fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
toLocallyBoundedMap (f : α → β) (hf : LipschitzWith K f) : LocallyBoundedMap α β :=
LocallyBoundedMap.ofMapBounded f fun _s hs =>
let ⟨C, hC⟩ := Metric.isBounded_iff.1 hs
Metric.isBounded_iff.2 ⟨K * C, forall_mem_image.2 fun _x hx => forall_mem_image.2 fun _y hy =>
hf.dist_le_mul_of_le (hC hx hy)⟩
@[simp] | def | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | toLocallyBoundedMap | A Lipschitz continuous map is a locally bounded map. |
coe_toLocallyBoundedMap (hf : LipschitzWith K f) : ⇑(hf.toLocallyBoundedMap f) = f :=
rfl | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | coe_toLocallyBoundedMap | null |
comap_cobounded_le (hf : LipschitzWith K f) :
comap f (Bornology.cobounded β) ≤ Bornology.cobounded α :=
(hf.toLocallyBoundedMap f).2 | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | comap_cobounded_le | null |
isBounded_image (hf : LipschitzWith K f) {s : Set α} (hs : IsBounded s) :
IsBounded (f '' s) :=
hs.image (toLocallyBoundedMap f hf) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | isBounded_image | The image of a bounded set under a Lipschitz map is bounded. |
diam_image_le (hf : LipschitzWith K f) (s : Set α) (hs : IsBounded s) :
Metric.diam (f '' s) ≤ K * Metric.diam s :=
Metric.diam_le_of_forall_dist_le (mul_nonneg K.coe_nonneg Metric.diam_nonneg) <|
forall_mem_image.2 fun _x hx =>
forall_mem_image.2 fun _y hy => hf.dist_le_mul_of_le <| Metric.dist_le_diam_of_mem hs hx hy | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | diam_image_le | null |
protected dist_left (y : α) : LipschitzWith 1 (dist · y) :=
LipschitzWith.mk_one fun _ _ => dist_dist_dist_le_left _ _ _ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist_left | null |
protected dist_right (x : α) : LipschitzWith 1 (dist x) :=
LipschitzWith.of_le_add fun _ _ => dist_triangle_right _ _ _ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist_right | null |
protected dist : LipschitzWith 2 (Function.uncurry <| @dist α _) := by
rw [← one_add_one_eq_two]
exact LipschitzWith.uncurry LipschitzWith.dist_left LipschitzWith.dist_right | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist | null |
dist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
dist (f^[n] x) (f^[n+1] x) ≤ dist x (f x) * (K : ℝ) ^ n := by
rw [iterate_succ, mul_comm]
simpa only [NNReal.coe_pow] using (hf.iterate n).dist_le_mul x (f x) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist_iterate_succ_le_geometric | null |
_root_.lipschitzWith_max : LipschitzWith 1 fun p : ℝ × ℝ => max p.1 p.2 :=
LipschitzWith.of_le_add fun _ _ => sub_le_iff_le_add'.1 <|
(le_abs_self _).trans (abs_max_sub_max_le_max _ _ _ _) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | _root_.lipschitzWith_max | null |
_root_.lipschitzWith_min : LipschitzWith 1 fun p : ℝ × ℝ => min p.1 p.2 :=
LipschitzWith.of_le_add fun _ _ => sub_le_iff_le_add'.1 <|
(le_abs_self _).trans (abs_min_sub_min_le_max _ _ _ _) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | _root_.lipschitzWith_min | null |
_root_.Real.lipschitzWith_toNNReal : LipschitzWith 1 Real.toNNReal := by
refine lipschitzWith_iff_dist_le_mul.mpr (fun x y ↦ ?_)
simpa only [NNReal.coe_one, dist_prod_same_right, one_mul, Real.dist_eq] using
lipschitzWith_iff_dist_le_mul.mp lipschitzWith_max (x, 0) (y, 0) | lemma | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | _root_.Real.lipschitzWith_toNNReal | null |
protected max (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
LipschitzWith (max Kf Kg) fun x => max (f x) (g x) := by
simpa only [(· ∘ ·), one_mul] using lipschitzWith_max.comp (hf.prodMk hg) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | max | null |
protected min (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
LipschitzWith (max Kf Kg) fun x => min (f x) (g x) := by
simpa only [(· ∘ ·), one_mul] using lipschitzWith_min.comp (hf.prodMk hg) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | min | null |
max_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max (f x) a := by
simpa only [max_eq_left (zero_le Kf)] using hf.max (LipschitzWith.const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | max_const | null |
const_max (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max a (f x) := by
simpa only [max_comm] using hf.max_const a | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | const_max | null |
min_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => min (f x) a := by
simpa only [max_eq_left (zero_le Kf)] using hf.min (LipschitzWith.const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | min_const | null |
const_min (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => min a (f x) := by
simpa only [min_comm] using hf.min_const a | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | const_min | null |
protected projIcc {a b : ℝ} (h : a ≤ b) : LipschitzWith 1 (projIcc a b h) :=
((LipschitzWith.id.const_min _).const_max _).subtype_mk _ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | projIcc | null |
LipschitzWith.properSpace {X Y : Type*} [PseudoMetricSpace X]
[PseudoMetricSpace Y] [ProperSpace Y] {f : X → Y} (hf : IsProperMap f)
{K : ℝ≥0} (hf' : LipschitzWith K f) : ProperSpace X :=
⟨fun x r ↦ (hf.isCompact_preimage (isCompact_closedBall (f x) (K * r))).of_isClosed_subset
Metric.isClosed_closedBall (hf'.mapsTo_closedBall x r).subset_preimage⟩ | lemma | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzWith.properSpace | The preimage of a proper space under a Lipschitz proper map is proper. |
protected of_dist_le' {K : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y) :
LipschitzOnWith (Real.toNNReal K) f s :=
of_dist_le_mul fun x hx y hy =>
le_trans (h x hx y hy) <| by gcongr; apply Real.le_coe_toNNReal | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_dist_le' | null |
protected mk_one (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y) :
LipschitzOnWith 1 f s :=
of_dist_le_mul <| by simpa only [NNReal.coe_one, one_mul] using h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | mk_one | null |
protected of_le_add_mul' {f : α → ℝ} (K : ℝ)
(h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y) : LipschitzOnWith (Real.toNNReal K) f s :=
have I : ∀ x ∈ s, ∀ y ∈ s, f x - f y ≤ K * dist x y := fun x hx y hy =>
sub_le_iff_le_add'.2 (h x hx y hy)
LipschitzOnWith.of_dist_le' fun x hx y hy =>
abs_sub_le_iff.2 ⟨I x hx y hy, dist_comm y x ▸ I y hy x hx⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add_mul' | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
doesn't assume `0≤K`. |
protected of_le_add_mul {f : α → ℝ} (K : ℝ≥0)
(h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y) : LipschitzOnWith K f s := by
simpa only [Real.toNNReal_coe] using LipschitzOnWith.of_le_add_mul' K h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add_mul | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
assumes `0≤K`. |
protected of_le_add {f : α → ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + dist x y) :
LipschitzOnWith 1 f s :=
LipschitzOnWith.of_le_add_mul 1 <| by simpa only [NNReal.coe_one, one_mul] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add | null |
protected le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : LipschitzOnWith K f s) {x : α} (hx : x ∈ s)
{y : α} (hy : y ∈ s) : f x ≤ f y + K * dist x y :=
sub_le_iff_le_add'.1 <| le_trans (le_abs_self _) <| h.dist_le_mul x hx y hy | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | le_add_mul | null |
protected iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y :=
⟨LipschitzOnWith.le_add_mul, LipschitzOnWith.of_le_add_mul K⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | iff_le_add_mul | null |
isBounded_image2 (f : α → β → γ) {K₁ K₂ : ℝ≥0} {s : Set α} {t : Set β}
(hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
(hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun a => f a b) s)
(hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) : Bornology.IsBounded (Set.image2 f s t) :=
Metric.isBounded_iff_ediam_ne_top.2 <|
ne_top_of_le_ne_top
(ENNReal.add_ne_top.mpr
⟨ENNReal.mul_ne_top ENNReal.coe_ne_top hs.ediam_ne_top,
ENNReal.mul_ne_top ENNReal.coe_ne_top ht.ediam_ne_top⟩)
(ediam_image2_le _ _ _ hf₁ hf₂) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | isBounded_image2 | null |
protected min (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
LocallyLipschitz (fun x => min (f x) (g x)) :=
lipschitzWith_min.locallyLipschitz.comp (hf.prodMk hg) | lemma | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | min | The minimum of locally Lipschitz functions is locally Lipschitz. |
protected max (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
LocallyLipschitz (fun x => max (f x) (g x)) :=
lipschitzWith_max.locallyLipschitz.comp (hf.prodMk hg) | lemma | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | max | The maximum of locally Lipschitz functions is locally Lipschitz. |
max_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max (f x) a :=
hf.max (LocallyLipschitz.const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | max_const | null |
const_max (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max a (f x) := by
simpa [max_comm] using (hf.max_const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | const_max | null |
min_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min (f x) a :=
hf.min (LocallyLipschitz.const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | min_const | null |
const_min (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min a (f x) := by
simpa [min_comm] using (hf.min_const a) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | const_min | null |
LipschitzOnWith.extend_real {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → ℝ, LipschitzWith K g ∧ EqOn f g s := by
/- An extension is given by `g y = Inf {f x + K * dist y x | x ∈ s}`. Taking `x = y`, one has
`g y ≤ f y` for `y ∈ s`, and the other inequality holds because `f` is `K`-Lipschitz, so that it
cannot counterbalance the growth of `K * dist y x`. One readily checks from the formula that
the extended function is also `K`-Lipschitz. -/
rcases eq_empty_or_nonempty s with (rfl | hs)
· exact ⟨fun _ => 0, (LipschitzWith.const _).weaken (zero_le _), eqOn_empty _ _⟩
have : Nonempty s := by simp only [hs, nonempty_coe_sort]
let g := fun y : α => iInf fun x : s => f x + K * dist y x
have B : ∀ y : α, BddBelow (range fun x : s => f x + K * dist y x) := fun y => by
rcases hs with ⟨z, hz⟩
refine ⟨f z - K * dist y z, ?_⟩
rintro w ⟨t, rfl⟩
dsimp
rw [sub_le_iff_le_add, add_assoc, ← mul_add, add_comm (dist y t)]
calc
f z ≤ f t + K * dist z t := hf.le_add_mul hz t.2
_ ≤ f t + K * (dist y z + dist y t) := by gcongr; apply dist_triangle_left
have E : EqOn f g s := fun x hx => by
refine le_antisymm (le_ciInf fun y => hf.le_add_mul hx y.2) ?_
simpa only [add_zero, Subtype.coe_mk, mul_zero, dist_self] using ciInf_le (B x) ⟨x, hx⟩
refine ⟨g, LipschitzWith.of_le_add_mul K fun x y => ?_, E⟩
rw [← sub_le_iff_le_add]
refine le_ciInf fun z => ?_
rw [sub_le_iff_le_add]
calc
g x ≤ f z + K * dist x z := ciInf_le (B x) _
_ ≤ f z + K * dist y z + K * dist x y := by
rw [add_assoc, ← mul_add, add_comm (dist y z)]
gcongr
apply dist_triangle | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzOnWith.extend_real | A function `f : α → ℝ` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz extension
to the whole space. |
LipschitzOnWith.extend_pi [Fintype ι] {f : α → ι → ℝ} {s : Set α}
{K : ℝ≥0} (hf : LipschitzOnWith K f s) : ∃ g : α → ι → ℝ, LipschitzWith K g ∧ EqOn f g s := by
have : ∀ i, ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s := fun i => by
have : LipschitzOnWith K (fun x : α => f x i) s :=
LipschitzOnWith.of_dist_le_mul fun x hx y hy =>
(dist_le_pi_dist _ _ i).trans (hf.dist_le_mul x hx y hy)
exact this.extend_real
choose g hg using this
refine ⟨fun x i => g i x, LipschitzWith.of_dist_le_mul fun x y => ?_, fun x hx ↦ ?_⟩
· exact (dist_pi_le_iff (mul_nonneg K.2 dist_nonneg)).2 fun i => (hg i).1.dist_le_mul x y
· ext1 i
exact (hg i).2 hx | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzOnWith.extend_pi | A function `f : α → (ι → ℝ)` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz
extension to the whole space. The same result for the space `ℓ^∞ (ι, ℝ)` over a possibly infinite
type `ι` is implemented in `LipschitzOnWith.extend_lp_infty`. |
IsSeparated (ε : ℝ≥0∞) (s : Set X) : Prop := s.Pairwise (ε < edist · ·) | def | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsSeparated | A set `s` is `ε`-separated if its elements are pairwise at distance at least `ε` from each
other. |
protected IsSeparated.empty : IsSeparated ε (∅ : Set X) := pairwise_empty _ | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsSeparated.empty | null |
protected IsSeparated.singleton : IsSeparated ε {x} := pairwise_singleton ..
@[simp] lemma IsSeparated.of_subsingleton (hs : s.Subsingleton) : IsSeparated ε s := hs.pairwise _
alias _root_.Set.Subsingleton.isSeparated := IsSeparated.of_subsingleton
nonrec lemma IsSeparated.anti (hεδ : ε ≤ δ) (hs : IsSeparated δ s) : IsSeparated ε s :=
hs.mono' fun _ _ ↦ hεδ.trans_lt | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsSeparated.singleton | null |
IsSeparated.subset (hst : s ⊆ t) (hs : IsSeparated ε t) : IsSeparated ε s := hs.mono hst | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsSeparated.subset | null |
isSeparated_insert :
IsSeparated ε (insert x s) ↔ IsSeparated ε s ∧ ∀ y ∈ s, x ≠ y → ε < edist x y :=
pairwise_insert_of_symmetric fun _ _ ↦ by simp [edist_comm] | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | isSeparated_insert | null |
isSeparated_insert_of_notMem (hx : x ∉ s) :
IsSeparated ε (insert x s) ↔ IsSeparated ε s ∧ ∀ y ∈ s, ε < edist x y :=
pairwise_insert_of_symmetric_of_notMem (fun _ _ ↦ by simp [edist_comm]) hx
@[deprecated (since := "2025-05-23")]
alias isSeparated_insert_of_not_mem := isSeparated_insert_of_notMem | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | isSeparated_insert_of_notMem | null |
protected IsSeparated.insert (hs : IsSeparated ε s) (h : ∀ y ∈ s, x ≠ y → ε < edist x y) :
IsSeparated ε (insert x s) := isSeparated_insert.2 ⟨hs, h⟩
/-! | lemma | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsSeparated.insert | null |
AreSeparated (s t : Set X) := ∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y | def | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | AreSeparated | Two sets in an (extended) metric space are called *metric separated* if the (extended) distance
between `x ∈ s` and `y ∈ t` is bounded from below by a positive constant. |
@[symm]
symm (h : AreSeparated s t) : AreSeparated t s :=
let ⟨r, r0, hr⟩ := h
⟨r, r0, fun y hy x hx => edist_comm x y ▸ hr x hx y hy⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | symm | null |
comm : AreSeparated s t ↔ AreSeparated t s := ⟨symm, symm⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | comm | null |
empty_left (s : Set X) : AreSeparated ∅ s :=
⟨1, one_ne_zero, fun _x => False.elim⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | empty_left | null |
empty_right (s : Set X) : AreSeparated s ∅ :=
(empty_left s).symm | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | empty_right | null |
protected disjoint (h : AreSeparated s t) : Disjoint s t :=
let ⟨r, r0, hr⟩ := h
Set.disjoint_left.mpr fun x hx1 hx2 => r0 <| by simpa using hr x hx1 x hx2 | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | disjoint | null |
subset_compl_right (h : AreSeparated s t) : s ⊆ tᶜ := fun _ hs ht =>
h.disjoint.le_bot ⟨hs, ht⟩
@[mono] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | subset_compl_right | null |
mono {s' t'} (hs : s ⊆ s') (ht : t ⊆ t') :
AreSeparated s' t' → AreSeparated s t := fun ⟨r, r0, hr⟩ =>
⟨r, r0, fun x hx y hy => hr x (hs hx) y (ht hy)⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | mono | null |
mono_left {s'} (h' : AreSeparated s' t) (hs : s ⊆ s') : AreSeparated s t :=
h'.mono hs Subset.rfl | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | mono_left | null |
mono_right {t'} (h' : AreSeparated s t') (ht : t ⊆ t') : AreSeparated s t :=
h'.mono Subset.rfl ht | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | mono_right | null |
union_left {s'} (h : AreSeparated s t) (h' : AreSeparated s' t) :
AreSeparated (s ∪ s') t := by
rcases h, h' with ⟨⟨r, r0, hr⟩, ⟨r', r0', hr'⟩⟩
refine ⟨min r r', ?_, fun x hx y hy => hx.elim ?_ ?_⟩
· rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0'
· exact fun hx => (min_le_left _ _).trans (hr _ hx _ hy)
· exact fun hx => (min_le_right _ _).trans (hr' _ hx _ hy)
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | union_left | null |
union_left_iff {s'} :
AreSeparated (s ∪ s') t ↔ AreSeparated s t ∧ AreSeparated s' t :=
⟨fun h => ⟨h.mono_left subset_union_left, h.mono_left subset_union_right⟩, fun h =>
h.1.union_left h.2⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | union_left_iff | null |
union_right {t'} (h : AreSeparated s t) (h' : AreSeparated s t') :
AreSeparated s (t ∪ t') :=
(h.symm.union_left h'.symm).symm
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | union_right | null |
union_right_iff {t'} :
AreSeparated s (t ∪ t') ↔ AreSeparated s t ∧ AreSeparated s t' :=
comm.trans <| union_left_iff.trans <| and_congr comm comm | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | union_right_iff | null |
finite_iUnion_left_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set X}
{t : Set X} : AreSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, AreSeparated (s i) t := by
induction I, hI using Set.Finite.induction_on with
| empty => simp
| insert _ _ hI => rw [biUnion_insert, forall_mem_insert, union_left_iff, hI]
alias ⟨_, finite_iUnion_left⟩ := finite_iUnion_left_iff | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | finite_iUnion_left_iff | null |
finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
{t : ι → Set X} : AreSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, AreSeparated s (t i) := by
simpa only [@comm _ _ s] using finite_iUnion_left_iff hI
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | finite_iUnion_right_iff | null |
finset_iUnion_left_iff {ι : Type*} {I : Finset ι} {s : ι → Set X} {t : Set X} :
AreSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, AreSeparated (s i) t :=
finite_iUnion_left_iff I.finite_toSet
alias ⟨_, finset_iUnion_left⟩ := finset_iUnion_left_iff
@[simp] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | finset_iUnion_left_iff | null |
finset_iUnion_right_iff {ι : Type*} {I : Finset ι} {s : Set X} {t : ι → Set X} :
AreSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, AreSeparated s (t i) :=
finite_iUnion_right_iff I.finite_toSet
alias ⟨_, finset_iUnion_right⟩ := finset_iUnion_right_iff | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/MetricSeparated.lean | finset_iUnion_right_iff | null |
eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
(mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
apply univ_mem'
rintro ⟨r, y⟩ hxy hyU i hi
simp only [mem_iInter, mem_compl_iff, not_imp_not, mem_preimage] at hxy
exact hyU _ (hxy _ hi)
intro i hi
rcases nhds_basis_closed_eball.mem_iff.1 ((hU i).mem_nhds <| hKU i hi) with ⟨R, hR₀, hR⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.mp hR₀ with ⟨r, hr₀, hrR⟩
filter_upwards [prod_mem_prod (eventually_lt_nhds hr₀)
(closedBall_mem_nhds x (tsub_pos_iff_lt.2 hrR))] with p hp z hz
apply hR
calc
edist z x ≤ edist z p.2 + edist p.2 x := edist_triangle _ _ _
_ ≤ p.1 + (R - p.1) := add_le_add hz <| le_trans hp.2 <| tsub_le_tsub_left hp.1.out.le _
_ = R := add_tsub_cancel_of_le (lt_trans (by exact hp.1) hrR).le | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | eventually_nhds_zero_forall_closedBall_subset | Let `K : ι → Set X` be a locally finite family of closed sets in an emetric space. Let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
`x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
`y ∈ K i`, then `EMetric.closedBall y r ⊆ U i`. |
exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
(hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∃ r : ℝ, ∀ᶠ y in 𝓝 x,
r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } := by
have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
(eventually_nhds_zero_forall_closedBall_subset hK hU hKU hfin x).curry
rcases this.exists_gt with ⟨r, hr0, hr⟩
refine ⟨r, hr.mono fun y hy => ⟨hr0, ?_⟩⟩
rwa [mem_preimage, mem_iInter₂] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_forall_closedBall_subset_aux₁ | null |
exists_forall_closedBall_subset_aux₂ (y : X) :
Convex ℝ
(Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i }) :=
(convex_Ioi _).inter <| OrdConnected.convex <| OrdConnected.preimage_ennreal_ofReal <|
ordConnected_iInter fun i => ordConnected_iInter fun (_ : y ∈ K i) =>
ordConnected_setOf_closedBall_subset y (U i) | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_forall_closedBall_subset_aux₂ | null |
exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧
∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i := by
simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_swap ι X] using
exists_continuous_forall_mem_convex_of_local_const exists_forall_closedBall_subset_aux₂
(exists_forall_closedBall_subset_aux₁ hK hU hKU hfin) | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_continuous_real_forall_closedBall_subset | Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i`. |
exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by
rcases exists_continuous_real_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ₀, hδ⟩
lift δ to C(X, ℝ≥0) using fun x => (hδ₀ x).le
refine ⟨δ, hδ₀, fun i x hi => ?_⟩
simpa only [← ENNReal.ofReal_coe_nnreal] using hδ i x hi | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_continuous_nnreal_forall_closedBall_subset | Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (δ x) ⊆ U i`. |
exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
⟨ContinuousMap.comp ⟨Coe.coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_continuous_eNNReal_forall_closedBall_subset | Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (δ x) ⊆ U i`. |
exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i := by
rcases EMetric.exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
refine ⟨δ, hδ0, fun i x hx => ?_⟩
rw [← emetric_closedBall_nnreal]
exact hδ i x hx | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_continuous_nnreal_forall_closedBall_subset | Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
`Metric.closedBall x (δ x) ⊆ U i`. |
exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i :=
let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedBall_subset hK hU hKU hfin
⟨ContinuousMap.comp ⟨Coe.coe, NNReal.continuous_coe⟩ δ, hδ₀, hδ⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Paracompact",
"Mathlib.Topology.Instances.ENNReal.Lemmas",
"Mathlib.Analysis.Convex.PartitionOfUnity"
] | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | exists_continuous_real_forall_closedBall_subset | Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have
`Metric.closedBall x (δ x) ⊆ U i`. |
private Perfect.small_diam_aux (hC : Perfect C) (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
let D := closure (EMetric.ball x (ε / 2) ∩ C)
Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
have : x ∈ EMetric.ball x (ε / 2) := by
apply EMetric.mem_ball_self
rw [ENNReal.div_pos_iff]
exact ⟨ne_of_gt ε_pos, by simp⟩
have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
refine ⟨this.1, this.2, ?_, ?_⟩
· rw [IsClosed.closure_subset_iff hC.closed]
apply inter_subset_right
rw [EMetric.diam_closure]
apply le_trans (EMetric.diam_mono inter_subset_left)
convert EMetric.diam_ball (x := x)
rw [mul_comm, ENNReal.div_mul_cancel] <;> norm_num | theorem | Topology | [
"Mathlib.Topology.Perfect",
"Mathlib.Topology.MetricSpace.Polish",
"Mathlib.Topology.MetricSpace.CantorScheme",
"Mathlib.Topology.Metrizable.Real"
] | Mathlib/Topology/MetricSpace/Perfect.lean | Perfect.small_diam_aux | null |
Perfect.small_diam_splitting (hC : Perfect C) (hnonempty : C.Nonempty) (ε_pos : 0 < ε) :
∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
(Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by
rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
obtain ⟨x₀, hx₀⟩ := non0
obtain ⟨x₁, hx₁⟩ := non1
rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
refine
⟨closure (EMetric.ball x₀ (ε / 2) ∩ D₀), closure (EMetric.ball x₁ (ε / 2) ∩ D₁),
⟨perf0', non0', sub0'.trans sub0, diam0⟩, ⟨perf1', non1', sub1'.trans sub1, diam1⟩, ?_⟩
apply Disjoint.mono _ _ hdisj <;> assumption
open CantorScheme | theorem | Topology | [
"Mathlib.Topology.Perfect",
"Mathlib.Topology.MetricSpace.Polish",
"Mathlib.Topology.MetricSpace.CantorScheme",
"Mathlib.Topology.Metrizable.Real"
] | Mathlib/Topology/MetricSpace/Perfect.lean | Perfect.small_diam_splitting | A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. |
Perfect.exists_nat_bool_injection
(hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f := by
obtain ⟨u, -, upos', hu⟩ := exists_seq_strictAnti_tendsto' (zero_lt_one' ℝ≥0∞)
have upos := fun n => (upos' n).1
let P := Subtype fun E : Set α => Perfect E ∧ E.Nonempty
choose C0 C1 h0 h1 hdisj using
fun {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε) =>
hC.small_diam_splitting hnonempty hε
let DP : List Bool → P := fun l => by
induction l with
| nil => exact ⟨C, ⟨hC, hnonempty⟩⟩
| cons a l ih =>
cases a
· use C0 ih.property.1 ih.property.2 (upos (l.length + 1))
exact ⟨(h0 _ _ _).1, (h0 _ _ _).2.1⟩
use C1 ih.property.1 ih.property.2 (upos (l.length + 1))
exact ⟨(h1 _ _ _).1, (h1 _ _ _).2.1⟩
let D : List Bool → Set α := fun l => (DP l).val
have hanti : ClosureAntitone D := by
refine Antitone.closureAntitone ?_ fun l => (DP l).property.1.closed
intro l a
cases a
· exact (h0 _ _ _).2.2.1
exact (h1 _ _ _).2.2.1
have hdiam : VanishingDiam D := by
intro x
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu
· simp
rw [eventually_atTop]
refine ⟨1, fun m (hm : 1 ≤ m) => ?_⟩
rw [Nat.one_le_iff_ne_zero] at hm
rcases Nat.exists_eq_succ_of_ne_zero hm with ⟨n, rfl⟩
dsimp
cases x n
· convert (h0 _ _ _).2.2.2
rw [PiNat.res_length]
convert (h1 _ _ _).2.2.2
rw [PiNat.res_length]
have hdisj' : CantorScheme.Disjoint D := by
rintro l (a | a) (b | b) hab <;> try contradiction
· exact hdisj _ _ _
exact (hdisj _ _ _).symm
have hdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).1 := fun {x} => by
rw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]
apply mem_univ
refine ⟨fun x => (inducedMap D).2 ⟨x, hdom⟩, ?_, ?_, ?_⟩
· rintro y ⟨x, rfl⟩
exact map_mem ⟨_, hdom⟩ 0
· apply hdiam.map_continuous.comp
fun_prop
... | theorem | Topology | [
"Mathlib.Topology.Perfect",
"Mathlib.Topology.MetricSpace.Polish",
"Mathlib.Topology.MetricSpace.CantorScheme",
"Mathlib.Topology.Metrizable.Real"
] | Mathlib/Topology/MetricSpace/Perfect.lean | Perfect.exists_nat_bool_injection | Any nonempty perfect set in a complete metric space admits a continuous injection
from the Cantor space, `ℕ → Bool`. |
IsClosed.exists_nat_bool_injection_of_not_countable {α : Type*} [TopologicalSpace α]
[PolishSpace α] {C : Set α} (hC : IsClosed C) (hunc : ¬C.Countable) :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Function.Injective f := by
letI := TopologicalSpace.upgradeIsCompletelyMetrizable α
obtain ⟨D, hD, Dnonempty, hDC⟩ := exists_perfect_nonempty_of_isClosed_of_not_countable hC hunc
obtain ⟨f, hfD, hf⟩ := hD.exists_nat_bool_injection Dnonempty
exact ⟨f, hfD.trans hDC, hf⟩ | theorem | Topology | [
"Mathlib.Topology.Perfect",
"Mathlib.Topology.MetricSpace.Polish",
"Mathlib.Topology.MetricSpace.CantorScheme",
"Mathlib.Topology.Metrizable.Real"
] | Mathlib/Topology/MetricSpace/Perfect.lean | IsClosed.exists_nat_bool_injection_of_not_countable | Any closed uncountable subset of a Polish space admits a continuous injection
from the Cantor space `ℕ → Bool`. |
cylinder (x : ∀ n, E n) (n : ℕ) : Set (∀ n, E n) :=
{ y | ∀ i, i < n → y i = x i } | def | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder | In a product space `Π n, E n`, then `firstDiff x y` is the first index at which `x` and `y`
differ. If `x = y`, then by convention we set `firstDiff x x = 0`. -/
irreducible_def firstDiff (x y : ∀ n, E n) : ℕ :=
if h : x ≠ y then Nat.find (ne_iff.1 h) else 0
theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) :
x (firstDiff x y) ≠ y (firstDiff x y) := by
rw [firstDiff_def, dif_pos h]
classical
exact Nat.find_spec (ne_iff.1 h)
theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n := by
rw [firstDiff_def] at hn
split_ifs at hn with h
· convert Nat.find_min (ne_iff.1 h) hn
simp
· exact (not_lt_zero' hn).elim
theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x := by
classical
simp only [firstDiff_def, ne_comm]
theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z := by
by_contra! H
rw [lt_min_iff] at H
refine apply_firstDiff_ne h ?_
calc
x (firstDiff x z) = y (firstDiff x z) := apply_eq_of_lt_firstDiff H.1
_ = z (firstDiff x z) := apply_eq_of_lt_firstDiff H.2
/-! ### Cylinders -/
/-- In a product space `Π n, E n`, the cylinder set of length `n` around `x`, denoted
`cylinder x n`, is the set of sequences `y` that coincide with `x` on the first `n` symbols, i.e.,
such that `y i = x i` for all `i < n`. |
cylinder_eq_pi (x : ∀ n, E n) (n : ℕ) :
cylinder x n = Set.pi (Finset.range n : Set ℕ) fun i : ℕ => {x i} := by
ext y
simp [cylinder]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder_eq_pi | null |
cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ := by simp [cylinder_eq_pi] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder_zero | null |
cylinder_anti (x : ∀ n, E n) {m n : ℕ} (h : m ≤ n) : cylinder x n ⊆ cylinder x m :=
fun _y hy i hi => hy i (hi.trans_le h)
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder_anti | null |
mem_cylinder_iff {x y : ∀ n, E n} {n : ℕ} : y ∈ cylinder x n ↔ ∀ i < n, y i = x i :=
Iff.rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | mem_cylinder_iff | null |
self_mem_cylinder (x : ∀ n, E n) (n : ℕ) : x ∈ cylinder x n := by simp | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | self_mem_cylinder | null |
mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n := by
constructor
· intro hy
apply Subset.antisymm
· intro z hz i hi
rw [← hy i hi]
exact hz i hi
· intro z hz i hi
rw [hy i hi]
exact hz i hi
· intro h
rw [← h]
exact self_mem_cylinder _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | mem_cylinder_iff_eq | null |
mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n := by
simp [eq_comm] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | mem_cylinder_comm | null |
mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : ℕ) :
x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
constructor
· intro h
by_contra!
exact apply_firstDiff_ne hne (h _ this)
· intro hi j hj
exact apply_eq_of_lt_firstDiff (hj.trans_le hi) | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | mem_cylinder_iff_le_firstDiff | null |
mem_cylinder_firstDiff (x y : ∀ n, E n) : x ∈ cylinder y (firstDiff x y) := fun _i hi =>
apply_eq_of_lt_firstDiff hi | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | mem_cylinder_firstDiff | null |
cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn : n ≤ firstDiff x y) :
cylinder x n = cylinder y n := by
rw [← mem_cylinder_iff_eq]
intro i hi
exact apply_eq_of_lt_firstDiff (hi.trans_le hn) | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder_eq_cylinder_of_le_firstDiff | null |
iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
⋃ k, cylinder (update x n k) (n + 1) = cylinder x n := by
ext y
simp only [mem_cylinder_iff, mem_iUnion]
constructor
· rintro ⟨k, hk⟩ i hi
simpa [hi.ne] using hk i (Nat.lt_succ_of_lt hi)
· intro H
refine ⟨y n, fun i hi => ?_⟩
rcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl)
· simp [H i h'i, h'i.ne]
· simp | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | iUnion_cylinder_update | null |
update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y ∈ cylinder x n :=
mem_cylinder_iff.2 fun i hi => by simp [hi.ne] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | update_mem_cylinder | null |
res (x : ℕ → α) : ℕ → List α
| 0 => nil
| Nat.succ n => x n :: res x n
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res | In the case where `E` has constant value `α`,
the cylinder `cylinder x n` can be identified with the element of `List α`
consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
We call this list `res x n`, the restriction of `x` to `n`. |
res_zero (x : ℕ → α) : res x 0 = @nil α :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res_zero | null |
res_succ (x : ℕ → α) (n : ℕ) : res x n.succ = x n :: res x n :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res_succ | null |
res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by induction n <;> simp [*] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res_length | null |
res_eq_res {x y : ℕ → α} {n : ℕ} :
res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m := by
constructor <;> intro h
· induction n with
| zero => simp
| succ n ih =>
intro m hm
rw [Nat.lt_succ_iff_lt_or_eq] at hm
simp only [res_succ, cons.injEq] at h
rcases hm with hm | hm
· exact ih h.2 hm
rw [hm]
exact h.1
· induction n with
| zero => simp
| succ n ih =>
simp only [res_succ, cons.injEq]
refine ⟨h (Nat.lt_succ_self _), ih fun m hm => ?_⟩
exact h (hm.trans (Nat.lt_succ_self _)) | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res_eq_res | The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`. |
res_injective : Injective (@res α) := by
intro x y h
ext n
apply res_eq_res.mp _ (Nat.lt_succ_self _)
rw [h] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | res_injective | null |
cylinder_eq_res (x : ℕ → α) (n : ℕ) :
cylinder x n = { y | res y n = res x n } := by
ext y
dsimp [cylinder]
rw [res_eq_res] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | cylinder_eq_res | `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`. |
protected dist : Dist (∀ n, E n) :=
⟨fun x y => if x ≠ y then (1 / 2 : ℝ) ^ firstDiff x y else 0⟩
attribute [local instance] PiNat.dist | def | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist | The distance function on a product space `Π n, E n`, given by `dist x y = (1/2)^n` where `n` is
the first index at which `x` and `y` differ. |
dist_eq_of_ne {x y : ∀ n, E n} (h : x ≠ y) : dist x y = (1 / 2 : ℝ) ^ firstDiff x y := by
simp [dist, h] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_eq_of_ne | null |
protected dist_self (x : ∀ n, E n) : dist x x = 0 := by simp [dist] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_self | null |
protected dist_comm (x y : ∀ n, E n) : dist x y = dist y x := by
classical
simp [dist, @eq_comm _ x y, firstDiff_comm] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_comm | null |
protected dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y := by
rcases eq_or_ne x y with (rfl | h)
· simp [dist]
· simp [dist, h] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_nonneg | null |
dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) := by
rcases eq_or_ne x z with (rfl | hxz)
· simp [PiNat.dist_self x, PiNat.dist_nonneg]
rcases eq_or_ne x y with (rfl | hxy)
· simp
rcases eq_or_ne y z with (rfl | hyz)
· simp
simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv₀, one_div, inv_pow, zero_lt_two, Ne,
not_false_iff, le_max_iff, pow_le_pow_iff_right₀, one_lt_two, pow_pos,
min_le_iff.1 (min_firstDiff_le x y z hxz)] | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_triangle_nonarch | null |
protected dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + dist y z :=
calc
dist x z ≤ max (dist x y) (dist y z) := dist_triangle_nonarch x y z
_ ≤ dist x y + dist y z := max_le_add_of_nonneg (PiNat.dist_nonneg _ _) (PiNat.dist_nonneg _ _) | theorem | Topology | [
"Mathlib.Topology.Algebra.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/PiNat.lean | dist_triangle | null |
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