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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dimH_univ_eq_finrank : dimH (univ : Set E) = finrank ℝ E :=
dimH_of_mem_nhds (@univ_mem _ (𝓝 0)) | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ_eq_finrank | null |
dimH_univ : dimH (univ : Set ℝ) = 1 := by
rw [dimH_univ_eq_finrank ℝ, Module.finrank_self, Nat.cast_one]
variable {E} | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_univ | null |
dimH_lt_top (s : Set E) : dimH s < ⊤ := by calc
dimH s ≤ dimH (univ : Set E) := dimH_mono (subset_univ s)
_ = finrank ℝ E := dimH_univ_eq_finrank E
_ < ⊤ := by simp | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_lt_top | The Hausdorff dimension of any set in a finite-dimensional real normed space is finite. |
dimH_ne_top (s : Set E) : dimH s ≠ ⊤ := (dimH_lt_top s).ne | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_ne_top | null |
hausdorffMeasure_of_finrank_lt [MeasurableSpace E] [BorelSpace E] {d : ℝ}
(hd : finrank ℝ E < d) : (μH[d] : Measure E) = 0 := by
lift d to ℝ≥0 using (Nat.cast_nonneg _).trans hd.le
rw [← measure_univ_eq_zero]
apply hausdorffMeasure_of_dimH_lt
rw [dimH_univ_eq_finrank]
exact mod_cast hd | lemma | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | hausdorffMeasure_of_finrank_lt | null |
dense_compl_of_dimH_lt_finrank {s : Set E} (hs : dimH s < finrank ℝ E) : Dense sᶜ := by
refine fun x => mem_closure_iff_nhds.2 fun t ht => nonempty_iff_ne_empty.2 fun he => hs.not_ge ?_
rw [← diff_eq, diff_eq_empty] at he
rw [← Real.dimH_of_mem_nhds ht]
exact dimH_mono he
/-! | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dense_compl_of_dimH_lt_finrank | null |
ContDiffOn.dimH_image_le {f : E → F} {s t : Set E} (hf : ContDiffOn ℝ 1 f s)
(hc : Convex ℝ s) (ht : t ⊆ s) : dimH (f '' t) ≤ dimH t :=
dimH_image_le_of_locally_lipschitzOn fun x hx =>
let ⟨C, u, hu, hf⟩ := (hf x (ht hx)).exists_lipschitzOnWith hc
⟨C, u, nhdsWithin_mono _ ht hu, hf⟩ | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | ContDiffOn.dimH_image_le | Let `f` be a function defined on a finite-dimensional real normed space. If `f` is `C¹`-smooth
on a convex set `s`, then the Hausdorff dimension of `f '' s` is less than or equal to the Hausdorff
dimension of `s`.
TODO: do we actually need `Convex ℝ s`? |
ContDiff.dimH_range_le {f : E → F} (h : ContDiff ℝ 1 f) : dimH (range f) ≤ finrank ℝ E :=
calc
dimH (range f) = dimH (f '' univ) := by rw [image_univ]
_ ≤ dimH (univ : Set E) := h.contDiffOn.dimH_image_le convex_univ Subset.rfl
_ = finrank ℝ E := Real.dimH_univ_eq_finrank E | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | ContDiff.dimH_range_le | The Hausdorff dimension of the range of a `C¹`-smooth function defined on a finite-dimensional
real normed space is at most the dimension of its domain as a vector space over `ℝ`. |
ContDiffOn.dense_compl_image_of_dimH_lt_finrank [FiniteDimensional ℝ F] {f : E → F}
{s t : Set E} (h : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s)
(htF : dimH t < finrank ℝ F) : Dense (f '' t)ᶜ :=
dense_compl_of_dimH_lt_finrank <| (h.dimH_image_le hc ht).trans_lt htF | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | ContDiffOn.dense_compl_image_of_dimH_lt_finrank | A particular case of Sard's Theorem. Let `f : E → F` be a map between finite-dimensional real
vector spaces. Suppose that `f` is `C¹` smooth on a convex set `s` of Hausdorff dimension strictly
less than the dimension of `F`. Then the complement of the image `f '' s` is dense in `F`. |
ContDiff.dense_compl_range_of_finrank_lt_finrank [FiniteDimensional ℝ F] {f : E → F}
(h : ContDiff ℝ 1 f) (hEF : finrank ℝ E < finrank ℝ F) : Dense (range f)ᶜ :=
dense_compl_of_dimH_lt_finrank <| h.dimH_range_le.trans_lt <| Nat.cast_lt.2 hEF | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | ContDiff.dense_compl_range_of_finrank_lt_finrank | A particular case of Sard's Theorem. If `f` is a `C¹` smooth map from a real vector space to a
real vector space `F` of strictly larger dimension, then the complement of the range of `f` is dense
in `F`. |
dimH_orthogonalProjection_le {𝕜 E : Type*} [RCLike 𝕜]
[NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E) [K.HasOrthogonalProjection] (s : Set E) :
dimH (K.orthogonalProjection '' s) ≤ dimH s :=
K.lipschitzWith_orthogonalProjection.dimH_image_le s | theorem | Topology | [
"Mathlib.Analysis.Calculus.ContDiff.RCLike",
"Mathlib.MeasureTheory.Measure.Hausdorff"
] | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | dimH_orthogonalProjection_le | The Hausdorff dimension of the orthogonal projection of a set `s` onto a subspace `K`
is less than or equal to the Hausdorff dimension of `s`. |
infEdist (x : α) (s : Set α) : ℝ≥0∞ :=
⨅ y ∈ s, edist x y
@[simp] | def | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist | The minimal edistance of a point to a set |
infEdist_empty : infEdist x ∅ = ∞ :=
iInf_emptyset | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_empty | null |
le_infEdist {d} : d ≤ infEdist x s ↔ ∀ y ∈ s, d ≤ edist x y := by
simp only [infEdist, le_iInf_iff] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | le_infEdist | null |
@[simp]
infEdist_union : infEdist x (s ∪ t) = infEdist x s ⊓ infEdist x t :=
iInf_union
@[simp] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_union | The edist to a union is the minimum of the edists |
infEdist_iUnion (f : ι → Set α) (x : α) : infEdist x (⋃ i, f i) = ⨅ i, infEdist x (f i) :=
iInf_iUnion f _ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_iUnion | null |
infEdist_biUnion {ι : Type*} (f : ι → Set α) (I : Set ι) (x : α) :
infEdist x (⋃ i ∈ I, f i) = ⨅ i ∈ I, infEdist x (f i) := by simp only [infEdist_iUnion] | lemma | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_biUnion | null |
@[simp]
infEdist_singleton : infEdist x {y} = edist x y :=
iInf_singleton | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_singleton | The edist to a singleton is the edistance to the single point of this singleton |
infEdist_le_edist_of_mem (h : y ∈ s) : infEdist x s ≤ edist x y :=
iInf₂_le y h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_le_edist_of_mem | The edist to a set is bounded above by the edist to any of its points |
infEdist_zero_of_mem (h : x ∈ s) : infEdist x s = 0 :=
nonpos_iff_eq_zero.1 <| @edist_self _ _ x ▸ infEdist_le_edist_of_mem h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_zero_of_mem | If a point `x` belongs to `s`, then its edist to `s` vanishes |
infEdist_anti (h : s ⊆ t) : infEdist x t ≤ infEdist x s :=
iInf_le_iInf_of_subset h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_anti | The edist is antitone with respect to inclusion. |
infEdist_lt_iff {r : ℝ≥0∞} : infEdist x s < r ↔ ∃ y ∈ s, edist x y < r := by
simp_rw [infEdist, iInf_lt_iff, exists_prop] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_lt_iff | The edist to a set is `< r` iff there exists a point in the set at edistance `< r` |
infEdist_le_infEdist_add_edist : infEdist x s ≤ infEdist y s + edist x y :=
calc
⨅ z ∈ s, edist x z ≤ ⨅ z ∈ s, edist y z + edist x y :=
iInf₂_mono fun _ _ => (edist_triangle _ _ _).trans_eq (add_comm _ _)
_ = (⨅ z ∈ s, edist y z) + edist x y := by simp only [ENNReal.iInf_add] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_le_infEdist_add_edist | The edist of `x` to `s` is bounded by the sum of the edist of `y` to `s` and
the edist from `x` to `y` |
infEdist_le_edist_add_infEdist : infEdist x s ≤ edist x y + infEdist y s := by
rw [add_comm]
exact infEdist_le_infEdist_add_edist | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_le_edist_add_infEdist | null |
edist_le_infEdist_add_ediam (hy : y ∈ s) : edist x y ≤ infEdist x s + diam s := by
simp_rw [infEdist, ENNReal.iInf_add]
refine le_iInf₂ fun i hi => ?_
calc
edist x y ≤ edist x i + edist i y := edist_triangle _ _ _
_ ≤ edist x i + diam s := add_le_add le_rfl (edist_le_diam_of_mem hi hy) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | edist_le_infEdist_add_ediam | null |
@[continuity, fun_prop]
continuous_infEdist : Continuous fun x => infEdist x s :=
continuous_of_le_add_edist 1 (by simp) <| by
simp only [one_mul, infEdist_le_infEdist_add_edist, forall₂_true_iff] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | continuous_infEdist | The edist to a set depends continuously on the point |
infEdist_closure : infEdist x (closure s) = infEdist x s := by
refine le_antisymm (infEdist_anti subset_closure) ?_
refine ENNReal.le_of_forall_pos_le_add fun ε εpos h => ?_
have ε0 : 0 < (ε / 2 : ℝ≥0∞) := by simpa [pos_iff_ne_zero] using εpos
have : infEdist x (closure s) < infEdist x (closure s) + ε / 2 :=
ENNReal.lt_add_right h.ne ε0.ne'
obtain ⟨y : α, ycs : y ∈ closure s, hy : edist x y < infEdist x (closure s) + ↑ε / 2⟩ :=
infEdist_lt_iff.mp this
obtain ⟨z : α, zs : z ∈ s, dyz : edist y z < ↑ε / 2⟩ := EMetric.mem_closure_iff.1 ycs (ε / 2) ε0
calc
infEdist x s ≤ edist x z := infEdist_le_edist_of_mem zs
_ ≤ edist x y + edist y z := edist_triangle _ _ _
_ ≤ infEdist x (closure s) + ε / 2 + ε / 2 := add_le_add (le_of_lt hy) (le_of_lt dyz)
_ = infEdist x (closure s) + ↑ε := by rw [add_assoc, ENNReal.add_halves] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_closure | The edist to a set and to its closure coincide |
mem_closure_iff_infEdist_zero : x ∈ closure s ↔ infEdist x s = 0 :=
⟨fun h => by
rw [← infEdist_closure]
exact infEdist_zero_of_mem h,
fun h =>
EMetric.mem_closure_iff.2 fun ε εpos => infEdist_lt_iff.mp <| by rwa [h]⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | mem_closure_iff_infEdist_zero | A point belongs to the closure of `s` iff its infimum edistance to this set vanishes |
mem_iff_infEdist_zero_of_closed (h : IsClosed s) : x ∈ s ↔ infEdist x s = 0 := by
rw [← mem_closure_iff_infEdist_zero, h.closure_eq] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | mem_iff_infEdist_zero_of_closed | Given a closed set `s`, a point belongs to `s` iff its infimum edistance to this set vanishes |
infEdist_pos_iff_notMem_closure {x : α} {E : Set α} :
0 < infEdist x E ↔ x ∉ closure E := by
rw [mem_closure_iff_infEdist_zero, pos_iff_ne_zero]
@[deprecated (since := "2025-05-23")]
alias infEdist_pos_iff_not_mem_closure := infEdist_pos_iff_notMem_closure | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_pos_iff_notMem_closure | The infimum edistance of a point to a set is positive if and only if the point is not in the
closure of the set. |
infEdist_closure_pos_iff_notMem_closure {x : α} {E : Set α} :
0 < infEdist x (closure E) ↔ x ∉ closure E := by
rw [infEdist_closure, infEdist_pos_iff_notMem_closure]
@[deprecated (since := "2025-05-23")]
alias infEdist_closure_pos_iff_not_mem_closure := infEdist_closure_pos_iff_notMem_closure | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_closure_pos_iff_notMem_closure | null |
exists_real_pos_lt_infEdist_of_notMem_closure {x : α} {E : Set α} (h : x ∉ closure E) :
∃ ε : ℝ, 0 < ε ∧ ENNReal.ofReal ε < infEdist x E := by
rw [← infEdist_pos_iff_notMem_closure, ENNReal.lt_iff_exists_real_btwn] at h
rcases h with ⟨ε, ⟨_, ⟨ε_pos, ε_lt⟩⟩⟩
exact ⟨ε, ⟨ENNReal.ofReal_pos.mp ε_pos, ε_lt⟩⟩
@[deprecated (since := "2025-05-23")]
alias exists_real_pos_lt_infEdist_of_not_mem_closure :=
exists_real_pos_lt_infEdist_of_notMem_closure | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_real_pos_lt_infEdist_of_notMem_closure | null |
disjoint_closedBall_of_lt_infEdist {r : ℝ≥0∞} (h : r < infEdist x s) :
Disjoint (closedBall x r) s := by
rw [disjoint_left]
intro y hy h'y
apply lt_irrefl (infEdist x s)
calc
infEdist x s ≤ edist x y := infEdist_le_edist_of_mem h'y
_ ≤ r := by rwa [mem_closedBall, edist_comm] at hy
_ < infEdist x s := h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | disjoint_closedBall_of_lt_infEdist | null |
infEdist_image (hΦ : Isometry Φ) : infEdist (Φ x) (Φ '' t) = infEdist x t := by
simp only [infEdist, iInf_image, hΦ.edist_eq]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_image | The infimum edistance is invariant under isometries |
infEdist_smul {M} [SMul M α] [IsIsometricSMul M α] (c : M) (x : α) (s : Set α) :
infEdist (c • x) (c • s) = infEdist x s :=
infEdist_image (isometry_smul _ _) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_smul | null |
_root_.IsOpen.exists_iUnion_isClosed {U : Set α} (hU : IsOpen U) :
∃ F : ℕ → Set α, (∀ n, IsClosed (F n)) ∧ (∀ n, F n ⊆ U) ∧ ⋃ n, F n = U ∧ Monotone F := by
obtain ⟨a, a_pos, a_lt_one⟩ : ∃ a : ℝ≥0∞, 0 < a ∧ a < 1 := exists_between zero_lt_one
let F := fun n : ℕ => (fun x => infEdist x Uᶜ) ⁻¹' Ici (a ^ n)
have F_subset : ∀ n, F n ⊆ U := fun n x hx ↦ by
by_contra h
have : infEdist x Uᶜ ≠ 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'
exact this (infEdist_zero_of_mem h)
refine ⟨F, fun n => IsClosed.preimage continuous_infEdist isClosed_Ici, F_subset, ?_, ?_⟩
· show ⋃ n, F n = U
refine Subset.antisymm (by simp only [iUnion_subset_iff, F_subset, forall_const]) fun x hx => ?_
have : x ∉ Uᶜ := by simpa using hx
rw [mem_iff_infEdist_zero_of_closed hU.isClosed_compl] at this
have B : 0 < infEdist x Uᶜ := by simpa [pos_iff_ne_zero] using this
have : Filter.Tendsto (fun n => a ^ n) atTop (𝓝 0) :=
ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one a_lt_one
rcases ((tendsto_order.1 this).2 _ B).exists with ⟨n, hn⟩
simp only [mem_iUnion]
exact ⟨n, hn.le⟩
show Monotone F
intro m n hmn x hx
simp only [F, mem_Ici, mem_preimage] at hx ⊢
apply le_trans (pow_le_pow_right_of_le_one' a_lt_one.le hmn) hx | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsOpen.exists_iUnion_isClosed | null |
_root_.IsCompact.exists_infEdist_eq_edist (hs : IsCompact s) (hne : s.Nonempty) (x : α) :
∃ y ∈ s, infEdist x s = edist x y := by
have A : Continuous fun y => edist x y := continuous_const.edist continuous_id
obtain ⟨y, ys, hy⟩ := hs.exists_isMinOn hne A.continuousOn
exact ⟨y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsCompact.exists_infEdist_eq_edist | null |
exists_pos_forall_lt_edist (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) :
∃ r : ℝ≥0, 0 < r ∧ ∀ x ∈ s, ∀ y ∈ t, (r : ℝ≥0∞) < edist x y := by
rcases s.eq_empty_or_nonempty with (rfl | hne)
· use 1
simp
obtain ⟨x, hx, h⟩ := hs.exists_isMinOn hne continuous_infEdist.continuousOn
have : 0 < infEdist x t :=
pos_iff_ne_zero.2 fun H => hst.le_bot ⟨hx, (mem_iff_infEdist_zero_of_closed ht).mpr H⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 this with ⟨r, h₀, hr⟩
exact ⟨r, ENNReal.coe_pos.mp h₀, fun y hy z hz => hr.trans_le <| le_infEdist.1 (h hy) z hz⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_pos_forall_lt_edist | null |
@[simp]
hausdorffEdist_self : hausdorffEdist s s = 0 := by
simp only [hausdorffEdist_def, sup_idem, ENNReal.iSup_eq_zero]
exact fun x hx => infEdist_zero_of_mem hx | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_self | The Hausdorff edistance between two sets is the smallest `r` such that each set
is contained in the `r`-neighborhood of the other one -/
irreducible_def hausdorffEdist {α : Type u} [PseudoEMetricSpace α] (s t : Set α) : ℝ≥0∞ :=
(⨆ x ∈ s, infEdist x t) ⊔ ⨆ y ∈ t, infEdist y s
section HausdorffEdist
variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] {x : α} {s t u : Set α} {Φ : α → β}
/-- The Hausdorff edistance of a set to itself vanishes. |
hausdorffEdist_comm : hausdorffEdist s t = hausdorffEdist t s := by
simp only [hausdorffEdist_def]; apply sup_comm | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_comm | The Hausdorff edistances of `s` to `t` and of `t` to `s` coincide. |
hausdorffEdist_le_of_infEdist {r : ℝ≥0∞} (H1 : ∀ x ∈ s, infEdist x t ≤ r)
(H2 : ∀ x ∈ t, infEdist x s ≤ r) : hausdorffEdist s t ≤ r := by
simp only [hausdorffEdist_def, sup_le_iff, iSup_le_iff]
exact ⟨H1, H2⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_le_of_infEdist | Bounding the Hausdorff edistance by bounding the edistance of any point
in each set to the other set |
hausdorffEdist_le_of_mem_edist {r : ℝ≥0∞} (H1 : ∀ x ∈ s, ∃ y ∈ t, edist x y ≤ r)
(H2 : ∀ x ∈ t, ∃ y ∈ s, edist x y ≤ r) : hausdorffEdist s t ≤ r := by
refine hausdorffEdist_le_of_infEdist (fun x xs ↦ ?_) (fun x xt ↦ ?_)
· rcases H1 x xs with ⟨y, yt, hy⟩
exact le_trans (infEdist_le_edist_of_mem yt) hy
· rcases H2 x xt with ⟨y, ys, hy⟩
exact le_trans (infEdist_le_edist_of_mem ys) hy | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_le_of_mem_edist | Bounding the Hausdorff edistance by exhibiting, for any point in each set,
another point in the other set at controlled distance |
infEdist_le_hausdorffEdist_of_mem (h : x ∈ s) : infEdist x t ≤ hausdorffEdist s t := by
rw [hausdorffEdist_def]
refine le_trans ?_ le_sup_left
exact le_iSup₂ (α := ℝ≥0∞) x h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_le_hausdorffEdist_of_mem | The distance to a set is controlled by the Hausdorff distance. |
exists_edist_lt_of_hausdorffEdist_lt {r : ℝ≥0∞} (h : x ∈ s) (H : hausdorffEdist s t < r) :
∃ y ∈ t, edist x y < r :=
infEdist_lt_iff.mp <|
calc
infEdist x t ≤ hausdorffEdist s t := infEdist_le_hausdorffEdist_of_mem h
_ < r := H | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_edist_lt_of_hausdorffEdist_lt | If the Hausdorff distance is `< r`, then any point in one of the sets has
a corresponding point at distance `< r` in the other set. |
infEdist_le_infEdist_add_hausdorffEdist :
infEdist x t ≤ infEdist x s + hausdorffEdist s t :=
ENNReal.le_of_forall_pos_le_add fun ε εpos h => by
have ε0 : (ε / 2 : ℝ≥0∞) ≠ 0 := by simpa [pos_iff_ne_zero] using εpos
have : infEdist x s < infEdist x s + ε / 2 :=
ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).1.ne ε0
obtain ⟨y : α, ys : y ∈ s, dxy : edist x y < infEdist x s + ↑ε / 2⟩ := infEdist_lt_iff.mp this
have : hausdorffEdist s t < hausdorffEdist s t + ε / 2 :=
ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).2.ne ε0
obtain ⟨z : α, zt : z ∈ t, dyz : edist y z < hausdorffEdist s t + ↑ε / 2⟩ :=
exists_edist_lt_of_hausdorffEdist_lt ys this
calc
infEdist x t ≤ edist x z := infEdist_le_edist_of_mem zt
_ ≤ edist x y + edist y z := edist_triangle _ _ _
_ ≤ infEdist x s + ε / 2 + (hausdorffEdist s t + ε / 2) := add_le_add dxy.le dyz.le
_ = infEdist x s + hausdorffEdist s t + ε := by
simp [ENNReal.add_halves, add_comm, add_left_comm] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_le_infEdist_add_hausdorffEdist | The distance from `x` to `s` or `t` is controlled in terms of the Hausdorff distance
between `s` and `t`. |
hausdorffEdist_image (h : Isometry Φ) :
hausdorffEdist (Φ '' s) (Φ '' t) = hausdorffEdist s t := by
simp only [hausdorffEdist_def, iSup_image, infEdist_image h] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_image | The Hausdorff edistance is invariant under isometries. |
hausdorffEdist_le_ediam (hs : s.Nonempty) (ht : t.Nonempty) :
hausdorffEdist s t ≤ diam (s ∪ t) := by
rcases hs with ⟨x, xs⟩
rcases ht with ⟨y, yt⟩
refine hausdorffEdist_le_of_mem_edist ?_ ?_
· intro z hz
exact ⟨y, yt, edist_le_diam_of_mem (subset_union_left hz) (subset_union_right yt)⟩
· intro z hz
exact ⟨x, xs, edist_le_diam_of_mem (subset_union_right hz) (subset_union_left xs)⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_le_ediam | The Hausdorff distance is controlled by the diameter of the union. |
hausdorffEdist_triangle : hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u := by
rw [hausdorffEdist_def]
simp only [sup_le_iff, iSup_le_iff]
constructor
· change ∀ x ∈ s, infEdist x u ≤ hausdorffEdist s t + hausdorffEdist t u
exact fun x xs =>
calc
infEdist x u ≤ infEdist x t + hausdorffEdist t u :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ hausdorffEdist s t + hausdorffEdist t u :=
add_le_add_right (infEdist_le_hausdorffEdist_of_mem xs) _
· change ∀ x ∈ u, infEdist x s ≤ hausdorffEdist s t + hausdorffEdist t u
exact fun x xu =>
calc
infEdist x s ≤ infEdist x t + hausdorffEdist t s :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ hausdorffEdist u t + hausdorffEdist t s :=
add_le_add_right (infEdist_le_hausdorffEdist_of_mem xu) _
_ = hausdorffEdist s t + hausdorffEdist t u := by simp [hausdorffEdist_comm, add_comm] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_triangle | The Hausdorff distance satisfies the triangle inequality. |
hausdorffEdist_zero_iff_closure_eq_closure :
hausdorffEdist s t = 0 ↔ closure s = closure t := by
simp only [hausdorffEdist_def, ENNReal.sup_eq_zero, ENNReal.iSup_eq_zero, ← subset_def,
← mem_closure_iff_infEdist_zero, subset_antisymm_iff, isClosed_closure.closure_subset_iff] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_zero_iff_closure_eq_closure | Two sets are at zero Hausdorff edistance if and only if they have the same closure. |
@[simp]
hausdorffEdist_self_closure : hausdorffEdist s (closure s) = 0 := by
rw [hausdorffEdist_zero_iff_closure_eq_closure, closure_closure] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_self_closure | The Hausdorff edistance between a set and its closure vanishes. |
@[simp]
hausdorffEdist_closure₁ : hausdorffEdist (closure s) t = hausdorffEdist s t := by
refine le_antisymm ?_ ?_
· calc
_ ≤ hausdorffEdist (closure s) s + hausdorffEdist s t := hausdorffEdist_triangle
_ = hausdorffEdist s t := by simp [hausdorffEdist_comm]
· calc
_ ≤ hausdorffEdist s (closure s) + hausdorffEdist (closure s) t := hausdorffEdist_triangle
_ = hausdorffEdist (closure s) t := by simp | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_closure₁ | Replacing a set by its closure does not change the Hausdorff edistance. |
@[simp]
hausdorffEdist_closure₂ : hausdorffEdist s (closure t) = hausdorffEdist s t := by
simp [@hausdorffEdist_comm _ _ s _] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_closure₂ | Replacing a set by its closure does not change the Hausdorff edistance. |
hausdorffEdist_closure : hausdorffEdist (closure s) (closure t) = hausdorffEdist s t := by
simp | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_closure | The Hausdorff edistance between sets or their closures is the same. |
hausdorffEdist_zero_iff_eq_of_closed (hs : IsClosed s) (ht : IsClosed t) :
hausdorffEdist s t = 0 ↔ s = t := by
rw [hausdorffEdist_zero_iff_closure_eq_closure, hs.closure_eq, ht.closure_eq] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_zero_iff_eq_of_closed | Two closed sets are at zero Hausdorff edistance if and only if they coincide. |
hausdorffEdist_empty (ne : s.Nonempty) : hausdorffEdist s ∅ = ∞ := by
rcases ne with ⟨x, xs⟩
have : infEdist x ∅ ≤ hausdorffEdist s ∅ := infEdist_le_hausdorffEdist_of_mem xs
simpa using this | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_empty | The Hausdorff edistance to the empty set is infinite. |
nonempty_of_hausdorffEdist_ne_top (hs : s.Nonempty) (fin : hausdorffEdist s t ≠ ⊤) :
t.Nonempty :=
t.eq_empty_or_nonempty.resolve_left fun ht ↦ fin (ht.symm ▸ hausdorffEdist_empty hs) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | nonempty_of_hausdorffEdist_ne_top | If a set is at finite Hausdorff edistance of a nonempty set, it is nonempty. |
empty_or_nonempty_of_hausdorffEdist_ne_top (fin : hausdorffEdist s t ≠ ⊤) :
(s = ∅ ∧ t = ∅) ∨ (s.Nonempty ∧ t.Nonempty) := by
rcases s.eq_empty_or_nonempty with hs | hs
· rcases t.eq_empty_or_nonempty with ht | ht
· exact Or.inl ⟨hs, ht⟩
· rw [hausdorffEdist_comm] at fin
exact Or.inr ⟨nonempty_of_hausdorffEdist_ne_top ht fin, ht⟩
· exact Or.inr ⟨hs, nonempty_of_hausdorffEdist_ne_top hs fin⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | empty_or_nonempty_of_hausdorffEdist_ne_top | null |
infDist (x : α) (s : Set α) : ℝ :=
ENNReal.toReal (infEdist x s) | def | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist | The minimal distance of a point to a set |
infDist_eq_iInf : infDist x s = ⨅ y : s, dist x y := by
rw [infDist, infEdist, iInf_subtype', ENNReal.toReal_iInf]
· simp only [dist_edist]
· finiteness | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_eq_iInf | null |
infDist_nonneg : 0 ≤ infDist x s := toReal_nonneg | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_nonneg | The minimal distance is always nonnegative |
@[simp]
infDist_empty : infDist x ∅ = 0 := by simp [infDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_empty | The minimal distance to the empty set is 0 (if you want to have the more reasonable
value `∞` instead, use `EMetric.infEdist`, which takes values in `ℝ≥0∞`) |
isGLB_infDist (hs : s.Nonempty) : IsGLB ((dist x ·) '' s) (infDist x s) := by
simpa [infDist_eq_iInf, sInf_image']
using isGLB_csInf (hs.image _) ⟨0, by simp [lowerBounds]⟩ | lemma | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | isGLB_infDist | null |
infEdist_ne_top (h : s.Nonempty) : infEdist x s ≠ ⊤ := by
rcases h with ⟨y, hy⟩
exact ne_top_of_le_ne_top (edist_ne_top _ _) (infEdist_le_edist_of_mem hy)
@[simp] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_ne_top | In a metric space, the minimal edistance to a nonempty set is finite. |
infEdist_eq_top_iff : infEdist x s = ∞ ↔ s = ∅ := by
rcases s.eq_empty_or_nonempty with rfl | hs <;> simp [*, Nonempty.ne_empty, infEdist_ne_top] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infEdist_eq_top_iff | null |
infDist_zero_of_mem (h : x ∈ s) : infDist x s = 0 := by
simp [infEdist_zero_of_mem h, infDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_zero_of_mem | The minimal distance of a point to a set containing it vanishes. |
@[simp]
infDist_singleton : infDist x {y} = dist x y := by simp [infDist, dist_edist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_singleton | The minimal distance to a singleton is the distance to the unique point in this singleton. |
infDist_le_dist_of_mem (h : y ∈ s) : infDist x s ≤ dist x y := by
rw [dist_edist, infDist]
exact ENNReal.toReal_mono (edist_ne_top _ _) (infEdist_le_edist_of_mem h) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_le_dist_of_mem | The minimal distance to a set is bounded by the distance to any point in this set. |
infDist_le_infDist_of_subset (h : s ⊆ t) (hs : s.Nonempty) : infDist x t ≤ infDist x s :=
ENNReal.toReal_mono (infEdist_ne_top hs) (infEdist_anti h) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_le_infDist_of_subset | The minimal distance is monotone with respect to inclusion. |
le_infDist {r : ℝ} (hs : s.Nonempty) : r ≤ infDist x s ↔ ∀ ⦃y⦄, y ∈ s → r ≤ dist x y := by
simp_rw [infDist, ← ENNReal.ofReal_le_iff_le_toReal (infEdist_ne_top hs), le_infEdist,
ENNReal.ofReal_le_iff_le_toReal (edist_ne_top _ _), ← dist_edist] | lemma | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | le_infDist | null |
infDist_lt_iff {r : ℝ} (hs : s.Nonempty) : infDist x s < r ↔ ∃ y ∈ s, dist x y < r := by
simp [← not_le, le_infDist hs] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_lt_iff | The minimal distance to a set `s` is `< r` iff there exists a point in `s` at distance `< r`. |
infDist_le_infDist_add_dist : infDist x s ≤ infDist y s + dist x y := by
rw [infDist, infDist, dist_edist]
refine ENNReal.toReal_le_add' infEdist_le_infEdist_add_edist ?_ (flip absurd (edist_ne_top _ _))
simp only [infEdist_eq_top_iff, imp_self] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_le_infDist_add_dist | The minimal distance from `x` to `s` is bounded by the distance from `y` to `s`, modulo
the distance between `x` and `y`. |
notMem_of_dist_lt_infDist (h : dist x y < infDist x s) : y ∉ s := fun hy =>
h.not_ge <| infDist_le_dist_of_mem hy
@[deprecated (since := "2025-05-23")] alias not_mem_of_dist_lt_infDist := notMem_of_dist_lt_infDist | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | notMem_of_dist_lt_infDist | null |
disjoint_ball_infDist : Disjoint (ball x (infDist x s)) s :=
disjoint_left.2 fun _y hy => notMem_of_dist_lt_infDist <| mem_ball'.1 hy | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | disjoint_ball_infDist | null |
ball_infDist_subset_compl : ball x (infDist x s) ⊆ sᶜ :=
(disjoint_ball_infDist (s := s)).subset_compl_right | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | ball_infDist_subset_compl | null |
ball_infDist_compl_subset : ball x (infDist x sᶜ) ⊆ s :=
ball_infDist_subset_compl.trans_eq (compl_compl s) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | ball_infDist_compl_subset | null |
disjoint_closedBall_of_lt_infDist {r : ℝ} (h : r < infDist x s) :
Disjoint (closedBall x r) s :=
disjoint_ball_infDist.mono_left <| closedBall_subset_ball h | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | disjoint_closedBall_of_lt_infDist | null |
dist_le_infDist_add_diam (hs : IsBounded s) (hy : y ∈ s) :
dist x y ≤ infDist x s + diam s := by
rw [infDist, diam, dist_edist]
exact toReal_le_add (edist_le_infEdist_add_ediam hy) (infEdist_ne_top ⟨y, hy⟩) hs.ediam_ne_top
variable (s) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | dist_le_infDist_add_diam | null |
lipschitz_infDist_pt : LipschitzWith 1 (infDist · s) :=
LipschitzWith.of_le_add fun _ _ => infDist_le_infDist_add_dist | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | lipschitz_infDist_pt | The minimal distance to a set is Lipschitz in point with constant 1 |
uniformContinuous_infDist_pt : UniformContinuous (infDist · s) :=
(lipschitz_infDist_pt s).uniformContinuous | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | uniformContinuous_infDist_pt | The minimal distance to a set is uniformly continuous in point |
@[continuity, fun_prop]
continuous_infDist_pt : Continuous (infDist · s) :=
(uniformContinuous_infDist_pt s).continuous
variable {s} | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | continuous_infDist_pt | The minimal distance to a set is continuous in point |
infDist_closure : infDist x (closure s) = infDist x s := by
simp [infDist, infEdist_closure] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_closure | The minimal distances to a set and its closure coincide. |
infDist_zero_of_mem_closure (hx : x ∈ closure s) : infDist x s = 0 := by
rw [← infDist_closure]
exact infDist_zero_of_mem hx | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_zero_of_mem_closure | If a point belongs to the closure of `s`, then its infimum distance to `s` equals zero.
The converse is true provided that `s` is nonempty, see `Metric.mem_closure_iff_infDist_zero`. |
mem_closure_iff_infDist_zero (h : s.Nonempty) : x ∈ closure s ↔ infDist x s = 0 := by
simp [mem_closure_iff_infEdist_zero, infDist, ENNReal.toReal_eq_zero_iff, infEdist_ne_top h] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | mem_closure_iff_infDist_zero | A point belongs to the closure of `s` iff its infimum distance to this set vanishes. |
infDist_pos_iff_notMem_closure (hs : s.Nonempty) :
x ∉ closure s ↔ 0 < infDist x s :=
(mem_closure_iff_infDist_zero hs).not.trans infDist_nonneg.lt_iff_ne'.symm
@[deprecated (since := "2025-05-23")]
alias infDist_pos_iff_not_mem_closure := infDist_pos_iff_notMem_closure | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_pos_iff_notMem_closure | null |
_root_.IsClosed.mem_iff_infDist_zero (h : IsClosed s) (hs : s.Nonempty) :
x ∈ s ↔ infDist x s = 0 := by rw [← mem_closure_iff_infDist_zero hs, h.closure_eq] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsClosed.mem_iff_infDist_zero | Given a closed set `s`, a point belongs to `s` iff its infimum distance to this set vanishes |
_root_.IsClosed.notMem_iff_infDist_pos (h : IsClosed s) (hs : s.Nonempty) :
x ∉ s ↔ 0 < infDist x s := by
simp [h.mem_iff_infDist_zero hs, infDist_nonneg.lt_iff_ne']
@[deprecated (since := "2025-05-23")]
alias _root_.IsClosed.not_mem_iff_infDist_pos := _root_.IsClosed.notMem_iff_infDist_pos | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsClosed.notMem_iff_infDist_pos | Given a closed set `s`, a point belongs to `s` iff its infimum distance to this set vanishes. |
continuousAt_inv_infDist_pt (h : x ∉ closure s) :
ContinuousAt (fun x ↦ (infDist x s)⁻¹) x := by
rcases s.eq_empty_or_nonempty with (rfl | hs)
· simp only [infDist_empty, continuousAt_const]
· refine (continuous_infDist_pt s).continuousAt.inv₀ ?_
rwa [Ne, ← mem_closure_iff_infDist_zero hs] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | continuousAt_inv_infDist_pt | null |
infDist_image (hΦ : Isometry Φ) : infDist (Φ x) (Φ '' t) = infDist x t := by
simp [infDist, infEdist_image hΦ] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_image | The infimum distance is invariant under isometries. |
infDist_inter_closedBall_of_mem (h : y ∈ s) :
infDist x (s ∩ closedBall x (dist y x)) = infDist x s := by
replace h : y ∈ s ∩ closedBall x (dist y x) := ⟨h, mem_closedBall.2 le_rfl⟩
refine le_antisymm ?_ (infDist_le_infDist_of_subset inter_subset_left ⟨y, h⟩)
refine not_lt.1 fun hlt => ?_
rcases (infDist_lt_iff ⟨y, h.1⟩).mp hlt with ⟨z, hzs, hz⟩
rcases le_or_gt (dist z x) (dist y x) with hle | hlt
· exact hz.not_ge (infDist_le_dist_of_mem ⟨hzs, hle⟩)
· rw [dist_comm z, dist_comm y] at hlt
exact (hlt.trans hz).not_ge (infDist_le_dist_of_mem h) | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infDist_inter_closedBall_of_mem | null |
_root_.IsCompact.exists_infDist_eq_dist (h : IsCompact s) (hne : s.Nonempty) (x : α) :
∃ y ∈ s, infDist x s = dist x y :=
let ⟨y, hys, hy⟩ := h.exists_infEdist_eq_edist hne x
⟨y, hys, by rw [infDist, dist_edist, hy]⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsCompact.exists_infDist_eq_dist | null |
_root_.IsClosed.exists_infDist_eq_dist [ProperSpace α] (h : IsClosed s) (hne : s.Nonempty)
(x : α) : ∃ y ∈ s, infDist x s = dist x y := by
rcases hne with ⟨z, hz⟩
rw [← infDist_inter_closedBall_of_mem hz]
set t := s ∩ closedBall x (dist z x)
have htc : IsCompact t := (isCompact_closedBall x (dist z x)).inter_left h
have htne : t.Nonempty := ⟨z, hz, mem_closedBall.2 le_rfl⟩
obtain ⟨y, ⟨hys, -⟩, hyd⟩ : ∃ y ∈ t, infDist x t = dist x y := htc.exists_infDist_eq_dist htne x
exact ⟨y, hys, hyd⟩ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | _root_.IsClosed.exists_infDist_eq_dist | null |
exists_mem_closure_infDist_eq_dist [ProperSpace α] (hne : s.Nonempty) (x : α) :
∃ y ∈ closure s, infDist x s = dist x y := by
simpa only [infDist_closure] using isClosed_closure.exists_infDist_eq_dist hne.closure x
/-! ### Distance of a point to a set as a function into `ℝ≥0`. -/ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | exists_mem_closure_infDist_eq_dist | null |
infNndist (x : α) (s : Set α) : ℝ≥0 :=
ENNReal.toNNReal (infEdist x s)
@[simp] | def | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | infNndist | The minimal distance of a point to a set as a `ℝ≥0` |
coe_infNndist : (infNndist x s : ℝ) = infDist x s :=
rfl | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | coe_infNndist | null |
lipschitz_infNndist_pt (s : Set α) : LipschitzWith 1 fun x => infNndist x s :=
LipschitzWith.of_le_add fun _ _ => infDist_le_infDist_add_dist | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | lipschitz_infNndist_pt | The minimal distance to a set (as `ℝ≥0`) is Lipschitz in point with constant 1 |
uniformContinuous_infNndist_pt (s : Set α) : UniformContinuous fun x => infNndist x s :=
(lipschitz_infNndist_pt s).uniformContinuous | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | uniformContinuous_infNndist_pt | The minimal distance to a set (as `ℝ≥0`) is uniformly continuous in point |
@[continuity, fun_prop]
continuous_infNndist_pt (s : Set α) : Continuous fun x => infNndist x s :=
(uniformContinuous_infNndist_pt s).continuous
/-! ### The Hausdorff distance as a function into `ℝ`. -/ | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | continuous_infNndist_pt | The minimal distance to a set (as `ℝ≥0`) is continuous in point |
hausdorffDist (s t : Set α) : ℝ :=
ENNReal.toReal (hausdorffEdist s t) | def | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist | The Hausdorff distance between two sets is the smallest nonnegative `r` such that each set is
included in the `r`-neighborhood of the other. If there is no such `r`, it is defined to
be `0`, arbitrarily. |
hausdorffDist_nonneg : 0 ≤ hausdorffDist s t := by simp [hausdorffDist] | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffDist_nonneg | The Hausdorff distance is nonnegative. |
hausdorffEdist_ne_top_of_nonempty_of_bounded (hs : s.Nonempty) (ht : t.Nonempty)
(bs : IsBounded s) (bt : IsBounded t) : hausdorffEdist s t ≠ ⊤ := by
rcases hs with ⟨cs, hcs⟩
rcases ht with ⟨ct, hct⟩
rcases bs.subset_closedBall ct with ⟨rs, hrs⟩
rcases bt.subset_closedBall cs with ⟨rt, hrt⟩
have : hausdorffEdist s t ≤ ENNReal.ofReal (max rs rt) := by
apply hausdorffEdist_le_of_mem_edist
· intro x xs
exists ct, hct
have : dist x ct ≤ max rs rt := le_trans (hrs xs) (le_max_left _ _)
rwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]
exact le_trans dist_nonneg this
· intro x xt
exists cs, hcs
have : dist x cs ≤ max rs rt := le_trans (hrt xt) (le_max_right _ _)
rwa [edist_dist, ENNReal.ofReal_le_ofReal_iff]
exact le_trans dist_nonneg this
exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top this | theorem | Topology | [
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Tactic.Finiteness"
] | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | hausdorffEdist_ne_top_of_nonempty_of_bounded | If two sets are nonempty and bounded in a metric space, they are at finite Hausdorff
edistance. |
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