fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
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| filename
stringlengths 20
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| symbolic_name
stringlengths 1
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| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y := rfl
@[to_additive (attr := simp)]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
dist_unop
| null |
dist_op (x y : α) : dist (op x) (op y) = dist x y := rfl
@[to_additive (attr := simp)]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
dist_op
| null |
nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y := rfl
@[to_additive (attr := simp)]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
nndist_unop
| null |
nndist_op (x y : α) : nndist (op x) (op y) = nndist x y := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
nndist_op
| null |
NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.dist_eq
| null |
NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) :=
eq_of_forall_ge_iff fun _ => by
simp only [max_le_iff, tsub_le_iff_right (α := ℝ≥0)]
simp only [← NNReal.coe_le_coe, coe_nndist, dist_eq, abs_sub_le_iff,
tsub_le_iff_right, NNReal.coe_add]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.nndist_eq
| null |
NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by
simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le']
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.nndist_zero_eq_val
| null |
NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z := by
rw [nndist_comm]
exact NNReal.nndist_zero_eq_val z
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.nndist_zero_eq_val'
| null |
NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b := by
suffices (a : ℝ) ≤ (b : ℝ) + dist a b by
rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]
rw [← sub_le_iff_le_add']
exact le_of_abs_le (dist_eq a b).ge
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.le_add_nndist
| null |
NNReal.ball_zero_eq_Ico' (c : ℝ≥0) :
Metric.ball (0 : ℝ≥0) c.toReal = Set.Ico 0 c := by ext x; simp
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.ball_zero_eq_Ico'
| null |
NNReal.ball_zero_eq_Ico (c : ℝ) :
Metric.ball (0 : ℝ≥0) c = Set.Ico 0 c.toNNReal := by
by_cases c_pos : 0 < c
· convert NNReal.ball_zero_eq_Ico' ⟨c, c_pos.le⟩
simp [Real.toNNReal, c_pos.le]
simp [not_lt.mp c_pos]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.ball_zero_eq_Ico
| null |
NNReal.closedBall_zero_eq_Icc' (c : ℝ≥0) :
Metric.closedBall (0 : ℝ≥0) c.toReal = Set.Icc 0 c := by ext x; simp
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.closedBall_zero_eq_Icc'
| null |
NNReal.closedBall_zero_eq_Icc {c : ℝ} (c_nn : 0 ≤ c) :
Metric.closedBall (0 : ℝ≥0) c = Set.Icc 0 c.toNNReal := by
convert NNReal.closedBall_zero_eq_Icc' ⟨c, c_nn⟩
simp [Real.toNNReal, c_nn]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
NNReal.closedBall_zero_eq_Icc
| null |
dist_eq (x y : ULift β) : dist x y = dist x.down y.down := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
dist_eq
| null |
nndist_eq (x y : ULift β) : nndist x y = nndist x.down y.down := rfl
@[simp] lemma dist_up_up (x y : β) : dist (ULift.up x) (ULift.up y) = dist x y := rfl
@[simp] lemma nndist_up_up (x y : β) : nndist (ULift.up x) (ULift.up y) = nndist x y := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
nndist_eq
| null |
Prod.pseudoMetricSpaceMax : PseudoMetricSpace (α × β) :=
let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
(fun x y : α × β => dist x.1 y.1 ⊔ dist x.2 y.2)
(fun _ _ => (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).ne) fun x y => by
simp only [dist_edist, ← ENNReal.toReal_max (edist_ne_top _ _) (edist_ne_top _ _),
Prod.edist_eq]
i.replaceBornology fun s => by
simp only [← isBounded_image_fst_and_snd, isBounded_iff_eventually, forall_mem_image, ←
eventually_and, ← forall_and, ← max_le_iff]
rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Prod.pseudoMetricSpaceMax
| null |
Prod.dist_eq {x y : α × β} : dist x y = max (dist x.1 y.1) (dist x.2 y.2) := rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Prod.dist_eq
| null |
dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ := by
simp [Prod.dist_eq]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
dist_prod_same_left
| null |
dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ := by
simp [Prod.dist_eq]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
dist_prod_same_right
| null |
ball_prod_same (x : α) (y : β) (r : ℝ) : ball x r ×ˢ ball y r = ball (x, y) r :=
ext fun z => by simp [Prod.dist_eq]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
ball_prod_same
| null |
closedBall_prod_same (x : α) (y : β) (r : ℝ) :
closedBall x r ×ˢ closedBall y r = closedBall (x, y) r := ext fun z => by simp [Prod.dist_eq]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
closedBall_prod_same
| null |
sphere_prod (x : α × β) (r : ℝ) :
sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r := by
obtain hr | rfl | hr := lt_trichotomy r 0
· simp [hr]
· cases x
simp_rw [← closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]
· ext ⟨x', y'⟩
simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,
max_eq_iff]
refine or_congr (and_congr_right ?_) (and_comm.trans (and_congr_left ?_))
all_goals rintro rfl; rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
sphere_prod
| null |
uniformContinuous_dist : UniformContinuous fun p : α × α => dist p.1 p.2 :=
Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨ε / 2, half_pos ε0, fun {a b} h =>
calc dist (dist a.1 a.2) (dist b.1 b.2) ≤ dist a.1 b.1 + dist a.2 b.2 :=
dist_dist_dist_le _ _ _ _
_ ≤ dist a b + dist a b := add_le_add (le_max_left _ _) (le_max_right _ _)
_ < ε / 2 + ε / 2 := add_lt_add h h
_ = ε := add_halves ε⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
uniformContinuous_dist
| null |
protected UniformContinuous.dist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun b => dist (f b) (g b) :=
uniformContinuous_dist.comp (hf.prodMk hg)
@[continuity]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
UniformContinuous.dist
| null |
continuous_dist : Continuous fun p : α × α ↦ dist p.1 p.2 := uniformContinuous_dist.continuous
@[continuity, fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
continuous_dist
| null |
protected Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => dist (f b) (g b) :=
continuous_dist.comp₂ hf hg
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Continuous.dist
| null |
protected Filter.Tendsto.dist {f g : β → α} {x : Filter β} {a b : α}
(hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
Tendsto (fun x => dist (f x) (g x)) x (𝓝 (dist a b)) :=
(continuous_dist.tendsto (a, b)).comp (hf.prodMk_nhds hg)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Filter.Tendsto.dist
| null |
continuous_iff_continuous_dist [TopologicalSpace β] {f : β → α} :
Continuous f ↔ Continuous fun x : β × β => dist (f x.1) (f x.2) :=
⟨fun h => h.fst'.dist h.snd', fun h =>
continuous_iff_continuousAt.2 fun _ => tendsto_iff_dist_tendsto_zero.2 <|
(h.comp (.prodMk_left _)).tendsto' _ _ <| dist_self _⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
continuous_iff_continuous_dist
| null |
uniformContinuous_nndist : UniformContinuous fun p : α × α => nndist p.1 p.2 :=
uniformContinuous_dist.subtype_mk _
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
uniformContinuous_nndist
| null |
protected UniformContinuous.nndist [UniformSpace β] {f g : β → α} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous fun b => nndist (f b) (g b) :=
uniformContinuous_nndist.comp (hf.prodMk hg)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
UniformContinuous.nndist
| null |
continuous_nndist : Continuous fun p : α × α => nndist p.1 p.2 :=
uniformContinuous_nndist.continuous
@[fun_prop]
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
continuous_nndist
| null |
protected Continuous.nndist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => nndist (f b) (g b) :=
continuous_nndist.comp₂ hf hg
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Continuous.nndist
| null |
protected Filter.Tendsto.nndist {f g : β → α} {x : Filter β} {a b : α}
(hf : Tendsto f x (𝓝 a)) (hg : Tendsto g x (𝓝 b)) :
Tendsto (fun x => nndist (f x) (g x)) x (𝓝 (nndist a b)) :=
(continuous_nndist.tendsto (a, b)).comp (hf.prodMk_nhds hg)
|
lemma
|
Topology
|
[
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.Pseudo.Defs",
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] |
Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
|
Filter.Tendsto.nndist
| null |
UniformSpace.ofDist_aux (ε : ℝ) (hε : 0 < ε) : ∃ δ > (0 : ℝ), ∀ x < δ, ∀ y < δ, x + y < ε :=
⟨ε / 2, half_pos hε, fun _x hx _y hy => add_halves ε ▸ add_lt_add hx hy⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
UniformSpace.ofDist_aux
| null |
UniformSpace.ofDist (dist : α → α → ℝ) (dist_self : ∀ x : α, dist x x = 0)
(dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : UniformSpace α :=
.ofFun dist dist_self dist_comm dist_triangle ofDist_aux
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
UniformSpace.ofDist
|
Construct a uniform structure from a distance function and metric space axioms
|
Bornology.ofDist {α : Type*} (dist : α → α → ℝ) (dist_comm : ∀ x y, dist x y = dist y x)
(dist_triangle : ∀ x y z, dist x z ≤ dist x y + dist y z) : Bornology α :=
Bornology.ofBounded { s : Set α | ∃ C, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C }
⟨0, fun _ hx _ => hx.elim⟩ (fun _ ⟨c, hc⟩ _ h => ⟨c, fun _ hx _ hy => hc (h hx) (h hy)⟩)
(fun s hs t ht => by
rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
· rwa [empty_union]
rcases t.eq_empty_or_nonempty with rfl | ⟨y, hy⟩
· rwa [union_empty]
rsuffices ⟨C, hC⟩ : ∃ C, ∀ z ∈ s ∪ t, dist x z ≤ C
· refine ⟨C + C, fun a ha b hb => (dist_triangle a x b).trans ?_⟩
simpa only [dist_comm] using add_le_add (hC _ ha) (hC _ hb)
rcases hs with ⟨Cs, hs⟩; rcases ht with ⟨Ct, ht⟩
refine ⟨max Cs (dist x y + Ct), fun z hz => hz.elim
(fun hz => (hs hx hz).trans (le_max_left _ _))
(fun hz => (dist_triangle x y z).trans <|
(add_le_add le_rfl (ht hy hz)).trans (le_max_right _ _))⟩)
fun z => ⟨dist z z, forall_eq.2 <| forall_eq.2 le_rfl⟩
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Bornology.ofDist
|
Construct a bornology from a distance function and metric space axioms.
|
@[ext]
Dist (α : Type*) where
/-- Distance between two points -/
dist : α → α → ℝ
export Dist (dist)
|
class
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Dist
|
The distance function (given an ambient metric space on `α`), which returns
a nonnegative real number `dist x y` given `x y : α`.
|
private dist_nonneg' {α} {x y : α} (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z) : 0 ≤ dist x y :=
have : 0 ≤ 2 * dist x y :=
calc 0 = dist x x := (dist_self _).symm
_ ≤ dist x y + dist y x := dist_triangle _ _ _
_ = 2 * dist x y := by rw [two_mul, dist_comm]
nonneg_of_mul_nonneg_right this two_pos
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_nonneg'
|
This is an internal lemma used inside the default of `PseudoMetricSpace.edist`.
|
PseudoMetricSpace (α : Type u) : Type u extends Dist α where
dist_self : ∀ x : α, dist x x = 0
dist_comm : ∀ x y : α, dist x y = dist y x
dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z
/-- Extended distance between two points -/
edist : α → α → ℝ≥0∞ := fun x y => ENNReal.ofNNReal ⟨dist x y, dist_nonneg' _ ‹_› ‹_› ‹_›⟩
edist_dist : ∀ x y : α, edist x y = ENNReal.ofReal (dist x y) := by
intro x y; exact ENNReal.coe_nnreal_eq _
toUniformSpace : UniformSpace α := .ofDist dist dist_self dist_comm dist_triangle
uniformity_dist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | dist p.1 p.2 < ε } := by intros; rfl
toBornology : Bornology α := Bornology.ofDist dist dist_comm dist_triangle
cobounded_sets : (Bornology.cobounded α).sets =
{ s | ∃ C : ℝ, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C } := by intros; rfl
|
class
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace
|
A pseudometric space is a type endowed with a `ℝ`-valued distance `dist` satisfying
reflexivity `dist x x = 0`, commutativity `dist x y = dist y x`, and the triangle inequality
`dist x z ≤ dist x y + dist y z`.
Note that we do not require `dist x y = 0 → x = y`. See metric spaces (`MetricSpace`) for the
similar class with that stronger assumption.
Any pseudometric space is a topological space and a uniform space (see `TopologicalSpace`,
`UniformSpace`), where the topology and uniformity come from the metric.
Note that a T1 pseudometric space is just a metric space.
We make the uniformity/topology part of the data instead of deriving it from the metric. This e.g.
ensures that we do not get a diamond when doing
`[PseudoMetricSpace α] [PseudoMetricSpace β] : TopologicalSpace (α × β)`:
The product metric and product topology agree, but not definitionally so.
See Note [forgetful inheritance].
|
@[ext]
PseudoMetricSpace.ext {α : Type*} {m m' : PseudoMetricSpace α}
(h : m.toDist = m'.toDist) : m = m' := by
let d := m.toDist
obtain ⟨_, _, _, _, hed, _, hU, _, hB⟩ := m
let d' := m'.toDist
obtain ⟨_, _, _, _, hed', _, hU', _, hB'⟩ := m'
obtain rfl : d = d' := h
congr
· ext x y : 2
rw [hed, hed']
· exact UniformSpace.ext (hU.trans hU'.symm)
· ext : 2
rw [← Filter.mem_sets, ← Filter.mem_sets, hB, hB']
variable [PseudoMetricSpace α]
attribute [instance] PseudoMetricSpace.toUniformSpace PseudoMetricSpace.toBornology
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.ext
|
Two pseudo metric space structures with the same distance function coincide.
|
PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ)
(dist_self : ∀ x : α, dist x x = 0) (dist_comm : ∀ x y : α, dist x y = dist y x)
(dist_triangle : ∀ x y z : α, dist x z ≤ dist x y + dist y z)
(H : ∀ s : Set α, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s) :
PseudoMetricSpace α :=
{ dist := dist
dist_self := dist_self
dist_comm := dist_comm
dist_triangle := dist_triangle
toUniformSpace :=
(UniformSpace.ofDist dist dist_self dist_comm dist_triangle).replaceTopology <|
TopologicalSpace.ext_iff.2 fun s ↦ (H s).trans <| forall₂_congr fun x _ ↦
((UniformSpace.hasBasis_ofFun (exists_gt (0 : ℝ)) dist dist_self dist_comm dist_triangle
UniformSpace.ofDist_aux).comap (Prod.mk x)).mem_iff.symm
uniformity_dist := rfl
toBornology := Bornology.ofDist dist dist_comm dist_triangle
cobounded_sets := rfl }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.ofDistTopology
|
Construct a pseudo-metric space structure whose underlying topological space structure
(definitionally) agrees which a pre-existing topology which is compatible with a given distance
function.
|
dist_self (x : α) : dist x x = 0 :=
PseudoMetricSpace.dist_self x
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_self
| null |
dist_comm (x y : α) : dist x y = dist y x :=
PseudoMetricSpace.dist_comm x y
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_comm
| null |
edist_dist (x y : α) : edist x y = ENNReal.ofReal (dist x y) :=
PseudoMetricSpace.edist_dist x y
@[bound]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_dist
| null |
dist_triangle (x y z : α) : dist x z ≤ dist x y + dist y z :=
PseudoMetricSpace.dist_triangle x y z
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle
| null |
dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y := by
rw [dist_comm z]; apply dist_triangle
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle_left
| null |
dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by
rw [dist_comm y]; apply dist_triangle
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle_right
| null |
dist_triangle4 (x y z w : α) : dist x w ≤ dist x y + dist y z + dist z w :=
calc
dist x w ≤ dist x z + dist z w := dist_triangle x z w
_ ≤ dist x y + dist y z + dist z w := by gcongr; apply dist_triangle x y z
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle4
| null |
dist_triangle4_left (x₁ y₁ x₂ y₂ : α) :
dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
rw [add_left_comm, dist_comm x₁, ← add_assoc]
apply dist_triangle4
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle4_left
| null |
dist_triangle4_right (x₁ y₁ x₂ y₂ : α) :
dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂ := by
rw [add_right_comm, dist_comm y₁]
apply dist_triangle4
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle4_right
| null |
dist_triangle8 (a b c d e f g h : α) : dist a h ≤ dist a b + dist b c + dist c d
+ dist d e + dist e f + dist f g + dist g h := by
apply le_trans (dist_triangle4 a f g h)
gcongr
apply le_trans (dist_triangle4 a d e f)
gcongr
exact dist_triangle4 a b c d
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_triangle8
| null |
swap_dist : Function.swap (@dist α _) = dist := by funext x y; exact dist_comm _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
swap_dist
| null |
abs_dist_sub_le (x y z : α) : |dist x z - dist y z| ≤ dist x y :=
abs_sub_le_iff.2
⟨sub_le_iff_le_add.2 (dist_triangle _ _ _), sub_le_iff_le_add.2 (dist_triangle_left _ _ _)⟩
@[simp, bound]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
abs_dist_sub_le
| null |
dist_nonneg {x y : α} : 0 ≤ dist x y :=
dist_nonneg' dist dist_self dist_comm dist_triangle
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_nonneg
| null |
@[positivity Dist.dist _ _]
evalDist : PositivityExt where eval {u α} _zα _pα e := do
match u, α, e with
| 0, ~q(ℝ), ~q(@Dist.dist $β $inst $a $b) =>
let _inst ← synthInstanceQ q(PseudoMetricSpace $β)
assertInstancesCommute
pure (.nonnegative q(dist_nonneg))
| _, _, _ => throwError "not dist"
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
evalDist
|
Extension for the `positivity` tactic: distances are nonnegative.
|
@[simp] abs_dist {a b : α} : |dist a b| = dist a b := abs_of_nonneg dist_nonneg
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
abs_dist
| null |
NNDist (α : Type*) where
/-- Nonnegative distance between two points -/
nndist : α → α → ℝ≥0
export NNDist (nndist)
|
class
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
NNDist
|
A version of `Dist` that takes value in `ℝ≥0`.
|
dist_nndist (x y : α) : dist x y = nndist x y := rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_nndist
|
Distance as a nonnegative real number. -/
instance (priority := 100) PseudoMetricSpace.toNNDist : NNDist α :=
⟨fun a b => ⟨dist a b, dist_nonneg⟩⟩
/-- Express `dist` in terms of `nndist`
|
coe_nndist (x y : α) : ↑(nndist x y) = dist x y := rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
coe_nndist
| null |
edist_nndist (x y : α) : edist x y = nndist x y := by
rw [edist_dist, dist_nndist, ENNReal.ofReal_coe_nnreal]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_nndist
|
Express `edist` in terms of `nndist`
|
nndist_edist (x y : α) : nndist x y = (edist x y).toNNReal := by
simp [edist_nndist]
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_edist
|
Express `nndist` in terms of `edist`
|
coe_nnreal_ennreal_nndist (x y : α) : ↑(nndist x y) = edist x y :=
(edist_nndist x y).symm
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
coe_nnreal_ennreal_nndist
| null |
edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c := by
rw [edist_nndist, ENNReal.coe_lt_coe]
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_lt_coe
| null |
edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
rw [edist_nndist, ENNReal.coe_le_coe]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_le_coe
| null |
edist_lt_top {α : Type*} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
(edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_lt_top
|
In a pseudometric space, the extended distance is always finite
|
@[simp] nndist_self (a : α) : nndist a a = 0 := NNReal.coe_eq_zero.1 (dist_self a)
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_self
|
In a pseudometric space, the extended distance is always finite -/
@[aesop (rule_sets := [finiteness]) safe apply]
theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
(edist_lt_top x y).ne
/-- `nndist x x` vanishes
|
dist_lt_coe {x y : α} {c : ℝ≥0} : dist x y < c ↔ nndist x y < c :=
Iff.rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_lt_coe
| null |
dist_le_coe {x y : α} {c : ℝ≥0} : dist x y ≤ c ↔ nndist x y ≤ c :=
Iff.rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_le_coe
| null |
edist_lt_ofReal {x y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r := by
rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_lt_ofReal
| null |
edist_le_ofReal {x y : α} {r : ℝ} (hr : 0 ≤ r) :
edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r := by
rw [edist_dist, ENNReal.ofReal_le_ofReal_iff hr]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
edist_le_ofReal
| null |
nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by
rw [dist_nndist, Real.toNNReal_coe]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_dist
|
Express `nndist` in terms of `dist`
|
nndist_comm (x y : α) : nndist x y = nndist y x := NNReal.eq <| dist_comm x y
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_comm
| null |
nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
dist_triangle _ _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_triangle
|
Triangle inequality for the nonnegative distance
|
nndist_triangle_left (x y z : α) : nndist x y ≤ nndist z x + nndist z y :=
dist_triangle_left _ _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_triangle_left
| null |
nndist_triangle_right (x y z : α) : nndist x y ≤ nndist x z + nndist y z :=
dist_triangle_right _ _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_triangle_right
| null |
dist_edist (x y : α) : dist x y = (edist x y).toReal := by
rw [edist_dist, ENNReal.toReal_ofReal dist_nonneg]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_edist
|
Express `dist` in terms of `edist`
|
ball (x : α) (ε : ℝ) : Set α :=
{ y | dist y x < ε }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball
|
`ball x ε` is the set of all points `y` with `dist y x < ε`
|
mem_ball : y ∈ ball x ε ↔ dist y x < ε :=
Iff.rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_ball
| null |
mem_ball' : y ∈ ball x ε ↔ dist x y < ε := by rw [dist_comm, mem_ball]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_ball'
| null |
pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
dist_nonneg.trans_lt hy
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
pos_of_mem_ball
| null |
mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by
rwa [mem_ball, dist_self]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_ball_self
| null |
nonempty_ball : (ball x ε).Nonempty ↔ 0 < ε :=
⟨fun ⟨_x, hx⟩ => pos_of_mem_ball hx, fun h => ⟨x, mem_ball_self h⟩⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nonempty_ball
| null |
ball_eq_empty : ball x ε = ∅ ↔ ε ≤ 0 := by
rw [← not_nonempty_iff_eq_empty, nonempty_ball, not_lt]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_eq_empty
| null |
ball_zero : ball x 0 = ∅ := by rw [ball_eq_empty]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_zero
| null |
exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' := by
simp only [mem_ball] at h ⊢
exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
exists_lt_mem_ball_of_mem_ball
|
If a point belongs to an open ball, then there is a strictly smaller radius whose ball also
contains it.
See also `exists_lt_subset_ball`.
|
ball_eq_ball (ε : ℝ) (x : α) :
UniformSpace.ball x { p | dist p.2 p.1 < ε } = Metric.ball x ε :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_eq_ball
| null |
ball_eq_ball' (ε : ℝ) (x : α) :
UniformSpace.ball x { p | dist p.1 p.2 < ε } = Metric.ball x ε := by
ext
simp [dist_comm, UniformSpace.ball]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_eq_ball'
| null |
iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ :=
iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x)
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
iUnion_ball_nat
| null |
iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ :=
iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
iUnion_ball_nat_succ
| null |
closedBall (x : α) (ε : ℝ) :=
{ y | dist y x ≤ ε }
@[simp] theorem mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε := Iff.rfl
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall
|
`closedBall x ε` is the set of all points `y` with `dist y x ≤ ε`
|
mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε := by rw [dist_comm, mem_closedBall]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closedBall'
| null |
sphere (x : α) (ε : ℝ) := { y | dist y x = ε }
@[simp] theorem mem_sphere : y ∈ sphere x ε ↔ dist y x = ε := Iff.rfl
|
def
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere
|
`sphere x ε` is the set of all points `y` with `dist y x = ε`
|
mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε := by rw [dist_comm, mem_sphere]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_sphere'
| null |
ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x :=
ne_of_mem_of_not_mem h <| by simpa using hε.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ne_of_mem_sphere
| null |
nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε :=
dist_nonneg.trans_eq hy
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nonneg_of_mem_sphere
| null |
sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
Set.eq_empty_iff_forall_notMem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_gt hε
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_eq_empty_of_neg
| null |
sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ :=
Set.eq_empty_iff_forall_notMem.mpr fun _ h => ne_of_mem_sphere h hε (Subsingleton.elim _ _)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_eq_empty_of_subsingleton
| null |
sphere_isEmpty_of_subsingleton [Subsingleton α] [NeZero ε] : IsEmpty (sphere x ε) := by
rw [sphere_eq_empty_of_subsingleton (NeZero.ne ε)]; infer_instance
|
instance
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_isEmpty_of_subsingleton
| null |
closedBall_eq_singleton_of_subsingleton [Subsingleton α] (h : 0 ≤ ε) :
closedBall x ε = {x} := by
ext x'
simpa [Subsingleton.allEq x x']
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_eq_singleton_of_subsingleton
| null |
ball_eq_singleton_of_subsingleton [Subsingleton α] (h : 0 < ε) : ball x ε = {x} := by
ext x'
simpa [Subsingleton.allEq x x']
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_eq_singleton_of_subsingleton
| null |
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