statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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dimH_univ (e : X ≃ᵢ Y) : dimH (univ : set X) = dimH (univ : set Y) | by rw [← e.dimH_preimage univ, preimage_univ] | lemma | isometry_equiv.dimH_univ | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_image (e : E ≃L[𝕜] F) (s : set E) : dimH (e '' s) = dimH s | le_antisymm (e.lipschitz.dimH_image_le s) $
by simpa only [e.symm_image_image] using e.symm.lipschitz.dimH_image_le (e '' s) | lemma | continuous_linear_equiv.dimH_image | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_preimage (e : E ≃L[𝕜] F) (s : set F) : dimH (e ⁻¹' s) = dimH s | by rw [← e.image_symm_eq_preimage, e.symm.dimH_image] | lemma | continuous_linear_equiv.dimH_preimage | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_univ (e : E ≃L[𝕜] F) : dimH (univ : set E) = dimH (univ : set F) | by rw [← e.dimH_preimage, preimage_univ] | lemma | continuous_linear_equiv.dimH_univ | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_ball_pi (x : ι → ℝ) {r : ℝ} (hr : 0 < r) :
dimH (metric.ball x r) = fintype.card ι | begin
casesI is_empty_or_nonempty ι,
{ rwa [dimH_subsingleton, eq_comm, nat.cast_eq_zero, fintype.card_eq_zero_iff],
exact λ x _ y _, subsingleton.elim x y },
{ rw ← ennreal.coe_nat,
have : μH[fintype.card ι] (metric.ball x r) = ennreal.of_real ((2 * r) ^ fintype.card ι),
by rw [hausdorff_measure_pi... | theorem | real.dimH_ball_pi | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_of_hausdorff_measure_ne_zero_ne_top",
"dimH_subsingleton",
"ennreal.coe_nat",
"ennreal.of_real",
"ennreal.of_real_ne_top",
"fintype.card",
"fintype.card_eq_zero_iff",
"is_empty_or_nonempty",
"metric.ball",
"nat.cast_eq_zero",
"nnreal.coe_nat_cast",
"pow_pos",
"real.volume_pi_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_ball_pi_fin {n : ℕ} (x : fin n → ℝ) {r : ℝ} (hr : 0 < r) :
dimH (metric.ball x r) = n | by rw [dimH_ball_pi x hr, fintype.card_fin] | theorem | real.dimH_ball_pi_fin | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"fintype.card_fin",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_univ_pi (ι : Type*) [fintype ι] : dimH (univ : set (ι → ℝ)) = fintype.card ι | by simp only [← metric.Union_ball_nat_succ (0 : ι → ℝ), dimH_Union,
dimH_ball_pi _ (nat.cast_add_one_pos _), supr_const] | theorem | real.dimH_univ_pi | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"dimH_Union",
"fintype",
"fintype.card",
"metric.Union_ball_nat_succ",
"nat.cast_add_one_pos",
"supr_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_univ_pi_fin (n : ℕ) : dimH (univ : set (fin n → ℝ)) = n | by rw [dimH_univ_pi, fintype.card_fin] | theorem | real.dimH_univ_pi_fin | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"fintype.card_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_of_mem_nhds {x : E} {s : set E} (h : s ∈ 𝓝 x) :
dimH s = finrank ℝ E | begin
have e : E ≃L[ℝ] (fin (finrank ℝ E) → ℝ),
from continuous_linear_equiv.of_finrank_eq (finite_dimensional.finrank_fin_fun ℝ).symm,
rw ← e.dimH_image,
refine le_antisymm _ _,
{ exact (dimH_mono (subset_univ _)).trans_eq (dimH_univ_pi_fin _) },
{ have : e '' s ∈ 𝓝 (e x), by { rw ← e.map_nhds_eq, exact... | theorem | real.dimH_of_mem_nhds | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"continuous_linear_equiv.of_finrank_eq",
"dimH",
"dimH_mono",
"finite_dimensional.finrank_fin_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_of_nonempty_interior {s : set E} (h : (interior s).nonempty) :
dimH s = finrank ℝ E | let ⟨x, hx⟩ := h in dimH_of_mem_nhds (mem_interior_iff_mem_nhds.1 hx) | theorem | real.dimH_of_nonempty_interior | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_univ_eq_finrank : dimH (univ : set E) = finrank ℝ E | dimH_of_mem_nhds (@univ_mem _ (𝓝 0)) | theorem | real.dimH_univ_eq_finrank | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimH_univ : dimH (univ : set ℝ) = 1 | by rw [dimH_univ_eq_finrank ℝ, finite_dimensional.finrank_self, nat.cast_one] | theorem | real.dimH_univ | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dimH",
"finite_dimensional.finrank_self",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_compl_of_dimH_lt_finrank {s : set E} (hs : dimH s < finrank ℝ E) : dense sᶜ | begin
refine λ x, mem_closure_iff_nhds.2 (λ t ht, nonempty_iff_ne_empty.2 $ λ he, hs.not_le _),
rw [← diff_eq, diff_eq_empty] at he,
rw [← real.dimH_of_mem_nhds ht],
exact dimH_mono he
end | theorem | dense_compl_of_dimH_lt_finrank | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"dense",
"dimH",
"dimH_mono",
"real.dimH_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.dimH_image_le {f : E → F} {s t : set E} (hf : cont_diff_on ℝ 1 f s)
(hc : convex ℝ s) (ht : t ⊆ s) :
dimH (f '' t) ≤ dimH t | dimH_image_le_of_locally_lipschitz_on $ λ x hx,
let ⟨C, u, hu, hf⟩ := (hf x (ht hx)).exists_lipschitz_on_with hc
in ⟨C, u, nhds_within_mono _ ht hu, hf⟩ | lemma | cont_diff_on.dimH_image_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"cont_diff_on",
"convex",
"dimH",
"dimH_image_le_of_locally_lipschitz_on",
"nhds_within_mono"
] | 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`? | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.dimH_range_le {f : E → F} (h : cont_diff 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.cont_diff_on.dimH_image_le convex_univ subset.rfl
... = finrank ℝ E : real.dimH_univ_eq_finrank E | lemma | cont_diff.dimH_range_le | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"cont_diff",
"convex_univ",
"dimH",
"real.dimH_univ_eq_finrank"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.dense_compl_image_of_dimH_lt_finrank [finite_dimensional ℝ F] {f : E → F}
{s t : set E} (h : cont_diff_on 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 | lemma | cont_diff_on.dense_compl_image_of_dimH_lt_finrank | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"cont_diff_on",
"convex",
"dense",
"dense_compl_of_dimH_lt_finrank",
"dimH",
"finite_dimensional"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.dense_compl_range_of_finrank_lt_finrank [finite_dimensional ℝ F] {f : E → F}
(h : cont_diff 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 | lemma | cont_diff.dense_compl_range_of_finrank_lt_finrank | topology.metric_space | src/topology/metric_space/hausdorff_dimension.lean | [
"analysis.calculus.cont_diff",
"measure_theory.measure.hausdorff"
] | [
"cont_diff",
"dense",
"dense_compl_of_dimH_lt_finrank",
"finite_dimensional"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist (x : α) (s : set α) : ℝ≥0∞ | ⨅ y ∈ s, edist x y | def | emetric.inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | The minimal edistance of a point to a set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_empty : inf_edist x ∅ = ∞ | infi_emptyset | lemma | emetric.inf_edist_empty | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_emptyset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inf_edist {d} : d ≤ inf_edist x s ↔ ∀ y ∈ s, d ≤ edist x y | by simp only [inf_edist, le_infi_iff] | lemma | emetric.le_inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"le_infi_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_union : inf_edist x (s ∪ t) = inf_edist x s ⊓ inf_edist x t | infi_union | lemma | emetric.inf_edist_union | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_union"
] | The edist to a union is the minimum of the edists | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_Union (f : ι → set α) (x : α) :
inf_edist x (⋃ i, f i) = ⨅ i, inf_edist x (f i) | infi_Union f _ | lemma | emetric.inf_edist_Union | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_singleton : inf_edist x {y} = edist x y | infi_singleton | lemma | emetric.inf_edist_singleton | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_singleton"
] | The edist to a singleton is the edistance to the single point of this singleton | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_edist_of_mem (h : y ∈ s) : inf_edist x s ≤ edist x y | infi₂_le _ h | lemma | emetric.inf_edist_le_edist_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi₂_le"
] | The edist to a set is bounded above by the edist to any of its points | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_zero_of_mem (h : x ∈ s) : inf_edist x s = 0 | nonpos_iff_eq_zero.1 $ @edist_self _ _ x ▸ inf_edist_le_edist_of_mem h | lemma | emetric.inf_edist_zero_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | If a point `x` belongs to `s`, then its edist to `s` vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_anti (h : s ⊆ t) : inf_edist x t ≤ inf_edist x s | infi_le_infi_of_subset h | lemma | emetric.inf_edist_anti | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_le_infi_of_subset"
] | The edist is antitone with respect to inclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_lt_iff {r : ℝ≥0∞} : inf_edist x s < r ↔ ∃ y ∈ s, edist x y < r | by simp_rw [inf_edist, infi_lt_iff] | lemma | emetric.inf_edist_lt_iff | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_lt_iff"
] | The edist to a set is `< r` iff there exists a point in the set at edistance `< r` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_inf_edist_add_edist : inf_edist x s ≤ inf_edist y s + edist x y | calc (⨅ z ∈ s, edist x z) ≤ ⨅ z ∈ s, edist y z + edist x y :
infi₂_mono $ λ z hz, (edist_triangle _ _ _).trans_eq (add_comm _ _)
... = (⨅ z ∈ s, edist y z) + edist x y : by simp only [ennreal.infi_add] | lemma | emetric.inf_edist_le_inf_edist_add_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.infi_add",
"infi₂_mono"
] | 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` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_edist_add_inf_edist : inf_edist x s ≤ edist x y + inf_edist y s | by { rw add_comm, exact inf_edist_le_inf_edist_add_edist } | lemma | emetric.inf_edist_le_edist_add_inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_le_inf_edist_add_ediam (hy : y ∈ s) : edist x y ≤ inf_edist x s + diam s | begin
simp_rw [inf_edist, ennreal.infi_add],
refine le_infi (λ i, le_infi (λ 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)
end | lemma | emetric.edist_le_inf_edist_add_ediam | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.infi_add",
"le_infi",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf_edist : continuous (λx, inf_edist x s) | continuous_of_le_add_edist 1 (by simp) $
by simp only [one_mul, inf_edist_le_inf_edist_add_edist, forall_2_true_iff] | lemma | emetric.continuous_inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"continuous",
"continuous_of_le_add_edist",
"forall_2_true_iff",
"one_mul"
] | The edist to a set depends continuously on the point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_closure : inf_edist x (closure s) = inf_edist x s | begin
refine le_antisymm (inf_edist_anti subset_closure) _,
refine ennreal.le_of_forall_pos_le_add (λε εpos h, _),
have ε0 : 0 < (ε / 2 : ℝ≥0∞) := by simpa [pos_iff_ne_zero] using εpos,
have : inf_edist x (closure s) < inf_edist x (closure s) + ε/2,
from ennreal.lt_add_right h.ne ε0.ne',
rcases inf_edist_... | lemma | emetric.inf_edist_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"ennreal.add_halves",
"ennreal.le_of_forall_pos_le_add",
"ennreal.lt_add_right",
"subset_closure"
] | The edist to a set and to its closure coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff_inf_edist_zero : x ∈ closure s ↔ inf_edist x s = 0 | ⟨λ h, by { rw ← inf_edist_closure, exact inf_edist_zero_of_mem h },
λ h, emetric.mem_closure_iff.2 $ λ ε εpos, inf_edist_lt_iff.mp $ by rwa h⟩ | lemma | emetric.mem_closure_iff_inf_edist_zero | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | A point belongs to the closure of `s` iff its infimum edistance to this set vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_iff_inf_edist_zero_of_closed (h : is_closed s) : x ∈ s ↔ inf_edist x s = 0 | begin
convert ← mem_closure_iff_inf_edist_zero,
exact h.closure_eq
end | lemma | emetric.mem_iff_inf_edist_zero_of_closed | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"is_closed"
] | Given a closed set `s`, a point belongs to `s` iff its infimum edistance to this set vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_pos_iff_not_mem_closure {x : α} {E : set α} :
0 < inf_edist x E ↔ x ∉ closure E | by rw [mem_closure_iff_inf_edist_zero, pos_iff_ne_zero] | lemma | emetric.inf_edist_pos_iff_not_mem_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_closure_pos_iff_not_mem_closure {x : α} {E : set α} :
0 < inf_edist x (closure E) ↔ x ∉ closure E | by rw [inf_edist_closure, inf_edist_pos_iff_not_mem_closure] | lemma | emetric.inf_edist_closure_pos_iff_not_mem_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_real_pos_lt_inf_edist_of_not_mem_closure {x : α} {E : set α} (h : x ∉ closure E) :
∃ (ε : ℝ), 0 < ε ∧ ennreal.of_real ε < inf_edist x E | begin
rw [← inf_edist_pos_iff_not_mem_closure, ennreal.lt_iff_exists_real_btwn] at h,
rcases h with ⟨ε, ⟨_, ⟨ε_pos, ε_lt⟩⟩⟩,
exact ⟨ε, ⟨ennreal.of_real_pos.mp ε_pos, ε_lt⟩⟩,
end | lemma | emetric.exists_real_pos_lt_inf_edist_of_not_mem_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"ennreal.lt_iff_exists_real_btwn",
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closed_ball_of_lt_inf_edist {r : ℝ≥0∞} (h : r < inf_edist x s) :
disjoint (closed_ball x r) s | begin
rw disjoint_left,
assume y hy h'y,
apply lt_irrefl (inf_edist x s),
calc inf_edist x s ≤ edist x y : inf_edist_le_edist_of_mem h'y
... ≤ r : by rwa [mem_closed_ball, edist_comm] at hy
... < inf_edist x s : h
end | lemma | emetric.disjoint_closed_ball_of_lt_inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_image (hΦ : isometry Φ) :
inf_edist (Φ x) (Φ '' t) = inf_edist x t | by simp only [inf_edist, infi_image, hΦ.edist_eq] | lemma | emetric.inf_edist_image | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"infi_image",
"isometry"
] | The infimum edistance is invariant under isometries | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_smul {M} [has_smul M α] [has_isometric_smul M α]
(c : M) (x : α) (s : set α) :
inf_edist (c • x) (c • s) = inf_edist x s | inf_edist_image (isometry_smul _ _) | lemma | emetric.inf_edist_smul | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"has_isometric_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_open.exists_Union_is_closed {U : set α} (hU : is_open U) :
∃ F : ℕ → set α, (∀ n, is_closed (F n)) ∧ (∀ n, F n ⊆ U) ∧ ((⋃ n, F n) = U) ∧ monotone F | begin
obtain ⟨a, a_pos, a_lt_one⟩ : ∃ (a : ℝ≥0∞), 0 < a ∧ a < 1 := exists_between zero_lt_one,
let F := λ (n : ℕ), (λ x, inf_edist x Uᶜ) ⁻¹' (Ici (a^n)),
have F_subset : ∀ n, F n ⊆ U,
{ assume n x hx,
have : inf_edist x Uᶜ ≠ 0 := ((ennreal.pow_pos a_pos _).trans_le hx).ne',
contrapose! this,
exact i... | lemma | is_open.exists_Union_is_closed | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.pow_pos",
"ennreal.tendsto_pow_at_top_nhds_0_of_lt_1",
"exists_between",
"filter.tendsto",
"forall_const",
"is_closed",
"is_closed.preimage",
"is_closed_Ici",
"is_open",
"monotone",
"pow_le_pow_of_le_one'",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.exists_inf_edist_eq_edist (hs : is_compact s) (hne : s.nonempty) (x : α) :
∃ y ∈ s, inf_edist x s = edist x y | begin
have A : continuous (λ y, edist x y) := continuous_const.edist continuous_id,
obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, ∀ z, z ∈ s → edist x y ≤ edist x z :=
hs.exists_forall_le hne A.continuous_on,
exact ⟨y, ys, le_antisymm (inf_edist_le_edist_of_mem ys) (by rwa le_inf_edist)⟩
end | lemma | is_compact.exists_inf_edist_eq_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"continuous",
"continuous_id",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_forall_lt_edist (hs : is_compact s) (ht : is_closed t) (hst : disjoint s t) :
∃ r : ℝ≥0, 0 < r ∧ ∀ (x ∈ s) (y ∈ t), (r : ℝ≥0∞) < edist x y | begin
rcases s.eq_empty_or_nonempty with rfl|hne, { use 1, simp },
obtain ⟨x, hx, h⟩ : ∃ x ∈ s, ∀ y ∈ s, inf_edist x t ≤ inf_edist y t :=
hs.exists_forall_le hne continuous_inf_edist.continuous_on,
have : 0 < inf_edist x t,
from pos_iff_ne_zero.2 (λ H, hst.le_bot ⟨hx, (mem_iff_inf_edist_zero_of_closed ht)... | lemma | emetric.exists_pos_forall_lt_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"disjoint",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hausdorff_edist {α : Type u} [pseudo_emetric_space α] (s t : set α) : ℝ≥0∞ | (⨆ x ∈ s, inf_edist x t) ⊔ (⨆ y ∈ t, inf_edist y s) | def | emetric.Hausdorff_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"pseudo_emetric_space"
] | The Hausdorff edistance between two sets is the smallest `r` such that each set
is contained in the `r`-neighborhood of the other one | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_def {α : Type u} [pseudo_emetric_space α] (s t : set α) :
Hausdorff_edist s t = (⨆ x ∈ s, inf_edist x t) ⊔ (⨆ y ∈ t, inf_edist y s) | by rw Hausdorff_edist | lemma | emetric.Hausdorff_edist_def | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Hausdorff_edist_self : Hausdorff_edist s s = 0 | begin
simp only [Hausdorff_edist_def, sup_idem, ennreal.supr_eq_zero],
exact λ x hx, inf_edist_zero_of_mem hx
end | lemma | emetric.Hausdorff_edist_self | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.supr_eq_zero",
"sup_idem"
] | The Hausdorff edistance of a set to itself vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_comm : Hausdorff_edist s t = Hausdorff_edist t s | by unfold Hausdorff_edist; apply sup_comm | lemma | emetric.Hausdorff_edist_comm | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"sup_comm"
] | The Haudorff edistances of `s` to `t` and of `t` to `s` coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_le_of_inf_edist {r : ℝ≥0∞}
(H1 : ∀x ∈ s, inf_edist x t ≤ r) (H2 : ∀x ∈ t, inf_edist x s ≤ r) :
Hausdorff_edist s t ≤ r | begin
simp only [Hausdorff_edist, sup_le_iff, supr_le_iff],
exact ⟨H1, H2⟩
end | lemma | emetric.Hausdorff_edist_le_of_inf_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"sup_le_iff",
"supr_le_iff"
] | Bounding the Hausdorff edistance by bounding the edistance of any point
in each set to the other set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_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) :
Hausdorff_edist s t ≤ r | begin
refine Hausdorff_edist_le_of_inf_edist _ _,
{ assume x xs,
rcases H1 x xs with ⟨y, yt, hy⟩,
exact le_trans (inf_edist_le_edist_of_mem yt) hy },
{ assume x xt,
rcases H2 x xt with ⟨y, ys, hy⟩,
exact le_trans (inf_edist_le_edist_of_mem ys) hy }
end | lemma | emetric.Hausdorff_edist_le_of_mem_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | Bounding the Hausdorff edistance by exhibiting, for any point in each set,
another point in the other set at controlled distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_Hausdorff_edist_of_mem (h : x ∈ s) : inf_edist x t ≤ Hausdorff_edist s t | begin
rw Hausdorff_edist_def,
refine le_trans _ le_sup_left,
exact le_supr₂ x h
end | lemma | emetric.inf_edist_le_Hausdorff_edist_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"le_sup_left",
"le_supr₂"
] | The distance to a set is controlled by the Hausdorff distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_edist_lt_of_Hausdorff_edist_lt {r : ℝ≥0∞} (h : x ∈ s)
(H : Hausdorff_edist s t < r) : ∃ y ∈ t, edist x y < r | inf_edist_lt_iff.mp $ calc
inf_edist x t ≤ Hausdorff_edist s t : inf_edist_le_Hausdorff_edist_of_mem h
... < r : H | lemma | emetric.exists_edist_lt_of_Hausdorff_edist_lt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | 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 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_inf_edist_add_Hausdorff_edist :
inf_edist x t ≤ inf_edist x s + Hausdorff_edist s t | ennreal.le_of_forall_pos_le_add $ λε εpos h, begin
have ε0 : (ε / 2 : ℝ≥0∞) ≠ 0 := by simpa [pos_iff_ne_zero] using εpos,
have : inf_edist x s < inf_edist x s + ε/2 :=
ennreal.lt_add_right (ennreal.add_lt_top.1 h).1.ne ε0,
rcases inf_edist_lt_iff.mp this with ⟨y, ys, dxy⟩,
-- y : α, ys : y ∈ s, dxy : edis... | lemma | emetric.inf_edist_le_inf_edist_add_Hausdorff_edist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.add_halves",
"ennreal.le_of_forall_pos_le_add",
"ennreal.lt_add_right"
] | The distance from `x` to `s` or `t` is controlled in terms of the Hausdorff distance
between `s` and `t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_image (h : isometry Φ) :
Hausdorff_edist (Φ '' s) (Φ '' t) = Hausdorff_edist s t | by simp only [Hausdorff_edist_def, supr_image, inf_edist_image h] | lemma | emetric.Hausdorff_edist_image | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"isometry",
"supr_image"
] | The Hausdorff edistance is invariant under eisometries | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_le_ediam (hs : s.nonempty) (ht : t.nonempty) :
Hausdorff_edist s t ≤ diam (s ∪ t) | begin
rcases hs with ⟨x, xs⟩,
rcases ht with ⟨y, yt⟩,
refine Hausdorff_edist_le_of_mem_edist _ _,
{ intros z hz,
exact ⟨y, yt, edist_le_diam_of_mem (subset_union_left _ _ hz) (subset_union_right _ _ yt)⟩ },
{ intros z hz,
exact ⟨x, xs, edist_le_diam_of_mem (subset_union_right _ _ hz) (subset_union_lef... | lemma | emetric.Hausdorff_edist_le_ediam | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | The Hausdorff distance is controlled by the diameter of the union | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_triangle : Hausdorff_edist s u ≤ Hausdorff_edist s t + Hausdorff_edist t u | begin
rw Hausdorff_edist_def,
simp only [sup_le_iff, supr_le_iff],
split,
show ∀x ∈ s, inf_edist x u ≤ Hausdorff_edist s t + Hausdorff_edist t u, from λx xs, calc
inf_edist x u ≤ inf_edist x t + Hausdorff_edist t u : inf_edist_le_inf_edist_add_Hausdorff_edist
... ≤ Hausdorff_edist s t + Hausdorff_edist ... | lemma | emetric.Hausdorff_edist_triangle | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"sup_le_iff",
"supr_le_iff"
] | The Hausdorff distance satisfies the triangular inequality | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_zero_iff_closure_eq_closure :
Hausdorff_edist s t = 0 ↔ closure s = closure t | calc Hausdorff_edist s t = 0 ↔ s ⊆ closure t ∧ t ⊆ closure s :
by simp only [Hausdorff_edist_def, ennreal.sup_eq_zero, ennreal.supr_eq_zero,
← mem_closure_iff_inf_edist_zero, subset_def]
... ↔ closure s = closure t :
⟨λ h, subset.antisymm (closure_minimal h.1 is_closed_closure)
(closure_minimal h.2 is_clos... | lemma | emetric.Hausdorff_edist_zero_iff_closure_eq_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"closure_minimal",
"ennreal.sup_eq_zero",
"ennreal.supr_eq_zero",
"is_closed_closure",
"subset_closure"
] | Two sets are at zero Hausdorff edistance if and only if they have the same closure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_self_closure : Hausdorff_edist s (closure s) = 0 | by rw [Hausdorff_edist_zero_iff_closure_eq_closure, closure_closure] | lemma | emetric.Hausdorff_edist_self_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"closure_closure"
] | The Hausdorff edistance between a set and its closure vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_closure₁ : Hausdorff_edist (closure s) t = Hausdorff_edist s t | begin
refine le_antisymm _ _,
{ calc _ ≤ Hausdorff_edist (closure s) s + Hausdorff_edist s t : Hausdorff_edist_triangle
... = Hausdorff_edist s t : by simp [Hausdorff_edist_comm] },
{ calc _ ≤ Hausdorff_edist s (closure s) + Hausdorff_edist (closure s) t :
Hausdorff_edist_triangle
... = Hausdorff_e... | lemma | emetric.Hausdorff_edist_closure₁ | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | Replacing a set by its closure does not change the Hausdorff edistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_closure₂ : Hausdorff_edist s (closure t) = Hausdorff_edist s t | by simp [@Hausdorff_edist_comm _ _ s _] | lemma | emetric.Hausdorff_edist_closure₂ | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | Replacing a set by its closure does not change the Hausdorff edistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_closure :
Hausdorff_edist (closure s) (closure t) = Hausdorff_edist s t | by simp | lemma | emetric.Hausdorff_edist_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | The Hausdorff edistance between sets or their closures is the same | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_zero_iff_eq_of_closed (hs : is_closed s) (ht : is_closed t) :
Hausdorff_edist s t = 0 ↔ s = t | by rw [Hausdorff_edist_zero_iff_closure_eq_closure, hs.closure_eq, ht.closure_eq] | lemma | emetric.Hausdorff_edist_zero_iff_eq_of_closed | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"is_closed"
] | Two closed sets are at zero Hausdorff edistance if and only if they coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_empty (ne : s.nonempty) : Hausdorff_edist s ∅ = ∞ | begin
rcases ne with ⟨x, xs⟩,
have : inf_edist x ∅ ≤ Hausdorff_edist s ∅ := inf_edist_le_Hausdorff_edist_of_mem xs,
simpa using this,
end | lemma | emetric.Hausdorff_edist_empty | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | The Haudorff edistance to the empty set is infinite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_of_Hausdorff_edist_ne_top (hs : s.nonempty) (fin : Hausdorff_edist s t ≠ ⊤) :
t.nonempty | t.eq_empty_or_nonempty.elim (λ ht, (fin $ ht.symm ▸ Hausdorff_edist_empty hs).elim) id | lemma | emetric.nonempty_of_Hausdorff_edist_ne_top | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | If a set is at finite Hausdorff edistance of a nonempty set, it is nonempty | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_or_nonempty_of_Hausdorff_edist_ne_top (fin : Hausdorff_edist s t ≠ ⊤) :
s = ∅ ∧ t = ∅ ∨ s.nonempty ∧ t.nonempty | begin
cases s.eq_empty_or_nonempty with hs hs,
{ cases t.eq_empty_or_nonempty with ht ht,
{ exact or.inl ⟨hs, ht⟩ },
{ rw Hausdorff_edist_comm at fin,
exact or.inr ⟨nonempty_of_Hausdorff_edist_ne_top ht fin, ht⟩ } },
{ exact or.inr ⟨hs, nonempty_of_Hausdorff_edist_ne_top hs fin⟩ }
end | lemma | emetric.empty_or_nonempty_of_Hausdorff_edist_ne_top | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_dist (x : α) (s : set α) : ℝ | ennreal.to_real (inf_edist x s) | def | metric.inf_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.to_real"
] | The minimal distance of a point to a set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_eq_infi : inf_dist x s = ⨅ y : s, dist x y | begin
rw [inf_dist, inf_edist, infi_subtype', ennreal.to_real_infi],
{ simp only [dist_edist],
refl },
{ exact λ _, edist_ne_top _ _ }
end | theorem | metric.inf_dist_eq_infi | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"edist_ne_top",
"ennreal.to_real_infi",
"infi_subtype'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_dist_nonneg : 0 ≤ inf_dist x s | by simp [inf_dist] | lemma | metric.inf_dist_nonneg | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | the minimal distance is always nonnegative | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_empty : inf_dist x ∅ = 0 | by simp [inf_dist] | lemma | metric.inf_dist_empty | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | the minimal distance to the empty set is 0 (if you want to have the more reasonable
value ∞ instead, use `inf_edist`, which takes values in ℝ≥0∞) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_ne_top (h : s.nonempty) : inf_edist x s ≠ ⊤ | begin
rcases h with ⟨y, hy⟩,
apply lt_top_iff_ne_top.1,
calc inf_edist x s ≤ edist x y : inf_edist_le_edist_of_mem hy
... < ⊤ : lt_top_iff_ne_top.2 (edist_ne_top _ _)
end | lemma | metric.inf_edist_ne_top | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"edist_ne_top"
] | In a metric space, the minimal edistance to a nonempty set is finite | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_zero_of_mem (h : x ∈ s) : inf_dist x s = 0 | by simp [inf_edist_zero_of_mem h, inf_dist] | lemma | metric.inf_dist_zero_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | The minimal distance of a point to a set containing it vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_singleton : inf_dist x {y} = dist x y | by simp [inf_dist, inf_edist, dist_edist] | lemma | metric.inf_dist_singleton | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist"
] | The minimal distance to a singleton is the distance to the unique point in this singleton | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_le_dist_of_mem (h : y ∈ s) : inf_dist x s ≤ dist x y | begin
rw [dist_edist, inf_dist,
ennreal.to_real_le_to_real (inf_edist_ne_top ⟨_, h⟩) (edist_ne_top _ _)],
exact inf_edist_le_edist_of_mem h
end | lemma | metric.inf_dist_le_dist_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"edist_ne_top",
"ennreal.to_real_le_to_real"
] | The minimal distance to a set is bounded by the distance to any point in this set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_le_inf_dist_of_subset (h : s ⊆ t) (hs : s.nonempty) :
inf_dist x t ≤ inf_dist x s | begin
have ht : t.nonempty := hs.mono h,
rw [inf_dist, inf_dist, ennreal.to_real_le_to_real (inf_edist_ne_top ht) (inf_edist_ne_top hs)],
exact inf_edist_anti h
end | lemma | metric.inf_dist_le_inf_dist_of_subset | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.to_real_le_to_real"
] | The minimal distance is monotonous with respect to inclusion | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_lt_iff {r : ℝ} (hs : s.nonempty) :
inf_dist x s < r ↔ ∃ y ∈ s, dist x y < r | by simp_rw [inf_dist, ← ennreal.lt_of_real_iff_to_real_lt (inf_edist_ne_top hs), inf_edist_lt_iff,
ennreal.lt_of_real_iff_to_real_lt (edist_ne_top _ _), ← dist_edist] | lemma | metric.inf_dist_lt_iff | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"edist_ne_top",
"ennreal.lt_of_real_iff_to_real_lt"
] | The minimal distance to a set is `< r` iff there exists a point in this set at distance `< r` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_le_inf_dist_add_dist : inf_dist x s ≤ inf_dist y s + dist x y | begin
cases s.eq_empty_or_nonempty with hs hs,
{ simp [hs, dist_nonneg] },
{ rw [inf_dist, inf_dist, dist_edist,
← ennreal.to_real_add (inf_edist_ne_top hs) (edist_ne_top _ _),
ennreal.to_real_le_to_real (inf_edist_ne_top hs)],
{ exact inf_edist_le_inf_edist_add_edist },
{ simp [ennreal.ad... | lemma | metric.inf_dist_le_inf_dist_add_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"dist_nonneg",
"edist_ne_top",
"ennreal.add_eq_top",
"ennreal.to_real_add",
"ennreal.to_real_le_to_real"
] | The minimal distance from `x` to `s` is bounded by the distance from `y` to `s`, modulo
the distance between `x` and `y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_of_dist_lt_inf_dist (h : dist x y < inf_dist x s) : y ∉ s | λ hy, h.not_le $ inf_dist_le_dist_of_mem hy | lemma | metric.not_mem_of_dist_lt_inf_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_ball_inf_dist : disjoint (ball x (inf_dist x s)) s | disjoint_left.2 $ λ y hy, not_mem_of_dist_lt_inf_dist $
calc dist x y = dist y x : dist_comm _ _
... < inf_dist x s : hy | lemma | metric.disjoint_ball_inf_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"disjoint",
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_inf_dist_subset_compl : ball x (inf_dist x s) ⊆ sᶜ | disjoint_ball_inf_dist.subset_compl_right | lemma | metric.ball_inf_dist_subset_compl | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_inf_dist_compl_subset : ball x (inf_dist x sᶜ) ⊆ s | ball_inf_dist_subset_compl.trans (compl_compl s).subset | lemma | metric.ball_inf_dist_compl_subset | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_closed_ball_of_lt_inf_dist {r : ℝ} (h : r < inf_dist x s) :
disjoint (closed_ball x r) s | disjoint_ball_inf_dist.mono_left $ closed_ball_subset_ball h | lemma | metric.disjoint_closed_ball_of_lt_inf_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_inf_dist_add_diam (hs : bounded s) (hy : y ∈ s) : dist x y ≤ inf_dist x s + diam s | begin
have A : inf_edist x s ≠ ∞, from inf_edist_ne_top ⟨y, hy⟩,
have B : emetric.diam s ≠ ∞, from hs.ediam_ne_top,
rw [inf_dist, diam, ← ennreal.to_real_add A B, dist_edist],
apply (ennreal.to_real_le_to_real _ _).2,
{ exact edist_le_inf_edist_add_ediam hy },
{ rw edist_dist, exact ennreal.of_real_ne_top }... | lemma | metric.dist_le_inf_dist_add_diam | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"edist_dist",
"emetric.diam",
"ennreal.of_real_ne_top",
"ennreal.to_real_add",
"ennreal.to_real_le_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_inf_dist_pt : lipschitz_with 1 (λx, inf_dist x s) | lipschitz_with.of_le_add $ λ x y, inf_dist_le_inf_dist_add_dist | lemma | metric.lipschitz_inf_dist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"lipschitz_with",
"lipschitz_with.of_le_add"
] | The minimal distance to a set is Lipschitz in point with constant 1 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_inf_dist_pt :
uniform_continuous (λx, inf_dist x s) | (lipschitz_inf_dist_pt s).uniform_continuous | lemma | metric.uniform_continuous_inf_dist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"uniform_continuous"
] | The minimal distance to a set is uniformly continuous in point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inf_dist_pt : continuous (λx, inf_dist x s) | (uniform_continuous_inf_dist_pt s).continuous | lemma | metric.continuous_inf_dist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"continuous"
] | The minimal distance to a set is continuous in point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_eq_closure : inf_dist x (closure s) = inf_dist x s | by simp [inf_dist, inf_edist_closure] | lemma | metric.inf_dist_eq_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure"
] | The minimal distance to a set and its closure coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_zero_of_mem_closure (hx : x ∈ closure s) : inf_dist x s = 0 | by { rw ← inf_dist_eq_closure, exact inf_dist_zero_of_mem hx } | lemma | metric.inf_dist_zero_of_mem_closure | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"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 `mem_closure_iff_inf_dist_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_iff_inf_dist_zero (h : s.nonempty) : x ∈ closure s ↔ inf_dist x s = 0 | by simp [mem_closure_iff_inf_edist_zero, inf_dist, ennreal.to_real_eq_zero_iff, inf_edist_ne_top h] | lemma | metric.mem_closure_iff_inf_dist_zero | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"ennreal.to_real_eq_zero_iff"
] | A point belongs to the closure of `s` iff its infimum distance to this set vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_closed.mem_iff_inf_dist_zero (h : is_closed s) (hs : s.nonempty) :
x ∈ s ↔ inf_dist x s = 0 | by rw [←mem_closure_iff_inf_dist_zero hs, h.closure_eq] | lemma | is_closed.mem_iff_inf_dist_zero | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"is_closed"
] | Given a closed set `s`, a point belongs to `s` iff its infimum distance to this set vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_closed.not_mem_iff_inf_dist_pos (h : is_closed s) (hs : s.nonempty) :
x ∉ s ↔ 0 < inf_dist x s | begin
rw ← not_iff_not,
push_neg,
simp [h.mem_iff_inf_dist_zero hs, le_antisymm_iff, inf_dist_nonneg],
end | lemma | is_closed.not_mem_iff_inf_dist_pos | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"is_closed",
"not_iff_not"
] | Given a closed set `s`, a point belongs to `s` iff its infimum distance to this set vanishes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_image (hΦ : isometry Φ) :
inf_dist (Φ x) (Φ '' t) = inf_dist x t | by simp [inf_dist, inf_edist_image hΦ] | lemma | metric.inf_dist_image | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"isometry"
] | The infimum distance is invariant under isometries | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_inter_closed_ball_of_mem (h : y ∈ s) :
inf_dist x (s ∩ closed_ball x (dist y x)) = inf_dist x s | begin
replace h : y ∈ s ∩ closed_ball x (dist y x) := ⟨h, mem_closed_ball.2 le_rfl⟩,
refine le_antisymm _ (inf_dist_le_inf_dist_of_subset (inter_subset_left _ _) ⟨y, h⟩),
refine not_lt.1 (λ hlt, _),
rcases (inf_dist_lt_iff ⟨y, h.1⟩).mp hlt with ⟨z, hzs, hz⟩,
cases le_or_lt (dist z x) (dist y x) with hle hlt,
... | lemma | metric.inf_dist_inter_closed_ball_of_mem | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.exists_inf_dist_eq_dist (h : is_compact s) (hne : s.nonempty) (x : α) :
∃ y ∈ s, inf_dist x s = dist x y | let ⟨y, hys, hy⟩ := h.exists_inf_edist_eq_edist hne x
in ⟨y, hys, by rw [inf_dist, dist_edist, hy]⟩ | lemma | is_compact.exists_inf_dist_eq_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"dist_edist",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_closed.exists_inf_dist_eq_dist [proper_space α]
(h : is_closed s) (hne : s.nonempty) (x : α) :
∃ y ∈ s, inf_dist x s = dist x y | begin
rcases hne with ⟨z, hz⟩,
rw ← inf_dist_inter_closed_ball_of_mem hz,
set t := s ∩ closed_ball x (dist z x),
have htc : is_compact t := (is_compact_closed_ball x (dist z x)).inter_left h,
have htne : t.nonempty := ⟨z, hz, mem_closed_ball.2 le_rfl⟩,
obtain ⟨y, ⟨hys, hyx⟩, hyd⟩ : ∃ y ∈ t, inf_dist x t = d... | lemma | is_closed.exists_inf_dist_eq_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"is_closed",
"is_compact",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_closure_inf_dist_eq_dist [proper_space α] (hne : s.nonempty) (x : α) :
∃ y ∈ closure s, inf_dist x s = dist x y | by simpa only [inf_dist_eq_closure] using is_closed_closure.exists_inf_dist_eq_dist hne.closure x | lemma | metric.exists_mem_closure_inf_dist_eq_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"closure",
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_nndist (x : α) (s : set α) : ℝ≥0 | ennreal.to_nnreal (inf_edist x s) | def | metric.inf_nndist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.to_nnreal"
] | The minimal distance of a point to a set as a `ℝ≥0` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inf_nndist : (inf_nndist x s : ℝ) = inf_dist x s | rfl | lemma | metric.coe_inf_nndist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_inf_nndist_pt (s : set α) : lipschitz_with 1 (λx, inf_nndist x s) | lipschitz_with.of_le_add $ λ x y, inf_dist_le_inf_dist_add_dist | lemma | metric.lipschitz_inf_nndist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"lipschitz_with",
"lipschitz_with.of_le_add"
] | The minimal distance to a set (as `ℝ≥0`) is Lipschitz in point with constant 1 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_inf_nndist_pt (s : set α) :
uniform_continuous (λx, inf_nndist x s) | (lipschitz_inf_nndist_pt s).uniform_continuous | lemma | metric.uniform_continuous_inf_nndist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"uniform_continuous"
] | The minimal distance to a set (as `ℝ≥0`) is uniformly continuous in point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_inf_nndist_pt (s : set α) : continuous (λx, inf_nndist x s) | (uniform_continuous_inf_nndist_pt s).continuous | lemma | metric.continuous_inf_nndist_pt | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"continuous"
] | The minimal distance to a set (as `ℝ≥0`) is continuous in point | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist (s t : set α) : ℝ | ennreal.to_real (Hausdorff_edist s t) | def | metric.Hausdorff_dist | topology.metric_space | src/topology/metric_space/hausdorff_distance.lean | [
"analysis.specific_limits.basic",
"topology.metric_space.isometric_smul",
"topology.metric_space.isometry",
"topology.instances.ennreal"
] | [
"ennreal.to_real"
] | 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 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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