fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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PseudoEMetricSpace.induced {α β} (f : α → β) (m : PseudoEMetricSpace β) :
PseudoEMetricSpace α where
edist x y := edist (f x) (f y)
edist_self _ := edist_self _
edist_comm _ _ := edist_comm _ _
edist_triangle _ _ _ := edist_triangle _ _ _
toUniformSpace := UniformSpace.comap f m.toUniformSpace
uniformit... | abbrev | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | PseudoEMetricSpace.induced | The extended pseudometric induced by a function taking values in a pseudoemetric space.
See note [reducible non-instances]. |
Subtype.edist_eq {p : α → Prop} (x y : Subtype p) : edist x y = edist (x : α) y := rfl | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | Subtype.edist_eq | Pseudoemetric space instance on subsets of pseudoemetric spaces -/
instance {α : Type*} {p : α → Prop} [PseudoEMetricSpace α] : PseudoEMetricSpace (Subtype p) :=
PseudoEMetricSpace.induced Subtype.val ‹_›
/-- The extended pseudodistance on a subset of a pseudoemetric space is the restriction of
the original pseudodi... |
@[simp]
Subtype.edist_mk_mk {p : α → Prop} {x y : α} (hx : p x) (hy : p y) :
edist (⟨x, hx⟩ : Subtype p) ⟨y, hy⟩ = edist x y :=
rfl | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | Subtype.edist_mk_mk | The extended pseudodistance on a subtype of a pseudoemetric space is the restriction of
the original pseudodistance, by definition. |
@[reducible] noncomputable PseudoEmetricSpace.ofEdistOfTopology {α : Type*} [TopologicalSpace α]
(d : α → α → ℝ≥0∞) (h_self : ∀ x, d x x = 0) (h_comm : ∀ x y, d x y = d y x)
(h_triangle : ∀ x y z, d x z ≤ d x y + d y z)
(h_basis : ∀ x, (𝓝 x).HasBasis (fun c ↦ 0 < c) (fun c ↦ {y | d x y < c})) :
PseudoE... | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | PseudoEmetricSpace.ofEdistOfTopology | Consider an extended distance on a topological space, for which the neighborhoods can be
expressed in terms of the distance. Then we define the emetric space structure associated to this
distance, with a topology defeq to the initial one. |
Prod.pseudoEMetricSpaceMax [PseudoEMetricSpace β] :
PseudoEMetricSpace (α × β) where
edist x y := edist x.1 y.1 ⊔ edist x.2 y.2
edist_self x := by simp
edist_comm x y := by simp [edist_comm]
edist_triangle _ _ _ :=
max_le (le_trans (edist_triangle _ _ _) (add_le_add (le_max_left _ _) (le_max_left _ _)))... | instance | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | Prod.pseudoEMetricSpaceMax | Pseudoemetric space instance on the multiplicative opposite of a pseudoemetric space. -/
@[to_additive
/-- Pseudoemetric space instance on the additive opposite of a pseudoemetric space. -/]
instance {α : Type*} [PseudoEMetricSpace α] : PseudoEMetricSpace αᵐᵒᵖ :=
PseudoEMetricSpace.induced unop ‹_›
@[to_additive]
th... |
Prod.edist_eq [PseudoEMetricSpace β] (x y : α × β) :
edist x y = max (edist x.1 y.1) (edist x.2 y.2) :=
rfl | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | Prod.edist_eq | null |
ball (x : α) (ε : ℝ≥0∞) : Set α :=
{ y | edist y x < ε }
@[simp] theorem mem_ball : y ∈ ball x ε ↔ edist y x < ε := Iff.rfl | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball | `EMetric.ball x ε` is the set of all points `y` with `edist y x < ε` |
mem_ball' : y ∈ ball x ε ↔ edist x y < ε := by rw [edist_comm, mem_ball] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_ball' | null |
closedBall (x : α) (ε : ℝ≥0∞) :=
{ y | edist y x ≤ ε }
@[simp] theorem mem_closedBall : y ∈ closedBall x ε ↔ edist y x ≤ ε := Iff.rfl | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | closedBall | `EMetric.closedBall x ε` is the set of all points `y` with `edist y x ≤ ε` |
mem_closedBall' : y ∈ closedBall x ε ↔ edist x y ≤ ε := by rw [edist_comm, mem_closedBall]
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_closedBall' | null |
closedBall_top (x : α) : closedBall x ∞ = univ :=
eq_univ_of_forall fun _ => mem_setOf.2 le_top | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | closedBall_top | null |
ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _ h => le_of_lt h.out | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_subset_closedBall | null |
pos_of_mem_ball (hy : y ∈ ball x ε) : 0 < ε :=
lt_of_le_of_lt (zero_le _) hy | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | pos_of_mem_ball | null |
mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by
rwa [mem_ball, edist_self] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_ball_self | null |
mem_closedBall_self : x ∈ closedBall x ε := by
rw [mem_closedBall, edist_self]; apply zero_le | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_closedBall_self | null |
mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_ball_comm | null |
mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by
rw [mem_closedBall', mem_closedBall]
@[gcongr] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_closedBall_comm | null |
ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun _y (yx : _ < ε₁) =>
lt_of_lt_of_le yx h
@[gcongr] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_subset_ball | null |
closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
fun _y (yx : _ ≤ ε₁) => le_trans yx h | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | closedBall_subset_closedBall | null |
ball_disjoint (h : ε₁ + ε₂ ≤ edist x y) : Disjoint (ball x ε₁) (ball y ε₂) :=
Set.disjoint_left.mpr fun z h₁ h₂ =>
(edist_triangle_left x y z).not_gt <| (ENNReal.add_lt_add h₁ h₂).trans_le h | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_disjoint | null |
ball_subset (h : edist x y + ε₁ ≤ ε₂) (h' : edist x y ≠ ∞) : ball x ε₁ ⊆ ball y ε₂ :=
fun z zx =>
calc
edist z y ≤ edist z x + edist x y := edist_triangle _ _ _
_ = edist x y + edist z x := add_comm _ _
_ < edist x y + ε₁ := ENNReal.add_lt_add_left h' zx
_ ≤ ε₂ := h | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_subset | null |
exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε := by
have : 0 < ε - edist y x := by simpa using h
refine ⟨ε - edist y x, this, ball_subset ?_ (ne_top_of_lt h)⟩
exact (add_tsub_cancel_of_le (mem_ball.mp h).le).le | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | exists_ball_subset_ball | null |
ball_eq_empty_iff : ball x ε = ∅ ↔ ε = 0 :=
eq_empty_iff_forall_notMem.trans
⟨fun h => le_bot_iff.1 (le_of_not_gt fun ε0 => h _ (mem_ball_self ε0)), fun ε0 _ h =>
not_lt_of_ge (le_of_eq ε0) (pos_of_mem_ball h)⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_eq_empty_iff | null |
ordConnected_setOf_closedBall_subset (x : α) (s : Set α) :
OrdConnected { r | closedBall x r ⊆ s } :=
⟨fun _ _ _ h₁ _ h₂ => (closedBall_subset_closedBall h₂.2).trans h₁⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ordConnected_setOf_closedBall_subset | null |
ordConnected_setOf_ball_subset (x : α) (s : Set α) : OrdConnected { r | ball x r ⊆ s } :=
⟨fun _ _ _ h₁ _ h₂ => (ball_subset_ball h₂.2).trans h₁⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ordConnected_setOf_ball_subset | null |
edistLtTopSetoid : Setoid α where
r x y := edist x y < ⊤
iseqv :=
⟨fun x => by rw [edist_self]; exact ENNReal.coe_lt_top,
fun h => by rwa [edist_comm], fun hxy hyz =>
lt_of_le_of_lt (edist_triangle _ _ _) (ENNReal.add_lt_top.2 ⟨hxy, hyz⟩)⟩
@[simp] | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edistLtTopSetoid | Relation “two points are at a finite edistance” is an equivalence relation. |
ball_zero : ball x 0 = ∅ := by rw [EMetric.ball_eq_empty_iff] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_zero | null |
nhds_basis_eball : (𝓝 x).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) (ball x) :=
nhds_basis_uniformity uniformity_basis_edist | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | nhds_basis_eball | null |
nhdsWithin_basis_eball : (𝓝[s] x).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => ball x ε ∩ s :=
nhdsWithin_hasBasis nhds_basis_eball s | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | nhdsWithin_basis_eball | null |
nhds_basis_closed_eball : (𝓝 x).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) (closedBall x) :=
nhds_basis_uniformity uniformity_basis_edist_le | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | nhds_basis_closed_eball | null |
nhdsWithin_basis_closed_eball :
(𝓝[s] x).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => closedBall x ε ∩ s :=
nhdsWithin_hasBasis nhds_basis_closed_eball s | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | nhdsWithin_basis_closed_eball | null |
nhds_eq : 𝓝 x = ⨅ ε > 0, 𝓟 (ball x ε) :=
nhds_basis_eball.eq_biInf | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | nhds_eq | null |
mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
nhds_basis_eball.mem_iff | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_nhds_iff | null |
mem_nhdsWithin_iff : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
nhdsWithin_basis_eball.mem_iff | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_nhdsWithin_iff | null |
tendsto_nhdsWithin_nhdsWithin {t : Set β} {a b} :
Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x⦄, x ∈ s → edist x a < δ → f x ∈ t ∧ edist (f x) b < ε :=
(nhdsWithin_basis_eball.tendsto_iff nhdsWithin_basis_eball).trans <|
forall₂_congr fun ε _ => exists_congr fun δ => and_congr_right fun _ =>
... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | tendsto_nhdsWithin_nhdsWithin | null |
tendsto_nhdsWithin_nhds {a b} :
Tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → edist x a < δ → edist (f x) b < ε := by
rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
simp only [mem_univ, true_and] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | tendsto_nhdsWithin_nhds | null |
tendsto_nhds_nhds {a b} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x⦄, edist x a < δ → edist (f x) b < ε :=
nhds_basis_eball.tendsto_iff nhds_basis_eball | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | tendsto_nhds_nhds | null |
isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
simp [isOpen_iff_nhds, mem_nhds_iff]
@[simp] theorem isOpen_ball : IsOpen (ball x ε) :=
isOpen_iff.2 fun _ => exists_ball_subset_ball | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | isOpen_iff | null |
isClosed_ball_top : IsClosed (ball x ⊤) :=
isOpen_compl_iff.1 <| isOpen_iff.2 fun _y hy =>
⟨⊤, ENNReal.coe_lt_top, fun _z hzy hzx =>
hy (edistLtTopSetoid.trans (edistLtTopSetoid.symm hzy) hzx)⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | isClosed_ball_top | null |
ball_mem_nhds (x : α) {ε : ℝ≥0∞} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
isOpen_ball.mem_nhds (mem_ball_self ε0) | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_mem_nhds | null |
closedBall_mem_nhds (x : α) {ε : ℝ≥0∞} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | closedBall_mem_nhds | null |
ball_prod_same [PseudoEMetricSpace β] (x : α) (y : β) (r : ℝ≥0∞) :
ball x r ×ˢ ball y r = ball (x, y) r :=
ext fun z => by simp [Prod.edist_eq] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | ball_prod_same | null |
closedBall_prod_same [PseudoEMetricSpace β] (x : α) (y : β) (r : ℝ≥0∞) :
closedBall x r ×ˢ closedBall y r = closedBall (x, y) r :=
ext fun z => by simp [Prod.edist_eq] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | closedBall_prod_same | null |
mem_closure_iff : x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, edist x y < ε :=
(mem_closure_iff_nhds_basis nhds_basis_eball).trans <| by simp only [mem_ball, edist_comm x] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_closure_iff | ε-characterization of the closure in pseudoemetric spaces |
tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, edist (u x) a < ε :=
nhds_basis_eball.tendsto_right_iff | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | tendsto_nhds | null |
tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (u n) a < ε :=
(atTop_basis.tendsto_iff nhds_basis_eball).trans <| by
simp only [true_and, mem_Ici, mem_ball] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | tendsto_atTop | null |
subset_countable_closure_of_almost_dense_set (s : Set α)
(hs : ∀ ε > 0, ∃ t : Set α, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε) :
∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
rcases s.eq_empty_or_nonempty with (rfl | ⟨x₀, hx₀⟩)
· exact ⟨∅, empty_subset _, countable_empty, empty_subset _⟩
choose! T hTc h... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | subset_countable_closure_of_almost_dense_set | For a set `s` in a pseudo emetric space, if for every `ε > 0` there exists a countable
set that is `ε`-dense in `s`, then there exists a countable subset `t ⊆ s` that is dense in `s`. |
EMetricSpace (α : Type u) : Type u extends PseudoEMetricSpace α where
eq_of_edist_eq_zero : ∀ {x y : α}, edist x y = 0 → x = y
@[ext] | class | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetricSpace | An extended metric space is a type endowed with a `ℝ≥0∞`-valued distance `edist` satisfying
`edist x y = 0 ↔ x = y`, commutativity `edist x y = edist y x`, and the triangle inequality
`edist x z ≤ edist x y + edist y z`.
See pseudo extended metric spaces (`PseudoEMetricSpace`) for the similar class with the
`edist x y... |
protected EMetricSpace.ext
{α : Type*} {m m' : EMetricSpace α} (h : m.toEDist = m'.toEDist) : m = m' := by
cases m
cases m'
congr
ext1
assumption
variable {γ : Type w} [EMetricSpace γ]
export EMetricSpace (eq_of_edist_eq_zero) | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetricSpace.ext | null |
@[simp]
edist_eq_zero {x y : γ} : edist x y = 0 ↔ x = y :=
⟨eq_of_edist_eq_zero, fun h => h ▸ edist_self _⟩
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_eq_zero | Characterize the equality of points by the vanishing of their extended distance |
zero_eq_edist {x y : γ} : 0 = edist x y ↔ x = y := eq_comm.trans edist_eq_zero | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | zero_eq_edist | null |
edist_le_zero {x y : γ} : edist x y ≤ 0 ↔ x = y :=
nonpos_iff_eq_zero.trans edist_eq_zero
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_le_zero | null |
edist_pos {x y : γ} : 0 < edist x y ↔ x ≠ y := by simp [← not_le]
@[simp] lemma EMetric.closedBall_zero (x : γ) : closedBall x 0 = {x} := by ext; simp | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_pos | null |
eq_of_forall_edist_le {x y : γ} (h : ∀ ε > 0, edist x y ≤ ε) : x = y :=
eq_of_edist_eq_zero (eq_of_le_of_forall_lt_imp_le_of_dense bot_le h) | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | eq_of_forall_edist_le | Two points coincide if their distance is `< ε` for all positive ε |
EMetricSpace.replaceUniformity {γ} [U : UniformSpace γ] (m : EMetricSpace γ)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : EMetricSpace γ where
edist := @edist _ m.toEDist
edist_self := edist_self
eq_of_edist_eq_zero := @eq_of_edist_eq_zero _ _
edist_comm := edist_comm
edist_triangle := edist_tria... | abbrev | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetricSpace.replaceUniformity | Auxiliary function to replace the uniformity on an emetric space with
a uniformity which is equal to the original one, but maybe not defeq.
This is useful if one wants to construct an emetric space with a
specified uniformity. See Note [forgetful inheritance] explaining why having definitionally
the right uniformity is... |
EMetricSpace.replaceTopology {γ} [T : TopologicalSpace γ] (m : EMetricSpace γ)
(H : T = m.toUniformSpace.toTopologicalSpace) : EMetricSpace γ where
edist := @edist _ m.toEDist
edist_self := edist_self
eq_of_edist_eq_zero := @eq_of_edist_eq_zero _ _
edist_comm := edist_comm
edist_triangle := edist_triangle... | abbrev | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetricSpace.replaceTopology | Auxiliary function to replace the topology on an emetric space with
a topology which is equal to the original one, but maybe not defeq.
This is useful if one wants to construct an emetric space with a
specified topology. See Note [forgetful inheritance] explaining why having definitionally
the right topology is often i... |
EMetricSpace.induced {γ β} (f : γ → β) (hf : Function.Injective f) (m : EMetricSpace β) :
EMetricSpace γ :=
{ PseudoEMetricSpace.induced f m.toPseudoEMetricSpace with
eq_of_edist_eq_zero := fun h => hf (edist_eq_zero.1 h) } | abbrev | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetricSpace.induced | The extended metric induced by an injective function taking values in an emetric space.
See Note [reducible non-instances]. |
uniformity_edist : 𝓤 γ = ⨅ ε > 0, 𝓟 { p : γ × γ | edist p.1 p.2 < ε } :=
PseudoEMetricSpace.uniformity_edist
/-! | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_edist | EMetric space instance on subsets of emetric spaces -/
instance {α : Type*} {p : α → Prop} [EMetricSpace α] : EMetricSpace (Subtype p) :=
EMetricSpace.induced Subtype.val Subtype.coe_injective ‹_›
/-- EMetric space instance on the multiplicative opposite of an emetric space. -/
@[to_additive /-- EMetric space instan... |
@[simp]
edist_ofMul (a b : X) : edist (ofMul a) (ofMul b) = edist a b :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_ofMul | null |
edist_ofAdd (a b : X) : edist (ofAdd a) (ofAdd b) = edist a b :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_ofAdd | null |
edist_toMul (a b : Additive X) : edist a.toMul b.toMul = edist a b :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_toMul | null |
edist_toAdd (a b : Multiplicative X) : edist a.toAdd b.toAdd = edist a b :=
rfl | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_toAdd | null |
@[simp]
edist_toDual (a b : X) : edist (toDual a) (toDual b) = edist a b :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_toDual | null |
edist_ofDual (a b : Xᵒᵈ) : edist (ofDual a) (ofDual b) = edist a b :=
rfl | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_ofDual | null |
noncomputable diam (s : Set α) :=
⨆ (x ∈ s) (y ∈ s), edist x y | def | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam | The diameter of a set in a pseudoemetric space, named `EMetric.diam` |
diam_eq_sSup (s : Set α) : diam s = sSup (image2 edist s s) := sSup_image2.symm | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_eq_sSup | null |
diam_le_iff {d : ℝ≥0∞} : diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d := by
simp only [diam, iSup_le_iff] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_le_iff | null |
diam_image_le_iff {d : ℝ≥0∞} {f : β → α} {s : Set β} :
diam (f '' s) ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) ≤ d := by
simp only [diam_le_iff, forall_mem_image] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_image_le_iff | null |
edist_le_of_diam_le {d} (hx : x ∈ s) (hy : y ∈ s) (hd : diam s ≤ d) : edist x y ≤ d :=
diam_le_iff.1 hd x hx y hy | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | edist_le_of_diam_le | null |
edist_le_diam_of_mem (hx : x ∈ s) (hy : y ∈ s) : edist x y ≤ diam s :=
edist_le_of_diam_le hx hy le_rfl | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | edist_le_diam_of_mem | If two points belong to some set, their edistance is bounded by the diameter of the set |
diam_le {d : ℝ≥0∞} (h : ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d) : diam s ≤ d :=
diam_le_iff.2 h | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_le | If the distance between any two points in a set is bounded by some constant, this constant
bounds the diameter. |
diam_subsingleton (hs : s.Subsingleton) : diam s = 0 :=
nonpos_iff_eq_zero.1 <| diam_le fun _x hx y hy => (hs hx hy).symm ▸ edist_self y ▸ le_rfl | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_subsingleton | The diameter of a subsingleton vanishes. |
@[simp]
diam_empty : diam (∅ : Set α) = 0 :=
diam_subsingleton subsingleton_empty | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_empty | The diameter of the empty set vanishes |
@[simp]
diam_singleton : diam ({x} : Set α) = 0 :=
diam_subsingleton subsingleton_singleton
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_singleton | The diameter of a singleton vanishes |
diam_one [One α] : diam (1 : Set α) = 0 :=
diam_singleton | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_one | null |
diam_iUnion_mem_option {ι : Type*} (o : Option ι) (s : ι → Set α) :
diam (⋃ i ∈ o, s i) = ⨆ i ∈ o, diam (s i) := by cases o <;> simp | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_iUnion_mem_option | null |
diam_insert : diam (insert x s) = max (⨆ y ∈ s, edist x y) (diam s) :=
eq_of_forall_ge_iff fun d => by
simp only [diam_le_iff, forall_mem_insert, edist_self, edist_comm x, max_le_iff, iSup_le_iff,
zero_le, true_and, forall_and, and_self_iff, ← and_assoc] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_insert | null |
diam_pair : diam ({x, y} : Set α) = edist x y := by
simp only [iSup_singleton, diam_insert, diam_singleton, ENNReal.max_zero_right] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_pair | null |
diam_triple : diam ({x, y, z} : Set α) = max (max (edist x y) (edist x z)) (edist y z) := by
simp only [diam_insert, iSup_insert, iSup_singleton, diam_singleton, ENNReal.max_zero_right] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_triple | null |
@[gcongr]
diam_mono {s t : Set α} (h : s ⊆ t) : diam s ≤ diam t :=
diam_le fun _x hx _y hy => edist_le_diam_of_mem (h hx) (h hy) | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_mono | The diameter is monotonous with respect to inclusion |
diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + edist x y + diam t := by
have A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ diam s + edist x y + diam t := fun a ha b hb =>
calc
edist a b ≤ edist a x + edist x y + edist y b := edist_triangle4 _ _ _ _
_ ≤ diam s + edist x y + diam t :... | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_union | The diameter of a union is controlled by the diameter of the sets, and the edistance
between two points in the sets. |
diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t := by
let ⟨x, ⟨xs, xt⟩⟩ := h
simpa using diam_union xs xt | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_union' | null |
diam_closedBall {r : ℝ≥0∞} : diam (closedBall x r) ≤ 2 * r :=
diam_le fun a ha b hb =>
calc
edist a b ≤ edist a x + edist b x := edist_triangle_right _ _ _
_ ≤ r + r := add_le_add ha hb
_ = 2 * r := (two_mul r).symm | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_closedBall | null |
diam_ball {r : ℝ≥0∞} : diam (ball x r) ≤ 2 * r :=
le_trans (diam_mono ball_subset_closedBall) diam_closedBall | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_ball | null |
diam_pi_le_of_le {X : β → Type*} [Fintype β] [∀ b, PseudoEMetricSpace (X b)]
{s : ∀ b : β, Set (X b)} {c : ℝ≥0∞} (h : ∀ b, diam (s b) ≤ c) : diam (Set.pi univ s) ≤ c := by
refine diam_le fun x hx y hy => edist_pi_le_iff.mpr ?_
rw [mem_univ_pi] at hx hy
exact fun b => diam_le_iff.1 (h b) (x b) (hx b) (y b) (hy... | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_pi_le_of_le | null |
diam_eq_zero_iff : diam s = 0 ↔ s.Subsingleton :=
⟨fun h _x hx _y hy => edist_le_zero.1 <| h ▸ edist_le_diam_of_mem hx hy, diam_subsingleton⟩ | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_eq_zero_iff | null |
diam_pos_iff : 0 < diam s ↔ s.Nontrivial := by
simp only [pos_iff_ne_zero, Ne, diam_eq_zero_iff, Set.not_subsingleton_iff] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_pos_iff | null |
diam_pos_iff' : 0 < diam s ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y := by
simp only [diam_pos_iff, Set.Nontrivial] | theorem | Topology | [
"Mathlib.Topology.EMetricSpace.Pi"
] | Mathlib/Topology/EMetricSpace/Diam.lean | diam_pos_iff' | null |
LipschitzWith (K : ℝ≥0) (f : α → β) := ∀ x y, edist (f x) (f y) ≤ K * edist x y | def | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LipschitzWith | A function `f` is **Lipschitz continuous** with constant `K ≥ 0` if for all `x, y`
we have `dist (f x) (f y) ≤ K * dist x y`. |
LipschitzOnWith (K : ℝ≥0) (f : α → β) (s : Set α) :=
∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → edist (f x) (f y) ≤ K * edist x y | def | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LipschitzOnWith | A function `f` is **Lipschitz continuous** with constant `K ≥ 0` **on `s`** if
for all `x, y` in `s` we have `dist (f x) (f y) ≤ K * dist x y`. |
LocallyLipschitz (f : α → β) : Prop := ∀ x, ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t | def | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LocallyLipschitz | `f : α → β` is called **locally Lipschitz continuous** iff every point `x`
has a neighbourhood on which `f` is Lipschitz. |
LocallyLipschitzOn (s : Set α) (f : α → β) : Prop :=
∀ ⦃x⦄, x ∈ s → ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t | def | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LocallyLipschitzOn | `f : α → β` is called **locally Lipschitz continuous** on `s` iff every point `x` of `s`
has a neighbourhood within `s` on which `f` is Lipschitz. |
@[simp]
lipschitzOnWith_empty (K : ℝ≥0) (f : α → β) : LipschitzOnWith K f ∅ := fun _ => False.elim
@[simp] lemma locallyLipschitzOn_empty (f : α → β) : LocallyLipschitzOn ∅ f := fun _ ↦ False.elim | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | lipschitzOnWith_empty | Every function is Lipschitz on the empty set (with any Lipschitz constant). |
LipschitzOnWith.mono (hf : LipschitzOnWith K f t) (h : s ⊆ t) : LipschitzOnWith K f s :=
fun _x x_in _y y_in => hf (h x_in) (h y_in) | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LipschitzOnWith.mono | Being Lipschitz on a set is monotone w.r.t. that set. |
LocallyLipschitzOn.mono (hf : LocallyLipschitzOn t f) (h : s ⊆ t) : LocallyLipschitzOn s f :=
fun x hx ↦ by obtain ⟨K, u, hu, hfu⟩ := hf (h hx); exact ⟨K, u, nhdsWithin_mono _ h hu, hfu⟩ | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LocallyLipschitzOn.mono | null |
@[simp] lipschitzOnWith_univ : LipschitzOnWith K f univ ↔ LipschitzWith K f := by
simp [LipschitzOnWith, LipschitzWith]
@[simp] lemma locallyLipschitzOn_univ : LocallyLipschitzOn univ f ↔ LocallyLipschitz f := by
simp [LocallyLipschitzOn, LocallyLipschitz] | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | lipschitzOnWith_univ | `f` is Lipschitz iff it is Lipschitz on the entire space. |
protected LocallyLipschitz.locallyLipschitzOn (h : LocallyLipschitz f) :
LocallyLipschitzOn s f := (locallyLipschitzOn_univ.2 h).mono s.subset_univ | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | LocallyLipschitz.locallyLipschitzOn | null |
lipschitzOnWith_iff_restrict : LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f) := by
simp [LipschitzOnWith, LipschitzWith] | theorem | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | lipschitzOnWith_iff_restrict | null |
lipschitzOnWith_restrict {t : Set s} :
LipschitzOnWith K (s.restrict f) t ↔ LipschitzOnWith K f (s ∩ Subtype.val '' t) := by
simp [LipschitzOnWith] | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | lipschitzOnWith_restrict | null |
locallyLipschitzOn_iff_restrict :
LocallyLipschitzOn s f ↔ LocallyLipschitz (s.restrict f) := by
simp only [LocallyLipschitzOn, LocallyLipschitz, SetCoe.forall',
lipschitzOnWith_restrict,
nhds_subtype_eq_comap_nhdsWithin, mem_comap]
congr! with x K
constructor
· rintro ⟨t, ht, hft⟩
exact ⟨_, ⟨t,... | lemma | Topology | [
"Mathlib.Algebra.Group.End",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/EMetricSpace/Lipschitz.lean | locallyLipschitzOn_iff_restrict | null |
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