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PseudoMetricSpace.replaceUniformity {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : PseudoMetricSpace α :=
{ m with
toUniformSpace := U
uniformity_dist := H.trans PseudoMetricSpace.uniformity_dist }
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceUniformity
|
Build a new pseudometric space from an old one where the bundled uniform structure is provably
(but typically non-definitionaly) equal to some given uniform structure.
See Note [forgetful inheritance].
See Note [reducible non-instances].
|
PseudoMetricSpace.replaceUniformity_eq {α} [U : UniformSpace α] (m : PseudoMetricSpace α)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by
ext
rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceUniformity_eq
| null |
PseudoMetricSpace.replaceTopology {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
(H : U = m.toUniformSpace.toTopologicalSpace) : PseudoMetricSpace γ :=
@PseudoMetricSpace.replaceUniformity γ (m.toUniformSpace.replaceTopology H) m rfl
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceTopology
|
Build a new pseudo metric space from an old one where the bundled topological structure is
provably (but typically non-definitionaly) equal to some given topological structure.
See Note [forgetful inheritance].
See Note [reducible non-instances].
|
PseudoMetricSpace.replaceTopology_eq {γ} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ)
(H : U = m.toUniformSpace.toTopologicalSpace) : m.replaceTopology H = m := by
ext
rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceTopology_eq
| null |
PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetricSpace α]
(dist : α → α → ℝ) (edist_ne_top : ∀ x y : α, edist x y ≠ ⊤)
(h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) : PseudoMetricSpace α where
dist := dist
dist_self x := by simp [h]
dist_comm x y := by simp [h, edist_comm]
dist_triangle x y z := by
simp only [h]
exact ENNReal.toReal_le_add (edist_triangle _ _ _) (edist_ne_top _ _) (edist_ne_top _ _)
edist := edist
edist_dist _ _ := by simp only [h, ENNReal.ofReal_toReal (edist_ne_top _ _)]
toUniformSpace := e.toUniformSpace
uniformity_dist := e.uniformity_edist.trans <| by
simpa only [ENNReal.coe_toNNReal (edist_ne_top _ _), h]
using (Metric.uniformity_edist_aux fun x y : α => (edist x y).toNNReal).symm
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoEMetricSpace.toPseudoMetricSpaceOfDist
|
One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the
distance is given separately, to be able to prescribe some expression which is not defeq to the
push-forward of the edistance to reals. See note [reducible non-instances].
|
PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [PseudoEMetricSpace α]
(h : ∀ x y : α, edist x y ≠ ⊤) : PseudoMetricSpace α :=
PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun _ _ =>
rfl
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoEMetricSpace.toPseudoMetricSpace
|
One gets a pseudometric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the pseudometric space and the emetric space.
|
PseudoMetricSpace.replaceBornology {α} [B : Bornology α] (m : PseudoMetricSpace α)
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
PseudoMetricSpace α :=
{ m with
toBornology := B
cobounded_sets := Set.ext <| compl_surjective.forall.2 fun s =>
(H s).trans <| by rw [isBounded_iff, mem_setOf_eq, compl_compl] }
|
abbrev
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceBornology
|
Build a new pseudometric space from an old one where the bundled bornology structure is provably
(but typically non-definitionaly) equal to some given bornology structure.
See Note [forgetful inheritance].
See Note [reducible non-instances].
|
PseudoMetricSpace.replaceBornology_eq {α} [m : PseudoMetricSpace α] [B : Bornology α]
(H : ∀ s, @IsBounded _ B s ↔ @IsBounded _ PseudoMetricSpace.toBornology s) :
PseudoMetricSpace.replaceBornology _ H = m := by
ext
rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.replaceBornology_eq
| null |
Real.pseudoMetricSpace : PseudoMetricSpace ℝ where
dist x y := |x - y|
dist_self := by simp [abs_zero]
dist_comm _ _ := abs_sub_comm _ _
dist_triangle _ _ _ := abs_sub_le _ _ _
|
instance
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.pseudoMetricSpace
|
Instantiate the reals as a pseudometric space.
|
Real.dist_eq (x y : ℝ) : dist x y = |x - y| := rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.dist_eq
| null |
Real.nndist_eq (x y : ℝ) : nndist x y = Real.nnabs (x - y) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.nndist_eq
| null |
Real.nndist_eq' (x y : ℝ) : nndist x y = Real.nnabs (y - x) :=
nndist_comm _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.nndist_eq'
| null |
Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| := by simp [Real.dist_eq]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.dist_0_eq_abs
| null |
Real.sub_le_dist (x y : ℝ) : x - y ≤ dist x y := by
rw [Real.dist_eq, le_abs]
exact Or.inl (le_refl _)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.sub_le_dist
| null |
Real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) :=
Set.ext fun y => by
rw [mem_ball, dist_comm, Real.dist_eq, abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add',
sub_lt_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.ball_eq_Ioo
| null |
Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by
ext y
rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
sub_le_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.closedBall_eq_Icc
| null |
Real.Ioo_eq_ball (x y : ℝ) : Ioo x y = ball ((x + y) / 2) ((y - x) / 2) := by
rw [Real.ball_eq_Ioo, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
← add_div, add_assoc, add_sub_cancel, add_self_div_two]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.Ioo_eq_ball
| null |
Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by
rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
← add_div, add_assoc, add_sub_cancel, add_self_div_two]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Real.Icc_eq_closedBall
| null |
Metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (fun p : α × α => dist p.1 p.2) (𝓝 (0 : ℝ)) := by
ext s
simp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff]
simp [subset_def, Real.dist_0_eq_abs]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.uniformity_eq_comap_nhds_zero
| null |
tendsto_uniformity_iff_dist_tendsto_zero {f : ι → α × α} {p : Filter ι} :
Tendsto f p (𝓤 α) ↔ Tendsto (fun x => dist (f x).1 (f x).2) p (𝓝 0) := by
rw [Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff, Function.comp_def]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_uniformity_iff_dist_tendsto_zero
| null |
Filter.Tendsto.congr_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
(h₁ : Tendsto f₁ p (𝓝 a)) (h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) :
Tendsto f₂ p (𝓝 a) :=
h₁.congr_uniformity <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
alias tendsto_of_tendsto_of_dist := Filter.Tendsto.congr_dist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Filter.Tendsto.congr_dist
| null |
tendsto_iff_of_dist {f₁ f₂ : ι → α} {p : Filter ι} {a : α}
(h : Tendsto (fun x => dist (f₁ x) (f₂ x)) p (𝓝 0)) : Tendsto f₁ p (𝓝 a) ↔ Tendsto f₂ p (𝓝 a) :=
Uniform.tendsto_congr <| tendsto_uniformity_iff_dist_tendsto_zero.2 h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_iff_of_dist
| null |
PseudoMetricSpace.dist_eq_of_dist_zero (x : α) {y z : α} (h : dist y z = 0) :
dist x y = dist x z :=
dist_comm y x ▸ dist_comm z x ▸ sub_eq_zero.1 (abs_nonpos_iff.1 (h ▸ abs_dist_sub_le y z x))
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
PseudoMetricSpace.dist_eq_of_dist_zero
| null |
dist_dist_dist_le_left (x y z : α) : dist (dist x z) (dist y z) ≤ dist x y :=
abs_dist_sub_le ..
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_dist_dist_le_left
| null |
dist_dist_dist_le_right (x y z : α) : dist (dist x y) (dist x z) ≤ dist y z := by
simpa only [dist_comm x] using dist_dist_dist_le_left y z x
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_dist_dist_le_right
| null |
dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y' :=
(dist_triangle _ _ _).trans <|
add_le_add (dist_dist_dist_le_left _ _ _) (dist_dist_dist_le_right _ _ _)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_dist_dist_le
| null |
nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a := by
simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
Function.comp_def, dist_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_comap_dist
| null |
tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
rw [← nhds_comap_dist a, tendsto_comap_iff, Function.comp_def]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_iff_dist_tendsto_zero
| null |
ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) :=
interior_maximal ball_subset_closedBall isOpen_ball
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_subset_interior_closedBall
| null |
mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
(mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closure_iff
|
ε-characterization of the closure in pseudometric spaces
|
mem_closure_range_iff {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by
simp only [mem_closure_iff, exists_range_iff]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closure_range_iff
| null |
mem_closure_range_iff_nat {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ n : ℕ, ∃ k : β, dist a (e k) < 1 / ((n : ℝ) + 1) :=
(mem_closure_iff_nhds_basis nhds_basis_ball_inv_nat_succ).trans <| by
simp only [mem_ball, dist_comm, exists_range_iff, forall_const]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closure_range_iff_nat
| null |
mem_of_closed' {s : Set α} (hs : IsClosed s) {a : α} :
a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε := by
simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_of_closed'
| null |
dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
forall_congr' fun x => by
simp only [mem_closure_iff, Set.Nonempty, mem_inter_iff, mem_ball', and_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dense_iff
| null |
dense_iff_iUnion_ball (s : Set α) : Dense s ↔ ∀ r > 0, ⋃ c ∈ s, ball c r = univ := by
simp_rw [eq_univ_iff_forall, mem_iUnion, exists_prop, mem_ball, Dense, mem_closure_iff,
forall_comm (α := α)]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dense_iff_iUnion_ball
| null |
denseRange_iff {f : β → α} : DenseRange f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r :=
forall_congr' fun x => by simp only [mem_closure_iff, exists_range_iff]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
denseRange_iff
| null |
@[simp] nndist_ofMul (a b : X) : nndist (ofMul a) (ofMul b) = nndist a b := rfl
@[simp] theorem nndist_ofAdd (a b : X) : nndist (ofAdd a) (ofAdd b) = nndist a b := rfl
@[simp] theorem nndist_toMul (a b : Additive X) : nndist a.toMul b.toMul = nndist a b := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_ofMul
| null |
nndist_toAdd (a b : Multiplicative X) : nndist a.toAdd b.toAdd = nndist a b := rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_toAdd
| null |
@[simp] nndist_toDual (a b : X) : nndist (toDual a) (toDual b) = nndist a b := rfl
@[simp] theorem nndist_ofDual (a b : Xᵒᵈ) : nndist (ofDual a) (ofDual b) = nndist a b := rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nndist_toDual
| null |
Real.singleton_eq_inter_Icc (b : ℝ) : {b} = ⋂ (r > 0), Icc (b - r) (b + r) := by
simp [Icc_eq_closedBall, biInter_basis_nhds Metric.nhds_basis_closedBall]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
Real.singleton_eq_inter_Icc
| null |
squeeze_zero' {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ᶠ t in t₀, 0 ≤ f t)
(hft : ∀ᶠ t in t₀, f t ≤ g t) (g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
squeeze_zero'
|
Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
general case.
|
squeeze_zero {α} {f g : α → ℝ} {t₀ : Filter α} (hf : ∀ t, 0 ≤ f t) (hft : ∀ t, f t ≤ g t)
(g0 : Tendsto g t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 0) :=
squeeze_zero' (Eventually.of_forall hf) (Eventually.of_forall hft) g0
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
squeeze_zero
|
Special case of the sandwich lemma; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case.
|
eventually_closedBall_subset {x : α} {u : Set α} (hu : u ∈ 𝓝 x) :
∀ᶠ r in 𝓝 (0 : ℝ), closedBall x r ⊆ u := by
obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ closedBall x ε ⊆ u := nhds_basis_closedBall.mem_iff.1 hu
have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos
filter_upwards [this] with _ hr using Subset.trans (closedBall_subset_closedBall hr) hε
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
eventually_closedBall_subset
|
If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
`Metric.closedBall x r` is contained in `u`.
|
tendsto_closedBall_smallSets (x : α) : Tendsto (closedBall x) (𝓝 0) (𝓝 x).smallSets :=
tendsto_smallSets_iff.2 fun _ ↦ eventually_closedBall_subset
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
tendsto_closedBall_smallSets
| null |
isClosed_closedBall : IsClosed (closedBall x ε) :=
isClosed_le (continuous_id.dist continuous_const) continuous_const
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
isClosed_closedBall
| null |
isClosed_sphere : IsClosed (sphere x ε) :=
isClosed_eq (continuous_id.dist continuous_const) continuous_const
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
isClosed_sphere
| null |
closure_closedBall : closure (closedBall x ε) = closedBall x ε :=
isClosed_closedBall.closure_eq
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
closure_closedBall
| null |
closure_sphere : closure (sphere x ε) = sphere x ε :=
isClosed_sphere.closure_eq
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
closure_sphere
| null |
closure_ball_subset_closedBall : closure (ball x ε) ⊆ closedBall x ε :=
closure_minimal ball_subset_closedBall isClosed_closedBall
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
closure_ball_subset_closedBall
| null |
frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε :=
frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
frontier_ball_subset_sphere
| null |
frontier_closedBall_subset_sphere : frontier (closedBall x ε) ⊆ sphere x ε :=
frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
frontier_closedBall_subset_sphere
| null |
closedBall_zero' (x : α) : closedBall x 0 = closure {x} :=
Subset.antisymm
(fun _y hy =>
mem_closure_iff.2 fun _ε ε0 => ⟨x, mem_singleton x, (mem_closedBall.1 hy).trans_lt ε0⟩)
(closure_minimal (singleton_subset_iff.2 (dist_self x).le) isClosed_closedBall)
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
closedBall_zero'
| null |
eventually_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
∀ᶠ r in 𝓝 (0 : ℝ), IsCompact (closedBall x r) := by
rcases exists_compact_mem_nhds x with ⟨s, s_compact, hs⟩
filter_upwards [eventually_closedBall_subset hs] with r hr
exact IsCompact.of_isClosed_subset s_compact isClosed_closedBall hr
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
eventually_isCompact_closedBall
| null |
exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
∃ r, 0 < r ∧ IsCompact (closedBall x r) := by
have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) :=
eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x)
simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
exists_isCompact_closedBall
| null |
biInter_gt_closedBall (x : α) (r : ℝ) : ⋂ r' > r, closedBall x r' = closedBall x r := by
ext
simp [forall_gt_imp_ge_iff_le_of_dense]
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
biInter_gt_closedBall
| null |
biInter_gt_ball (x : α) (r : ℝ) : ⋂ r' > r, ball x r' = closedBall x r := by
ext
simp [forall_gt_iff_le]
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
biInter_gt_ball
| null |
biUnion_lt_ball (x : α) (r : ℝ) : ⋃ r' < r, ball x r' = ball x r := by
ext
rw [← not_iff_not]
simp [forall_lt_imp_le_iff_le_of_dense]
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
biUnion_lt_ball
| null |
biUnion_lt_closedBall (x : α) (r : ℝ) : ⋃ r' < r, closedBall x r' = ball x r := by
ext
rw [← not_iff_not]
simp [forall_lt_iff_le]
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
biUnion_lt_closedBall
| null |
lebesgue_number_lemma_of_metric {s : Set α} {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, dist_comm]
using uniformity_basis_dist.lebesgue_number_lemma hs hc₁ hc₂
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
lebesgue_number_lemma_of_metric
| null |
lebesgue_number_lemma_of_metric_sUnion {s : Set α} {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_metric hs (by simpa) hc₂
|
theorem
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Constructions",
"Mathlib.Topology.Order.DenselyOrdered",
"Mathlib.Topology.UniformSpace.Compact"
] |
Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
|
lebesgue_number_lemma_of_metric_sUnion
| null |
pseudoMetricSpacePi : PseudoMetricSpace (∀ b, X b) := by
/- we construct the instance from the pseudoemetric space instance to avoid checking again that
the uniformity is the same as the product uniformity, but we register nevertheless a nice
formula for the distance -/
let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
(fun f g : ∀ b, X b => ((sup univ fun b => nndist (f b) (g b) : ℝ≥0) : ℝ))
(fun f g => ((Finset.sup_lt_iff bot_lt_top).2 fun b _ => edist_lt_top _ _).ne)
(fun f g => by
simp only [edist_pi_def, edist_nndist, ← ENNReal.coe_finset_sup, ENNReal.coe_toReal])
refine i.replaceBornology fun s => ?_
simp only [isBounded_iff_eventually, ← forall_isBounded_image_eval_iff,
forall_mem_image, ← Filter.eventually_all, @dist_nndist (X _)]
refine eventually_congr ((eventually_ge_atTop 0).mono fun C hC ↦ ?_)
lift C to ℝ≥0 using hC
refine ⟨fun H x hx y hy ↦ NNReal.coe_le_coe.2 <| Finset.sup_le fun b _ ↦ H b hx hy,
fun H b x hx y hy ↦ NNReal.coe_le_coe.2 ?_⟩
simpa only using Finset.sup_le_iff.1 (NNReal.coe_le_coe.1 <| H hx hy) b (Finset.mem_univ b)
|
instance
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
pseudoMetricSpacePi
|
A finite product of pseudometric spaces is a pseudometric space, with the sup distance.
|
nndist_pi_def (f g : ∀ b, X b) : nndist f g = sup univ fun b => nndist (f b) (g b) := rfl
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_def
| null |
dist_pi_def (f g : ∀ b, X b) : dist f g = (sup univ fun b => nndist (f b) (g b) : ℝ≥0) := rfl
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_def
| null |
nndist_pi_le_iff {f g : ∀ b, X b} {r : ℝ≥0} :
nndist f g ≤ r ↔ ∀ b, nndist (f b) (g b) ≤ r := by simp [nndist_pi_def]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_le_iff
| null |
nndist_pi_lt_iff {f g : ∀ b, X b} {r : ℝ≥0} (hr : 0 < r) :
nndist f g < r ↔ ∀ b, nndist (f b) (g b) < r := by
simp [nndist_pi_def, Finset.sup_lt_iff hr]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_lt_iff
| null |
nndist_pi_eq_iff {f g : ∀ b, X b} {r : ℝ≥0} (hr : 0 < r) :
nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ b, nndist (f b) (g b) ≤ r := by
rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_eq_iff
| null |
dist_pi_lt_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 < r) :
dist f g < r ↔ ∀ b, dist (f b) (g b) < r := by
lift r to ℝ≥0 using hr.le
exact nndist_pi_lt_iff hr
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_lt_iff
| null |
dist_pi_le_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 ≤ r) :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by
lift r to ℝ≥0 using hr
exact nndist_pi_le_iff
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_le_iff
| null |
dist_pi_eq_iff {f g : ∀ b, X b} {r : ℝ} (hr : 0 < r) :
dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r := by
lift r to ℝ≥0 using hr.le
simp_rw [← coe_nndist, NNReal.coe_inj, nndist_pi_eq_iff hr, NNReal.coe_le_coe]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_eq_iff
| null |
dist_pi_le_iff' [Nonempty β] {f g : ∀ b, X b} {r : ℝ} :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by
by_cases hr : 0 ≤ r
· exact dist_pi_le_iff hr
· exact iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h =>
hr <| dist_nonneg.trans <| h <| Classical.arbitrary _
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_le_iff'
| null |
dist_pi_const_le (a b : α) : (dist (fun _ : β => a) fun _ => b) ≤ dist a b :=
(dist_pi_le_iff dist_nonneg).2 fun _ => le_rfl
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_const_le
| null |
nndist_pi_const_le (a b : α) : (nndist (fun _ : β => a) fun _ => b) ≤ nndist a b :=
nndist_pi_le_iff.2 fun _ => le_rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_const_le
| null |
dist_pi_const [Nonempty β] (a b : α) : (dist (fun _ : β => a) fun _ => b) = dist a b := by
simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_pi_const
| null |
nndist_pi_const [Nonempty β] (a b : α) : (nndist (fun _ : β => a) fun _ => b) = nndist a b :=
NNReal.eq <| dist_pi_const a b
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_pi_const
| null |
nndist_le_pi_nndist (f g : ∀ b, X b) (b : β) : nndist (f b) (g b) ≤ nndist f g := by
rw [← ENNReal.coe_le_coe, ← edist_nndist, ← edist_nndist]
exact edist_le_pi_edist f g b
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
nndist_le_pi_nndist
| null |
dist_le_pi_dist (f g : ∀ b, X b) (b : β) : dist (f b) (g b) ≤ dist f g := by
simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
dist_le_pi_dist
| null |
ball_pi (x : ∀ b, X b) {r : ℝ} (hr : 0 < r) :
ball x r = Set.pi univ fun b => ball (x b) r := by
ext p
simp [dist_pi_lt_iff hr]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
ball_pi
|
An open ball in a product space is a product of open balls. See also `ball_pi'`
for a version assuming `Nonempty β` instead of `0 < r`.
|
ball_pi' [Nonempty β] (x : ∀ b, X b) (r : ℝ) :
ball x r = Set.pi univ fun b => ball (x b) r :=
(lt_or_ge 0 r).elim (ball_pi x) fun hr => by simp [ball_eq_empty.2 hr]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
ball_pi'
|
An open ball in a product space is a product of open balls. See also `ball_pi`
for a version assuming `0 < r` instead of `Nonempty β`.
|
closedBall_pi (x : ∀ b, X b) {r : ℝ} (hr : 0 ≤ r) :
closedBall x r = Set.pi univ fun b => closedBall (x b) r := by
ext p
simp [dist_pi_le_iff hr]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
closedBall_pi
|
A closed ball in a product space is a product of closed balls. See also `closedBall_pi'`
for a version assuming `Nonempty β` instead of `0 ≤ r`.
|
closedBall_pi' [Nonempty β] (x : ∀ b, X b) (r : ℝ) :
closedBall x r = Set.pi univ fun b => closedBall (x b) r :=
(le_or_gt 0 r).elim (closedBall_pi x) fun hr => by simp [closedBall_eq_empty.2 hr]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
closedBall_pi'
|
A closed ball in a product space is a product of closed balls. See also `closedBall_pi`
for a version assuming `0 ≤ r` instead of `Nonempty β`.
|
sphere_pi (x : ∀ b, X b) {r : ℝ} (h : 0 < r ∨ Nonempty β) :
sphere x r = (⋃ i : β, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r := by
obtain hr | rfl | hr := lt_trichotomy r 0
· simp [hr]
· rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]
letI := h.resolve_left (lt_irrefl _)
inhabit β
refine subset_iUnion_of_subset default ?_
intro x hx
replace hx := hx.le
rw [dist_pi_le_iff le_rfl] at hx
exact le_antisymm (hx default) dist_nonneg
· ext
simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
sphere_pi
|
A sphere in a product space is a union of spheres on each component restricted to the closed
ball.
|
Fin.nndist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
[∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
nndist (i.insertNth x f) (i.insertNth y g) = max (nndist x y) (nndist f g) :=
eq_of_forall_ge_iff fun c => by simp [nndist_pi_le_iff, i.forall_iff_succAbove]
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
Fin.nndist_insertNth_insertNth
| null |
Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
[∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by
simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.EMetricSpace.Pi",
"Mathlib.Topology.MetricSpace.Pseudo.Defs"
] |
Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
|
Fin.dist_insertNth_insertNth
| null |
dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by
simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_left_le_of_mem_uIcc
| null |
dist_right_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist y z ≤ dist x z := by
simpa only [dist_comm _ z] using abs_sub_right_of_mem_uIcc h
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_right_le_of_mem_uIcc
| null |
dist_le_of_mem_uIcc {x y x' y' : ℝ} (hx : x ∈ uIcc x' y') (hy : y ∈ uIcc x' y') :
dist x y ≤ dist x' y' :=
abs_sub_le_of_uIcc_subset_uIcc <| uIcc_subset_uIcc (by rwa [uIcc_comm]) (by rwa [uIcc_comm])
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_le_of_mem_uIcc
| null |
dist_le_of_mem_Icc {x y x' y' : ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
dist x y ≤ y' - x' := by
simpa only [Real.dist_eq, abs_of_nonpos (sub_nonpos.2 <| hx.1.trans hx.2), neg_sub] using
Real.dist_le_of_mem_uIcc (Icc_subset_uIcc hx) (Icc_subset_uIcc hy)
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_le_of_mem_Icc
| null |
dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
dist x y ≤ 1 := by simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
variable [Fintype ι] {x y x' y' : ι → ℝ}
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_le_of_mem_Icc_01
| null |
dist_le_of_mem_pi_Icc (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') : dist x y ≤ dist x' y' := by
refine (dist_pi_le_iff dist_nonneg).2 fun b =>
(Real.dist_le_of_mem_uIcc ?_ ?_).trans (dist_le_pi_dist x' y' b) <;> refine Icc_subset_uIcc ?_
exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Order.Group.Pointwise.Interval",
"Mathlib.Topology.MetricSpace.Pseudo.Pi"
] |
Mathlib/Topology/MetricSpace/Pseudo/Real.lean
|
dist_le_of_mem_pi_Icc
| null |
IsUltrametricDist (X : Type*) [Dist X] : Prop where
dist_triangle_max : ∀ x y z : X, dist x z ≤ max (dist x y) (dist y z)
open Metric
variable [PseudoMetricSpace X] [IsUltrametricDist X] (x y z : X) (r s : ℝ)
|
class
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
IsUltrametricDist
|
The `dist : X → X → ℝ` respects the ultrametric inequality
of `dist(x, z) ≤ max (dist(x,y)) (dist(y,z))`.
|
dist_triangle_max : dist x z ≤ max (dist x y) (dist y z) :=
IsUltrametricDist.dist_triangle_max x y z
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
dist_triangle_max
| null |
dist_eq_max_of_dist_ne_dist (h : dist x y ≠ dist y z) :
dist x z = max (dist x y) (dist y z) := by
apply le_antisymm (dist_triangle_max x y z)
rcases h.lt_or_gt with h | h
· rw [max_eq_right h.le]
apply (le_max_iff.mp <| dist_triangle_max y x z).resolve_left
simpa only [not_le, dist_comm x y] using h
· rw [max_eq_left h.le, dist_comm x y, dist_comm x z]
apply (le_max_iff.mp <| dist_triangle_max y z x).resolve_left
simpa only [not_le, dist_comm x y] using h
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
dist_eq_max_of_dist_ne_dist
|
All triangles are isosceles in an ultrametric space.
|
subtype (p : X → Prop) : IsUltrametricDist (Subtype p) :=
⟨fun _ _ _ ↦ by simpa [Subtype.dist_eq] using dist_triangle_max _ _ _⟩
|
instance
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
subtype
| null |
ball_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ ball x r) : ball x r = ball y r := by
ext a
simp_rw [mem_ball] at h ⊢
constructor <;> intro h' <;>
exact (dist_triangle_max _ _ _).trans_lt (max_lt h' (dist_comm x _ ▸ h))
@[deprecated (since := "2025-08-16")] alias mem_ball_iff := mem_ball_comm
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
ball_eq_of_mem
| null |
ball_subset_trichotomy :
ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s) := by
wlog hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r (lt_of_not_ge hrs).le
· refine Set.disjoint_or_nonempty_inter (ball x r) (ball y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
have hx := ball_subset_ball hrs (x := x)
rwa [ball_eq_of_mem hyz |>.trans (ball_eq_of_mem <| hx hxz).symm]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
ball_subset_trichotomy
| null |
ball_eq_or_disjoint :
ball x r = ball y r ∨ Disjoint (ball x r) (ball y r) := by
refine Set.disjoint_or_nonempty_inter (ball x r) (ball y r) |>.symm.imp (fun h ↦ ?_) id
have h₁ := ball_eq_of_mem <| Set.inter_subset_left h.some_mem
have h₂ := ball_eq_of_mem <| Set.inter_subset_right h.some_mem
exact h₁.trans h₂.symm
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
ball_eq_or_disjoint
| null |
closedBall_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ closedBall x r) :
closedBall x r = closedBall y r := by
ext
simp_rw [mem_closedBall] at h ⊢
constructor <;> intro h' <;>
exact (dist_triangle_max _ _ _).trans (max_le h' (dist_comm x _ ▸ h))
@[deprecated (since := "2025-08-16")] alias mem_closedBall_iff := mem_closedBall_comm
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
closedBall_eq_of_mem
| null |
closedBall_subset_trichotomy :
closedBall x r ⊆ closedBall y s ∨ closedBall y s ⊆ closedBall x r ∨
Disjoint (closedBall x r) (closedBall y s) := by
wlog hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r (lt_of_not_ge hrs).le
· refine Set.disjoint_or_nonempty_inter (closedBall x r) (closedBall y s) |>.symm.imp
(fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
have hx := closedBall_subset_closedBall hrs (x := x)
rwa [closedBall_eq_of_mem hyz |>.trans (closedBall_eq_of_mem <| hx hxz).symm]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
closedBall_subset_trichotomy
| null |
isClosed_ball (x : X) (r : ℝ) : IsClosed (ball x r) := by
cases le_or_gt r 0 with
| inl hr =>
simp [ball_eq_empty.mpr hr]
| inr h =>
rw [← isOpen_compl_iff, isOpen_iff]
simp only [Set.mem_compl_iff, gt_iff_lt]
intro y hy
cases ball_eq_or_disjoint x y r with
| inl hd =>
rw [hd] at hy
simp [h.not_ge] at hy
| inr hd =>
use r
simp [h, ← Set.le_iff_subset, le_compl_iff_disjoint_left, hd]
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isClosed_ball
| null |
isClopen_ball : IsClopen (ball x r) := ⟨isClosed_ball x r, isOpen_ball⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isClopen_ball
| null |
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