statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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uniformity_basis_dist_inv_nat_succ :
(𝓤 α).has_basis (λ _, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 < 1 / (↑n+1) }) | metric.mk_uniformity_basis (λ n _, div_pos zero_lt_one $ nat.cast_add_one_pos n)
(λ ε ε0, (exists_nat_one_div_lt ε0).imp $ λ n hn, ⟨trivial, le_of_lt hn⟩) | theorem | metric.uniformity_basis_dist_inv_nat_succ | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"div_pos",
"exists_nat_one_div_lt",
"metric.mk_uniformity_basis",
"nat.cast_add_one_pos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_dist_inv_nat_pos :
(𝓤 α).has_basis (λ n:ℕ, 0<n) (λ n:ℕ, {p:α×α | dist p.1 p.2 < 1 / ↑n }) | metric.mk_uniformity_basis (λ n hn, div_pos zero_lt_one $ nat.cast_pos.2 hn)
(λ ε ε0, let ⟨n, hn⟩ := exists_nat_one_div_lt ε0 in ⟨n+1, nat.succ_pos n,
by exact_mod_cast hn.le⟩) | theorem | metric.uniformity_basis_dist_inv_nat_pos | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"div_pos",
"exists_nat_one_div_lt",
"metric.mk_uniformity_basis",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).has_basis (λ n:ℕ, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 < r ^ n }) | metric.mk_uniformity_basis (λ n hn, pow_pos h0 _)
(λ ε ε0, let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 in ⟨n, trivial, hn.le⟩) | theorem | metric.uniformity_basis_dist_pow | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_pow_lt_of_lt_one",
"metric.mk_uniformity_basis",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
(𝓤 α).has_basis (λ r : ℝ, 0 < r ∧ r < R) (λ r, {p : α × α | dist p.1 p.2 < r}) | metric.mk_uniformity_basis (λ r, and.left) $ λ r hr,
⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 $ or.inr (half_lt_self hR)⟩,
min_le_left _ _⟩ | theorem | metric.uniformity_basis_dist_lt | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"half_pos",
"metric.mk_uniformity_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x (hx : p x), f x ≤ ε) :
(𝓤 α).has_basis p (λ x, {p:α×α | dist p.1 p.2 ≤ f x}) | begin
refine ⟨λ s, uniformity_basis_dist.mem_iff.trans _⟩,
split,
{ rintros ⟨ε, ε₀, hε⟩,
rcases exists_between ε₀ with ⟨ε', hε'⟩,
rcases hf ε' hε'.1 with ⟨i, hi, H⟩,
exact ⟨i, hi, λ x (hx : _ ≤ _), hε $ lt_of_le_of_lt (le_trans hx H) hε'.2⟩ },
{ exact λ ⟨i, hi, H⟩, ⟨f i, hf₀ i hi, λ x (hx : _ < _), ... | theorem | metric.mk_uniformity_basis_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_between"
] | Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
form a basis of `𝓤 α`.
Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
More can be easily added if needed in t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_basis_dist_le :
(𝓤 α).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {p:α×α | dist p.1 p.2 ≤ ε}) | metric.mk_uniformity_basis_le (λ _, id) (λ ε ε₀, ⟨ε, ε₀, le_refl ε⟩) | theorem | metric.uniformity_basis_dist_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.mk_uniformity_basis_le"
] | Contant size closed neighborhoods of the diagonal form a basis
of the uniformity filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).has_basis (λ n:ℕ, true) (λ n:ℕ, {p:α×α | dist p.1 p.2 ≤ r ^ n }) | metric.mk_uniformity_basis_le (λ n hn, pow_pos h0 _)
(λ ε ε0, let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1 in ⟨n, trivial, hn.le⟩) | theorem | metric.uniformity_basis_dist_le_pow | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_pow_lt_of_lt_one",
"metric.mk_uniformity_basis_le",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_uniformity_dist {s : set (α×α)} :
s ∈ 𝓤 α ↔ (∃ε>0, ∀{a b:α}, dist a b < ε → (a, b) ∈ s) | uniformity_basis_dist.mem_uniformity_iff | theorem | metric.mem_uniformity_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mem_uniformity {ε:ℝ} (ε0 : 0 < ε) :
{p:α×α | dist p.1 p.2 < ε} ∈ 𝓤 α | mem_uniformity_dist.2 ⟨ε, ε0, λ a b, id⟩ | theorem | metric.dist_mem_uniformity | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | A constant size neighborhood of the diagonal is an entourage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_iff [pseudo_metric_space β] {f : α → β} :
uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0,
∀{a b:α}, dist a b < δ → dist (f a) (f b) < ε | uniformity_basis_dist.uniform_continuous_iff uniformity_basis_dist | theorem | metric.uniform_continuous_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"uniform_continuous",
"uniform_continuous_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on_iff [pseudo_metric_space β] {f : α → β} {s : set α} :
uniform_continuous_on f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x y ∈ s, dist x y < δ → dist (f x) (f y) < ε | metric.uniformity_basis_dist.uniform_continuous_on_iff metric.uniformity_basis_dist | lemma | metric.uniform_continuous_on_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.uniformity_basis_dist",
"pseudo_metric_space",
"uniform_continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_on_iff_le [pseudo_metric_space β] {f : α → β} {s : set α} :
uniform_continuous_on f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε | metric.uniformity_basis_dist_le.uniform_continuous_on_iff metric.uniformity_basis_dist_le | lemma | metric.uniform_continuous_on_iff_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"metric.uniformity_basis_dist_le",
"pseudo_metric_space",
"uniform_continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_iff [pseudo_metric_space β] {f : α → β} :
uniform_embedding f ↔ function.injective f ∧ uniform_continuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ | begin
simp only [uniformity_basis_dist.uniform_embedding_iff uniformity_basis_dist, exists_prop],
refl
end | theorem | metric.uniform_embedding_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_prop",
"pseudo_metric_space",
"uniform_continuous",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
controlled_of_uniform_embedding [pseudo_metric_space β] {f : α → β} :
uniform_embedding f →
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
(∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ) | begin
assume h,
exact ⟨uniform_continuous_iff.1 (uniform_embedding_iff.1 h).2.1, (uniform_embedding_iff.1 h).2.2⟩
end | theorem | metric.controlled_of_uniform_embedding | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"uniform_embedding"
] | If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded_iff {s : set α} :
totally_bounded s ↔ ∀ ε > 0, ∃t : set α, t.finite ∧ s ⊆ ⋃y∈t, ball y ε | ⟨λ H ε ε0, H _ (dist_mem_uniformity ε0),
λ H r ru, let ⟨ε, ε0, hε⟩ := mem_uniformity_dist.1 ru,
⟨t, ft, h⟩ := H ε ε0 in
⟨t, ft, h.trans $ Union₂_mono $ λ y yt z, hε⟩⟩ | theorem | metric.totally_bounded_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_of_finite_discretization {s : set α}
(H : ∀ε > (0 : ℝ), ∃ (β : Type u) (_ : fintype β) (F : s → β),
∀x y, F x = F y → dist (x:α) y < ε) :
totally_bounded s | begin
cases s.eq_empty_or_nonempty with hs hs,
{ rw hs, exact totally_bounded_empty },
rcases hs with ⟨x0, hx0⟩,
haveI : inhabited s := ⟨⟨x0, hx0⟩⟩,
refine totally_bounded_iff.2 (λ ε ε0, _),
rcases H ε ε0 with ⟨β, fβ, F, hF⟩,
resetI,
let Finv := function.inv_fun F,
refine ⟨range (subtype.val ∘ Finv), ... | lemma | metric.totally_bounded_of_finite_discretization | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"fintype",
"set.mem_Union",
"set.mem_range",
"totally_bounded",
"totally_bounded_empty"
] | A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
space from finitely many data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_approx_of_totally_bounded {s : set α} (hs : totally_bounded s) :
∀ ε > 0, ∃ t ⊆ s, set.finite t ∧ s ⊆ ⋃y∈t, ball y ε | begin
intros ε ε_pos,
rw totally_bounded_iff_subset at hs,
exact hs _ (dist_mem_uniformity ε_pos),
end | theorem | metric.finite_approx_of_totally_bounded | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"set.finite",
"totally_bounded",
"totally_bounded_iff_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} {p' : filter β} :
tendsto_uniformly_on_filter F f p p' ↔
∀ ε > 0, ∀ᶠ (n : ι × β) in (p ×ᶠ p'), dist (f n.snd) (F n.fst n.snd) < ε | begin
refine ⟨λ H ε hε, H _ (dist_mem_uniformity hε), λ H u hu, _⟩,
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩,
refine (H ε εpos).mono (λ n hn, hε hn),
end | lemma | metric.tendsto_uniformly_on_filter_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"tendsto_uniformly_on_filter"
] | Expressing uniform convergence using `dist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_on_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_locally_uniformly_on F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε | begin
refine ⟨λ H ε hε, H _ (dist_mem_uniformity hε), λ H u hu x hx, _⟩,
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩,
rcases H ε εpos x hx with ⟨t, ht, Ht⟩,
exact ⟨t, ht, Ht.mono (λ n hs x hx, hε (hs x hx))⟩
end | lemma | metric.tendsto_locally_uniformly_on_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"tendsto_locally_uniformly_on",
"topological_space"
] | Expressing locally uniform convergence on a set using `dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :
tendsto_uniformly_on F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε | begin
refine ⟨λ H ε hε, H _ (dist_mem_uniformity hε), λ H u hu, _⟩,
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩,
exact (H ε εpos).mono (λ n hs x hx, hε (hs x hx))
end | lemma | metric.tendsto_uniformly_on_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"tendsto_uniformly_on"
] | Expressing uniform convergence on a set using `dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_locally_uniformly_iff {ι : Type*} [topological_space β]
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_locally_uniformly F f p ↔
∀ ε > 0, ∀ (x : β), ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε | by simp only [← tendsto_locally_uniformly_on_univ, tendsto_locally_uniformly_on_iff,
nhds_within_univ, mem_univ, forall_const, exists_prop] | lemma | metric.tendsto_locally_uniformly_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_prop",
"filter",
"forall_const",
"nhds_within_univ",
"tendsto_locally_uniformly",
"tendsto_locally_uniformly_on_univ",
"topological_space"
] | Expressing locally uniform convergence using `dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_iff {ι : Type*}
{F : ι → β → α} {f : β → α} {p : filter ι} :
tendsto_uniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε | by { rw [← tendsto_uniformly_on_univ, tendsto_uniformly_on_iff], simp } | lemma | metric.tendsto_uniformly_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"tendsto_uniformly",
"tendsto_uniformly_on_univ"
] | Expressing uniform convergence using `dist`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_iff {f : filter α} :
cauchy f ↔ ne_bot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x y ∈ t, dist x y < ε | uniformity_basis_dist.cauchy_iff | lemma | metric.cauchy_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"cauchy",
"cauchy_iff",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_ball : (𝓝 x).has_basis (λ ε:ℝ, 0 < ε) (ball x) | nhds_basis_uniformity uniformity_basis_dist | theorem | metric.nhds_basis_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ε>0, ball x ε ⊆ s | nhds_basis_ball.mem_iff | theorem | metric.mem_nhds_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_iff {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ε>0, ∀ ⦃y⦄, dist y x < ε → p y | mem_nhds_iff | theorem | metric.eventually_nhds_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eventually_nhds_iff",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_iff_ball {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε>0, ∀ y ∈ ball x ε, p y | mem_nhds_iff | lemma | metric.eventually_nhds_iff_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_prod_nhds_iff {f : filter ι} {x₀ : α} {p : ι × α → Prop}:
(∀ᶠ x in f ×ᶠ 𝓝 x₀, p x) ↔ ∃ (pa : ι → Prop) (ha : ∀ᶠ i in f, pa i) (ε > 0),
∀ {i}, pa i → ∀ {x}, dist x x₀ < ε → p (i, x) | begin
simp_rw [eventually_prod_iff, metric.eventually_nhds_iff],
refine exists_congr (λ q, exists_congr $ λ hq, _),
split,
{ rintro ⟨r, ⟨ε, hε, hεr⟩, hp⟩, exact ⟨ε, hε, λ i hi x hx, hp hi $ hεr hx⟩ },
{ rintro ⟨ε, hε, hp⟩, exact ⟨λ x, dist x x₀ < ε, ⟨ε, hε, λ y, id⟩, @hp⟩ }
end | lemma | metric.eventually_prod_nhds_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"metric.eventually_nhds_iff"
] | A version of `filter.eventually_prod_iff` where the second filter consists of neighborhoods
in a pseudo-metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_nhds_prod_iff {ι α} [pseudo_metric_space α] {f : filter ι} {x₀ : α}
{p : α × ι → Prop}:
(∀ᶠ x in 𝓝 x₀ ×ᶠ f, p x) ↔ ∃ (ε > (0 : ℝ)) (pa : ι → Prop) (ha : ∀ᶠ i in f, pa i) ,
∀ {x}, dist x x₀ < ε → ∀ {i}, pa i → p (x, i) | begin
rw [eventually_swap_iff, metric.eventually_prod_nhds_iff],
split; { rintro ⟨a1, a2, a3, a4, a5⟩, refine ⟨a3, a4, a1, a2, λ b1 b2 b3 b4, a5 b4 b2⟩ }
end | lemma | metric.eventually_nhds_prod_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"metric.eventually_prod_nhds_iff",
"pseudo_metric_space"
] | A version of `filter.eventually_prod_iff` where the first filter consists of neighborhoods
in a pseudo-metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_closed_ball : (𝓝 x).has_basis (λ ε:ℝ, 0 < ε) (closed_ball x) | nhds_basis_uniformity uniformity_basis_dist_le | theorem | metric.nhds_basis_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_ball_inv_nat_succ :
(𝓝 x).has_basis (λ _, true) (λ n:ℕ, ball x (1 / (↑n+1))) | nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ | theorem | metric.nhds_basis_ball_inv_nat_succ | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_ball_inv_nat_pos :
(𝓝 x).has_basis (λ n, 0<n) (λ n:ℕ, ball x (1 / ↑n)) | nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos | theorem | metric.nhds_basis_ball_inv_nat_pos | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).has_basis (λ n, true) (λ n:ℕ, ball x (r ^ n)) | nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1) | theorem | metric.nhds_basis_ball_pow | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_basis_closed_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).has_basis (λ n, true) (λ n:ℕ, closed_ball x (r ^ n)) | nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1) | theorem | metric.nhds_basis_closed_ball_pow | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff : is_open s ↔ ∀x∈s, ∃ε>0, ball x ε ⊆ s | by simp only [is_open_iff_mem_nhds, mem_nhds_iff] | theorem | metric.is_open_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_open",
"is_open_iff_mem_nhds",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_ball : is_open (ball x ε) | is_open_iff.2 $ λ y, exists_ball_subset_ball | theorem | metric.is_open_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x | is_open_ball.mem_nhds (mem_ball_self ε0) | theorem | metric.ball_mem_nhds | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closed_ball x ε ∈ 𝓝 x | mem_of_superset (ball_mem_nhds x ε0) ball_subset_closed_ball | theorem | metric.closed_ball_mem_nhds | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) :
closed_ball c ε ∈ 𝓝 x | mem_of_superset (is_open_ball.mem_nhds h) ball_subset_closed_ball | theorem | metric.closed_ball_mem_nhds_of_mem | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_basis_ball {s : set α} :
(𝓝[s] x).has_basis (λ ε:ℝ, 0 < ε) (λ ε, ball x ε ∩ s) | nhds_within_has_basis nhds_basis_ball s | theorem | metric.nhds_within_basis_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_within_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_iff {t : set α} : s ∈ 𝓝[t] x ↔ ∃ε>0, ball x ε ∩ t ⊆ s | nhds_within_basis_ball.mem_iff | theorem | metric.mem_nhds_within_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_nhds_within [pseudo_metric_space β] {t : set β} {f : α → β} {a b} :
tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε | (nhds_within_basis_ball.tendsto_iff nhds_within_basis_ball).trans $
forall₂_congr $ λ ε hε, exists₂_congr $ λ δ hδ,
forall_congr $ λ x, by simp; itauto | theorem | metric.tendsto_nhds_within_nhds_within | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists₂_congr",
"forall₂_congr",
"itauto",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_nhds [pseudo_metric_space β] {f : α → β} {a b} :
tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → dist (f x) b < ε | by { rw [← nhds_within_univ b, tendsto_nhds_within_nhds_within],
simp only [mem_univ, true_and] } | theorem | metric.tendsto_nhds_within_nhds | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nhds_within_univ",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_nhds [pseudo_metric_space β] {f : α → β} {a b} :
tendsto f (𝓝 a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, dist x a < δ → dist (f x) b < ε | nhds_basis_ball.tendsto_iff nhds_basis_ball | theorem | metric.tendsto_nhds_nhds | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff [pseudo_metric_space β] {f : α → β} {a : α} :
continuous_at f a ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, dist x a < δ → dist (f x) (f a) < ε | by rw [continuous_at, tendsto_nhds_nhds] | theorem | metric.continuous_at_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_at",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff [pseudo_metric_space β] {f : α → β} {a : α} {s : set α} :
continuous_within_at f s a ↔
∀ ε > 0, ∃ δ > 0, ∀{x:α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε | by rw [continuous_within_at, tendsto_nhds_within_nhds] | theorem | metric.continuous_within_at_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_within_at",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff [pseudo_metric_space β] {f : α → β} {s : set α} :
continuous_on f s ↔
∀ (b ∈ s) (ε > 0), ∃ δ > 0, ∀a ∈ s, dist a b < δ → dist (f a) (f b) < ε | by simp [continuous_on, continuous_within_at_iff] | theorem | metric.continuous_on_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_on",
"continuous_on_iff",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff [pseudo_metric_space β] {f : α → β} :
continuous f ↔
∀b (ε > 0), ∃ δ > 0, ∀a, dist a b < δ → dist (f a) (f b) < ε | continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds_nhds | theorem | metric.continuous_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds {f : filter β} {u : β → α} {a : α} :
tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε | nhds_basis_ball.tendsto_right_iff | theorem | metric.tendsto_nhds | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff' [topological_space β] {f : β → α} {b : β} :
continuous_at f b ↔
∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε | by rw [continuous_at, tendsto_nhds] | theorem | metric.continuous_at_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_at",
"tendsto_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff' [topological_space β] {f : β → α} {b : β} {s : set β} :
continuous_within_at f s b ↔
∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε | by rw [continuous_within_at, tendsto_nhds] | theorem | metric.continuous_within_at_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_within_at",
"tendsto_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff' [topological_space β] {f : β → α} {s : set β} :
continuous_on f s ↔
∀ (b ∈ s) (ε > 0), ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε | by simp [continuous_on, continuous_within_at_iff'] | theorem | metric.continuous_on_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_on",
"continuous_on_iff'",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff' [topological_space β] {f : β → α} :
continuous f ↔ ∀a (ε > 0), ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε | continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds | theorem | metric.continuous_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous",
"tendsto_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top [nonempty β] [semilattice_sup β] {u : β → α} {a : α} :
tendsto u at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) a < ε | (at_top_basis.tendsto_iff nhds_basis_ball).trans $
by { simp only [exists_prop, true_and], refl } | theorem | metric.tendsto_at_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_prop",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top' [nonempty β] [semilattice_sup β] [no_max_order β] {u : β → α} {a : α} :
tendsto u at_top (𝓝 a) ↔ ∀ε>0, ∃N, ∀n>N, dist (u n) a < ε | (at_top_basis_Ioi.tendsto_iff nhds_basis_ball).trans $
by { simp only [exists_prop, true_and], refl } | theorem | metric.tendsto_at_top' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"exists_prop",
"no_max_order",
"semilattice_sup"
] | A variant of `tendsto_at_top` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_singleton_iff {α : Type*} [pseudo_metric_space α] {x : α} :
is_open ({x} : set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x | by simp [is_open_iff, subset_singleton_iff, mem_ball] | lemma | metric.is_open_singleton_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_open",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ball_inter_eq_singleton_of_mem_discrete [discrete_topology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, metric.ball x ε ∩ s = {x} | nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hx | lemma | metric.exists_ball_inter_eq_singleton_of_mem_discrete | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"discrete_topology",
"metric.ball"
] | Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
centered at `x` and intersecting `s` only at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_closed_ball_inter_eq_singleton_of_discrete [discrete_topology s] {x : α} (hx : x ∈ s) :
∃ ε > 0, metric.closed_ball x ε ∩ s = {x} | nhds_basis_closed_ball.exists_inter_eq_singleton_of_mem_discrete hx | lemma | metric.exists_closed_ball_inter_eq_singleton_of_discrete | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"discrete_topology",
"metric.closed_ball"
] | Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
of positive radius centered at `x` and intersecting `s` only at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.dense.exists_dist_lt {s : set α} (hs : dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
∃ y ∈ s, dist x y < ε | begin
have : (ball x ε).nonempty, by simp [hε],
simpa only [mem_ball'] using hs.exists_mem_open is_open_ball this
end | lemma | dense.exists_dist_lt | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.dense_range.exists_dist_lt {β : Type*} {f : β → α} (hf : dense_range f)
(x : α) {ε : ℝ} (hε : 0 < ε) :
∃ y, dist x (f y) < ε | exists_range_iff.1 (hf.exists_dist_lt x hε) | lemma | dense_range.exists_dist_lt | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_metric.uniformity_basis_edist :
(𝓤 α).has_basis (λ ε:ℝ≥0∞, 0 < ε) (λ ε, {p | edist p.1 p.2 < ε}) | ⟨begin
intro t,
refine mem_uniformity_dist.trans ⟨_, _⟩; rintro ⟨ε, ε0, Hε⟩,
{ use [ennreal.of_real ε, ennreal.of_real_pos.2 ε0],
rintros ⟨a, b⟩,
simp only [edist_dist, ennreal.of_real_lt_of_real_iff ε0],
exact Hε },
{ rcases ennreal.lt_iff_exists_real_btwn.1 ε0 with ⟨ε', _, ε0', hε⟩,
rw [ennrea... | lemma | pseudo_metric.uniformity_basis_edist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_dist",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff",
"ennreal.of_real_pos"
] | Expressing the uniformity in terms of `edist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.uniformity_edist : 𝓤 α = (⨅ ε>0, 𝓟 {p:α×α | edist p.1 p.2 < ε}) | pseudo_metric.uniformity_basis_edist.eq_binfi | theorem | metric.uniformity_edist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_metric_space.to_pseudo_emetric_space : pseudo_emetric_space α | { edist := edist,
edist_self := by simp [edist_dist],
edist_comm := by simp only [edist_dist, dist_comm]; simp,
edist_triangle := assume x y z, begin
simp only [edist_dist, ← ennreal.of_real_add, dist_nonneg],
rw ennreal.of_real_le_of_real_iff _,
{ exact dist_trian... | instance | pseudo_metric_space.to_pseudo_emetric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm",
"dist_nonneg",
"dist_triangle",
"edist_dist",
"ennreal.of_real_add",
"ennreal.of_real_le_of_real_iff",
"metric.uniformity_edist",
"pseudo_emetric_space",
"uniformity_edist"
] | A pseudometric space induces a pseudoemetric space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.eball_top_eq_univ (x : α) :
emetric.ball x ∞ = set.univ | set.eq_univ_iff_forall.mpr (λ y, edist_lt_top y x) | lemma | metric.eball_top_eq_univ | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_lt_top",
"emetric.ball"
] | In a pseudometric space, an open ball of infinite radius is the whole space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.emetric_ball {x : α} {ε : ℝ} : emetric.ball x (ennreal.of_real ε) = ball x ε | begin
ext y,
simp only [emetric.mem_ball, mem_ball, edist_dist],
exact ennreal.of_real_lt_of_real_iff_of_nonneg dist_nonneg
end | lemma | metric.emetric_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nonneg",
"edist_dist",
"emetric.ball",
"emetric.mem_ball",
"ennreal.of_real",
"ennreal.of_real_lt_of_real_iff_of_nonneg"
] | Balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : emetric.ball x ε = ball x ε | by { convert metric.emetric_ball, simp } | lemma | metric.emetric_ball_nnreal | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.ball",
"metric.emetric_ball"
] | Balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.emetric_closed_ball {x : α} {ε : ℝ} (h : 0 ≤ ε) :
emetric.closed_ball x (ennreal.of_real ε) = closed_ball x ε | by ext y; simp [edist_dist]; rw ennreal.of_real_le_of_real_iff h | lemma | metric.emetric_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_dist",
"emetric.closed_ball",
"ennreal.of_real",
"ennreal.of_real_le_of_real_iff"
] | Closed balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.emetric_closed_ball_nnreal {x : α} {ε : ℝ≥0} :
emetric.closed_ball x ε = closed_ball x ε | by { convert metric.emetric_closed_ball ε.2, simp } | lemma | metric.emetric_closed_ball_nnreal | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"emetric.closed_ball",
"metric.emetric_closed_ball"
] | Closed balls defined using the distance or the edistance coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.emetric_ball_top (x : α) : emetric.ball x ⊤ = univ | eq_univ_of_forall $ λ y, edist_lt_top _ _ | lemma | metric.emetric_ball_top | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"edist_lt_top",
"emetric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.inseparable_iff {x y : α} : inseparable x y ↔ dist x y = 0 | by rw [emetric.inseparable_iff, edist_nndist, dist_nndist, ennreal.coe_eq_zero,
nnreal.coe_eq_zero] | lemma | metric.inseparable_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_nndist",
"edist_nndist",
"emetric.inseparable_iff",
"ennreal.coe_eq_zero",
"inseparable",
"nnreal.coe_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_metric_space.replace_uniformity {α} [U : uniform_space α] (m : pseudo_metric_space α)
(H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
pseudo_metric_space α | { dist := @dist _ m.to_has_dist,
dist_self := dist_self,
dist_comm := dist_comm,
dist_triangle := dist_triangle,
edist := edist,
edist_dist := edist_dist,
to_uniform_space := U,
uniformity_dist := H.trans pseudo_metric_space.uniformity_dist } | def | pseudo_metric_space.replace_uniformity | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"edist_dist",
"pseudo_metric_space",
"uniform_space"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.replace_uniformity_eq {α} [U : uniform_space α]
(m : pseudo_metric_space α) (H : 𝓤[U] = 𝓤[pseudo_emetric_space.to_uniform_space]) :
m.replace_uniformity H = m | by { ext, refl } | lemma | pseudo_metric_space.replace_uniformity_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_metric_space.replace_topology {γ} [U : topological_space γ]
(m : pseudo_metric_space γ) (H : U = m.to_uniform_space.to_topological_space) :
pseudo_metric_space γ | @pseudo_metric_space.replace_uniformity γ (m.to_uniform_space.replace_topology H) m rfl | def | pseudo_metric_space.replace_topology | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"pseudo_metric_space.replace_uniformity",
"topological_space"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.replace_topology_eq {γ} [U : topological_space γ]
(m : pseudo_metric_space γ) (H : U = m.to_uniform_space.to_topological_space) :
m.replace_topology H = m | by { ext, refl } | lemma | pseudo_metric_space.replace_topology_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"pseudo_metric_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_emetric_space.to_pseudo_metric_space_of_dist {α : Type u} [e : pseudo_emetric_space α]
(dist : α → α → ℝ)
(edist_ne_top : ∀x y: α, edist x y ≠ ⊤)
(h : ∀x y, dist x y = ennreal.to_real (edist x y)) :
pseudo_metric_space α | let m : pseudo_metric_space α :=
{ dist := dist,
dist_self := λx, by simp [h],
dist_comm := λx y, by simp [h, pseudo_emetric_space.edist_comm],
dist_triangle := λx y z, begin
simp only [h],
rw [← ennreal.to_real_add (edist_ne_top _ _) (edist_ne_top _ _),
ennreal.to_real_le_t... | def | pseudo_emetric_space.to_pseudo_metric_space_of_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm",
"dist_self",
"dist_triangle",
"edist_dist",
"edist_ne_top",
"ennreal.add_eq_top",
"ennreal.of_real_to_real",
"ennreal.to_real",
"ennreal.to_real_add",
"ennreal.to_real_le_to_real",
"metric.uniformity_edist",
"pseudo_emetric_space",
"pseudo_metric_space",
"uniformity_pseudoedis... | 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 s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_emetric_space.to_pseudo_metric_space {α : Type u} [e : pseudo_emetric_space α]
(h : ∀x y: α, edist x y ≠ ⊤) : pseudo_metric_space α | pseudo_emetric_space.to_pseudo_metric_space_of_dist
(λx y, ennreal.to_real (edist x y)) h (λx y, rfl) | def | pseudo_emetric_space.to_pseudo_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ennreal.to_real",
"pseudo_emetric_space",
"pseudo_emetric_space.to_pseudo_metric_space_of_dist",
"pseudo_metric_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.replace_bornology {α} [B : bornology α] (m : pseudo_metric_space α)
(H : ∀ s, @is_bounded _ B s ↔ @is_bounded _ pseudo_metric_space.to_bornology s) :
pseudo_metric_space α | { to_bornology := B,
cobounded_sets := set.ext $ compl_surjective.forall.2 $ λ s, (H s).trans $
by rw [is_bounded_iff, mem_set_of_eq, compl_compl],
.. m } | def | pseudo_metric_space.replace_bornology | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology",
"compl_compl",
"pseudo_metric_space",
"set.ext"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.replace_bornology_eq {α} [m : pseudo_metric_space α] [B : bornology α]
(H : ∀ s, @is_bounded _ B s ↔ @is_bounded _ pseudo_metric_space.to_bornology s) :
pseudo_metric_space.replace_bornology _ H = m | by { ext, refl } | lemma | pseudo_metric_space.replace_bornology_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology",
"pseudo_metric_space",
"pseudo_metric_space.replace_bornology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.complete_of_convergent_controlled_sequences (B : ℕ → real) (hB : ∀n, 0 < B n)
(H : ∀u : ℕ → α, (∀N n m : ℕ, N ≤ n → N ≤ m → dist (u n) (u m) < B N) →
∃x, tendsto u at_top (𝓝 x)) :
complete_space α | uniform_space.complete_of_convergent_controlled_sequences
(λ n, {p:α×α | dist p.1 p.2 < B n}) (λ n, dist_mem_uniformity $ hB n) H | theorem | metric.complete_of_convergent_controlled_sequences | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"complete_space",
"real",
"uniform_space.complete_of_convergent_controlled_sequences"
] | A very useful criterion to show that a space is complete is to show that all sequences
which satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m ≥ N` are
converging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to
`0`, which makes it possible to use arguments of converging ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.complete_of_cauchy_seq_tendsto :
(∀ u : ℕ → α, cauchy_seq u → ∃a, tendsto u at_top (𝓝 a)) → complete_space α | emetric.complete_of_cauchy_seq_tendsto | theorem | metric.complete_of_cauchy_seq_tendsto | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"cauchy_seq",
"complete_space",
"emetric.complete_of_cauchy_seq_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.pseudo_metric_space : pseudo_metric_space ℝ | { dist := λx y, |x - y|,
dist_self := by simp [abs_zero],
dist_comm := assume x y, abs_sub_comm _ _,
dist_triangle := assume x y z, abs_sub_le _ _ _ } | instance | real.pseudo_metric_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_sub_comm",
"abs_sub_le",
"abs_zero",
"dist_comm",
"dist_self",
"dist_triangle",
"pseudo_metric_space"
] | Instantiate the reals as a pseudometric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.dist_eq (x y : ℝ) : dist x y = |x - y| | rfl | theorem | real.dist_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.nndist_eq (x y : ℝ) : nndist x y = real.nnabs (x - y) | rfl | theorem | real.nndist_eq | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"real.nnabs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.nndist_eq' (x y : ℝ) : nndist x y = real.nnabs (y - x) | nndist_comm _ _ | theorem | real.nndist_eq' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nndist_comm",
"real.nnabs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x| | by simp [real.dist_eq] | theorem | real.dist_0_eq_abs | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.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 | theorem | real.dist_left_le_of_mem_uIcc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.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 | theorem | real.dist_right_le_of_mem_uIcc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.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) | theorem | real.dist_le_of_mem_uIcc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.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) | theorem | real.dist_le_of_mem_Icc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_of_nonpos",
"real.dist_eq",
"real.dist_le_of_mem_uIcc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.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 | theorem | real.dist_le_of_mem_Icc_01 | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"real.dist_le_of_mem_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.ball_eq_Ioo (x r : ℝ) : ball x r = Ioo (x - r) (x + r) | set.ext $ λ y, by rw [mem_ball, dist_comm, real.dist_eq,
abs_sub_lt_iff, mem_Ioo, ← sub_lt_iff_lt_add', sub_lt_comm] | lemma | real.ball_eq_Ioo | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_sub_lt_iff",
"dist_comm",
"real.dist_eq",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.closed_ball_eq_Icc {x r : ℝ} : closed_ball x r = Icc (x - r) (x + r) | by ext y; rw [mem_closed_ball, dist_comm, real.dist_eq,
abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add', sub_le_comm] | lemma | real.closed_ball_eq_Icc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_sub_le_iff",
"dist_comm",
"real.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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', add_self_div_two, ← add_div,
add_assoc, add_sub_cancel'_right, add_self_div_two] | theorem | real.Ioo_eq_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"add_div",
"add_self_div_two",
"real.ball_eq_Ioo",
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.Icc_eq_closed_ball (x y : ℝ) : Icc x y = closed_ball ((x + y) / 2) ((y - x) / 2) | by rw [real.closed_ball_eq_Icc, ← sub_div, add_comm, ← sub_add,
add_sub_cancel', add_self_div_two, ← add_div,
add_assoc, add_sub_cancel'_right, add_self_div_two] | theorem | real.Icc_eq_closed_ball | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"add_div",
"add_self_div_two",
"real.closed_ball_eq_Icc",
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_Icc (a b : α) : totally_bounded (Icc a b) | is_compact_Icc.totally_bounded | lemma | totally_bounded_Icc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_Ico (a b : α) : totally_bounded (Ico a b) | totally_bounded_subset Ico_subset_Icc_self (totally_bounded_Icc a b) | lemma | totally_bounded_Ico | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded",
"totally_bounded_Icc",
"totally_bounded_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_Ioc (a b : α) : totally_bounded (Ioc a b) | totally_bounded_subset Ioc_subset_Icc_self (totally_bounded_Icc a b) | lemma | totally_bounded_Ioc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded",
"totally_bounded_Icc",
"totally_bounded_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_Ioo (a b : α) : totally_bounded (Ioo a b) | totally_bounded_subset Ioo_subset_Icc_self (totally_bounded_Icc a b) | lemma | totally_bounded_Ioo | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"totally_bounded",
"totally_bounded_Icc",
"totally_bounded_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (nhds 0)) : tendsto f t₀ (𝓝 0) | tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft | lemma | squeeze_zero' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"nhds",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the
general case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 | squeeze_zero | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter",
"squeeze_zero'"
] | Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`
and `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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