace_iff | null |
eq_toGHSpace {p : NonemptyCompacts ℓ_infty_ℝ} : ⟦p⟧ = toGHSpace p :=
eq_toGHSpace_iff.2 ⟨fun x => x, isometry_subtype_coe, Subtype.range_coe⟩ | 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 | eq_toGHSpace | null |
repGHSpaceMetricSpace {p : GHSpace} : MetricSpace p.Rep :=
inferInstanceAs <| MetricSpace p.out | instance | 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 | repGHSpaceMetricSpace | null |
rep_gHSpace_compactSpace {p : GHSpace} : CompactSpace p.Rep :=
inferInstanceAs <| CompactSpace p.out | instance | 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 | rep_gHSpace_compactSpace | null |
rep_gHSpace_nonempty {p : GHSpace} : Nonempty p.Rep :=
inferInstanceAs <| Nonempty p.out | instance | 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 | rep_gHSpace_nonempty | null |
GHSpace.toGHSpace_rep (p : GHSpace) : toGHSpace p.Rep = p := by
change toGHSpace (Quot.out p : NonemptyCompacts ℓ_infty_ℝ) = p
rw [← eq_toGHSpace]
exact Quot.out_eq p | 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 | GHSpace.toGHSpace_rep | null |
toGHSpace_eq_toGHSpace_iff_isometryEquiv {X : Type u} [MetricSpace X] [CompactSpace X]
[Nonempty X] {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
toGHSpace X = toGHSpace Y ↔ Nonempty (X ≃ᵢ Y) :=
⟨by
simp only [toGHSpace]
rw [Quotient.eq]
rintro ⟨e⟩
have I :
(NonemptyCompacts.kuratowskiEmbedding X ≃ᵢ NonemptyCompacts.kuratowskiEmbedding Y) =
(range (kuratowskiEmbedding X) ≃ᵢ range (kuratowskiEmbedding Y)) := by
dsimp only [NonemptyCompacts.kuratowskiEmbedding]; rfl
have f := (kuratowskiEmbedding.isometry X).isometryEquivOnRange
have g := (kuratowskiEmbedding.isometry Y).isometryEquivOnRange.symm
exact ⟨f.trans <| (cast I e).trans g⟩, by
rintro ⟨e⟩
simp only [toGHSpace]
have f := (kuratowskiEmbedding.isometry X).isometryEquivOnRange.symm
have g := (kuratowskiEmbedding.isometry Y).isometryEquivOnRange
have I :
(range (kuratowskiEmbedding X) ≃ᵢ range (kuratowskiEmbedding Y)) =
(NonemptyCompacts.kuratowskiEmbedding X ≃ᵢ NonemptyCompacts.kuratowskiEmbedding Y) := by
dsimp only [NonemptyCompacts.kuratowskiEmbedding]; rfl
rw [Quotient.eq]
exact ⟨cast I ((f.trans e).trans g)⟩⟩ | 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 | toGHSpace_eq_toGHSpace_iff_isometryEquiv | Two nonempty compact spaces have the same image in `GHSpace` if and only if they are
isometric. |
ghDist (X : Type u) (Y : Type v) [MetricSpace X] [Nonempty X] [CompactSpace X] [MetricSpace Y]
[Nonempty Y] [CompactSpace Y] : ℝ :=
dist (toGHSpace X) (toGHSpace Y) | 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 | ghDist | Distance on `GHSpace`: the distance between two nonempty compact spaces is the infimum
Hausdorff distance between isometric copies of the two spaces in a metric space. For the definition,
we only consider embeddings in `ℓ^∞(ℝ)`, but we will prove below that it works for all spaces. -/
instance : Dist GHSpace where
dist x y := sInf <| (fun p : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ =>
hausdorffDist (p.1 : Set ℓ_infty_ℝ) p.2) '' { a | ⟦a⟧ = x } ×ˢ { b | ⟦b⟧ = y }
/-- The Gromov-Hausdorff distance between two nonempty compact metric spaces, equal by definition to
the distance of the equivalence classes of these spaces in the Gromov-Hausdorff space. |
dist_ghDist (p q : GHSpace) : dist p q = ghDist p.Rep q.Rep := by
rw [ghDist, p.toGHSpace_rep, q.toGHSpace_rep] | 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 | dist_ghDist | null |
ghDist_le_hausdorffDist {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] {γ : Type w} [MetricSpace γ]
{Φ : X → γ} {Ψ : Y → γ} (ha : Isometry Φ) (hb : Isometry Ψ) :
ghDist X Y ≤ hausdorffDist (range Φ) (range Ψ) := by
/- For the proof, we want to embed `γ` in `ℓ^∞(ℝ)`, to say that the Hausdorff distance is realized
in `ℓ^∞(ℝ)` and therefore bounded below by the Gromov-Hausdorff-distance. However, `γ` is not
separable in general. We restrict to the union of the images of `X` and `Y` in `γ`, which is
separable and therefore embeddable in `ℓ^∞(ℝ)`. -/
rcases exists_mem_of_nonempty X with ⟨xX, _⟩
let s : Set γ := range Φ ∪ range Ψ
let Φ' : X → Subtype s := fun y => ⟨Φ y, mem_union_left _ (mem_range_self _)⟩
let Ψ' : Y → Subtype s := fun y => ⟨Ψ y, mem_union_right _ (mem_range_self _)⟩
have IΦ' : Isometry Φ' := fun x y => ha x y
have IΨ' : Isometry Ψ' := fun x y => hb x y
have : IsCompact s := (isCompact_range ha.continuous).union (isCompact_range hb.continuous)
let _ : MetricSpace (Subtype s) := by infer_instance
have : CompactSpace (Subtype s) := ⟨isCompact_iff_isCompact_univ.1 ‹IsCompact s›⟩
have ΦΦ' : Φ = Subtype.val ∘ Φ' := rfl
have ΨΨ' : Ψ = Subtype.val ∘ Ψ' := rfl
have : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ') := by
rw [ΦΦ', ΨΨ', range_comp, range_comp]
exact hausdorffDist_image isometry_subtype_coe
rw [this]
let F := kuratowskiEmbedding (Subtype s)
have : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ') :=
hausdorffDist_image (kuratowskiEmbedding.isometry _)
rw [← this]
let A : NonemptyCompacts ℓ_infty_ℝ :=
⟨⟨F '' range Φ',
(isCompact_range IΦ'.continuous).image (kuratowskiEmbedding.isometry _).continuous⟩,
(range_nonempty _).image _⟩
let B : NonemptyCompacts ℓ_infty_ℝ :=
⟨⟨F '' range Ψ',
(isCompact_range IΨ'.continuous).image (kuratowskiEmbedding.isometry _).continuous⟩,
(range_nonempty _).image _⟩
have AX : ⟦A⟧ = toGHSpace X := by
rw [eq_toGHSpace_iff]
exact ⟨fun x => F (Φ' x), (kuratowskiEmbedding.isometry _).comp IΦ', range_comp _ _⟩
have BY : ⟦B⟧ = toGHSpace Y := by
rw [eq_toGHSpace_iff]
exact ⟨fun x => F (Ψ' x), (kuratowskiEmbedding.isometry _).comp IΨ', range_comp _ _⟩
refine csInf_le ⟨0, ?_⟩ ?_
· simp only [lowerBounds, mem_image, mem_prod, mem_setOf_eq, Prod.exists, and_imp,
forall_exists_index]
intro t _ _ _ _ ht
rw [← ht]
exact hausdorffDist_nonneg
apply (mem_image _ _ _).2
exists (⟨A, B⟩ : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ) | 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 | ghDist_le_hausdorffDist | The Gromov-Hausdorff distance between two spaces is bounded by the Hausdorff distance
of isometric copies of the spaces, in any metric space. |
hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y := by
inhabit X; inhabit Y
/- we only need to check the inequality `≤`, as the other one follows from the previous lemma.
As the Gromov-Hausdorff distance is an infimum, we need to check that the Hausdorff distance
in the optimal coupling is smaller than the Hausdorff distance of any coupling.
First, we check this for couplings which already have small Hausdorff distance: in this
case, the induced "distance" on `X ⊕ Y` belongs to the candidates family introduced in the
definition of the optimal coupling, and the conclusion follows from the optimality
of the optimal coupling within this family.
-/
have A :
∀ p q : NonemptyCompacts ℓ_infty_ℝ,
⟦p⟧ = toGHSpace X →
⟦q⟧ = toGHSpace Y →
hausdorffDist (p : Set ℓ_infty_ℝ) q < diam (univ : Set X) + 1 + diam (univ : Set Y) →
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤
hausdorffDist (p : Set ℓ_infty_ℝ) q := by
intro p q hp hq bound
rcases eq_toGHSpace_iff.1 hp with ⟨Φ, ⟨Φisom, Φrange⟩⟩
rcases eq_toGHSpace_iff.1 hq with ⟨Ψ, ⟨Ψisom, Ψrange⟩⟩
have I : diam (range Φ ∪ range Ψ) ≤ 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by
rcases exists_mem_of_nonempty X with ⟨xX, _⟩
have : ∃ y ∈ range Ψ, dist (Φ xX) y < diam (univ : Set X) + 1 + diam (univ : Set Y) := by
rw [Ψrange]
have : Φ xX ∈ (p : Set _) := Φrange ▸ (mem_range_self _)
exact
exists_dist_lt_of_hausdorffDist_lt this bound
(hausdorffEdist_ne_top_of_nonempty_of_bounded p.nonempty q.nonempty
p.isCompact.isBounded q.isCompact.isBounded)
rcases this with ⟨y, hy, dy⟩
rcases mem_range.1 hy with ⟨z, hzy⟩
rw [← hzy] at dy
have DΦ : diam (range Φ) = diam (univ : Set X) := Φisom.diam_range
have DΨ : diam (range Ψ) = diam (univ : Set Y) := Ψisom.diam_range
calc
diam (range Φ ∪ range Ψ) ≤ diam (range Φ) + dist (Φ xX) (Ψ z) + diam (range Ψ) :=
diam_union (mem_range_self _) (mem_range_self _)
_ ≤
diam (univ : Set X) + (diam (univ : Set X) + 1 + diam (univ : Set Y)) +
diam (univ : Set Y) := by
rw [DΦ, DΨ]
gcongr
_ = 2 * diam (univ : Set X) + 1 + 2 * diam (univ : Set Y) := by ring
let f : X ⊕ Y → ℓ_infty_ℝ := fun x =>
match x with
| inl y => Φ y
| inr z => Ψ z
let F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
have Fgood : F ∈ candidates X Y := by
... | 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 | hausdorffDist_optimal | The optimal coupling constructed above realizes exactly the Gromov-Hausdorff distance,
essentially by design. |
ghDist_eq_hausdorffDist (X : Type u) [MetricSpace X] [CompactSpace X] [Nonempty X]
(Y : Type v) [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
∃ Φ : X → ℓ_infty_ℝ,
∃ Ψ : Y → ℓ_infty_ℝ,
Isometry Φ ∧ Isometry Ψ ∧ ghDist X Y = hausdorffDist (range Φ) (range Ψ) := by
let F := kuratowskiEmbedding (OptimalGHCoupling X Y)
let Φ := F ∘ optimalGHInjl X Y
let Ψ := F ∘ optimalGHInjr X Y
refine ⟨Φ, Ψ, ?_, ?_, ?_⟩
· exact (kuratowskiEmbedding.isometry _).comp (isometry_optimalGHInjl X Y)
· exact (kuratowskiEmbedding.isometry _).comp (isometry_optimalGHInjr X Y)
· rw [← image_univ, ← image_univ, image_comp F, image_univ, image_comp F (optimalGHInjr X Y),
image_univ, ← hausdorffDist_optimal]
exact (hausdorffDist_image (kuratowskiEmbedding.isometry _)).symm | 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 | ghDist_eq_hausdorffDist | The Gromov-Hausdorff distance can also be realized by a coupling in `ℓ^∞(ℝ)`, by embedding
the optimal coupling through its Kuratowski embedding. |
TopologicalSpace.NonemptyCompacts.toGHSpace {X : Type u} [MetricSpace X]
(p : NonemptyCompacts X) : GromovHausdorff.GHSpace :=
GromovHausdorff.toGHSpace p | 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 | TopologicalSpace.NonemptyCompacts.toGHSpace | The Gromov-Hausdorff distance defines a genuine distance on the Gromov-Hausdorff space. -/
instance : MetricSpace GHSpace where
dist := dist
dist_self x := by
rcases exists_rep x with ⟨y, hy⟩
refine le_antisymm ?_ ?_
· apply csInf_le
· exact ⟨0, by rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg⟩
· simp only [mem_image, mem_prod, mem_setOf_eq, Prod.exists]
exists y, y
simpa only [and_self_iff, hausdorffDist_self_zero, eq_self_iff_true, and_true]
· apply le_csInf
· exact Set.Nonempty.image _ <| Set.Nonempty.prod ⟨y, hy⟩ ⟨y, hy⟩
· rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg
dist_comm x y := by
have A :
(fun p : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ =>
hausdorffDist (p.1 : Set ℓ_infty_ℝ) p.2) ''
{ a | ⟦a⟧ = x } ×ˢ { b | ⟦b⟧ = y } =
(fun p : NonemptyCompacts ℓ_infty_ℝ × NonemptyCompacts ℓ_infty_ℝ =>
hausdorffDist (p.1 : Set ℓ_infty_ℝ) p.2) ∘
Prod.swap ''
{ a | ⟦a⟧ = x } ×ˢ { b | ⟦b⟧ = y } := by
funext
simp only [comp_apply, Prod.fst_swap, Prod.snd_swap]
congr
simp only [hausdorffDist_comm]
simp only [dist, A, image_comp, image_swap_prod]
eq_of_dist_eq_zero {x} {y} hxy := by
/- To show that two spaces at zero distance are isometric,
we argue that the distance is realized by some coupling.
In this coupling, the two spaces are at zero Hausdorff distance,
i.e., they coincide. Therefore, the original spaces are isometric. -/
rcases ghDist_eq_hausdorffDist x.Rep y.Rep with ⟨Φ, Ψ, Φisom, Ψisom, DΦΨ⟩
rw [← dist_ghDist, hxy] at DΦΨ
have : range Φ = range Ψ := by
have hΦ : IsCompact (range Φ) := isCompact_range Φisom.continuous
have hΨ : IsCompact (range Ψ) := isCompact_range Ψisom.continuous
apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm
· exact hΦ.isClosed
· exact hΨ.isClosed
· exact hausdorffEdist_ne_top_of_nonempty_of_bounded (range_nonempty _) (range_nonempty _)
hΦ.isBounded hΨ.isBounded
have T : (range Ψ ≃ᵢ y.Rep) = (range Φ ≃ᵢ y.Rep) := by rw [this]
have eΨ := cast T Ψisom.isometryEquivOnRange.symm
have e := Φisom.isometryEquivOnRange.trans eΨ
rw [← x.toGHSpace_rep, ← y.toGHSpace_rep, toGHSpace_eq_toGHSpace_iff_isometryEquiv]
exact ⟨e⟩
dist_triangle x y z := by
/- To show the triangular inequality between `X`, `Y` and `Z`,
realize an optimal coupling between `X` and `Y` in a space `γ1`,
and an optimal coupling between `Y` and `Z` in a space `γ2`.
Then, glue these metric spaces along `Y`. We get a new space `γ`
in which `X` and `Y` are optimally coupled, as well as `Y` and `Z`.
Apply the triangle inequality for the Hausdorff distance in `γ`
to conclude. -/
let X := x.Rep
let Y := y.Rep
let Z := z.Rep
let γ1 := OptimalGHCoupling X Y
let γ2 := OptimalGHCoupling Y Z
let Φ : Y → γ1 := optimalGHInjr X Y
have hΦ : Isometry Φ := isometry_optimalGHInjr X Y
let Ψ : Y → γ2 := optimalGHInjl Y Z
have hΨ : Isometry Ψ := isometry_optimalGHInjl Y Z
have Comm : toGlueL hΦ hΨ ∘ optimalGHInjr X Y = toGlueR hΦ hΨ ∘ optimalGHInjl Y Z :=
toGlue_commute hΦ hΨ
calc
dist x z = dist (toGHSpace X) (toGHSpace Z) := by
rw [x.toGHSpace_rep, z.toGHSpace_rep]
_ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
(range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) :=
(ghDist_le_hausdorffDist ((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y))
((toGlueR_isometry hΦ hΨ).comp (isometry_optimalGHInjr Y Z)))
_ ≤ hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjl X Y))
(range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y)) +
hausdorffDist (range (toGlueL hΦ hΨ ∘ optimalGHInjr X Y))
(range (toGlueR hΦ hΨ ∘ optimalGHInjr Y Z)) := by
refine hausdorffDist_triangle <| hausdorffEdist_ne_top_of_nonempty_of_bounded
(range_nonempty _) (range_nonempty _) ?_ ?_
· exact (isCompact_range (Isometry.continuous
((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjl X Y)))).isBounded
· exact (isCompact_range (Isometry.continuous
((toGlueL_isometry hΦ hΨ).comp (isometry_optimalGHInjr X Y)))).isBounded
_ = hausdorffDist (toGlueL hΦ hΨ '' range (optimalGHInjl X Y))
(toGlueL hΦ hΨ '' range (optimalGHInjr X Y)) +
hausdorffDist (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z))
(toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) := by
simp only [← range_comp, Comm]
_ = hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) := by
rw [hausdorffDist_image (toGlueL_isometry hΦ hΨ),
hausdorffDist_image (toGlueR_isometry hΦ hΨ)]
_ = dist (toGHSpace X) (toGHSpace Y) + dist (toGHSpace Y) (toGHSpace Z) := by
rw [hausdorffDist_optimal, hausdorffDist_optimal, ghDist, ghDist]
_ = dist x y + dist y z := by rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
end GHSpace --section
end GromovHausdorff
/-- In particular, nonempty compacts of a metric space map to `GHSpace`.
We register this in the `TopologicalSpace` namespace to take advantage
of the notation `p.toGHSpace`. |
ghDist_le_nonemptyCompacts_dist (p q : NonemptyCompacts X) :
dist p.toGHSpace q.toGHSpace ≤ dist p q := by
have ha : Isometry ((↑) : p → X) := isometry_subtype_coe
have hb : Isometry ((↑) : q → X) := isometry_subtype_coe
have A : dist p q = hausdorffDist (p : Set X) q := rfl
have I : ↑p = range ((↑) : p → X) := Subtype.range_coe_subtype.symm
have J : ↑q = range ((↑) : q → X) := Subtype.range_coe_subtype.symm
rw [A, I, J]
exact ghDist_le_hausdorffDist ha hb | 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 | ghDist_le_nonemptyCompacts_dist | null |
toGHSpace_lipschitz :
LipschitzWith 1 (NonemptyCompacts.toGHSpace : NonemptyCompacts X → GHSpace) :=
LipschitzWith.mk_one ghDist_le_nonemptyCompacts_dist | 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 | toGHSpace_lipschitz | null |
toGHSpace_continuous :
Continuous (NonemptyCompacts.toGHSpace : NonemptyCompacts X → GHSpace) :=
toGHSpace_lipschitz.continuous | 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 | toGHSpace_continuous | null |
ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ := by
refine le_of_forall_pos_le_add fun δ δ0 => ?_
rcases exists_mem_of_nonempty X with ⟨xX, _⟩
rcases hs xX with ⟨xs, hxs, Dxs⟩
have sne : s.Nonempty := ⟨xs, hxs⟩
let _ : Nonempty s := sne.to_subtype
have : 0 ≤ ε₂ := le_trans (abs_nonneg _) (H ⟨xs, hxs⟩ ⟨xs, hxs⟩)
have : ∀ p q : s, |dist p q - dist (Φ p) (Φ q)| ≤ 2 * (ε₂ / 2 + δ) := fun p q =>
calc
|dist p q - dist (Φ p) (Φ q)| ≤ ε₂ := H p q
_ ≤ 2 * (ε₂ / 2 + δ) := by linarith
let _ : MetricSpace (X ⊕ Y) :=
glueMetricApprox (fun x : s => (x : X)) (fun x => Φ x) (ε₂ / 2 + δ) (by linarith) this
let Fl := @Sum.inl X Y
let Fr := @Sum.inr X Y
have Il : Isometry Fl := Isometry.of_dist_eq fun x y => rfl
have Ir : Isometry Fr := Isometry.of_dist_eq fun x y => rfl
/- The proof goes as follows : the `ghDist` is bounded by the Hausdorff distance of the images
in the coupling, which is bounded (using the triangular inequality) by the sum of the Hausdorff
distances of `X` and `s` (in the coupling or, equivalently in the original space), of `s` and
`Φ s`, and of `Φ s` and `Y` (in the coupling or, equivalently, in the original space).
The first term is bounded by `ε₁`, by `ε₁`-density. The third one is bounded by `ε₃`.
And the middle one is bounded by `ε₂/2` as in the coupling the points `x` and `Φ x` are
at distance `ε₂/2` by construction of the coupling (in fact `ε₂/2 + δ` where `δ` is an
arbitrarily small positive constant where positivity is used to ensure that the coupling
is really a metric space and not a premetric space on `X ⊕ Y`). -/
have : ghDist X Y ≤ hausdorffDist (range Fl) (range Fr) := ghDist_le_hausdorffDist Il Ir
have :
hausdorffDist (range Fl) (range Fr) ≤
hausdorffDist (range Fl) (Fl '' s) + hausdorffDist (Fl '' s) (range Fr) :=
have B : IsBounded (range Fl) := (isCompact_range Il.continuous).isBounded
hausdorffDist_triangle
(hausdorffEdist_ne_top_of_nonempty_of_bounded (range_nonempty _) (sne.image _) B
(B.subset (image_subset_range _ _)))
have :
hausdorffDist (Fl '' s) (range Fr) ≤
hausdorffDist (Fl '' s) (Fr '' range Φ) + hausdorffDist (Fr '' range Φ) (range Fr) :=
have B : IsBounded (range Fr) := (isCompact_range Ir.continuous).isBounded
hausdorffDist_triangle'
(hausdorffEdist_ne_top_of_nonempty_of_bounded ((range_nonempty _).image _) (range_nonempty _)
(B.subset (image_subset_range _ _)) B)
have : hausdorffDist (range Fl) (Fl '' s) ≤ ε₁ := by
rw [← image_univ, hausdorffDist_image Il]
have : 0 ≤ ε₁ := le_trans dist_nonneg Dxs
refine hausdorffDist_le_of_mem_dist this (fun x _ => hs x) fun x _ =>
⟨x, mem_univ _, by simpa only [dist_self]⟩
have : hausdorffDist (Fl '' s) (Fr '' range Φ) ≤ ε₂ / 2 + δ := by
refine hausdorffDist_le_of_mem_dist (by linarith) ?_ ?_
· intro x' hx'
... | 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 | ghDist_le_of_approx_subsets | If there are subsets which are `ε₁`-dense and `ε₃`-dense in two spaces, and
isometric up to `ε₂`, then the Gromov-Hausdorff distance between the spaces is bounded by
`ε₁ + ε₂/2 + ε₃`. |
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