statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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Hausdorff_dist_nonneg : 0 ≤ Hausdorff_dist s t | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_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 Hausdorff distance is nonnegative | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_edist_ne_top_of_nonempty_of_bounded (hs : s.nonempty) (ht : t.nonempty)
(bs : bounded s) (bt : bounded t) : Hausdorff_edist s t ≠ ⊤ | begin
rcases hs with ⟨cs, hcs⟩,
rcases ht with ⟨ct, hct⟩,
rcases (bounded_iff_subset_ball ct).1 bs with ⟨rs, hrs⟩,
rcases (bounded_iff_subset_ball cs).1 bt with ⟨rt, hrt⟩,
have : Hausdorff_edist s t ≤ ennreal.of_real (max rs rt),
{ apply Hausdorff_edist_le_of_mem_edist,
{ assume x xs,
existsi [ct,... | lemma | metric.Hausdorff_edist_ne_top_of_nonempty_of_bounded | 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_nonneg",
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_le_of_real_iff",
"ennreal.of_real_ne_top",
"ne_top_of_le_ne_top"
] | If two sets are nonempty and bounded in a metric space, they are at finite Hausdorff
edistance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_self_zero : Hausdorff_dist s s = 0 | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_dist_self_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"
] | [] | The Hausdorff distance between a set and itself is zero | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_comm : Hausdorff_dist s t = Hausdorff_dist t s | by simp [Hausdorff_dist, Hausdorff_edist_comm] | lemma | metric.Hausdorff_dist_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"
] | [] | The Hausdorff distance from `s` to `t` and from `t` to `s` coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_empty : Hausdorff_dist s ∅ = 0 | begin
cases s.eq_empty_or_nonempty with h h,
{ simp [h] },
{ simp [Hausdorff_dist, Hausdorff_edist_empty h] }
end | lemma | metric.Hausdorff_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 Hausdorff distance to the empty set vanishes (if you want to have the more reasonable
value ∞ instead, use `Hausdorff_edist`, which takes values in ℝ≥0∞) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_empty' : Hausdorff_dist ∅ s = 0 | by simp [Hausdorff_dist_comm] | lemma | metric.Hausdorff_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 Hausdorff distance to the empty set vanishes (if you want to have the more reasonable
value ∞ instead, use `Hausdorff_edist`, which takes values in ℝ≥0∞) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_le_of_inf_dist {r : ℝ} (hr : 0 ≤ r)
(H1 : ∀x ∈ s, inf_dist x t ≤ r) (H2 : ∀x ∈ t, inf_dist x s ≤ r) :
Hausdorff_dist s t ≤ r | begin
by_cases h1 : Hausdorff_edist s t = ⊤,
{ rwa [Hausdorff_dist, h1, ennreal.top_to_real] },
cases s.eq_empty_or_nonempty with hs hs,
{ rwa [hs, Hausdorff_dist_empty'] },
cases t.eq_empty_or_nonempty with ht ht,
{ rwa [ht, Hausdorff_dist_empty] },
have : Hausdorff_edist s t ≤ ennreal.of_real r,
{ app... | lemma | metric.Hausdorff_dist_le_of_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.of_real",
"ennreal.of_real_ne_top",
"ennreal.to_real_le_to_real",
"ennreal.to_real_of_real",
"ennreal.top_to_real"
] | Bounding the Hausdorff distance by bounding the distance of any point
in each set to the other set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_le_of_mem_dist {r : ℝ} (hr : 0 ≤ r)
(H1 : ∀x ∈ s, ∃y ∈ t, dist x y ≤ r) (H2 : ∀x ∈ t, ∃y ∈ s, dist x y ≤ r) :
Hausdorff_dist s t ≤ r | begin
apply Hausdorff_dist_le_of_inf_dist hr,
{ assume x xs,
rcases H1 x xs with ⟨y, yt, hy⟩,
exact le_trans (inf_dist_le_dist_of_mem yt) hy },
{ assume x xt,
rcases H2 x xt with ⟨y, ys, hy⟩,
exact le_trans (inf_dist_le_dist_of_mem ys) hy }
end | lemma | metric.Hausdorff_dist_le_of_mem_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"
] | [] | Bounding the Hausdorff distance by exhibiting, for any point in each set,
another point in the other set at controlled distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_le_diam (hs : s.nonempty) (bs : bounded s) (ht : t.nonempty) (bt : bounded t) :
Hausdorff_dist s t ≤ diam (s ∪ t) | begin
rcases hs with ⟨x, xs⟩,
rcases ht with ⟨y, yt⟩,
refine Hausdorff_dist_le_of_mem_dist diam_nonneg _ _,
{ exact λz hz, ⟨y, yt, dist_le_diam_of_mem (bounded_union.2 ⟨bs, bt⟩)
(subset_union_left _ _ hz) (subset_union_right _ _ yt)⟩ },
{ exact λz hz, ⟨x, xs, dist_le_diam_of_mem (bounded_union.2 ⟨bs, b... | lemma | metric.Hausdorff_dist_le_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"
] | [] | The Hausdorff distance is controlled by the diameter of the union | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_le_Hausdorff_dist_of_mem (hx : x ∈ s) (fin : Hausdorff_edist s t ≠ ⊤) :
inf_dist x t ≤ Hausdorff_dist s t | begin
have ht : t.nonempty := nonempty_of_Hausdorff_edist_ne_top ⟨x, hx⟩ fin,
rw [Hausdorff_dist, inf_dist, ennreal.to_real_le_to_real (inf_edist_ne_top ht) fin],
exact inf_edist_le_Hausdorff_edist_of_mem hx
end | lemma | metric.inf_dist_le_Hausdorff_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"
] | [
"ennreal.to_real_le_to_real"
] | The distance to a set is controlled by the Hausdorff distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_dist_lt_of_Hausdorff_dist_lt {r : ℝ} (h : x ∈ s) (H : Hausdorff_dist s t < r)
(fin : Hausdorff_edist s t ≠ ⊤) : ∃y∈t, dist x y < r | begin
have r0 : 0 < r := lt_of_le_of_lt (Hausdorff_dist_nonneg) H,
have : Hausdorff_edist s t < ennreal.of_real r,
{ rwa [Hausdorff_dist, ← ennreal.to_real_of_real (le_of_lt r0),
ennreal.to_real_lt_to_real fin (ennreal.of_real_ne_top)] at H },
rcases exists_edist_lt_of_Hausdorff_edist_lt h this with ⟨y, h... | lemma | metric.exists_dist_lt_of_Hausdorff_dist_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"
] | [
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff",
"ennreal.of_real_ne_top",
"ennreal.to_real_lt_to_real",
"ennreal.to_real_of_real"
] | If the Hausdorff distance is `<r`, then any point in one of the sets is at distance
`<r` of a point in the other set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_dist_lt_of_Hausdorff_dist_lt' {r : ℝ} (h : y ∈ t) (H : Hausdorff_dist s t < r)
(fin : Hausdorff_edist s t ≠ ⊤) : ∃x∈s, dist x y < r | begin
rw Hausdorff_dist_comm at H,
rw Hausdorff_edist_comm at fin,
simpa [dist_comm] using exists_dist_lt_of_Hausdorff_dist_lt h H fin
end | lemma | metric.exists_dist_lt_of_Hausdorff_dist_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"
] | [
"dist_comm"
] | If the Hausdorff distance is `<r`, then any point in one of the sets is at distance
`<r` of a point in the other set | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_dist_le_inf_dist_add_Hausdorff_dist (fin : Hausdorff_edist s t ≠ ⊤) :
inf_dist x t ≤ inf_dist x s + Hausdorff_dist s t | begin
rcases empty_or_nonempty_of_Hausdorff_edist_ne_top fin with ⟨hs,ht⟩|⟨hs,ht⟩,
{ simp only [hs, ht, Hausdorff_dist_empty, inf_dist_empty, zero_add] },
rw [inf_dist, inf_dist, Hausdorff_dist, ← ennreal.to_real_add (inf_edist_ne_top hs) fin,
ennreal.to_real_le_to_real (inf_edist_ne_top ht)],
{ exact inf... | lemma | metric.inf_dist_le_inf_dist_add_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_add",
"ennreal.to_real_le_to_real"
] | The infimum distance to `s` and `t` are the same, up to the Hausdorff distance
between `s` and `t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_image (h : isometry Φ) :
Hausdorff_dist (Φ '' s) (Φ '' t) = Hausdorff_dist s t | by simp [Hausdorff_dist, Hausdorff_edist_image h] | lemma | metric.Hausdorff_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 Hausdorff distance is invariant under isometries | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_triangle (fin : Hausdorff_edist s t ≠ ⊤) :
Hausdorff_dist s u ≤ Hausdorff_dist s t + Hausdorff_dist t u | begin
by_cases Hausdorff_edist s u = ⊤,
{ calc Hausdorff_dist s u = 0 + 0 : by simp [Hausdorff_dist, h]
... ≤ Hausdorff_dist s t + Hausdorff_dist t u :
add_le_add (Hausdorff_dist_nonneg) (Hausdorff_dist_nonneg) },
{ have Dtu : Hausdorff_edist t u < ⊤ := calc
Hausdorff_edist t u ≤ Hausdor... | lemma | metric.Hausdorff_dist_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"
] | [
"ennreal.add_eq_top",
"ennreal.to_real_add",
"ennreal.to_real_le_to_real"
] | The Hausdorff distance satisfies the triangular inequality | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_triangle' (fin : Hausdorff_edist t u ≠ ⊤) :
Hausdorff_dist s u ≤ Hausdorff_dist s t + Hausdorff_dist t u | begin
rw Hausdorff_edist_comm at fin,
have I : Hausdorff_dist u s ≤ Hausdorff_dist u t + Hausdorff_dist t s :=
Hausdorff_dist_triangle fin,
simpa [add_comm, Hausdorff_dist_comm] using I
end | lemma | metric.Hausdorff_dist_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"
] | [] | The Hausdorff distance satisfies the triangular inequality | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_self_closure : Hausdorff_dist s (closure s) = 0 | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_dist_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"
] | The Hausdorff distance between a set and its closure vanish | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_closure₁ : Hausdorff_dist (closure s) t = Hausdorff_dist s t | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_dist_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 distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_closure₂ : Hausdorff_dist s (closure t) = Hausdorff_dist s t | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_dist_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 distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_closure :
Hausdorff_dist (closure s) (closure t) = Hausdorff_dist s t | by simp [Hausdorff_dist] | lemma | metric.Hausdorff_dist_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 distance between two sets and their closures coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Hausdorff_dist_zero_iff_closure_eq_closure (fin : Hausdorff_edist s t ≠ ⊤) :
Hausdorff_dist s t = 0 ↔ closure s = closure t | by simp [Hausdorff_edist_zero_iff_closure_eq_closure.symm, Hausdorff_dist,
ennreal.to_real_eq_zero_iff, fin] | lemma | metric.Hausdorff_dist_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",
"ennreal.to_real_eq_zero_iff"
] | Two sets are at zero Hausdorff distance if and only if they have the same closures | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_closed.Hausdorff_dist_zero_iff_eq (hs : is_closed s) (ht : is_closed t)
(fin : Hausdorff_edist s t ≠ ⊤) : Hausdorff_dist s t = 0 ↔ s = t | by simp [←Hausdorff_edist_zero_iff_eq_of_closed hs ht, Hausdorff_dist,
ennreal.to_real_eq_zero_iff, fin] | lemma | is_closed.Hausdorff_dist_zero_iff_eq | 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_eq_zero_iff",
"is_closed"
] | Two closed sets are at zero Hausdorff distance if and only if they coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening (δ : ℝ) (E : set α) : set α | {x : α | inf_edist x E < ennreal.of_real δ} | def | metric.thickening | 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.of_real"
] | The (open) `δ`-thickening `thickening δ E` of a subset `E` in a pseudo emetric space consists
of those points that are at distance less than `δ` from some point of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_thickening_iff_inf_edist_lt : x ∈ thickening δ s ↔ inf_edist x s < ennreal.of_real δ | iff.rfl | lemma | metric.mem_thickening_iff_inf_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"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_eq_preimage_inf_edist (δ : ℝ) (E : set α) :
thickening δ E = (λ x, inf_edist x E) ⁻¹' (Iio (ennreal.of_real δ)) | rfl | lemma | metric.thickening_eq_preimage_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"
] | [
"ennreal.of_real"
] | The (open) thickening equals the preimage of an open interval under `inf_edist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_thickening {δ : ℝ} {E : set α} : is_open (thickening δ E) | continuous.is_open_preimage continuous_inf_edist _ is_open_Iio | lemma | metric.is_open_thickening | 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_open",
"is_open_Iio"
] | The (open) thickening is an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_empty (δ : ℝ) : thickening δ (∅ : set α) = ∅ | by simp only [thickening, set_of_false, inf_edist_empty, not_top_lt] | lemma | metric.thickening_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"
] | [
"not_top_lt"
] | The (open) thickening of the empty set is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_of_nonpos (hδ : δ ≤ 0) (s : set α) : thickening δ s = ∅ | eq_empty_of_forall_not_mem $ λ x, ((ennreal.of_real_of_nonpos hδ).trans_le bot_le).not_lt | lemma | metric.thickening_of_nonpos | 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"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : set α) :
thickening δ₁ E ⊆ thickening δ₂ E | preimage_mono (Iio_subset_Iio (ennreal.of_real_le_of_real hle)) | lemma | metric.thickening_mono | 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.of_real_le_of_real"
] | The (open) thickening `thickening δ E` of a fixed subset `E` is an increasing function of the
thickening radius `δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_subset_of_subset (δ : ℝ) {E₁ E₂ : set α} (h : E₁ ⊆ E₂) :
thickening δ E₁ ⊆ thickening δ E₂ | λ _ hx, lt_of_le_of_lt (inf_edist_anti h) hx | lemma | metric.thickening_subset_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"
] | [] | The (open) thickening `thickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_thickening_iff_exists_edist_lt {δ : ℝ} (E : set α) (x : α) :
x ∈ thickening δ E ↔ ∃ z ∈ E, edist x z < ennreal.of_real δ | inf_edist_lt_iff | lemma | metric.mem_thickening_iff_exists_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"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_thickening_subset (E : set α) {δ : ℝ} :
frontier (thickening δ E) ⊆ {x : α | inf_edist x E = ennreal.of_real δ} | frontier_lt_subset_eq continuous_inf_edist continuous_const | lemma | metric.frontier_thickening_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"
] | [
"continuous_const",
"ennreal.of_real",
"frontier",
"frontier_lt_subset_eq"
] | The frontier of the (open) thickening of a set is contained in an `inf_edist` level set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier_thickening_disjoint (A : set α) :
pairwise (disjoint on (λ (r : ℝ), frontier (thickening r A))) | begin
refine (pairwise_disjoint_on _).2 (λ r₁ r₂ hr, _),
cases le_total r₁ 0 with h₁ h₁,
{ simp [thickening_of_nonpos h₁] },
refine ((disjoint_singleton.2 $ λ h, hr.ne _).preimage _).mono
(frontier_thickening_subset _) (frontier_thickening_subset _),
apply_fun ennreal.to_real at h,
rwa [ennreal.to_real_... | lemma | metric.frontier_thickening_disjoint | 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",
"ennreal.to_real",
"ennreal.to_real_of_real",
"frontier",
"pairwise",
"pairwise_disjoint_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_thickening_iff {E : set X} {x : X} :
x ∈ thickening δ E ↔ (∃ z ∈ E, dist x z < δ) | begin
have key_iff : ∀ (z : X), edist x z < ennreal.of_real δ ↔ dist x z < δ,
{ intros z,
rw dist_edist,
have d_lt_top : edist x z < ∞, by simp only [edist_dist, ennreal.of_real_lt_top],
have key := (@ennreal.of_real_lt_of_real_iff_of_nonneg
((edist x z).to_real) δ (ennreal.to_real_nonne... | lemma | metric.mem_thickening_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_dist",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff_of_nonneg",
"ennreal.of_real_lt_top",
"ennreal.of_real_to_real",
"ennreal.to_real_nonneg"
] | A point in a metric space belongs to the (open) `δ`-thickening of a subset `E` if and only if
it is at distance less than `δ` from some point of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_singleton (δ : ℝ) (x : X) :
thickening δ ({x} : set X) = ball x δ | by { ext, simp [mem_thickening_iff] } | lemma | metric.thickening_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset_thickening {x : X} {E : set X} (hx : x ∈ E) (δ : ℝ) : ball x δ ⊆ thickening δ E | subset.trans (by simp) (thickening_subset_of_subset δ $ singleton_subset_iff.mpr hx) | lemma | metric.ball_subset_thickening | 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 | |
thickening_eq_bUnion_ball {δ : ℝ} {E : set X} :
thickening δ E = ⋃ x ∈ E, ball x δ | by { ext x, rw mem_Union₂, exact mem_thickening_iff } | lemma | metric.thickening_eq_bUnion_ball | 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 (open) `δ`-thickening `thickening δ E` of a subset `E` in a metric space equals the
union of balls of radius `δ` centered at points of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded.thickening {δ : ℝ} {E : set X} (h : bounded E) :
bounded (thickening δ E) | begin
refine bounded_iff_mem_bounded.2 (λ x hx, _),
rcases h.subset_ball x with ⟨R, hR⟩,
refine (bounded_iff_subset_ball x).2 ⟨R + δ, _⟩,
assume y hy,
rcases mem_thickening_iff.1 hy with ⟨z, zE, hz⟩,
calc dist y x ≤ dist z x + dist y z : by { rw add_comm, exact dist_triangle _ _ _ }
... ≤ R + δ : add_le_a... | lemma | metric.bounded.thickening | 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_triangle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening (δ : ℝ) (E : set α) : set α | {x : α | inf_edist x E ≤ ennreal.of_real δ} | def | metric.cthickening | 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.of_real"
] | The closed `δ`-thickening `cthickening δ E` of a subset `E` in a pseudo emetric space consists
of those points that are at infimum distance at most `δ` from `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_cthickening_iff : x ∈ cthickening δ s ↔ inf_edist x s ≤ ennreal.of_real δ | iff.rfl | lemma | metric.mem_cthickening_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"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cthickening_of_edist_le (x y : α) (δ : ℝ) (E : set α) (h : y ∈ E)
(h' : edist x y ≤ ennreal.of_real δ) :
x ∈ cthickening δ E | (inf_edist_le_edist_of_mem h).trans h' | lemma | metric.mem_cthickening_of_edist_le | 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.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cthickening_of_dist_le {α : Type*} [pseudo_metric_space α]
(x y : α) (δ : ℝ) (E : set α) (h : y ∈ E) (h' : dist x y ≤ δ) :
x ∈ cthickening δ E | begin
apply mem_cthickening_of_edist_le x y δ E h,
rw edist_dist,
exact ennreal.of_real_le_of_real h',
end | lemma | metric.mem_cthickening_of_dist_le | 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_dist",
"ennreal.of_real_le_of_real",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_eq_preimage_inf_edist (δ : ℝ) (E : set α) :
cthickening δ E = (λ x, inf_edist x E) ⁻¹' (Iic (ennreal.of_real δ)) | rfl | lemma | metric.cthickening_eq_preimage_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"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_cthickening {δ : ℝ} {E : set α} : is_closed (cthickening δ E) | is_closed.preimage continuous_inf_edist is_closed_Iic | lemma | metric.is_closed_cthickening | 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_closed.preimage",
"is_closed_Iic"
] | The closed thickening is a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_empty (δ : ℝ) : cthickening δ (∅ : set α) = ∅ | by simp only [cthickening, ennreal.of_real_ne_top, set_of_false, inf_edist_empty, top_le_iff] | lemma | metric.cthickening_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"
] | [
"ennreal.of_real_ne_top",
"top_le_iff"
] | The closed thickening of the empty set is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : set α) :
cthickening δ E = closure E | by { ext x, simp [mem_closure_iff_inf_edist_zero, cthickening, ennreal.of_real_eq_zero.2 hδ] } | lemma | metric.cthickening_of_nonpos | 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 | |
cthickening_zero (E : set α) : cthickening 0 E = closure E | cthickening_of_nonpos le_rfl E | lemma | metric.cthickening_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",
"le_rfl"
] | The closed thickening with radius zero is the closure of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_max_zero (δ : ℝ) (E : set α) : cthickening (max 0 δ) E = cthickening δ E | by cases le_total δ 0; simp [cthickening_of_nonpos, *] | lemma | metric.cthickening_max_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : set α) :
cthickening δ₁ E ⊆ cthickening δ₂ E | preimage_mono (Iic_subset_Iic.mpr (ennreal.of_real_le_of_real hle)) | lemma | metric.cthickening_mono | 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.of_real_le_of_real"
] | The closed thickening `cthickening δ E` of a fixed subset `E` is an increasing function of
the thickening radius `δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_singleton {α : Type*} [pseudo_metric_space α]
(x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
cthickening δ ({x} : set α) = closed_ball x δ | by { ext y, simp [cthickening, edist_dist, ennreal.of_real_le_of_real_iff hδ] } | lemma | metric.cthickening_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"
] | [
"edist_dist",
"ennreal.of_real_le_of_real_iff",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_subset_cthickening_singleton {α : Type*} [pseudo_metric_space α]
(x : α) (δ : ℝ) :
closed_ball x δ ⊆ cthickening δ ({x} : set α) | begin
rcases lt_or_le δ 0 with hδ|hδ,
{ simp only [closed_ball_eq_empty.mpr hδ, empty_subset] },
{ simp only [cthickening_singleton x hδ] }
end | lemma | metric.closed_ball_subset_cthickening_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"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_subset_of_subset (δ : ℝ) {E₁ E₂ : set α} (h : E₁ ⊆ E₂) :
cthickening δ E₁ ⊆ cthickening δ E₂ | λ _ hx, le_trans (inf_edist_anti h) hx | lemma | metric.cthickening_subset_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"
] | [] | The closed thickening `cthickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_subset_thickening {δ₁ : ℝ≥0} {δ₂ : ℝ} (hlt : (δ₁ : ℝ) < δ₂) (E : set α) :
cthickening δ₁ E ⊆ thickening δ₂ E | λ _ hx, lt_of_le_of_lt hx ((ennreal.of_real_lt_of_real_iff (lt_of_le_of_lt δ₁.prop hlt)).mpr hlt) | lemma | metric.cthickening_subset_thickening | 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.of_real_lt_of_real_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_subset_thickening' {δ₁ δ₂ : ℝ} (δ₂_pos : 0 < δ₂) (hlt : δ₁ < δ₂) (E : set α) :
cthickening δ₁ E ⊆ thickening δ₂ E | λ _ hx, lt_of_le_of_lt hx ((ennreal.of_real_lt_of_real_iff δ₂_pos).mpr hlt) | lemma | metric.cthickening_subset_thickening' | 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.of_real_lt_of_real_iff"
] | The closed thickening `cthickening δ₁ E` is contained in the open thickening `thickening δ₂ E`
if the radius of the latter is positive and larger. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_subset_cthickening (δ : ℝ) (E : set α) :
thickening δ E ⊆ cthickening δ E | by { intros x hx, rw [thickening, mem_set_of_eq] at hx, exact hx.le, } | lemma | metric.thickening_subset_cthickening | 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 open thickening `thickening δ E` is contained in the closed thickening `cthickening δ E`
with the same radius. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_subset_cthickening_of_le {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : set α) :
thickening δ₁ E ⊆ cthickening δ₂ E | (thickening_subset_cthickening δ₁ E).trans (cthickening_mono hle E) | lemma | metric.thickening_subset_cthickening_of_le | 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 | |
bounded.cthickening {α : Type*} [pseudo_metric_space α] {δ : ℝ} {E : set α} (h : bounded E) :
bounded (cthickening δ E) | begin
have : bounded (thickening (max (δ + 1) 1) E) := h.thickening,
apply bounded.mono _ this,
exact cthickening_subset_thickening' (zero_lt_one.trans_le (le_max_right _ _))
((lt_add_one _).trans_le (le_max_left _ _)) _
end | lemma | metric.bounded.cthickening | 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"
] | [
"lt_add_one",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_subset_interior_cthickening (δ : ℝ) (E : set α) :
thickening δ E ⊆ interior (cthickening δ E) | (subset_interior_iff_is_open.mpr (is_open_thickening)).trans
(interior_mono (thickening_subset_cthickening δ E)) | lemma | metric.thickening_subset_interior_cthickening | 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"
] | [
"interior",
"interior_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_thickening_subset_cthickening (δ : ℝ) (E : set α) :
closure (thickening δ E) ⊆ cthickening δ E | (closure_mono (thickening_subset_cthickening δ E)).trans is_closed_cthickening.closure_subset | lemma | metric.closure_thickening_subset_cthickening | 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_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_subset_cthickening (δ : ℝ) (E : set α) :
closure E ⊆ cthickening δ E | by { rw ← cthickening_of_nonpos (min_le_right δ 0), exact cthickening_mono (min_le_left δ 0) E, } | lemma | metric.closure_subset_cthickening | 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 closed thickening of a set contains the closure of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : set α) :
closure E ⊆ thickening δ E | by { rw ← cthickening_zero, exact cthickening_subset_thickening' δ_pos δ_pos E, } | lemma | metric.closure_subset_thickening | 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 (open) thickening of a set contains the closure of the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : set α) :
E ⊆ thickening δ E | (@subset_closure _ _ E).trans (closure_subset_thickening δ_pos E) | lemma | metric.self_subset_thickening | 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"
] | [
"subset_closure"
] | A set is contained in its own (open) thickening. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_subset_cthickening {δ : ℝ} (E : set α) :
E ⊆ cthickening δ E | subset_closure.trans (closure_subset_cthickening δ E) | lemma | metric.self_subset_cthickening | 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"
] | [] | A set is contained in its own closed thickening. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_mem_nhds_set (E : set α) {δ : ℝ} (hδ : 0 < δ) : thickening δ E ∈ 𝓝ˢ E | is_open_thickening.mem_nhds_set.2 $ self_subset_thickening hδ E | lemma | metric.thickening_mem_nhds_set | 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 | |
cthickening_mem_nhds_set (E : set α) {δ : ℝ} (hδ : 0 < δ) : cthickening δ E ∈ 𝓝ˢ E | mem_of_superset (thickening_mem_nhds_set E hδ) (thickening_subset_cthickening _ _) | lemma | metric.cthickening_mem_nhds_set | 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 | |
thickening_union (δ : ℝ) (s t : set α) :
thickening δ (s ∪ t) = thickening δ s ∪ thickening δ t | by simp_rw [thickening, inf_edist_union, inf_eq_min, min_lt_iff, set_of_or] | lemma | metric.thickening_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"
] | [
"inf_eq_min",
"min_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_union (δ : ℝ) (s t : set α) :
cthickening δ (s ∪ t) = cthickening δ s ∪ cthickening δ t | by simp_rw [cthickening, inf_edist_union, inf_eq_min, min_le_iff, set_of_or] | lemma | metric.cthickening_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"
] | [
"inf_eq_min",
"min_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_Union (δ : ℝ) (f : ι → set α) :
thickening δ (⋃ i, f i) = ⋃ i, thickening δ (f i) | by simp_rw [thickening, inf_edist_Union, infi_lt_iff, set_of_exists] | lemma | metric.thickening_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_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_cthickening_le (ε : ℝ≥0) : emetric.diam (cthickening ε s) ≤ emetric.diam s + 2 * ε | begin
refine diam_le (λ x hx y hy, ennreal.le_of_forall_pos_le_add $ λ δ hδ _, _),
rw [mem_cthickening_iff, ennreal.of_real_coe_nnreal] at hx hy,
have hε : (ε : ℝ≥0∞) < ε + ↑(δ / 2) :=
ennreal.coe_lt_coe.2 (lt_add_of_pos_right _ $ half_pos hδ),
rw [ennreal.coe_div two_ne_zero, ennreal.coe_two] at hε,
repl... | lemma | metric.ediam_cthickening_le | 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_triangle_right",
"emetric.diam",
"ennreal.coe_div",
"ennreal.coe_two",
"ennreal.le_of_forall_pos_le_add",
"ennreal.mul_div_cancel'",
"ennreal.of_real_coe_nnreal",
"ennreal.two_ne_top",
"half_pos",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_thickening_le (ε : ℝ≥0) : emetric.diam (thickening ε s) ≤ emetric.diam s + 2 * ε | (emetric.diam_mono $ thickening_subset_cthickening _ _).trans $ ediam_cthickening_le _ | lemma | metric.ediam_thickening_le | 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"
] | [
"emetric.diam",
"emetric.diam_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_cthickening_le {α : Type*} [pseudo_metric_space α] (s : set α) (hε : 0 ≤ ε) :
diam (cthickening ε s) ≤ diam s + 2 * ε | begin
by_cases hs : bounded (cthickening ε s),
{ replace hs := hs.mono (self_subset_cthickening _),
lift ε to ℝ≥0 using hε,
have : (2 : ℝ≥0∞) * ε ≠ ⊤ := by simp [ennreal.mul_eq_top],
refine (ennreal.to_real_mono (ennreal.add_ne_top.2 ⟨hs.ediam_ne_top, this⟩) $
ediam_cthickening_le ε).trans_eq _,
... | lemma | metric.diam_cthickening_le | 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.mul_eq_top",
"ennreal.to_real_add",
"ennreal.to_real_mono",
"lift",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_thickening_le {α : Type*} [pseudo_metric_space α] (s : set α) (hε : 0 ≤ ε) :
diam (thickening ε s) ≤ diam s + 2 * ε | begin
by_cases hs : bounded s,
{ exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans
(diam_cthickening_le _ hε) },
obtain rfl | hε := hε.eq_or_lt,
{ simp [thickening_of_nonpos, diam_nonneg] },
{ rw diam_eq_zero_of_unbounded (mt (bounded.mono $ self_subset_thickening hε _) hs),
... | lemma | metric.diam_thickening_le | 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_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thickening_closure : thickening δ (closure s) = thickening δ s | by simp_rw [thickening, inf_edist_closure] | lemma | metric.thickening_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 | |
cthickening_closure : cthickening δ (closure s) = cthickening δ s | by simp_rw [cthickening, inf_edist_closure] | lemma | metric.cthickening_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 | |
_root_.disjoint.exists_thickenings (hst : disjoint s t) (hs : is_compact s)
(ht : is_closed t) :
∃ δ, 0 < δ ∧ disjoint (thickening δ s) (thickening δ t) | begin
obtain ⟨r, hr, h⟩ := exists_pos_forall_lt_edist hs ht hst,
refine ⟨r / 2, half_pos (nnreal.coe_pos.2 hr), _⟩,
rw disjoint_iff_inf_le,
rintro z ⟨hzs, hzt⟩,
rw mem_thickening_iff_exists_edist_lt at hzs hzt,
rw [← nnreal.coe_two, ← nnreal.coe_div, ennreal.of_real_coe_nnreal] at hzs hzt,
obtain ⟨x, hx, ... | lemma | disjoint.exists_thickenings | 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"
] | [
"add_halves",
"disjoint",
"disjoint_iff_inf_le",
"edist_triangle_left",
"ennreal.coe_add",
"ennreal.of_real_coe_nnreal",
"half_pos",
"is_closed",
"is_compact",
"nnreal.coe_div",
"nnreal.coe_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.disjoint.exists_cthickenings (hst : disjoint s t) (hs : is_compact s)
(ht : is_closed t) :
∃ δ, 0 < δ ∧ disjoint (cthickening δ s) (cthickening δ t) | begin
obtain ⟨δ, hδ, h⟩ := hst.exists_thickenings hs ht,
refine ⟨δ / 2, half_pos hδ, h.mono _ _⟩;
exact (cthickening_subset_thickening' hδ (half_lt_self hδ) _),
end | lemma | disjoint.exists_cthickenings | 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",
"half_pos",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.exists_cthickening_subset_open (hs : is_compact s) (ht : is_open t)
(hst : s ⊆ t) :
∃ δ, 0 < δ ∧ cthickening δ s ⊆ t | (hst.disjoint_compl_right.exists_cthickenings hs ht.is_closed_compl).imp $ λ δ h,
⟨h.1, disjoint_compl_right_iff_subset.1 $ h.2.mono_right $ self_subset_cthickening _⟩ | lemma | is_compact.exists_cthickening_subset_open | 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_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.exists_thickening_subset_open (hs : is_compact s) (ht : is_open t)
(hst : s ⊆ t) :
∃ δ, 0 < δ ∧ thickening δ s ⊆ t | let ⟨δ, h₀, hδ⟩ := hs.exists_cthickening_subset_open ht hst
in ⟨δ, h₀, (thickening_subset_cthickening _ _).trans hδ⟩ | lemma | is_compact.exists_thickening_subset_open | 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_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_set_thickening {K : set α} (hK : is_compact K) :
(𝓝ˢ K).has_basis (λ δ : ℝ, 0 < δ) (λ δ, thickening δ K) | (has_basis_nhds_set K).to_has_basis' (λ U hU, hK.exists_thickening_subset_open hU.1 hU.2) $
λ _, thickening_mem_nhds_set K | lemma | metric.has_basis_nhds_set_thickening | 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_basis_nhds_set",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_set_cthickening {K : set α} (hK : is_compact K) :
(𝓝ˢ K).has_basis (λ δ : ℝ, 0 < δ) (λ δ, cthickening δ K) | (has_basis_nhds_set K).to_has_basis' (λ U hU, hK.exists_cthickening_subset_open hU.1 hU.2) $
λ _, cthickening_mem_nhds_set K | lemma | metric.has_basis_nhds_set_cthickening | 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_basis_nhds_set",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_eq_Inter_cthickening' {δ : ℝ}
(s : set ℝ) (hsδ : s ⊆ Ioi δ) (hs : ∀ ε, δ < ε → (s ∩ (Ioc δ ε)).nonempty) (E : set α) :
cthickening δ E = ⋂ ε ∈ s, cthickening ε E | begin
apply subset.antisymm,
{ exact subset_Inter₂ (λ _ hε, cthickening_mono (le_of_lt (hsδ hε)) E), },
{ unfold thickening cthickening,
intros x hx,
simp only [mem_Inter, mem_set_of_eq] at *,
apply ennreal.le_of_forall_pos_le_add,
intros η η_pos _,
rcases hs (δ + η) (lt_add_of_pos_right _ (nn... | lemma | metric.cthickening_eq_Inter_cthickening' | 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.coe_nnreal_eq",
"ennreal.le_of_forall_pos_le_add",
"ennreal.of_real_add_le",
"ennreal.of_real_le_of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cthickening_eq_Inter_cthickening {δ : ℝ} (E : set α) :
cthickening δ E = ⋂ (ε : ℝ) (h : δ < ε), cthickening ε E | begin
apply cthickening_eq_Inter_cthickening' (Ioi δ) rfl.subset,
simp_rw inter_eq_right_iff_subset.mpr Ioc_subset_Ioi_self,
exact λ _ hε, nonempty_Ioc.mpr hε,
end | lemma | metric.cthickening_eq_Inter_cthickening | 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 | |
cthickening_eq_Inter_thickening' {δ : ℝ} (δ_nn : 0 ≤ δ)
(s : set ℝ) (hsδ : s ⊆ Ioi δ) (hs : ∀ ε, δ < ε → (s ∩ (Ioc δ ε)).nonempty) (E : set α) :
cthickening δ E = ⋂ ε ∈ s, thickening ε E | begin
refine (subset_Inter₂ $ λ ε hε, _).antisymm _,
{ obtain ⟨ε', hsε', hε'⟩ := hs ε (hsδ hε),
have ss := cthickening_subset_thickening' (lt_of_le_of_lt δ_nn hε'.1) hε'.1 E,
exact ss.trans (thickening_mono hε'.2 E), },
{ rw cthickening_eq_Inter_cthickening' s hsδ hs E,
exact Inter₂_mono (λ ε hε, thic... | lemma | metric.cthickening_eq_Inter_thickening' | 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 | |
cthickening_eq_Inter_thickening {δ : ℝ} (δ_nn : 0 ≤ δ) (E : set α) :
cthickening δ E = ⋂ (ε : ℝ) (h : δ < ε), thickening ε E | begin
apply cthickening_eq_Inter_thickening' δ_nn (Ioi δ) rfl.subset,
simp_rw inter_eq_right_iff_subset.mpr Ioc_subset_Ioi_self,
exact λ _ hε, nonempty_Ioc.mpr hε,
end | lemma | metric.cthickening_eq_Inter_thickening | 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 | |
cthickening_eq_Inter_thickening'' (δ : ℝ) (E : set α) :
cthickening δ E = ⋂ (ε : ℝ) (h : max 0 δ < ε), thickening ε E | by { rw [←cthickening_max_zero, cthickening_eq_Inter_thickening], exact le_max_left _ _ } | lemma | metric.cthickening_eq_Inter_thickening'' | 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 | |
closure_eq_Inter_cthickening' (E : set α)
(s : set ℝ) (hs : ∀ ε, 0 < ε → (s ∩ (Ioc 0 ε)).nonempty) :
closure E = ⋂ δ ∈ s, cthickening δ E | begin
by_cases hs₀ : s ⊆ Ioi 0,
{ rw ← cthickening_zero, apply cthickening_eq_Inter_cthickening' _ hs₀ hs, },
obtain ⟨δ, hδs, δ_nonpos⟩ := not_subset.mp hs₀,
rw [set.mem_Ioi, not_lt] at δ_nonpos,
apply subset.antisymm,
{ exact subset_Inter₂ (λ ε _, closure_subset_cthickening ε E), },
{ rw ← cthickening_of... | lemma | metric.closure_eq_Inter_cthickening' | 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",
"set.mem_Ioi"
] | The closure of a set equals the intersection of its closed thickenings of positive radii
accumulating at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_Inter_cthickening (E : set α) :
closure E = ⋂ (δ : ℝ) (h : 0 < δ), cthickening δ E | by { rw ← cthickening_zero, exact cthickening_eq_Inter_cthickening E, } | lemma | metric.closure_eq_Inter_cthickening | 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 closure of a set equals the intersection of its closed thickenings of positive radii. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_Inter_thickening' (E : set α)
(s : set ℝ) (hs₀ : s ⊆ Ioi 0) (hs : ∀ ε, 0 < ε → (s ∩ (Ioc 0 ε)).nonempty) :
closure E = ⋂ δ ∈ s, thickening δ E | by { rw ← cthickening_zero, apply cthickening_eq_Inter_thickening' le_rfl _ hs₀ hs, } | lemma | metric.closure_eq_Inter_thickening' | 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",
"le_rfl"
] | The closure of a set equals the intersection of its open thickenings of positive radii
accumulating at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_eq_Inter_thickening (E : set α) :
closure E = ⋂ (δ : ℝ) (h : 0 < δ), thickening δ E | by { rw ← cthickening_zero, exact cthickening_eq_Inter_thickening rfl.ge E, } | lemma | metric.closure_eq_Inter_thickening | 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 closure of a set equals the intersection of its (open) thickenings of positive radii. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier_cthickening_subset (E : set α) {δ : ℝ} :
frontier (cthickening δ E) ⊆ {x : α | inf_edist x E = ennreal.of_real δ} | frontier_le_subset_eq continuous_inf_edist continuous_const | lemma | metric.frontier_cthickening_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"
] | [
"continuous_const",
"ennreal.of_real",
"frontier",
"frontier_le_subset_eq"
] | The frontier of the closed thickening of a set is contained in an `inf_edist` level set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ball_subset_cthickening {α : Type*} [pseudo_metric_space α]
{x : α} {E : set α} (hx : x ∈ E) (δ : ℝ) :
closed_ball x δ ⊆ cthickening δ E | begin
refine (closed_ball_subset_cthickening_singleton _ _).trans (cthickening_subset_of_subset _ _),
simpa using hx,
end | lemma | metric.closed_ball_subset_cthickening | 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_metric_space"
] | The closed ball of radius `δ` centered at a point of `E` is included in the closed
thickening of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_subset_Union_closed_ball_of_lt
{α : Type*} [pseudo_metric_space α] (E : set α) {δ δ' : ℝ} (hδ₀ : 0 < δ') (hδδ' : δ < δ') :
cthickening δ E ⊆ ⋃ x ∈ E, closed_ball x δ' | begin
refine (cthickening_subset_thickening' hδ₀ hδδ' E).trans (λ x hx, _),
obtain ⟨y, hy₁, hy₂⟩ := mem_thickening_iff.mp hx,
exact mem_Union₂.mpr ⟨y, hy₁, hy₂.le⟩,
end | lemma | metric.cthickening_subset_Union_closed_ball_of_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"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.cthickening_eq_bUnion_closed_ball
{α : Type*} [pseudo_metric_space α] {δ : ℝ} {E : set α} (hE : is_compact E) (hδ : 0 ≤ δ) :
cthickening δ E = ⋃ x ∈ E, closed_ball x δ | begin
rcases eq_empty_or_nonempty E with rfl|hne,
{ simp only [cthickening_empty, Union_false, Union_empty] },
refine subset.antisymm (λ x hx, _) (Union₂_subset $ λ x hx, closed_ball_subset_cthickening hx _),
obtain ⟨y, yE, hy⟩ : ∃ y ∈ E, inf_edist x E = edist x y :=
hE.exists_inf_edist_eq_edist hne _,
ha... | lemma | is_compact.cthickening_eq_bUnion_closed_ball | 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_dist",
"ennreal.of_real",
"ennreal.of_real_le_of_real_iff",
"is_compact",
"pseudo_metric_space"
] | The closed thickening of a compact set `E` is the union of the balls `closed_ball x δ` over
`x ∈ E`.
See also `metric.cthickening_eq_bUnion_closed_ball`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_eq_bUnion_closed_ball
{α : Type*} [pseudo_metric_space α] [proper_space α] (E : set α) (hδ : 0 ≤ δ) :
cthickening δ E = ⋃ x ∈ closure E, closed_ball x δ | begin
rcases eq_empty_or_nonempty E with rfl|hne,
{ simp only [cthickening_empty, Union_false, Union_empty, closure_empty], },
rw ← cthickening_closure,
refine subset.antisymm (λ x hx, _) (Union₂_subset $ λ x hx, closed_ball_subset_cthickening hx _),
obtain ⟨y, yE, hy⟩ : ∃ y ∈ closure E, inf_dist x (closure E... | lemma | metric.cthickening_eq_bUnion_closed_ball | 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_empty",
"ennreal.of_real",
"ennreal.of_real_le_of_real_iff",
"ennreal.of_real_to_real_le",
"proper_space",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_closed.cthickening_eq_bUnion_closed_ball
{α : Type*} [pseudo_metric_space α] [proper_space α] {E : set α} (hE : is_closed E) (hδ : 0 ≤ δ) :
cthickening δ E = ⋃ x ∈ E, closed_ball x δ | by rw [cthickening_eq_bUnion_closed_ball E hδ, hE.closure_eq] | lemma | is_closed.cthickening_eq_bUnion_closed_ball | 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",
"proper_space",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_le_inf_edist_cthickening_add :
inf_edist x s ≤ inf_edist x (cthickening δ s) + ennreal.of_real δ | begin
refine le_of_forall_lt' (λ r h, _),
simp_rw [←lt_tsub_iff_right, inf_edist_lt_iff, mem_cthickening_iff] at h,
obtain ⟨y, hy, hxy⟩ := h,
exact inf_edist_le_edist_add_inf_edist.trans_lt ((ennreal.add_lt_add_of_lt_of_le
(hy.trans_lt ennreal.of_real_lt_top).ne hxy hy).trans_le
(tsub_add_cancel_of_le $... | lemma | metric.inf_edist_le_inf_edist_cthickening_add | 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_lt_add_of_lt_of_le",
"ennreal.of_real",
"ennreal.of_real_lt_top",
"le_of_forall_lt'",
"tsub_add_cancel_of_le"
] | For the equality, see `inf_edist_cthickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_edist_le_inf_edist_thickening_add :
inf_edist x s ≤ inf_edist x (thickening δ s) + ennreal.of_real δ | inf_edist_le_inf_edist_cthickening_add.trans $
add_le_add_right (inf_edist_anti $ thickening_subset_cthickening _ _) _ | lemma | metric.inf_edist_le_inf_edist_thickening_add | 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.of_real"
] | For the equality, see `inf_edist_thickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_thickening_subset (ε δ : ℝ) (s : set α) :
thickening ε (thickening δ s) ⊆ thickening (ε + δ) s | begin
obtain hε | hε := le_total ε 0,
{ simp only [thickening_of_nonpos hε, empty_subset] },
obtain hδ | hδ := le_total δ 0,
{ simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset] },
intros x,
simp_rw [mem_thickening_iff_exists_edist_lt, ennreal.of_real_add hε hδ],
exact λ ⟨y, ⟨z, hz, hy⟩, ... | lemma | metric.thickening_thickening_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.add_lt_add",
"ennreal.of_real_add"
] | For the equality, see `thickening_thickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thickening_cthickening_subset (ε : ℝ) (hδ : 0 ≤ δ) (s : set α) :
thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s | begin
obtain hε | hε := le_total ε 0,
{ simp only [thickening_of_nonpos hε, empty_subset] },
intro x,
simp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, ←inf_edist_lt_iff,
ennreal.of_real_add hε hδ],
rintro ⟨y, hy, hxy⟩,
exact inf_edist_le_edist_add_inf_edist.trans_lt
(ennreal.add_lt_... | lemma | metric.thickening_cthickening_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.add_lt_add_of_lt_of_le",
"ennreal.of_real_add",
"ennreal.of_real_lt_top"
] | For the equality, see `thickening_cthickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cthickening_thickening_subset (hε : 0 ≤ ε) (δ : ℝ) (s : set α) :
cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s | begin
obtain hδ | hδ := le_total δ 0,
{ simp only [thickening_of_nonpos hδ, cthickening_empty, empty_subset] },
intro x,
simp_rw [mem_cthickening_iff, ennreal.of_real_add hε hδ],
exact λ hx, inf_edist_le_inf_edist_thickening_add.trans (add_le_add_right hx _),
end | lemma | metric.cthickening_thickening_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.of_real_add"
] | For the equality, see `cthickening_thickening`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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