statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : LipschitzWith K f) (x y) :
f x ≤ f y + K * dist x y | sub_le_iff_le_add'.1 <| le_trans (le_abs_self _) <| h.dist_le_mul x y | theorem | LipschitzWith.le_add_mul | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
LipschitzWith K f ↔ ∀ x y, f x ≤ f y + K * dist x y | ⟨LipschitzWith.le_add_mul, LipschitzWith.of_le_add_mul K⟩ | theorem | LipschitzWith.iff_le_add_mul | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.of_le_add_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nndist_le (hf : LipschitzWith K f) (x y : α) : nndist (f x) (f y) ≤ K * nndist x y | hf.dist_le_mul x y | theorem | LipschitzWith.nndist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_mul_of_le (hf : LipschitzWith K f) (hr : dist x y ≤ r) : dist (f x) (f y) ≤ K * r | (hf.dist_le_mul x y).trans <| by gcongr | theorem | LipschitzWith.dist_le_mul_of_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapsTo_closedBall (hf : LipschitzWith K f) (x : α) (r : ℝ) :
MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) (K * r)) | fun _y hy =>
hf.dist_le_mul_of_le hy | theorem | LipschitzWith.mapsTo_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"Metric.closedBall"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_lt_mul_of_lt (hf : LipschitzWith K f) (hK : K ≠ 0) (hr : dist x y < r) :
dist (f x) (f y) < K * r | (hf.dist_le_mul x y).trans_lt <| by gcongr | theorem | LipschitzWith.dist_lt_mul_of_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapsTo_ball (hf : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ) :
MapsTo f (Metric.ball x r) (Metric.ball (f x) (K * r)) | fun _y hy =>
hf.dist_lt_mul_of_lt hK hy | theorem | LipschitzWith.mapsTo_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"Metric.ball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)⟩ | def | LipschitzWith.toLocallyBoundedMap | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LocallyBoundedMap",
"LocallyBoundedMap.ofMapBounded"
] | A Lipschitz continuous map is a locally bounded map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_toLocallyBoundedMap (hf : LipschitzWith K f) : ⇑(hf.toLocallyBoundedMap f) = f | rfl | theorem | LipschitzWith.coe_toLocallyBoundedMap | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_cobounded_le (hf : LipschitzWith K f) :
comap f (Bornology.cobounded β) ≤ Bornology.cobounded α | (hf.toLocallyBoundedMap f).2 | theorem | LipschitzWith.comap_cobounded_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_image (hf : LipschitzWith K f) {s : Set α} (hs : IsBounded s) :
IsBounded (f '' s) | hs.image (toLocallyBoundedMap f hf) | theorem | LipschitzWith.isBounded_image | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"Set"
] | The image of a bounded set under a Lipschitz map is bounded. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | LipschitzWith.diam_image_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"Metric.diam",
"Metric.diam_le_of_forall_dist_le",
"Metric.diam_nonneg",
"Metric.dist_le_diam_of_mem",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_left (y : α) : LipschitzWith 1 (dist · y) | LipschitzWith.mk_one fun _ _ => dist_dist_dist_le_left _ _ _ | theorem | LipschitzWith.dist_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.mk_one",
"dist_dist_dist_le_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_right (x : α) : LipschitzWith 1 (dist x) | LipschitzWith.of_le_add fun _ _ => dist_triangle_right _ _ _ | theorem | LipschitzWith.dist_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.of_le_add",
"dist_triangle_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist : LipschitzWith 2 (Function.uncurry <| @dist α _) | by
rw [← one_add_one_eq_two]
exact LipschitzWith.uncurry LipschitzWith.dist_left LipschitzWith.dist_right | theorem | LipschitzWith.dist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.dist_left",
"LipschitzWith.dist_right",
"LipschitzWith.uncurry",
"one_add_one_eq_two"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.dist_iterate_succ_le_geometric | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"NNReal.coe_pow",
"mul_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | lipschitzWith_max | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.of_le_add",
"abs_max_sub_max_le_max",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | lipschitzWith_min | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.of_le_add",
"abs_min_sub_min_le_max",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | Real.lipschitzWith_toNNReal | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"NNReal.coe_one",
"Real.dist_eq",
"Real.toNNReal",
"dist_prod_same_right",
"lipschitzWith_max",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.max | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.min | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
max_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max (f x) a | by
simpa using hf.max (LipschitzWith.const a) | theorem | LipschitzWith.max_const | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.const_max | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
min_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => min (f x) a | by
simpa using hf.min (LipschitzWith.const a) | theorem | LipschitzWith.min_const | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"LipschitzWith.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.const_min | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith",
"min_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
projIcc {a b : ℝ} (h : a ≤ b) : LipschitzWith 1 (projIcc a b h) | ((LipschitzWith.id.const_min _).const_max _).subtype_mk _ | theorem | LipschitzWith.projIcc | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzWith.properSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"IsProperMap",
"LipschitzWith",
"Metric.isClosed_closedBall",
"ProperSpace",
"PseudoMetricSpace"
] | The preimage of a proper space under a Lipschitz proper map is proper. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | LipschitzOnWith.of_dist_le' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"Real.le_coe_toNNReal",
"Real.toNNReal",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzOnWith.mk_one | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"NNReal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzOnWith.of_le_add_mul' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"LipschitzOnWith.of_dist_le'",
"Real.toNNReal",
"dist_comm"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
doesn't assume `0≤K`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | LipschitzOnWith.of_le_add_mul | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"LipschitzOnWith.of_le_add_mul'",
"Real.toNNReal_coe"
] | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
assumes `0≤K`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | LipschitzOnWith.of_le_add | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"LipschitzOnWith.of_le_add_mul",
"NNReal.coe_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzOnWith.le_add_mul | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzOnWith.iff_le_add_mul | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"LipschitzOnWith.of_le_add_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | LipschitzOnWith.isBounded_image2 | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"Bornology.IsBounded",
"ENNReal.coe_ne_top",
"ENNReal.mul_ne_top",
"LipschitzOnWith",
"Set",
"Set.image2",
"ne_top_of_le_ne_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
min (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
LocallyLipschitz (fun x => min (f x) (g x)) | lipschitzWith_min.locallyLipschitz.comp (hf.prodMk hg) | lemma | LocallyLipschitz.min | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz"
] | The minimum of locally Lipschitz functions is locally Lipschitz. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
max (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
LocallyLipschitz (fun x => max (f x) (g x)) | lipschitzWith_max.locallyLipschitz.comp (hf.prodMk hg) | lemma | LocallyLipschitz.max | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz"
] | The maximum of locally Lipschitz functions is locally Lipschitz. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
max_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max (f x) a | hf.max (LocallyLipschitz.const a) | theorem | LocallyLipschitz.max_const | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz",
"LocallyLipschitz.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
const_max (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max a (f x) | by
simpa [max_comm] using (hf.max_const a) | theorem | LocallyLipschitz.const_max | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
min_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min (f x) a | hf.min (LocallyLipschitz.const a) | theorem | LocallyLipschitz.min_const | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz",
"LocallyLipschitz.const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
const_min (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min a (f x) | by
simpa [min_comm] using (hf.min_const a) | theorem | LocallyLipschitz.const_min | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LocallyLipschitz",
"min_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 functi... | theorem | LipschitzOnWith.extend_real | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"LipschitzOnWith",
"LipschitzWith",
"LipschitzWith.const",
"LipschitzWith.of_le_add_mul",
"Set",
"Subtype.coe_mk",
"ciInf_le",
"dist_self",
"dist_triangle",
"dist_triangle_left",
"le_antisymm",
"le_ciInf"
] | A function `f : α → ℝ` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz extension
to the whole space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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
r... | theorem | LipschitzOnWith.extend_pi | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Lipschitz.lean | [] | [
"Fintype",
"LipschitzOnWith",
"LipschitzWith",
"Set",
"dist_le_pi_dist",
"dist_nonneg",
"dist_pi_le_iff",
"trans"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsSeparated (ε : ℝ≥0∞) (s : Set X) : Prop | s.Pairwise (ε < edist · ·) | def | Metric.IsSeparated | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set"
] | A set `s` is `ε`-separated if the extended distance between any two distinct
elements is strictly greater than `ε`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isSeparated_iff_setRelIsSeparated :
IsSeparated ε s ↔ SetRel.IsSeparated {(x, y) | edist x y ≤ ε} s | by
simp [IsSeparated, SetRel.IsSeparated] | lemma | Metric.isSeparated_iff_setRelIsSeparated | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"SetRel.IsSeparated"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.empty : IsSeparated ε (∅ : Set X) | pairwise_empty _ | lemma | Metric.IsSeparated.empty | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.singleton : IsSeparated ε {x} | pairwise_singleton .. | lemma | Metric.IsSeparated.singleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.of_subsingleton (hs : s.Subsingleton) : IsSeparated ε s | hs.pairwise _
alias _root_.Set.Subsingleton.isSeparated := IsSeparated.of_subsingleton | lemma | Metric.IsSeparated.of_subsingleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.anti (hεδ : ε ≤ δ) (hs : IsSeparated δ s) : IsSeparated ε s | hs.mono' fun _ _ ↦ hεδ.trans_lt | lemma | Metric.IsSeparated.anti | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.subset (hst : s ⊆ t) (hs : IsSeparated ε t) : IsSeparated ε s | hs.mono hst | lemma | Metric.IsSeparated.subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.isSeparated_insert | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | lemma | Metric.isSeparated_insert_of_notMem | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.IsSeparated.insert | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isSeparated_zero {X : Type*} [EMetricSpace X] (s : Set X) : IsSeparated 0 s | by
simp [IsSeparated, Set.Pairwise] | lemma | Metric.isSeparated_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"EMetricSpace",
"Set",
"Set.Pairwise"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsSeparated.image_antilipschitz {ε K₁ : ℝ≥0} {f : X → Y}
(hs : IsSeparated ε s) (hf : AntilipschitzWith K₁ f) (hK₁ : 0 < K₁) :
IsSeparated ↑(ε / K₁) (f '' s) | by
rintro x' ⟨x, hx, rfl⟩ y' ⟨y, hy, rfl⟩ hne
have hmul : (↑ε : ℝ≥0∞) < edist (f x) (f y) * ↑K₁ :=
lt_of_lt_of_le (hs hx hy (by grind)) (by rw [mul_comm]; exact hf x y)
exact ENNReal.coe_div hK₁.ne' ▸ ENNReal.div_lt_of_lt_mul hmul | lemma | Metric.IsSeparated.image_antilipschitz | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"AntilipschitzWith",
"ENNReal.coe_div",
"ENNReal.div_lt_of_lt_mul",
"lt_of_lt_of_le",
"mul_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AreSeparated (s t : Set X) | ∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y | def | Metric.AreSeparated | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.AreSeparated.symm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comm : AreSeparated s t ↔ AreSeparated t s | ⟨symm, symm⟩ | theorem | Metric.AreSeparated.comm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
empty_left (s : Set X) : AreSeparated ∅ s | ⟨1, one_ne_zero, fun _x => False.elim⟩ | theorem | Metric.AreSeparated.empty_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
empty_right (s : Set X) : AreSeparated s ∅ | (empty_left s).symm | theorem | Metric.AreSeparated.empty_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.disjoint | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subset_compl_right (h : AreSeparated s t) : s ⊆ tᶜ | fun _ hs ht =>
h.disjoint.le_bot ⟨hs, ht⟩ | theorem | Metric.AreSeparated.subset_compl_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.mono | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mono_left {s'} (h' : AreSeparated s' t) (hs : s ⊆ s') : AreSeparated s t | h'.mono hs Subset.rfl | theorem | Metric.AreSeparated.mono_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mono_right {t'} (h' : AreSeparated s t') (ht : t ⊆ t') : AreSeparated s t | h'.mono Subset.rfl ht | theorem | Metric.AreSeparated.mono_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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) | theorem | Metric.AreSeparated.union_left | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"lt_min",
"min_le_left",
"min_le_right",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.union_left_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
union_right {t'} (h : AreSeparated s t) (h' : AreSeparated s t') :
AreSeparated s (t ∪ t') | (h.symm.union_left h'.symm).symm | theorem | Metric.AreSeparated.union_right | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.union_right_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.finite_iUnion_left_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set",
"Set.Finite.induction_on"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Metric.AreSeparated.finite_iUnion_right_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Set",
"comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Metric.AreSeparated.finset_iUnion_left_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Finset",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.AreSeparated.finset_iUnion_right_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/MetricSeparated.lean | [] | [
"Finset",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds_zero_forall_closedEBall_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 → closedEBall p.2 p.1 ⊆ U i | by
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedEBall 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_iIn... | theorem | Metric.eventually_nhds_zero_forall_closedEBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"IsClosed",
"IsOpen",
"LocallyFinite",
"add_tsub_cancel_of_le",
"le_trans",
"lt_trans",
"not_imp_not",
"tsub_le_tsub_left"
] | 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 `Metric.closedEBall y r ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_forall_closedEBall_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 | closedEBall y r ⊆ U i } | by
have := (ENNReal.continuous_ofReal.tendsto' 0 0 ENNReal.ofReal_zero).eventually
(eventually_nhds_zero_forall_closedEBall_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 | Metric.exists_forall_closedEBall_subset_aux₁ | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"ENNReal.ofReal",
"ENNReal.ofReal_zero",
"IsClosed",
"IsOpen",
"LocallyFinite"
] | Auxiliary lemma for `exists_continuous_real_forall_closedEBall_subset`
and its smooth counterpart. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_forall_closedEBall_subset_aux₂ (y : X) :
Convex ℝ
(Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedEBall 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_closedEBall_subset y (U i) | theorem | Metric.exists_forall_closedEBall_subset_aux₂ | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"Convex",
"ENNReal.ofReal",
"convex_Ioi"
] | Auxiliary lemma for `exists_continuous_real_forall_closedEBall_subset`
and its smooth counterpart. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_continuous_real_forall_closedEBall_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, closedEBall x (ENNReal.ofReal <| δ x) ⊆ U i | by
simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_comm ι X] using!
exists_continuous_forall_mem_convex_of_local_const exists_forall_closedEBall_subset_aux₂
(exists_forall_closedEBall_subset_aux₁ hK hU hKU hfin) | theorem | Metric.exists_continuous_real_forall_closedEBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"ENNReal.ofReal",
"IsClosed",
"IsOpen",
"LocallyFinite",
"exists_continuous_forall_mem_convex_of_local_const"
] | 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 `Metric.closedEBall x (ENNReal.o... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_continuous_nnreal_forall_closedEBall_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, closedEBall x (δ x) ⊆ U i | by
rcases exists_continuous_real_forall_closedEBall_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 | Metric.exists_continuous_nnreal_forall_closedEBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"ENNReal.ofReal_coe_nnreal",
"IsClosed",
"IsOpen",
"LocallyFinite"
] | 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 `Metric.closedEBall x (δ x) ⊆ ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_continuous_ennreal_forall_closedEBall_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, closedEBall x (δ x) ⊆ U i | let ⟨δ, hδ₀, hδ⟩ := exists_continuous_nnreal_forall_closedEBall_subset hK hU hKU hfin
⟨ContinuousMap.comp ⟨Coe.coe, ENNReal.continuous_coe⟩ δ, fun x => ENNReal.coe_pos.2 (hδ₀ x), hδ⟩ | theorem | Metric.exists_continuous_ennreal_forall_closedEBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"IsClosed",
"IsOpen",
"LocallyFinite"
] | 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 `Metric.closedEBall x (δ x) ⊆... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 Metric.exists_continuous_nnreal_forall_closedEBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
refine ⟨δ, hδ0, fun i x hx => ?_⟩
rw [← Metric.closedEBall_coe]
exact hδ i x hx | theorem | Metric.exists_continuous_nnreal_forall_closedBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"IsClosed",
"IsOpen",
"LocallyFinite",
"Metric.closedEBall_coe",
"Metric.exists_continuous_nnreal_forall_closedEBall_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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.exists_continuous_real_forall_closedBall_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | [] | [
"IsClosed",
"IsOpen",
"LocallyFinite"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Perfect.small_diam_aux (hC : Perfect C) (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
let D | closure (Metric.eball x (ε / 2) ∩ C)
Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ Metric.ediam D ≤ ε := by
have : x ∈ Metric.eball x (ε / 2) := by
apply Metric.mem_eball_self
rw [ENNReal.div_pos_iff]
exact ⟨ne_of_gt ε_pos, by simp⟩
have := hC.closure_nhds_inter x xC this Metric.isOpen_eball
refine ⟨this.1, th... | theorem | Perfect.small_diam_aux | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Perfect.lean | [] | [
"ENNReal.div_mul_cancel",
"ENNReal.div_pos_iff",
"IsClosed.closure_subset_iff",
"Metric.eball",
"Metric.ediam",
"Metric.ediam_closure",
"Metric.ediam_eball_le",
"Metric.ediam_mono",
"Metric.isOpen_eball",
"Metric.mem_eball_self",
"Perfect",
"closure",
"le_trans",
"mul_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Perfect.small_diam_splitting (hC : Perfect C) (hnonempty : C.Nonempty) (ε_pos : 0 < ε) :
∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ Metric.ediam C₀ ≤ ε) ∧
(Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ Metric.ediam 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
⟨clos... | theorem | Perfect.small_diam_splitting | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Perfect.lean | [] | [
"Disjoint",
"Disjoint.mono",
"Metric.eball",
"Metric.ediam",
"Perfect",
"Set",
"closure"
] | A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_s... | theorem | Perfect.exists_nat_bool_injection | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Perfect.lean | [] | [
"CantorScheme.Disjoint",
"CompleteSpace",
"Continuous",
"Perfect",
"PiNat.res_length",
"Set",
"Subtype.val_inj",
"exists_seq_strictAnti_tendsto'",
"symm",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'",
"zero_lt_one'"
] | Any nonempty perfect set in a complete metric space admits a continuous injection
from the Cantor space, `ℕ → Bool`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | IsClosed.exists_nat_bool_injection_of_not_countable | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Perfect.lean | [] | [
"Continuous",
"IsClosed",
"PolishSpace",
"Set",
"TopologicalSpace",
"TopologicalSpace.upgradeIsCompletelyMetrizable",
"exists_perfect_nonempty_of_isClosed_of_not_countable"
] | Any closed uncountable subset of a Polish space admits a continuous injection
from the Cantor space `ℕ → Bool`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | PiNat.apply_firstDiff_ne | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [
"Nat.find_spec"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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
aesop | theorem | PiNat.apply_eq_of_lt_firstDiff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x | by
classical
simp only [firstDiff_def, ne_comm] | theorem | PiNat.firstDiff_comm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | PiNat.min_firstDiff_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [
"lt_min_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cylinder (x : ∀ n, E n) (n : ℕ) : Set (∀ n, E n) | { y | ∀ i, i < n → y i = x i } | def | PiNat.cylinder | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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] | theorem | PiNat.cylinder_eq_pi | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [
"Finset.range",
"Set",
"Set.pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ | by simp [cylinder_eq_pi] | theorem | PiNat.cylinder_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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) | theorem | PiNat.cylinder_anti | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_cylinder_iff {x y : ∀ n, E n} {n : ℕ} : y ∈ cylinder x n ↔ ∀ i < n, y i = x i | Iff.rfl | theorem | PiNat.mem_cylinder_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_mem_cylinder (x : ∀ n, E n) (n : ℕ) : x ∈ cylinder x n | by simp | theorem | PiNat.self_mem_cylinder | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | PiNat.mem_cylinder_iff_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n | by
simp [eq_comm] | theorem | PiNat.mem_cylinder_comm | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | PiNat.mem_cylinder_iff_le_firstDiff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_cylinder_firstDiff (x y : ∀ n, E n) : x ∈ cylinder y (firstDiff x y) | fun _i hi =>
apply_eq_of_lt_firstDiff hi | theorem | PiNat.mem_cylinder_firstDiff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | PiNat.cylinder_eq_cylinder_of_le_firstDiff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/PiNat.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.