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 ⌀ |
|---|---|---|---|---|---|---|
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... | 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_m... | 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... | 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)
h... | 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, a... | 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_l... | 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,... | 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, .in... | 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_i... | 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 ⟨mi... | 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_e... | 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 unbounde... |
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 (l... | 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_tri... | 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 dista... |
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 arbitr... |
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, ... | 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, ?_⟩... | 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_c... | 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... |
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).isUnifo... | 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 ?_... | 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
... | 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_d... | 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 ?_... | 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 usin... | 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 := (kuratowskiEmbedd... | 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 :
(NonemptyCompac... | 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
di... |
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... | 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 othe... | 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 (O... | 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_no... |
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... | 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_n... | 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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