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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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 }
/-- The product of two emetric spaces, with the max distance, is an extended
metric spaces. We make sure that the uniform structure thus constructed is the one
corresponding to the product of uniform spaces, to avoid diamond problems. |
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_iff.1 hx ε ε0 with ⟨y, hyt, hxy⟩
exact mem_iUnion₂.2 ⟨y, hyt, mem_closedBall.2 hxy.le⟩
exact subset_countable_closure_of_almost_dense_set _ this | 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, Subtype.forall] | 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_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_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 n hmn
let v i := u (π i)
calc
∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
= ∑ i ∈ Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_congr rfl fun i hi ↦ by
rw [Finset.mem_Ico] at hi
simp only [v, π, projIcc_of_mem hmn ⟨hi.1, hi.2.le⟩,
projIcc_of_mem hmn ⟨hi.1.trans i.le_succ, hi.2⟩]
_ ≤ ∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_mono_set _ (Nat.Iio_eq_range ▸ Finset.Ico_subset_Iio_self)
_ ≤ eVariationOn f s :=
sum_le _ _ (fun i j h ↦ hu (π i).2 (π j).2 (monotone_projIcc hmn h)) fun i ↦ us _ (π i).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_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
| 0 => hxy
| (_ + 1) => le_rfl
have us : ∀ i, u i ∈ s := fun
| 0 => hy
| (_ + 1) => hx
simpa only [Finset.sum_range_one] using sum_le f 1 hu us | 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.sum_eq_zero fun i _ => h _ (us i.succ) _ (us i) | 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 ∧ ∀ i, u i ∈ s },
v < ∑ i ∈ Finset.range p.1, edist (f ((p.2 : ℕ → α) (i + 1))) (f ((p.2 : ℕ → α) i)) :=
lt_iSup_iff.mp hv
have : Tendsto (fun j => ∑ i ∈ Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p
(𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)))) := by
apply tendsto_finset_sum
exact fun i _ => Tendsto.edist (Ffs (u i.succ) (us i.succ)) (Ffs (u i) (us i))
exact (this.eventually_const_lt hlt).mono fun i h => h.trans_le (sum_le (F i) n um us) | 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_index, and_imp,
forall_apply_eq_imp_iff₂, tendstoUniformlyOn_singleton_iff_tendsto] using @tendsto_id _ (𝓝 f) | 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 [← tendstoUniformlyOn_singleton_iff_tendsto]
exact fun x xs => (this s rfl).mono (singleton_subset_iff.mpr xs) | 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]
exact edist_le f 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.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)) := by
rcases le_or_gt (u n) x with (h | h)
· let v i := if i ≤ n then u i else x
have vs : ∀ i, v i ∈ s := fun i ↦ by
simp only [v]
split_ifs
· exact us i
· exact hx
have hv : Monotone v := by
refine monotone_nat_of_le_succ fun i => ?_
simp only [v]
rcases lt_trichotomy i n with (hi | rfl | hi)
· have : i + 1 ≤ n := Nat.succ_le_of_lt hi
simp only [hi.le, this, if_true]
exact hu (Nat.le_succ i)
· simp only [le_refl, if_true, add_le_iff_nonpos_right, Nat.le_zero, Nat.one_ne_zero,
if_false, h]
· have A : ¬i ≤ n := hi.not_ge
have B : ¬i + 1 ≤ n := fun h => A (i.le_succ.trans h)
simp only [A, B, if_false, le_rfl]
refine ⟨v, n + 2, hv, vs, (mem_image _ _ _).2 ⟨n + 1, ?_, ?_⟩, ?_⟩
· rw [mem_Iio]; exact Nat.lt_succ_self (n + 1)
· have : ¬n + 1 ≤ n := Nat.not_succ_le_self n
simp only [v, this, ite_eq_right_iff, IsEmpty.forall_iff]
· calc
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) =
∑ i ∈ Finset.range n, edist (f (v (i + 1))) (f (v i)) := by
apply Finset.sum_congr rfl fun i hi => ?_
simp only [Finset.mem_range] at hi
have : i + 1 ≤ n := Nat.succ_le_of_lt hi
simp only [v, hi.le, this, if_true]
_ ≤ ∑ j ∈ Finset.range (n + 2), edist (f (v (j + 1))) (f (v j)) := by
gcongr
apply Nat.le_add_right
have exists_N : ∃ N, N ≤ n ∧ x < u N := ⟨n, le_rfl, h⟩
let N := Nat.find exists_N
have hN : N ≤ n ∧ x < u N := Nat.find_spec exists_N
let w : ℕ → α := fun i => if i < N then u i else if i = N then x else u (i - 1)
have ws : ∀ i, w i ∈ s := by grind
have hw : Monotone w := by
apply monotone_nat_of_le_succ fun i => ?_
dsimp only [w]
rcases lt_trichotomy (i + 1) N with (hi | hi | hi)
· have : i < N := Nat.lt_of_le_of_lt (Nat.le_succ i) hi
simp only [hi, this, if_true]
exact hu (Nat.le_succ _)
· have A : i < N := hi ▸ i.lt_succ_self
... | 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 = ∅
· simp [ht]
have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ t } :=
nonempty_monotone_mem (nonempty_iff_ne_empty.2 ht)
refine ENNReal.iSup_add_iSup_le ?_
/- We start from two sequences `u` and `v` along `s` and `t` respectively, and we build a new
sequence `w` along `s ∪ t` by juxtaposing them. Its variation is larger than the sum of the
variations. -/
rintro ⟨n, ⟨u, hu, us⟩⟩ ⟨m, ⟨v, hv, vt⟩⟩
let w i := if i ≤ n then u i else v (i - (n + 1))
have wst : ∀ i, w i ∈ s ∪ t := by
intro i
by_cases hi : i ≤ n
· simp [w, hi, us]
· simp [w, hi, vt]
have hw : Monotone w := by
intro i j hij
dsimp only [w]
split_ifs with h_1 h_2 h_2
· exact hu hij
· apply h _ (us _) _ (vt _)
· exfalso; exact h_1 (hij.trans h_2)
· apply hv (tsub_le_tsub hij le_rfl)
calc
((∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) +
∑ i ∈ Finset.range m, edist (f (v (i + 1))) (f (v i))) =
(∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i))) +
∑ i ∈ Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) := by
dsimp only [w]
congr 1
· grind [Finset.sum_congr]
· grind
_ = (∑ i ∈ Finset.range n, edist (f (w (i + 1))) (f (w i))) +
∑ i ∈ Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
congr 1
rw [Finset.range_eq_Ico]
convert Finset.sum_Ico_add (fun i : ℕ => edist (f (w (i + 1))) (f (w i))) 0 m (n + 1)
using 3 <;> abel
_ ≤ ∑ i ∈ Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
rw [← Finset.sum_union]
· gcongr
rintro i hi
simp only [Finset.mem_union, Finset.mem_range, Finset.mem_Ico] at hi ⊢
cutsat
· refine Finset.disjoint_left.2 fun i hi h'i => ?_
... | 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, m, hv, vst, xv, huv⟩ : ∃ (v : ℕ → α) (m : ℕ),
Monotone v ∧ (∀ i, v i ∈ s ∪ t) ∧ 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)) :=
eVariationOn.add_point f (mem_union_left t hs.1) u hu ust n
obtain ⟨N, hN, Nx⟩ : ∃ N, N < m ∧ v N = x := xv
calc
(∑ j ∈ Finset.range n, edist (f (u (j + 1))) (f (u j))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j)) :=
huv
_ = (∑ j ∈ Finset.Ico 0 N, edist (f (v (j + 1))) (f (v j))) +
∑ j ∈ Finset.Ico N m, edist (f (v (j + 1))) (f (v j)) := by
rw [Finset.range_eq_Ico, Finset.sum_Ico_consecutive _ (zero_le _) hN.le]
_ ≤ eVariationOn f s + eVariationOn f t := by
refine add_le_add ?_ ?_
· apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_
rcases vst i with (h | h); · exact h
have : v i = x := by
apply le_antisymm
· rw [← Nx]; exact hv hi.2
· exact ht.2 h
rw [this]
exact hs.1
· apply sum_le_of_monotoneOn_Icc _ (hv.monotoneOn _) fun i hi => ?_
rcases vst i with (h | h); swap; · exact h
have : v i = x := by
apply le_antisymm
· exact hs.2 h
· rw [← Nx]; exact hv hi.1
rw [this]
exact ht.1 | 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 : IsLeast (s ∩ Icc b c) b :=
⟨⟨hb, le_rfl, hbc⟩, inter_subset_right.trans Icc_subset_Ici_self⟩
rw [← eVariationOn.union f A B, ← inter_union_distrib_left, Icc_union_Icc_eq_Icc hab hbc] | 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_singleton]
| succ n ih =>
by_cases hn₀ : n = 0
· simp [hn₀]
rw [← Icc_add_Icc (b := E n)]
· rw [← ih (by intros; apply hn <;> omega)]
apply Finset.sum_range_succ
· apply hE; omega
· apply hE; omega
· apply hn <;> omega | 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 [right_eq_inter]
gcongr <;> (apply hI; rw [Finset.mem_range] at hi; omega)
· simp | 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
⟨n, fun i => φ (u <| n - i), fun x y xy => hφ (ut _) (ut _) (hu <| Nat.sub_le_sub_left xy n),
fun i => φst (ut _)⟩
le_rfl
rw [Finset.mem_range] at hx
dsimp only [Subtype.coe_mk, Function.comp_apply]
rw [edist_comm]
congr 4 <;> omega | 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 <|
(subsingleton_of_subsingleton.image _).anti (surjOn_image φ t)
let ψ := φ.invFunOn t
have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image φ t).rightInvOn_invFunOn
have ψts : MapsTo ψ (φ '' t) t := (surjOn_image φ t).mapsTo_invFunOn
have hψ : MonotoneOn ψ (φ '' t) := Function.monotoneOn_of_rightInvOn_of_mapsTo hφ ψφs ψts
change eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t
rw [← eq_of_eqOn (ψφs.comp_left : EqOn (f ∘ φ ∘ ψ) (f ∘ id) (φ '' t))]
exact comp_le_of_monotoneOn _ ψ hψ ψts | 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_left)
apply le_antisymm
· rintro _ ⟨⟨u, us, rfl⟩, vφx, vφy⟩
rcases le_total x u with (xu | ux)
· rcases le_total u y with (uy | yu)
· exact ⟨u, ⟨us, ⟨xu, uy⟩⟩, rfl⟩
· rw [le_antisymm vφy (hφ hy us yu)]
exact ⟨y, ⟨hy, ⟨h, le_rfl⟩⟩, rfl⟩
· rw [← le_antisymm vφx (hφ us hx ux)]
exact ⟨x, ⟨hx, ⟨le_rfl, h⟩⟩, rfl⟩
· rintro _ ⟨u, ⟨⟨hu, xu, uy⟩, rfl⟩⟩
exact ⟨⟨u, hu, rfl⟩, ⟨hφ hx hu xu, hφ hu hy uy⟩⟩
· rw [eVariationOn.subsingleton, eVariationOn.subsingleton]
exacts [(Set.subsingleton_Icc_of_ge (hφ hy hx h)).anti Set.inter_subset_right,
(Set.subsingleton_Icc_of_ge h).anti Set.inter_subset_right] | 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 <| (subsingleton_of_subsingleton.image _).anti
(surjOn_image φ t)
let ψ := φ.invFunOn t
have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image φ t).rightInvOn_invFunOn
have ψts := (surjOn_image φ t).mapsTo_invFunOn
have hψ : AntitoneOn ψ (φ '' t) := Function.antitoneOn_of_rightInvOn_of_mapsTo hφ ψφs ψts
change eVariationOn (f ∘ id) (φ '' t) ≤ eVariationOn (f ∘ φ) t
rw [← eq_of_eqOn (ψφs.comp_left : EqOn (f ∘ φ ∘ ψ) (f ∘ id) (φ '' t))]
exact comp_le_of_antitoneOn _ ψ hψ ψts
open OrderDual | 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, ENNReal.ofReal (f (u (i + 1)) - f (u i)) := by
refine Finset.sum_congr rfl fun i hi => ?_
simp only [Finset.mem_range] at hi
rw [edist_dist, Real.dist_eq, abs_of_nonneg]
exact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))
_ = ENNReal.ofReal (∑ i ∈ Finset.range n, (f (u (i + 1)) - f (u i))) := by
rw [ENNReal.ofReal_sum_of_nonneg]
intro i _
exact sub_nonneg_of_le (hf (us i).1 (us (i + 1)).1 (hu (Nat.le_succ _)))
_ = ENNReal.ofReal (f (u n) - f (u 0)) := by rw [Finset.sum_range_sub fun i => f (u i)]
_ ≤ ENNReal.ofReal (f b - f a) := by
apply ENNReal.ofReal_le_ofReal
exact sub_le_sub (hf (us n).1 bs (us n).2.2) (hf as (us 0).1 (us 0).2.1) | 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, if_neg ba.not_ge] | 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) ?_ ?_ ha hb hc
· rintro x y _xs _ys
simp only [variationOnFromTo.eq_neg_swap f s y x, add_neg_cancel]
· rintro x y z xy yz xs ys zs
rw [variationOnFromTo.eq_of_le f s xy, variationOnFromTo.eq_of_le f s yz,
variationOnFromTo.eq_of_le f s (xy.trans yz),
← ENNReal.toReal_add (hf x y xs ys) (hf y z ys zs), eVariationOn.Icc_add_Icc f xy yz ys]
variable {f s} in | 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]
· apply le_antisymm _ (zero_le _)
rw [← ENNReal.ofReal_zero, ← h, variationOnFromTo.eq_of_le f s h',
ENNReal.ofReal_toReal (hf a b ha hb)]
apply eVariationOn.edist_le
exacts [⟨ha, ⟨le_rfl, h'⟩⟩, ⟨hb, ⟨h', le_rfl⟩⟩] | 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, ENNReal.toReal_eq_zero_iff, or_iff_left (hf a b ha hb),
eVariationOn.eq_zero_iff] | 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_zero, ENNReal.toReal_eq_zero_iff,
or_iff_left (hf b a hb ha), eVariationOn.eq_zero_iff] | 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]
exact variationOnFromTo.eq_zero_iff_of_le hf ha hb ab
· rw [uIcc_of_ge ba]
exact variationOnFromTo.eq_zero_iff_of_ge hf ha hb ba
variable {f} {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 |
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_abs_self _
_ = dist (f b) (f c) := by rw [dist_comm, Real.dist_eq]
_ ≤ variationOnFromTo f s b c := by
rw [variationOnFromTo.eq_of_le f s bc, dist_edist]
apply ENNReal.toReal_mono (hf b c bs cs)
apply eVariationOn.edist_le f
exacts [⟨bs, le_rfl, bc⟩, ⟨cs, bc, le_rfl⟩]
_ = variationOnFromTo f s a c - variationOnFromTo f s a b := by
rw [← variationOnFromTo.add hf as bs cs, add_sub_cancel_left] | 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, variationOnFromTo.eq_of_le _ _ (hφ hx hy h),
eVariationOn.comp_inter_Icc_eq_of_monotoneOn f φ hφ hx hy]
· rw [variationOnFromTo.eq_of_ge _ _ h, variationOnFromTo.eq_of_ge _ _ (hφ hy hx h),
eVariationOn.comp_inter_Icc_eq_of_monotoneOn f φ hφ hy hx] | 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.monotoneOn _,
(sub_zero f).symm⟩
· exact ⟨_, _, variationOnFromTo.monotoneOn h cs, variationOnFromTo.sub_self_monotoneOn h cs,
(sub_sub_cancel _ _).symm⟩
/-! ## Lipschitz functions and bounded variation -/ | 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)))) ≤
∑ i ∈ Finset.range n, C * edist (g (u (i + 1))) (g (u i)) :=
Finset.sum_le_sum fun i _ => h (hg (us _)) (hg (us _))
_ = C * ∑ i ∈ Finset.range n, edist (g (u (i + 1))) (g (u i)) := by rw [Finset.mul_sum]
_ ≤ C * eVariationOn g s := mul_le_mul_left' (eVariationOn.sum_le _ _ hu us) _ | 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) (h x y xs ys) | 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 Topology Filter NNReal ENNReal Pointwise | 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.half_pos ε0.ne', fun _ h₁ _ h₂ =>
(ENNReal.add_lt_add h₁ h₂).trans_eq (ENNReal.add_halves _)⟩ | 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) (fun c ↦ {y | edist x y < c})) :
UniformSpace α :=
.ofFunOfHasBasis edist edist_self edist_comm edist_triangle (fun ε ε0 =>
⟨ε / 2, ENNReal.half_pos ε0.ne', fun _ h₁ _ h₂ =>
(ENNReal.add_lt_add h₁ h₂).trans_eq (ENNReal.add_halves _)⟩) basis | 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
uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } := by rfl
attribute [instance] PseudoEMetricSpace.toUniformSpace
/- Pseudoemetric spaces are less common than metric spaces. Therefore, we work in a dedicated
namespace, while notions associated to metric spaces are mostly in the root namespace. -/ | 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 (`EMetricSpace`) for
the similar class with that stronger assumption.
Any pseudo extended metric space is a topological space and a uniform space (see `TopologicalSpace`,
`UniformSpace`), where the topology and uniformity come from the metric.
Note that a T1 pseudo extended metric space is just an extended metric space.
We make the uniformity/topology part of the data instead of deriving it from the metric. This e.g.
ensures that we do not get a diamond when doing
`[PseudoEMetricSpace α] [PseudoEMetricSpace β] : TopologicalSpace (α × β)`:
The product metric and product topology agree, but not definitionally so.
See Note [forgetful inheritance]. |
@[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)
variable [PseudoEMetricSpace α]
export PseudoEMetricSpace (edist_self edist_comm edist_triangle)
attribute [simp] edist_self | 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⟩
rcases hf ε ε₀ with ⟨i, hi, H⟩
exact ⟨i, hi, fun x hx => hε <| lt_of_lt_of_le hx.out H⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩ | 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, hε⟩
rcases exists_between ε₀ with ⟨ε', hε'⟩
rcases hf ε' hε'.1 with ⟨i, hi, H⟩
exact ⟨i, hi, fun x hx => hε <| lt_of_le_of_lt (le_trans hx.out H) hε'.2⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x hx => H (le_of_lt hx.out)⟩ | 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⟩, min_le_left _ _⟩ | 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 _ _) hδ.2⟩, min_le_left _ _⟩ | 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, 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_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
uniformity_edist := H.trans (@PseudoEMetricSpace.uniformity_edist α _) | 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 uniformity is often important.
See note [reducible non-instances]. |
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