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 ⌀ |
|---|---|---|---|---|---|---|
Prod.emetricSpaceMax [EMetricSpace β] : EMetricSpace (γ × β) :=
.ofT0PseudoEMetricSpace _ | instance | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | Prod.emetricSpaceMax | If a `PseudoEMetricSpace` is a T₀ space, then it is an `EMetricSpace`. -/
-- TODO: make it an instance?
abbrev EMetricSpace.ofT0PseudoEMetricSpace (α : Type*) [PseudoEMetricSpace α] [T0Space α] :
EMetricSpace α :=
{ ‹PseudoEMetricSpace α› with
eq_of_edist_eq_zero := fun h => (EMetric.inseparable_iff.2 h).eq }... |
countable_closure_of_compact {s : Set γ} (hs : IsCompact s) :
∃ t, t ⊆ s ∧ t.Countable ∧ s = closure t := by
rcases subset_countable_closure_of_compact hs with ⟨t, hts, htc, hsub⟩
exact ⟨t, hts, htc, hsub.antisymm (closure_minimal hts hs.isClosed)⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | countable_closure_of_compact | A compact set in an emetric space is separable, i.e., it is the closure of a countable set. |
@[simp] SeparationQuotient.edist_mk [PseudoEMetricSpace X] (x y : X) :
edist (mk x) (mk y) = edist x y :=
rfl
open SeparationQuotient in | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | SeparationQuotient.edist_mk | null |
IsSeparable.exists_countable_dense_subset
{s : Set α} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
have : ∀ ε > 0, ∃ t : Set α, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε := fun ε ε0 => by
rcases hs with ⟨t, htc, hst⟩
refine ⟨t, htc, hst.trans fun x hx => ?_⟩
rcases mem_closure... | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | IsSeparable.exists_countable_dense_subset | If a set `s` is separable in a (pseudo extended) metric space, then it admits a countable dense
subset. This is not obvious, as the countable set whose closure covers `s` given by the definition
of separability does not need in general to be contained in `s`. |
IsSeparable.separableSpace {s : Set α} (hs : IsSeparable s) :
SeparableSpace s := by
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
lift t to Set s using hts
refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
rwa [IsInducing.subtypeVal.dense_iff, Sub... | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | IsSeparable.separableSpace | If a set `s` is separable, then the corresponding subtype is separable in a (pseudo extended)
metric space. This is not obvious, as the countable set whose closure covers `s` does not need in
general to be contained in `s`. |
lebesgue_number_lemma_of_emetric {ι : Sort*} {c : ι → Set α} (hs : IsCompact s)
(hc₁ : ∀ i, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i, c i) : ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i := by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm]
using uniformity_basis_edist.lebesgue_number_lemma hs hc₁ hc₂ | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric | null |
lebesgue_number_lemma_of_emetric_nhds' {c : (x : α) → x ∈ s → Set α} (hs : IsCompact s)
(hc : ∀ x hx, c x hx ∈ 𝓝 x) : ∃ δ > 0, ∀ x ∈ s, ∃ y : s, ball x δ ⊆ c y y.2 := by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm]
using uniformity_basis_edist.lebesgue_number_lemma_nhds' hs hc | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric_nhds' | null |
lebesgue_number_lemma_of_emetric_nhds {c : α → Set α} (hs : IsCompact s)
(hc : ∀ x ∈ s, c x ∈ 𝓝 x) : ∃ δ > 0, ∀ x ∈ s, ∃ y, ball x δ ⊆ c y := by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm]
using uniformity_basis_edist.lebesgue_number_lemma_nhds hs hc | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric_nhds | null |
lebesgue_number_lemma_of_emetric_nhdsWithin' {c : (x : α) → x ∈ s → Set α}
(hs : IsCompact s) (hc : ∀ x hx, c x hx ∈ 𝓝[s] x) :
∃ δ > 0, ∀ x ∈ s, ∃ y : s, ball x δ ∩ s ⊆ c y y.2 := by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm]
using uniformity_basis_edist.lebesgue_number_lemma_nhd... | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric_nhdsWithin' | null |
lebesgue_number_lemma_of_emetric_nhdsWithin {c : α → Set α} (hs : IsCompact s)
(hc : ∀ x ∈ s, c x ∈ 𝓝[s] x) : ∃ δ > 0, ∀ x ∈ s, ∃ y, ball x δ ∩ s ⊆ c y := by
simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm]
using uniformity_basis_edist.lebesgue_number_lemma_nhdsWithin hs hc | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric_nhdsWithin | null |
lebesgue_number_lemma_of_emetric_sUnion {c : Set (Set α)} (hs : IsCompact s)
(hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by
rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lemma_of_emetric hs (by simpa) hc₂ | theorem | Topology | [
"Mathlib.Algebra.Order.BigOperators.Group.Finset",
"Mathlib.Algebra.Order.Interval.Finset.SuccPred",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Order.Interval.Finset.Nat",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.LocallyUniformConvergence",... | Mathlib/Topology/EMetricSpace/Basic.lean | lebesgue_number_lemma_of_emetric_sUnion | null |
noncomputable eVariationOn (f : α → E) (s : Set α) : ℝ≥0∞ :=
⨆ p : ℕ × { u : ℕ → α // Monotone u ∧ ∀ i, u i ∈ s },
∑ i ∈ Finset.range p.1, edist (f (p.2.1 (i + 1))) (f (p.2.1 i)) | def | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eVariationOn | The (extended-real-valued) variation of a function `f` on a set `s` inside a linear order is
the supremum of the sum of `edist (f (u (i+1))) (f (u i))` over all finite increasing
sequences `u` in `s`. |
BoundedVariationOn (f : α → E) (s : Set α) :=
eVariationOn f s ≠ ∞ | def | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | BoundedVariationOn | A function has bounded variation on a set `s` if its total variation there is finite. |
LocallyBoundedVariationOn (f : α → E) (s : Set α) :=
∀ a b, a ∈ s → b ∈ s → BoundedVariationOn f (s ∩ Icc a b)
/-! ## Basic computations of variation -/ | def | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LocallyBoundedVariationOn | A function has locally bounded variation on a set `s` if, given any interval `[a, b]` with
endpoints in `s`, then the function has finite variation on `s ∩ [a, b]`. |
nonempty_monotone_mem {s : Set α} (hs : s.Nonempty) :
Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } := by
obtain ⟨x, hx⟩ := hs
exact ⟨⟨fun _ => x, fun i j _ => le_rfl, fun _ => hx⟩⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | nonempty_monotone_mem | null |
eq_of_edist_zero_on {f f' : α → E} {s : Set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) :
eVariationOn f s = eVariationOn f' s := by
dsimp only [eVariationOn]
congr 1 with p : 1
congr 1 with i : 1
rw [edist_congr_right (h <| p.snd.prop.2 (i + 1)), edist_congr_left (h <| p.snd.prop.2 i)] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_of_edist_zero_on | null |
eq_of_eqOn {f f' : α → E} {s : Set α} (h : EqOn f f' s) :
eVariationOn f s = eVariationOn f' s :=
eq_of_edist_zero_on fun x xs => by rw [h xs, edist_self] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_of_eqOn | null |
sum_le (f : α → E) {s : Set α} (n : ℕ) {u : ℕ → α} (hu : Monotone u) (us : ∀ i, u i ∈ s) :
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s :=
le_iSup_of_le ⟨n, u, hu, us⟩ le_rfl | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sum_le | null |
sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α}
(hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) :
(∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by
rcases le_total n m with hnm | hmn
· simp [Finset.Ico_eq_empty_of_le hnm]
let π := projIcc m... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sum_le_of_monotoneOn_Icc | null |
sum_le_of_monotoneOn_Iic (f : α → E) {s : Set α} {n : ℕ} {u : ℕ → α}
(hu : MonotoneOn u (Iic n)) (us : ∀ i ≤ n, u i ∈ s) :
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by
simpa using sum_le_of_monotoneOn_Icc f (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi ↦ us i hi.2 | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sum_le_of_monotoneOn_Iic | null |
mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s := by
apply iSup_le _
rintro ⟨n, ⟨u, hu, ut⟩⟩
exact sum_le f n hu fun i => hst (ut i) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | mono | null |
_root_.BoundedVariationOn.mono {f : α → E} {s : Set α} (h : BoundedVariationOn f s)
{t : Set α} (ht : t ⊆ s) : BoundedVariationOn f t :=
ne_top_of_le_ne_top h (eVariationOn.mono f ht) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | _root_.BoundedVariationOn.mono | null |
_root_.BoundedVariationOn.locallyBoundedVariationOn {f : α → E} {s : Set α}
(h : BoundedVariationOn f s) : LocallyBoundedVariationOn f s := fun _ _ _ _ =>
h.mono inter_subset_left | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | _root_.BoundedVariationOn.locallyBoundedVariationOn | null |
edist_le (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
edist (f x) (f y) ≤ eVariationOn f s := by
wlog hxy : y ≤ x generalizing x y
· rw [edist_comm]
exact this hy hx (le_of_not_ge hxy)
let u : ℕ → α := fun n => if n = 0 then y else x
have hu : Monotone u := monotone_nat_of_le_succ fun
... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | edist_le | null |
eq_zero_iff (f : α → E) {s : Set α} :
eVariationOn f s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0 := by
constructor
· rintro h x xs y ys
rw [← le_zero_iff, ← h]
exact edist_le f xs ys
· rintro h
dsimp only [eVariationOn]
rw [ENNReal.iSup_eq_zero]
rintro ⟨n, u, um, us⟩
exact Finset.s... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_zero_iff | null |
constant_on {f : α → E} {s : Set α} (hf : (f '' s).Subsingleton) :
eVariationOn f s = 0 := by
rw [eq_zero_iff]
rintro x xs y ys
rw [hf ⟨x, xs, rfl⟩ ⟨y, ys, rfl⟩, edist_self]
@[simp] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | constant_on | null |
protected subsingleton (f : α → E) {s : Set α} (hs : s.Subsingleton) :
eVariationOn f s = 0 :=
constant_on (hs.image f) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | subsingleton | null |
lowerSemicontinuous_aux {ι : Type*} {F : ι → α → E} {p : Filter ι} {f : α → E} {s : Set α}
(Ffs : ∀ x ∈ s, Tendsto (fun i => F i x) p (𝓝 (f x))) {v : ℝ≥0∞} (hv : v < eVariationOn f s) :
∀ᶠ n : ι in p, v < eVariationOn (F n) s := by
obtain ⟨⟨n, ⟨u, um, us⟩⟩, hlt⟩ :
∃ p : ℕ × { u : ℕ → α // Monotone u ∧ ∀ ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | lowerSemicontinuous_aux | null |
protected lowerSemicontinuous (s : Set α) :
LowerSemicontinuous fun f : α →ᵤ[s.image singleton] E => eVariationOn f s := fun f ↦ by
apply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun α E (s.image singleton)) id (𝓝 f) f s _
simpa only [UniformOnFun.tendsto_iff_tendstoUniformlyOn, mem_image, forall_exists_inde... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | lowerSemicontinuous | The map `(eVariationOn · s)` is lower semicontinuous for pointwise convergence *on `s`*.
Pointwise convergence on `s` is encoded here as uniform convergence on the family consisting of the
singletons of elements of `s`. |
lowerSemicontinuous_uniformOn (s : Set α) :
LowerSemicontinuous fun f : α →ᵤ[{s}] E => eVariationOn f s := fun f ↦ by
apply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun α E {s}) id (𝓝 f) f s _
have := @tendsto_id _ (𝓝 f)
rw [UniformOnFun.tendsto_iff_tendstoUniformlyOn] at this
simp_rw [← tendstoUniforml... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | lowerSemicontinuous_uniformOn | The map `(eVariationOn · s)` is lower semicontinuous for uniform convergence on `s`. |
_root_.BoundedVariationOn.dist_le {E : Type*} [PseudoMetricSpace E] {f : α → E}
{s : Set α} (h : BoundedVariationOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
dist (f x) (f y) ≤ (eVariationOn f s).toReal := by
rw [← ENNReal.ofReal_le_ofReal_iff ENNReal.toReal_nonneg, ENNReal.ofReal_toReal h, ← edist_dist]
e... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | _root_.BoundedVariationOn.dist_le | null |
_root_.BoundedVariationOn.sub_le {f : α → ℝ} {s : Set α} (h : BoundedVariationOn f s)
{x y : α} (hx : x ∈ s) (hy : y ∈ s) : f x - f y ≤ (eVariationOn f s).toReal := by
apply (le_abs_self _).trans
rw [← Real.dist_eq]
exact h.dist_le hx hy | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | _root_.BoundedVariationOn.sub_le | null |
add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
(us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j)) :... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | add_point | Consider a monotone function `u` parameterizing some points of a set `s`. Given `x ∈ s`, then
one can find another monotone function `v` parameterizing the same points as `u`, with `x` added.
In particular, the variation of a function along `u` is bounded by its variation along `v`. |
add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) := by
by_cases hs : s = ∅
· simp [hs]
have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } :=
nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs)
by_cases ht : t = ∅
· s... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | add_le_union | The variation of a function on the union of two sets `s` and `t`, with `s` to the left of `t`,
bounds the sum of the variations along `s` and `t`. |
union (f : α → E) {s t : Set α} {x : α} (hs : IsGreatest s x) (ht : IsLeast t x) :
eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t := by
classical
apply le_antisymm _ (eVariationOn.add_le_union f fun a ha b hb => le_trans (hs.2 ha) (ht.2 hb))
apply iSup_le _
rintro ⟨n, ⟨u, hu, ust⟩⟩
obtain ⟨v... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | union | If a set `s` is to the left of a set `t`, and both contain the boundary point `x`, then
the variation of `f` along `s ∪ t` is the sum of the variations. |
Icc_add_Icc (f : α → E) {s : Set α} {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) (hb : b ∈ s) :
eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c) := by
have A : IsGreatest (s ∩ Icc a b) b :=
⟨⟨hb, hab, le_rfl⟩, inter_subset_right.trans Icc_subset_Iic_self⟩
have B : IsLeas... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | Icc_add_Icc | null |
sum (f : α → E) {s : Set α} {E : ℕ → α} (hE : Monotone E) {n : ℕ}
(hn : ∀ i, 0 < i → i < n → E i ∈ s) :
∑ i ∈ Finset.range n, eVariationOn f (s ∩ Icc (E i) (E (i + 1))) =
eVariationOn f (s ∩ Icc (E 0) (E n)) := by
induction n with
| zero => simp [eVariationOn.subsingleton f Subsingleton.inter_singleto... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sum | null |
sum' (f : α → E) {I : ℕ → α} (hI : Monotone I) {n : ℕ} :
∑ i ∈ Finset.range n, eVariationOn f (Icc (I i) (I (i + 1)))
= eVariationOn f (Icc (I 0) (I n)) := by
convert sum f hI (s := Icc (I 0) (I n)) (n := n)
(hn := by intros; rw [mem_Icc]; constructor <;> (apply hI; omega) ) with i hi
· simp only [righ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sum' | null |
comp_le_of_monotoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
(φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s :=
iSup_le fun ⟨n, u, hu, ut⟩ =>
le_iSup_of_le ⟨n, φ ∘ u, fun x y xy => hφ (ut x) (ut y) (hu xy), fun i => φst (ut i)⟩ le_rfl | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_le_of_monotoneOn | null |
comp_le_of_antitoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
(φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s := by
refine iSup_le ?_
rintro ⟨n, u, hu, ut⟩
rw [← Finset.sum_range_reflect]
refine (Finset.sum_congr rfl fun x hx => ?_).trans_le <| le_iSup_of_le
... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_le_of_antitoneOn | null |
comp_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) :
eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) := by
apply le_antisymm (comp_le_of_monotoneOn f φ hφ (mapsTo_image φ t))
cases isEmpty_or_nonempty β
· convert zero_le (_ : ℝ≥0∞)
exact eVariationOn.subsingleton f <|
(... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_eq_of_monotoneOn | null |
comp_inter_Icc_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
{x y : β} (hx : x ∈ t) (hy : y ∈ t) :
eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y)) := by
rcases le_total x y with (h | h)
· convert comp_eq_of_monotoneOn f φ (hφ.mono Set.inter_subset_... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_inter_Icc_eq_of_monotoneOn | null |
comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) := by
apply le_antisymm (comp_le_of_antitoneOn f φ hφ (mapsTo_image φ t))
cases isEmpty_or_nonempty β
· convert zero_le (_ : ℝ≥0∞)
exact eVariationOn.subsingleton f <| (subsin... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_eq_of_antitoneOn | null |
comp_ofDual (f : α → E) (s : Set α) :
eVariationOn (f ∘ ofDual) (ofDual ⁻¹' s) = eVariationOn f s := by
convert comp_eq_of_antitoneOn f ofDual fun _ _ _ _ => id
simp only [Equiv.image_preimage] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_ofDual | null |
MonotoneOn.eVariationOn_le {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) {a b : α}
(as : a ∈ s) (bs : b ∈ s) : eVariationOn f (s ∩ Icc a b) ≤ ENNReal.ofReal (f b - f a) := by
apply iSup_le _
rintro ⟨n, ⟨u, hu, us⟩⟩
calc
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) =
∑ i ∈ Finset.range n... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | MonotoneOn.eVariationOn_le | null |
MonotoneOn.locallyBoundedVariationOn {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) :
LocallyBoundedVariationOn f s := fun _ _ as bs =>
((hf.eVariationOn_le as bs).trans_lt ENNReal.ofReal_lt_top).ne | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | MonotoneOn.locallyBoundedVariationOn | null |
noncomputable variationOnFromTo (f : α → E) (s : Set α) (a b : α) : ℝ :=
if a ≤ b then (eVariationOn f (s ∩ Icc a b)).toReal else -(eVariationOn f (s ∩ Icc b a)).toReal | def | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | variationOnFromTo | The **signed** variation of `f` on the interval `Icc a b` intersected with the set `s`,
squashed to a real (therefore only really meaningful if the variation is finite) |
protected self (a : α) : variationOnFromTo f s a a = 0 := by
dsimp only [variationOnFromTo]
rw [if_pos le_rfl, Icc_self, eVariationOn.subsingleton, ENNReal.toReal_zero]
exact fun x hx y hy => hx.2.trans hy.2.symm | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | self | null |
protected nonneg_of_le {a b : α} (h : a ≤ b) : 0 ≤ variationOnFromTo f s a b := by
simp only [variationOnFromTo, if_pos h, ENNReal.toReal_nonneg] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | nonneg_of_le | null |
protected eq_neg_swap (a b : α) :
variationOnFromTo f s a b = -variationOnFromTo f s b a := by
rcases lt_trichotomy a b with (ab | rfl | ba)
· simp only [variationOnFromTo, if_pos ab.le, if_neg ab.not_ge, neg_neg]
· simp only [variationOnFromTo.self, neg_zero]
· simp only [variationOnFromTo, if_pos ba.le, i... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_neg_swap | null |
protected nonpos_of_ge {a b : α} (h : b ≤ a) : variationOnFromTo f s a b ≤ 0 := by
rw [variationOnFromTo.eq_neg_swap]
exact neg_nonpos_of_nonneg (variationOnFromTo.nonneg_of_le f s h) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | nonpos_of_ge | null |
protected eq_of_le {a b : α} (h : a ≤ b) :
variationOnFromTo f s a b = (eVariationOn f (s ∩ Icc a b)).toReal :=
if_pos h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_of_le | null |
protected eq_of_ge {a b : α} (h : b ≤ a) :
variationOnFromTo f s a b = -(eVariationOn f (s ∩ Icc b a)).toReal := by
rw [variationOnFromTo.eq_neg_swap, neg_inj, variationOnFromTo.eq_of_le f s h] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_of_ge | null |
protected add {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b c : α}
(ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c := by
symm
refine additive_of_isTotal (· ≤ · : α → α → Prop) (variationOnFromTo f s) (· ∈ s) ?_ ?_... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | add | null |
protected edist_zero_of_eq_zero (hf : LocallyBoundedVariationOn f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : variationOnFromTo f s a b = 0) :
edist (f a) (f b) = 0 := by
wlog h' : a ≤ b
· rw [edist_comm]
apply this hf hb ha _ (le_of_not_ge h')
rw [variationOnFromTo.eq_neg_swap, h, neg_zero]
· app... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | edist_zero_of_eq_zero | null |
protected eq_left_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
{a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0 := by
simp only [← variationOnFromTo.add hf ha hb hc, left_eq_add] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_left_iff | null |
protected eq_zero_iff_of_le {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (ab : a ≤ b) :
variationOnFromTo f s a b = 0 ↔
∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc a b) ⦃y⦄ (_hy : y ∈ s ∩ Icc a b), edist (f x) (f y) = 0 := by
rw [variationOnFromTo.eq_of_le _ _ ab, ENNRea... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_zero_iff_of_le | null |
protected eq_zero_iff_of_ge {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
{a b : α} (ha : a ∈ s) (hb : b ∈ s) (ba : b ≤ a) :
variationOnFromTo f s a b = 0 ↔
∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc b a) ⦃y⦄ (_hy : y ∈ s ∩ Icc b a), edist (f x) (f y) = 0 := by
rw [variationOnFromTo.eq_of_ge _ _ ba, neg_eq... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_zero_iff_of_ge | null |
protected eq_zero_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α}
(ha : a ∈ s) (hb : b ∈ s) :
variationOnFromTo f s a b = 0 ↔
∀ ⦃x⦄ (_hx : x ∈ s ∩ uIcc a b) ⦃y⦄ (_hy : y ∈ s ∩ uIcc a b), edist (f x) (f y) = 0 := by
rcases le_total a b with (ab | ba)
· rw [uIcc_of_le ab]
... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | eq_zero_iff | null |
protected monotoneOn (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
MonotoneOn (variationOnFromTo f s a) s := by
rintro b bs c cs bc
rw [← variationOnFromTo.add hf as bs cs]
exact le_add_of_nonneg_right (variationOnFromTo.nonneg_of_le f s bc) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | monotoneOn | null |
protected antitoneOn (hf : LocallyBoundedVariationOn f s) {b : α} (bs : b ∈ s) :
AntitoneOn (fun a => variationOnFromTo f s a b) s := by
rintro a as c cs ac
dsimp only
rw [← variationOnFromTo.add hf as cs bs]
exact le_add_of_nonneg_left (variationOnFromTo.nonneg_of_le f s ac) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | antitoneOn | null |
protected sub_self_monotoneOn {f : α → ℝ} {s : Set α} (hf : LocallyBoundedVariationOn f s)
{a : α} (as : a ∈ s) : MonotoneOn (variationOnFromTo f s a - f) s := by
rintro b bs c cs bc
rw [Pi.sub_apply, Pi.sub_apply, le_sub_iff_add_le, add_comm_sub, ← le_sub_iff_add_le']
calc
f c - f b ≤ |f c - f b| := le_a... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | sub_self_monotoneOn | null |
protected comp_eq_of_monotoneOn {β : Type*} [LinearOrder β] (f : α → E) {t : Set β}
(φ : β → α) (hφ : MonotoneOn φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
variationOnFromTo (f ∘ φ) t x y = variationOnFromTo f (φ '' t) (φ x) (φ y) := by
rcases le_total x y with (h | h)
· rw [variationOnFromTo.eq_of_le _ _ h... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | comp_eq_of_monotoneOn | null |
LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn {f : α → ℝ} {s : Set α}
(h : LocallyBoundedVariationOn f s) :
∃ p q : α → ℝ, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - q := by
rcases eq_empty_or_nonempty s with (rfl | ⟨c, cs⟩)
· exact ⟨f, 0, subsingleton_empty.monotoneOn _, subsingleton_empty.mono... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn | If a real-valued function has bounded variation on a set, then it is a difference of monotone
functions there. |
LipschitzOnWith.comp_eVariationOn_le {f : E → F} {C : ℝ≥0} {t : Set E}
(h : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t) :
eVariationOn (f ∘ g) s ≤ C * eVariationOn g s := by
apply iSup_le _
rintro ⟨n, ⟨u, hu, us⟩⟩
calc
(∑ i ∈ Finset.range n, edist (f (g (u (i + 1)))) (f (g (u i)... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzOnWith.comp_eVariationOn_le | null |
LipschitzOnWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
(hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
(h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s :=
ne_top_of_le_ne_top (by finiteness) (hf.comp_eVariationOn_le hg) | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzOnWith.comp_boundedVariationOn | null |
LipschitzOnWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0} {t : Set E}
(hf : LipschitzOnWith C f t) {g : α → E} {s : Set α} (hg : MapsTo g s t)
(h : LocallyBoundedVariationOn g s) : LocallyBoundedVariationOn (f ∘ g) s :=
fun x y xs ys =>
hf.comp_boundedVariationOn (hg.mono_left inter_subset_left) ... | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzOnWith.comp_locallyBoundedVariationOn | null |
LipschitzWith.comp_boundedVariationOn {f : E → F} {C : ℝ≥0} (hf : LipschitzWith C f)
{g : α → E} {s : Set α} (h : BoundedVariationOn g s) : BoundedVariationOn (f ∘ g) s :=
hf.lipschitzOnWith.comp_boundedVariationOn (mapsTo_univ _ _) h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzWith.comp_boundedVariationOn | null |
LipschitzWith.comp_locallyBoundedVariationOn {f : E → F} {C : ℝ≥0}
(hf : LipschitzWith C f) {g : α → E} {s : Set α} (h : LocallyBoundedVariationOn g s) :
LocallyBoundedVariationOn (f ∘ g) s :=
hf.lipschitzOnWith.comp_locallyBoundedVariationOn (mapsTo_univ _ _) h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzWith.comp_locallyBoundedVariationOn | null |
LipschitzOnWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} {s : Set ℝ}
(hf : LipschitzOnWith C f s) : LocallyBoundedVariationOn f s :=
hf.comp_locallyBoundedVariationOn (mapsTo_id _)
(@monotoneOn_id ℝ _ s).locallyBoundedVariationOn | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzOnWith.locallyBoundedVariationOn | null |
LipschitzWith.locallyBoundedVariationOn {f : ℝ → E} {C : ℝ≥0} (hf : LipschitzWith C f)
(s : Set ℝ) : LocallyBoundedVariationOn f s :=
hf.lipschitzOnWith.locallyBoundedVariationOn | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Tactic.Finiteness",
"Mathlib.Topology.Semicontinuous",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | LipschitzWith.locallyBoundedVariationOn | null |
uniformity_dist_of_mem_uniformity [LT β] {U : Filter (α × α)} (z : β)
(D : α → α → β) (H : ∀ s, s ∈ U ↔ ∃ ε > z, ∀ {a b : α}, D a b < ε → (a, b) ∈ s) :
U = ⨅ ε > z, 𝓟 { p : α × α | D p.1 p.2 < ε } :=
HasBasis.eq_biInf ⟨fun s => by simp only [H, subset_def, Prod.forall, mem_setOf]⟩
open scoped Uniformity Topo... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_dist_of_mem_uniformity | Characterizing uniformities associated to a (generalized) distance function `D`
in terms of the elements of the uniformity. |
@[ext]
EDist (α : Type*) where
/-- Extended distance between two points -/
edist : α → α → ℝ≥0∞
export EDist (edist) | class | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EDist | `EDist α` means that `α` is equipped with an extended distance. |
@[reducible] uniformSpaceOfEDist (edist : α → α → ℝ≥0∞) (edist_self : ∀ x : α, edist x x = 0)
(edist_comm : ∀ x y : α, edist x y = edist y x)
(edist_triangle : ∀ x y z : α, edist x z ≤ edist x y + edist y z) : UniformSpace α :=
.ofFun edist edist_self edist_comm edist_triangle fun ε ε0 =>
⟨ε / 2, ENNReal.... | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformSpaceOfEDist | Creating a uniform space from an extended distance. |
@[reducible] noncomputable uniformSpaceOfEDistOfHasBasis [TopologicalSpace α]
(edist : α → α → ℝ≥0∞)
(edist_self : ∀ x : α, edist x x = 0)
(edist_comm : ∀ x y : α, edist x y = edist y x)
(edist_triangle : ∀ x y z : α, edist x z ≤ edist x y + edist y z)
(basis : ∀ x, (𝓝 x).HasBasis (fun c ↦ 0 < c) (... | def | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformSpaceOfEDistOfHasBasis | Creating a uniform space from an extended distance. We assume that
there is a preexisting topology, for which the neighborhoods can be expressed using the distance,
and we make sure that the uniform space structure we construct has a topology which is defeq
to the original one. |
PseudoEMetricSpace (α : Type u) : Type u extends EDist α where
edist_self : ∀ x : α, edist x x = 0
edist_comm : ∀ x y : α, edist x y = edist y x
edist_triangle : ∀ x y z : α, edist x z ≤ edist x y + edist y z
toUniformSpace : UniformSpace α := uniformSpaceOfEDist edist edist_self edist_comm edist_triangle
un... | class | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | PseudoEMetricSpace | A pseudo extended metric space is a type endowed with a `ℝ≥0∞`-valued distance `edist`
satisfying reflexivity `edist x x = 0`, commutativity `edist x y = edist y x`, and the triangle
inequality `edist x z ≤ edist x y + edist y z`.
Note that we do not require `edist x y = 0 → x = y`. See extended metric spaces (`EMetri... |
@[ext]
protected PseudoEMetricSpace.ext {α : Type*} {m m' : PseudoEMetricSpace α}
(h : m.toEDist = m'.toEDist) : m = m' := by
obtain ⟨_, _, _, U, hU⟩ := m; rename EDist α => ed
obtain ⟨_, _, _, U', hU'⟩ := m'; rename EDist α => ed'
congr 1
exact UniformSpace.ext (((show ed = ed' from h) ▸ hU).trans hU'.symm... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | PseudoEMetricSpace.ext | Two pseudo emetric space structures with the same edistance function coincide. |
edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y := by
rw [edist_comm z]; apply edist_triangle | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_triangle_left | Triangle inequality for the extended distance |
edist_triangle_right (x y z : α) : edist x y ≤ edist x z + edist y z := by
rw [edist_comm y]; apply edist_triangle | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_triangle_right | null |
edist_congr_right {x y z : α} (h : edist x y = 0) : edist x z = edist y z := by
apply le_antisymm
· rw [← zero_add (edist y z), ← h]
apply edist_triangle
· rw [edist_comm] at h
rw [← zero_add (edist x z), ← h]
apply edist_triangle | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_congr_right | null |
edist_congr_left {x y z : α} (h : edist x y = 0) : edist z x = edist z y := by
rw [edist_comm z x, edist_comm z y]
apply edist_congr_right h | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_congr_left | null |
edist_congr {w x y z : α} (hl : edist w x = 0) (hr : edist y z = 0) :
edist w y = edist x z :=
(edist_congr_right hl).trans (edist_congr_left hr) | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_congr | null |
edist_triangle4 (x y z t : α) : edist x t ≤ edist x y + edist y z + edist z t :=
calc
edist x t ≤ edist x z + edist z t := edist_triangle x z t
_ ≤ edist x y + edist y z + edist z t := add_le_add_right (edist_triangle x y z) _ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_triangle4 | null |
uniformity_pseudoedist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | 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_pseudoedist | Reformulation of the uniform structure in terms of the extended distance |
uniformSpace_edist :
‹PseudoEMetricSpace α›.toUniformSpace =
uniformSpaceOfEDist edist edist_self edist_comm edist_triangle :=
UniformSpace.ext uniformity_pseudoedist | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformSpace_edist | null |
uniformity_basis_edist :
(𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p : α × α | edist p.1 p.2 < ε } :=
(@uniformSpace_edist α _).symm ▸ UniformSpace.hasBasis_ofFun ⟨1, one_pos⟩ _ _ _ _ _ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist | null |
mem_uniformity_edist {s : Set (α × α)} :
s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ {a b : α}, edist a b < ε → (a, b) ∈ s :=
uniformity_basis_edist.mem_uniformity_iff | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | mem_uniformity_edist | Characterization of the elements of the uniformity in terms of the extended distance |
protected EMetric.mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ≥0∞}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
(𝓤 α).HasBasis p fun x => { p : α × α | edist p.1 p.2 < f x } := by
refine ⟨fun s => uniformity_basis_edist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀, hε... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetric.mk_uniformity_basis | Given `f : β → ℝ≥0∞`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_edist`, `uniformity_basis_edist'`,
`uniformity_basis_edist_nnreal`, and `uniformity_basis_edist_inv_nat`. |
protected EMetric.mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ≥0∞}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
(𝓤 α).HasBasis p fun x => { p : α × α | edist p.1 p.2 ≤ f x } := by
refine ⟨fun s => uniformity_basis_edist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀,... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | EMetric.mk_uniformity_basis_le | Given `f : β → ℝ≥0∞`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_edist_le` and `uniformity_basis_edist_le'`. |
uniformity_basis_edist_le :
(𝓤 α).HasBasis (fun ε : ℝ≥0∞ => 0 < ε) fun ε => { p : α × α | edist p.1 p.2 ≤ ε } :=
EMetric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_le | null |
uniformity_basis_edist' (ε' : ℝ≥0∞) (hε' : 0 < ε') :
(𝓤 α).HasBasis (fun ε : ℝ≥0∞ => ε ∈ Ioo 0 ε') fun ε => { p : α × α | edist p.1 p.2 < ε } :=
EMetric.mk_uniformity_basis (fun _ => And.left) fun ε ε₀ =>
let ⟨δ, hδ⟩ := exists_between hε'
⟨min ε δ, ⟨lt_min ε₀ hδ.1, lt_of_le_of_lt (min_le_right _ _) hδ.2⟩... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist' | null |
uniformity_basis_edist_le' (ε' : ℝ≥0∞) (hε' : 0 < ε') :
(𝓤 α).HasBasis (fun ε : ℝ≥0∞ => ε ∈ Ioo 0 ε') fun ε => { p : α × α | edist p.1 p.2 ≤ ε } :=
EMetric.mk_uniformity_basis_le (fun _ => And.left) fun ε ε₀ =>
let ⟨δ, hδ⟩ := exists_between hε'
⟨min ε δ, ⟨lt_min ε₀ hδ.1, lt_of_le_of_lt (min_le_right _ _)... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_le' | null |
uniformity_basis_edist_nnreal :
(𝓤 α).HasBasis (fun ε : ℝ≥0 => 0 < ε) fun ε => { p : α × α | edist p.1 p.2 < ε } :=
EMetric.mk_uniformity_basis (fun _ => ENNReal.coe_pos.2) fun _ε ε₀ =>
let ⟨δ, hδ⟩ := ENNReal.lt_iff_exists_nnreal_btwn.1 ε₀
⟨δ, ENNReal.coe_pos.1 hδ.1, le_of_lt hδ.2⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_nnreal | null |
uniformity_basis_edist_nnreal_le :
(𝓤 α).HasBasis (fun ε : ℝ≥0 => 0 < ε) fun ε => { p : α × α | edist p.1 p.2 ≤ ε } :=
EMetric.mk_uniformity_basis_le (fun _ => ENNReal.coe_pos.2) fun _ε ε₀ =>
let ⟨δ, hδ⟩ := ENNReal.lt_iff_exists_nnreal_btwn.1 ε₀
⟨δ, ENNReal.coe_pos.1 hδ.1, le_of_lt hδ.2⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_nnreal_le | null |
uniformity_basis_edist_inv_nat :
(𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | edist p.1 p.2 < (↑n)⁻¹ } :=
EMetric.mk_uniformity_basis (fun n _ ↦ ENNReal.inv_pos.2 <| ENNReal.natCast_ne_top n) fun _ε ε₀ ↦
let ⟨n, hn⟩ := ENNReal.exists_inv_nat_lt (ne_of_gt ε₀)
⟨n, trivial, le_of_lt hn⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_inv_nat | null |
uniformity_basis_edist_inv_two_pow :
(𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | edist p.1 p.2 < 2⁻¹ ^ n } :=
EMetric.mk_uniformity_basis (fun _ _ ↦ ENNReal.pow_pos (ENNReal.inv_pos.2 ENNReal.ofNat_ne_top) _)
fun _ε ε₀ ↦
let ⟨n, hn⟩ := ENNReal.exists_inv_two_pow_lt (ne_of_gt ε₀)
⟨n, tri... | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformity_basis_edist_inv_two_pow | null |
edist_mem_uniformity {ε : ℝ≥0∞} (ε0 : 0 < ε) : { p : α × α | edist p.1 p.2 < ε } ∈ 𝓤 α :=
mem_uniformity_edist.2 ⟨ε, ε0, id⟩ | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | edist_mem_uniformity | Fixed size neighborhoods of the diagonal belong to the uniform structure |
uniformContinuousOn_iff [PseudoEMetricSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔
∀ ε > 0, ∃ δ > 0, ∀ {a}, a ∈ s → ∀ {b}, b ∈ s → edist a b < δ → edist (f a) (f b) < ε :=
uniformity_basis_edist.uniformContinuousOn_iff uniformity_basis_edist | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformContinuousOn_iff | ε-δ characterization of uniform continuity on a set for pseudoemetric spaces |
uniformContinuous_iff [PseudoEMetricSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε :=
uniformity_basis_edist.uniformContinuous_iff uniformity_basis_edist | theorem | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | uniformContinuous_iff | ε-δ characterization of uniform continuity on pseudoemetric spaces |
PseudoEMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoEMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoEMetricSpace α where
edist := @edist _ m.toEDist
edist_self := edist_self
edist_comm := edist_comm
edist_triangle := edist_triangle
toUniformSpace := U
uni... | abbrev | Topology | [
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.OfFun"
] | Mathlib/Topology/EMetricSpace/Defs.lean | PseudoEMetricSpace.replaceUniformity | Auxiliary function to replace the uniformity on a pseudoemetric space with
a uniformity which is equal to the original one, but maybe not defeq.
This is useful if one wants to construct a pseudoemetric space with a
specified uniformity. See Note [forgetful inheritance] explaining why having definitionally
the right uni... |
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