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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|---|---|---|---|---|---|---|---|---|---|---|
metric.uniformity_eq_comap_nhds_zero :
𝓤 α = comap (λp:α×α, dist p.1 p.2) (𝓝 (0 : ℝ)) | by { ext s,
simp [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff, subset_def, real.dist_0_eq_abs] } | theorem | metric.uniformity_eq_comap_nhds_zero | 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_0_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_iff_tendsto_dist_at_top_0 [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ tendsto (λ (n : β × β), dist (u n.1) (u n.2)) at_top (𝓝 0) | by rw [cauchy_seq_iff_tendsto, metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff,
prod.map_def] | lemma | cauchy_seq_iff_tendsto_dist_at_top_0 | 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",
"cauchy_seq_iff_tendsto",
"metric.uniformity_eq_comap_nhds_zero",
"prod.map_def",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformity_iff_dist_tendsto_zero {ι : Type*} {f : ι → α × α} {p : filter ι} :
tendsto f p (𝓤 α) ↔ tendsto (λ x, dist (f x).1 (f x).2) p (𝓝 0) | by rw [metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff] | lemma | tendsto_uniformity_iff_dist_tendsto_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",
"metric.uniformity_eq_comap_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.congr_dist {ι : Type*} {f₁ f₂ : ι → α} {p : filter ι} {a : α}
(h₁ : tendsto f₁ p (𝓝 a)) (h : tendsto (λ 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 | lemma | filter.tendsto.congr_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_of_dist {ι : Type*} {f₁ f₂ : ι → α} {p : filter ι} {a : α}
(h : tendsto (λ 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 | lemma | tendsto_iff_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"
] | [
"filter",
"uniform.tendsto_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_closed_ball_subset {x : α} {u : set α} (hu : u ∈ 𝓝 x) :
∀ᶠ r in 𝓝 (0 : ℝ), closed_ball x r ⊆ u | begin
obtain ⟨ε, εpos, hε⟩ : ∃ ε (hε : 0 < ε), closed_ball x ε ⊆ u :=
nhds_basis_closed_ball.mem_iff.1 hu,
have : Iic ε ∈ 𝓝 (0 : ℝ) := Iic_mem_nhds εpos,
filter_upwards [this] with _ hr using subset.trans (closed_ball_subset_closed_ball hr) hε,
end | lemma | eventually_closed_ball_subset | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"Iic_mem_nhds"
] | If `u` is a neighborhood of `x`, then for small enough `r`, the closed ball
`closed_ball x r` is contained in `u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.cauchy_seq_iff {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀m n≥N, dist (u m) (u n) < ε | uniformity_basis_dist.cauchy_seq_iff | theorem | metric.cauchy_seq_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_seq"
] | In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
the distance between its elements is arbitrarily small | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.cauchy_seq_iff' {u : β → α} :
cauchy_seq u ↔ ∀ε>0, ∃N, ∀n≥N, dist (u n) (u N) < ε | uniformity_basis_dist.cauchy_seq_iff' | theorem | metric.cauchy_seq_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_seq"
] | A variation around the pseudometric characterization of Cauchy sequences | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
metric.uniform_cauchy_seq_on_iff {γ : Type*}
{F : β → γ → α} {s : set γ} :
uniform_cauchy_seq_on F at_top s ↔
∀ ε : ℝ, ε > 0 → ∃ (N : β), ∀ m : β, m ≥ N → ∀ n : β, n ≥ N → ∀ x : γ, x ∈ s →
dist (F m x) (F n x) < ε | begin
split,
{ intros h ε hε,
let u := { a : α × α | dist a.fst a.snd < ε },
have hu : u ∈ 𝓤 α := metric.mem_uniformity_dist.mpr ⟨ε, hε, (λ a b, by simp)⟩,
rw ←@filter.eventually_at_top_prod_self' _ _ _
(λ m, ∀ x : γ, x ∈ s → dist (F m.fst x) (F m.snd x) < ε),
specialize h u hu,
rw prod_a... | theorem | metric.uniform_cauchy_seq_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.eventually_at_top_prod_self'",
"uniform_cauchy_seq_on"
] | In a pseudometric space, unifom Cauchy sequences are characterized by the fact that, eventually,
the distance between all its elements is uniformly, arbitrarily small | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
(h : ∀ n m : β, n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : tendsto b at_top (𝓝 0)) :
cauchy_seq s | metric.cauchy_seq_iff'.2 $ λ ε ε0,
(h₀.eventually (gt_mem_nhds ε0)).exists.imp $ λ N hN n hn,
calc dist (s n) (s N) = dist (s N) (s n) : dist_comm _ _
... ≤ b N : h _ _ hn
... < ε : hN | lemma | cauchy_seq_of_le_tendsto_0' | 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",
"dist_comm",
"gt_mem_nhds"
] | If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
and `b` converges to zero, then `s` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
(h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : tendsto b at_top (𝓝 0)) :
cauchy_seq s | cauchy_seq_of_le_tendsto_0' b (λ n m hnm, h _ _ _ le_rfl hnm) h₀ | lemma | cauchy_seq_of_le_tendsto_0 | 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",
"cauchy_seq_of_le_tendsto_0'",
"le_rfl"
] | If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
and `b` converges to zero, then `s` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_bdd {u : ℕ → α} (hu : cauchy_seq u) :
∃ R > 0, ∀ m n, dist (u m) (u n) < R | begin
rcases metric.cauchy_seq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩,
rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R,
{ exact ⟨_, add_pos R0 R0, λ m n,
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩ },
let R := finset.sup (finset.range N) (λ n, nndist (u n) (u N)),
refine... | theorem | cauchy_seq_bdd | 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",
"dist_triangle_right",
"finset.le_sup",
"finset.range",
"finset.sup",
"zero_lt_one"
] | A Cauchy sequence on the natural numbers is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_iff_le_tendsto_0 {s : ℕ → α} : cauchy_seq s ↔ ∃ b : ℕ → ℝ,
(∀ n, 0 ≤ b n) ∧
(∀ n m N : ℕ, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) ∧
tendsto b at_top (𝓝 0) | ⟨λ hs, begin
/- `s` is a Cauchy sequence. The sequence `b` will be constructed by taking
the supremum of the distances between `s n` and `s m` for `n m ≥ N`.
First, we prove that all these distances are bounded, as otherwise the Sup
would not make sense. -/
let S := λ N, (λ(p : ℕ × ℕ), dist (s p.1) (s p.2)) '... | lemma | cauchy_seq_iff_le_tendsto_0 | 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_nonneg",
"bdd_above",
"cSup_le",
"cauchy_seq",
"cauchy_seq_bdd",
"cauchy_seq_of_le_tendsto_0",
"dist_self",
"half_pos",
"le_cSup",
"real.dist_0_eq_abs"
] | Yet another metric characterization of Cauchy sequences on integers. This one is often the
most efficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space.induced {α β} (f : α → β)
(m : pseudo_metric_space β) : pseudo_metric_space α | { dist := λ x y, dist (f x) (f y),
dist_self := λ x, dist_self _,
dist_comm := λ x y, dist_comm _ _,
dist_triangle := λ x y z, dist_triangle _ _ _,
edist := λ x y, edist (f x) (f y),
edist_dist := λ x y, edist_dist _ _,
to_uniform_space := uniform_... | def | pseudo_metric_space.induced | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bornology.induced",
"compl_compl",
"dist_comm",
"dist_self",
"dist_triangle",
"edist_dist",
"filter.mem_sets",
"pseudo_metric_space",
"set.ext",
"uniform_space.comap"
] | Pseudometric space structure pulled back by a function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inducing.comap_pseudo_metric_space {α β} [topological_space α] [pseudo_metric_space β]
{f : α → β} (hf : inducing f) : pseudo_metric_space α | (pseudo_metric_space.induced f ‹_›).replace_topology hf.induced | def | inducing.comap_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"
] | [
"inducing",
"pseudo_metric_space",
"pseudo_metric_space.induced",
"topological_space"
] | Pull back a pseudometric space structure by an inducing map. This is a version of
`pseudo_metric_space.induced` useful in case if the domain already has a `topological_space`
structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.comap_pseudo_metric_space {α β} [uniform_space α] [pseudo_metric_space β]
(f : α → β) (h : uniform_inducing f) : pseudo_metric_space α | (pseudo_metric_space.induced f ‹_›).replace_uniformity h.comap_uniformity.symm | def | uniform_inducing.comap_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"
] | [
"pseudo_metric_space",
"pseudo_metric_space.induced",
"uniform_inducing",
"uniform_space"
] | Pull back a pseudometric space structure by a uniform inducing map. This is a version of
`pseudo_metric_space.induced` useful in case if the domain already has a `uniform_space`
structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.pseudo_metric_space {p : α → Prop} : pseudo_metric_space (subtype p) | pseudo_metric_space.induced coe ‹_› | instance | subtype.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"
] | [
"pseudo_metric_space",
"pseudo_metric_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.dist_eq {p : α → Prop} (x y : subtype p) : dist x y = dist (x : α) y | rfl | theorem | subtype.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 | |
subtype.nndist_eq {p : α → Prop} (x y : subtype p) : nndist x y = nndist (x : α) y | rfl | theorem | subtype.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y | rfl | theorem | mul_opposite.dist_unop | 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_op (x y : α) : dist (op x) (op y) = dist x y | rfl | theorem | mul_opposite.dist_op | 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 | |
nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y | rfl | theorem | mul_opposite.nndist_unop | 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 | |
nndist_op (x y : α) : nndist (op x) (op y) = nndist x y | rfl | theorem | mul_opposite.nndist_op | 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 | |
nnreal.dist_eq (a b : ℝ≥0) : dist a b = |(a:ℝ) - b| | rfl | lemma | nnreal.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 | |
nnreal.nndist_eq (a b : ℝ≥0) :
nndist a b = max (a - b) (b - a) | begin
wlog h : b ≤ a,
{ rw [nndist_comm, max_comm], exact this b a (le_of_not_le h) },
rw [← nnreal.coe_eq, ← dist_nndist, nnreal.dist_eq, tsub_eq_zero_iff_le.2 h,
max_eq_left (zero_le $ a - b), ← nnreal.coe_sub h, abs_of_nonneg (a - b).coe_nonneg],
end | lemma | nnreal.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"
] | [
"abs_of_nonneg",
"dist_nndist",
"nndist_comm",
"nnreal.coe_eq",
"nnreal.coe_sub",
"nnreal.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z | by simp only [nnreal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_le'] | lemma | nnreal.nndist_zero_eq_val | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nnreal.nndist_eq",
"tsub_zero",
"zero_le'",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z | by { rw nndist_comm, exact nnreal.nndist_zero_eq_val z, } | lemma | nnreal.nndist_zero_eq_val' | 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",
"nnreal.nndist_zero_eq_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b | begin
suffices : (a : ℝ) ≤ (b : ℝ) + (dist a b),
{ exact nnreal.coe_le_coe.mp this, },
linarith [le_of_abs_le (by refl : abs (a-b : ℝ) ≤ (dist a b))],
end | lemma | nnreal.le_add_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"le_of_abs_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.dist_eq (x y : ulift β) : dist x y = dist x.down y.down | rfl | lemma | ulift.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 | |
ulift.nndist_eq (x y : ulift β) : nndist x y = nndist x.down y.down | rfl | lemma | ulift.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.dist_up_up (x y : β) : dist (ulift.up x) (ulift.up y) = dist x y | rfl | lemma | ulift.dist_up_up | 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 | |
ulift.nndist_up_up (x y : β) : nndist (ulift.up x) (ulift.up y) = nndist x y | rfl | lemma | ulift.nndist_up_up | 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 | |
prod.pseudo_metric_space_max :
pseudo_metric_space (α × β) | (pseudo_emetric_space.to_pseudo_metric_space_of_dist
(λ x y : α × β, dist x.1 y.1 ⊔ dist x.2 y.2)
(λ x y, (max_lt (edist_lt_top _ _) (edist_lt_top _ _)).ne)
(λ x y, by simp only [sup_eq_max, dist_edist,
← ennreal.to_real_max (edist_ne_top _ _) (edist_ne_top _ _), prod.edist_eq]))
.replace_bornology $
λ ... | instance | prod.pseudo_metric_space_max | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"edist_lt_top",
"edist_ne_top",
"ennreal.to_real_max",
"forall_and_distrib",
"max_le_iff",
"prod.edist_eq",
"pseudo_emetric_space.to_pseudo_metric_space_of_dist",
"pseudo_metric_space",
"sup_eq_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.dist_eq {x y : α × β} :
dist x y = max (dist x.1 y.1) (dist x.2 y.2) | rfl | lemma | prod.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 | |
dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ | by simp [prod.dist_eq, dist_nonneg] | lemma | dist_prod_same_left | 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",
"prod.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_prod_same_right {x₁ x₂ : α} {y : β} : dist (x₁, y) (x₂, y) = dist x₁ x₂ | by simp [prod.dist_eq, dist_nonneg] | lemma | dist_prod_same_right | 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",
"prod.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_prod_same (x : α) (y : β) (r : ℝ) :
ball x r ×ˢ ball y r = ball (x, y) r | ext $ λ z, by simp [prod.dist_eq] | theorem | ball_prod_same | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"prod.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_prod_same (x : α) (y : β) (r : ℝ) :
closed_ball x r ×ˢ closed_ball y r = closed_ball (x, y) r | ext $ λ z, by simp [prod.dist_eq] | theorem | closed_ball_prod_same | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"prod.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sphere_prod (x : α × β) (r : ℝ) :
sphere x r = sphere x.1 r ×ˢ closed_ball x.2 r ∪ closed_ball x.1 r ×ˢ sphere x.2 r | begin
obtain hr | rfl | hr := lt_trichotomy r 0,
{ simp [hr], },
{ cases x,
simp_rw [←closed_ball_eq_sphere_of_nonpos le_rfl, union_self, closed_ball_prod_same] },
{ ext ⟨x', y'⟩,
simp_rw [set.mem_union, set.mem_prod, metric.mem_closed_ball, metric.mem_sphere,
prod.dist_eq, max_eq_iff],
refine... | theorem | sphere_prod | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"and_congr_left",
"closed_ball_prod_same",
"le_rfl",
"max_eq_iff",
"metric.mem_closed_ball",
"metric.mem_sphere",
"prod.dist_eq",
"set.mem_prod",
"set.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_dist : uniform_continuous (λp:α×α, dist p.1 p.2) | metric.uniform_continuous_iff.2 (λ ε ε0, ⟨ε/2, half_pos ε0,
begin
suffices,
{ intros p q h, cases p with p₁ p₂, cases q with q₁ q₂,
cases max_lt_iff.1 h with h₁ h₂, clear h,
dsimp at h₁ h₂ ⊢,
rw real.dist_eq,
refine abs_sub_lt_iff.2 ⟨_, _⟩,
{ revert p₁ p₂ q₁ q₂ h₁ h₂, exact this },
{ apply t... | theorem | uniform_continuous_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"abs_dist_sub_le",
"add_halves",
"dist_comm",
"half_pos",
"real.dist_eq",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.dist [uniform_space β] {f g : β → α}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
uniform_continuous (λb, dist (f b) (g b)) | uniform_continuous_dist.comp (hf.prod_mk hg) | theorem | uniform_continuous.dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_dist : continuous (λp:α×α, dist p.1 p.2) | uniform_continuous_dist.continuous | theorem | continuous_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.dist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, dist (f b) (g b)) | continuous_dist.comp (hf.prod_mk hg : _) | theorem | continuous.dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.dist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, dist (f x) (g x)) x (𝓝 (dist a b)) | (continuous_dist.tendsto (a, b)).comp (hf.prod_mk_nhds hg) | theorem | filter.tendsto.dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_comap_dist (a : α) : (𝓝 (0 : ℝ)).comap (λa', dist a' a) = 𝓝 a | by simp only [@nhds_eq_comap_uniformity α, metric.uniformity_eq_comap_nhds_zero,
comap_comap, (∘), dist_comm] | lemma | nhds_comap_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",
"metric.uniformity_eq_comap_nhds_zero",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_dist_tendsto_zero {f : β → α} {x : filter β} {a : α} :
(tendsto f x (𝓝 a)) ↔ (tendsto (λb, dist (f b) a) x (𝓝 0)) | by rw [← nhds_comap_dist a, tendsto_comap_iff] | lemma | tendsto_iff_dist_tendsto_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_comap_dist"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_continuous_dist [topological_space β] {f : β → α} :
continuous f ↔ continuous (λ x : β × β, dist (f x.1) (f x.2)) | ⟨λ h, (h.comp continuous_fst).dist (h.comp continuous_snd), λ h, continuous_iff_continuous_at.2 $
λ x, tendsto_iff_dist_tendsto_zero.2 $
(h.comp (continuous_id.prod_mk continuous_const)).tendsto' _ _ $ dist_self _⟩ | lemma | continuous_iff_continuous_dist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous",
"continuous_const",
"continuous_fst",
"continuous_snd",
"dist_self",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_nndist : uniform_continuous (λp:α×α, nndist p.1 p.2) | uniform_continuous_dist.subtype_mk _ | lemma | uniform_continuous_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.nndist [uniform_space β] {f g : β → α} (hf : uniform_continuous f)
(hg : uniform_continuous g) :
uniform_continuous (λ b, nndist (f b) (g b)) | uniform_continuous_nndist.comp (hf.prod_mk hg) | lemma | uniform_continuous.nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_nndist : continuous (λp:α×α, nndist p.1 p.2) | uniform_continuous_nndist.continuous | lemma | continuous_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.nndist [topological_space β] {f g : β → α}
(hf : continuous f) (hg : continuous g) : continuous (λb, nndist (f b) (g b)) | continuous_nndist.comp (hf.prod_mk hg : _) | lemma | continuous.nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.nndist {f g : β → α} {x : filter β} {a b : α}
(hf : tendsto f x (𝓝 a)) (hg : tendsto g x (𝓝 b)) :
tendsto (λx, nndist (f x) (g x)) x (𝓝 (nndist a b)) | (continuous_nndist.tendsto (a, b)).comp (hf.prod_mk_nhds hg) | theorem | filter.tendsto.nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_ball : is_closed (closed_ball x ε) | is_closed_le (continuous_id.dist continuous_const) continuous_const | theorem | metric.is_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"
] | [
"continuous_const",
"is_closed",
"is_closed_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_sphere : is_closed (sphere x ε) | is_closed_eq (continuous_id.dist continuous_const) continuous_const | lemma | metric.is_closed_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_const",
"is_closed",
"is_closed_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_closed_ball : closure (closed_ball x ε) = closed_ball x ε | is_closed_ball.closure_eq | theorem | metric.closure_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"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_sphere : closure (sphere x ε) = sphere x ε | is_closed_sphere.closure_eq | theorem | metric.closure_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_ball_subset_closed_ball : closure (ball x ε) ⊆ closed_ball x ε | closure_minimal ball_subset_closed_ball is_closed_ball | theorem | metric.closure_ball_subset_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"
] | [
"closure",
"closure_minimal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_ball_subset_sphere : frontier (ball x ε) ⊆ sphere x ε | frontier_lt_subset_eq (continuous_id.dist continuous_const) continuous_const | theorem | metric.frontier_ball_subset_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_const",
"frontier",
"frontier_lt_subset_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_closed_ball_subset_sphere : frontier (closed_ball x ε) ⊆ sphere x ε | frontier_le_subset_eq (continuous_id.dist continuous_const) continuous_const | theorem | metric.frontier_closed_ball_subset_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_const",
"frontier",
"frontier_le_subset_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_subset_interior_closed_ball : ball x ε ⊆ interior (closed_ball x ε) | interior_maximal ball_subset_closed_ball is_open_ball | theorem | metric.ball_subset_interior_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"
] | [
"interior",
"interior_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | metric.mem_closure_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"dist_comm",
"mem_closure_iff",
"mem_closure_iff_nhds_basis"
] | ε-characterization of the closure in pseudometric spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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] | lemma | metric.mem_closure_range_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | metric.mem_closure_range_iff_nat | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"dist_comm",
"forall_const",
"mem_closure_iff_nhds_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_closed' {s : set α} (hs : is_closed 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 | metric.mem_of_closed' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_closed",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_zero' (x : α) : closed_ball x 0 = closure {x} | subset.antisymm
(λ y hy, mem_closure_iff.2 $ λ ε ε0, ⟨x, mem_singleton x, (mem_closed_ball.1 hy).trans_lt ε0⟩)
(closure_minimal (singleton_subset_iff.2 (dist_self x).le) is_closed_ball) | lemma | metric.closed_ball_zero' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closure",
"closure_minimal",
"dist_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_iff {s : set α} :
dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).nonempty | forall_congr $ λ x, by simp only [mem_closure_iff, set.nonempty, exists_prop, mem_inter_iff,
mem_ball', and_comm] | lemma | metric.dense_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dense",
"exists_prop",
"mem_closure_iff",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_iff {f : β → α} :
dense_range f ↔ ∀ x, ∀ r > 0, ∃ y, dist x (f y) < r | forall_congr $ λ x, by simp only [mem_closure_iff, exists_range_iff] | lemma | metric.dense_range_iff | 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",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.topological_space.is_separable.separable_space {s : set α} (hs : is_separable s) :
separable_space s | begin
classical,
rcases eq_empty_or_nonempty s with rfl|⟨⟨x₀, x₀s⟩⟩,
{ apply_instance },
rcases hs with ⟨c, hc, h'c⟩,
haveI : encodable c := hc.to_encodable,
obtain ⟨u, -, u_pos, u_lim⟩ : ∃ (u : ℕ → ℝ), strict_anti u ∧ (∀ (n : ℕ), 0 < u n) ∧
tendsto u at_top (𝓝 0) := exists_seq_strict_anti_tendsto (0 :... | lemma | topological_space.is_separable.separable_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"add_halves",
"dist_triangle",
"encodable",
"exists_seq_strict_anti_tendsto",
"half_pos",
"is_separable",
"metric.ball",
"strict_anti"
] | If a set `s` is separable, then the corresponding subtype is separable in a metric space.
This is not obvious, as the countable set whose closure covers `s` does not need in general to
be contained in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.inducing.is_separable_preimage {f : β → α} [topological_space β]
(hf : inducing f) {s : set α} (hs : is_separable s) :
is_separable (f ⁻¹' s) | begin
haveI : second_countable_topology s,
{ haveI : separable_space s := hs.separable_space,
exact uniform_space.second_countable_of_separable _ },
let g : f ⁻¹' s → s := cod_restrict (f ∘ coe) s (λ x, x.2),
have : inducing g := (hf.comp inducing_coe).cod_restrict _,
haveI : second_countable_topology (f ... | lemma | inducing.is_separable_preimage | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"continuous_subtype_coe",
"inducing",
"inducing_coe",
"is_separable",
"subtype.range_coe_subtype",
"topological_space",
"uniform_space.second_countable_of_separable"
] | The preimage of a separable set by an inducing map is separable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.embedding.is_separable_preimage {f : β → α} [topological_space β]
(hf : embedding f) {s : set α} (hs : is_separable s) :
is_separable (f ⁻¹' s) | hf.to_inducing.is_separable_preimage hs | lemma | embedding.is_separable_preimage | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"embedding",
"is_separable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_on.is_separable_image [topological_space β] {f : α → β} {s : set α}
(hf : continuous_on f s) (hs : is_separable s) :
is_separable (f '' s) | begin
rw show f '' s = s.restrict f '' univ, by ext ; simp,
exact (is_separable_univ_iff.2 hs.separable_space).image
(continuous_on_iff_continuous_restrict.1 hf),
end | lemma | continuous_on.is_separable_image | 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",
"is_separable",
"topological_space"
] | If a map is continuous on a separable set `s`, then the image of `s` is also separable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_metric_space_pi : pseudo_metric_space (Πb, π b) | begin
/- 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 -/
refine (pseudo_emetric_space.to_pseudo_metric_space_of_dist
(λf g : Π b, π b, ((sup univ... | instance | pseudo_metric_space_pi | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"bot_lt_top",
"dist_nndist",
"edist_lt_top",
"edist_nndist",
"ennreal.coe_finset_sup",
"ennreal.coe_to_real",
"finset.mem_univ",
"finset.sup_lt_iff",
"function.eval_apply",
"lift",
"pseudo_emetric_space.to_pseudo_metric_space_of_dist",
"pseudo_metric_space"
] | A finite product of pseudometric spaces is a pseudometric space, with the sup distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nndist_pi_def (f g : Πb, π b) : nndist f g = sup univ (λb, nndist (f b) (g b)) | nnreal.eq rfl | lemma | nndist_pi_def | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_def (f g : Πb, π b) :
dist f g = (sup univ (λb, nndist (f b) (g b)) : ℝ≥0) | rfl | lemma | dist_pi_def | 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 | |
nndist_pi_le_iff {f g : Πb, π b} {r : ℝ≥0} :
nndist f g ≤ r ↔ ∀b, nndist (f b) (g b) ≤ r | by simp [nndist_pi_def] | lemma | nndist_pi_le_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"nndist_pi_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_pi_lt_iff {f g : Πb, π 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 (show ⊥ < r, from hr)] | lemma | nndist_pi_lt_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"finset.sup_lt_iff",
"nndist_pi_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_pi_eq_iff {f g : Π b, π 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 | begin
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 (λ b, _),
apply (and_iff_right $ h _).symm,
end | lemma | nndist_pi_eq_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"and.congr_left_iff",
"eq_iff_le_not_lt",
"nndist_pi_le_iff",
"nndist_pi_lt_iff",
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_lt_iff {f g : Πb, π b} {r : ℝ} (hr : 0 < r) :
dist f g < r ↔ ∀b, dist (f b) (g b) < r | begin
lift r to ℝ≥0 using hr.le,
exact nndist_pi_lt_iff hr,
end | lemma | dist_pi_lt_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"lift",
"nndist_pi_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_le_iff {f g : Πb, π b} {r : ℝ} (hr : 0 ≤ r) :
dist f g ≤ r ↔ ∀b, dist (f b) (g b) ≤ r | begin
lift r to ℝ≥0 using hr,
exact nndist_pi_le_iff
end | lemma | dist_pi_le_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"lift",
"nndist_pi_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_eq_iff {f g : Πb, π b} {r : ℝ} (hr : 0 < r) :
dist f g = r ↔ (∃ i, dist (f i) (g i) = r) ∧ ∀ b, dist (f b) (g b) ≤ r | begin
lift r to ℝ≥0 using hr.le,
simp_rw [←coe_nndist, nnreal.coe_eq, nndist_pi_eq_iff hr, nnreal.coe_le_coe],
end | lemma | dist_pi_eq_iff | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"lift",
"nndist_pi_eq_iff",
"nnreal.coe_eq",
"nnreal.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_le_iff' [nonempty β] {f g : Π b, π b} {r : ℝ} :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r | begin
by_cases hr : 0 ≤ r,
{ exact dist_pi_le_iff hr },
{ exact iff_of_false (λ h, hr $ dist_nonneg.trans h)
(λ h, hr $ dist_nonneg.trans $ h $ classical.arbitrary _) }
end | lemma | dist_pi_le_iff' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"classical.arbitrary",
"dist_pi_le_iff",
"iff_of_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_const_le (a b : α) : dist (λ _ : β, a) (λ _, b) ≤ dist a b | (dist_pi_le_iff dist_nonneg).2 $ λ _, le_rfl | lemma | dist_pi_const_le | 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",
"dist_pi_le_iff",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_pi_const_le (a b : α) : nndist (λ _ : β, a) (λ _, b) ≤ nndist a b | nndist_pi_le_iff.2 $ λ _, le_rfl | lemma | nndist_pi_const_le | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_pi_const [nonempty β] (a b : α) : dist (λ x : β, a) (λ _, b) = dist a b | by simpa only [dist_edist] using congr_arg ennreal.to_real (edist_pi_const a b) | lemma | dist_pi_const | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_edist",
"edist_pi_const",
"ennreal.to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_pi_const [nonempty β] (a b : α) :
nndist (λ x : β, a) (λ _, b) = nndist a b | nnreal.eq $ dist_pi_const a b | lemma | nndist_pi_const | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_pi_const",
"nnreal.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_le_pi_nndist (f g : Πb, π b) (b : β) : nndist (f b) (g b) ≤ nndist f g | by { rw [nndist_pi_def], exact finset.le_sup (finset.mem_univ b) } | lemma | nndist_le_pi_nndist | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"finset.le_sup",
"finset.mem_univ",
"nndist_pi_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_pi_dist (f g : Πb, π 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 | dist_le_pi_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_nndist",
"nndist_le_pi_nndist",
"nnreal.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_pi (x : Πb, π b) {r : ℝ} (hr : 0 < r) :
ball x r = set.pi univ (λ b, ball (x b) r) | by { ext p, simp [dist_pi_lt_iff hr] } | lemma | ball_pi | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_pi_lt_iff",
"set.pi"
] | An open ball in a product space is a product of open balls. See also `metric.ball_pi'`
for a version assuming `nonempty β` instead of `0 < r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ball_pi' [nonempty β] (x : Π b, π b) (r : ℝ) :
ball x r = set.pi univ (λ b, ball (x b) r) | (lt_or_le 0 r).elim (ball_pi x) $ λ hr, by simp [ball_eq_empty.2 hr] | lemma | ball_pi' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"ball_pi",
"set.pi"
] | An open ball in a product space is a product of open balls. See also `metric.ball_pi`
for a version assuming `0 < r` instead of `nonempty β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ball_pi (x : Πb, π b) {r : ℝ} (hr : 0 ≤ r) :
closed_ball x r = set.pi univ (λ b, closed_ball (x b) r) | by { ext p, simp [dist_pi_le_iff hr] } | lemma | closed_ball_pi | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_pi_le_iff",
"set.pi"
] | A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi'`
for a version assuming `nonempty β` instead of `0 ≤ r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_ball_pi' [nonempty β] (x : Π b, π b) (r : ℝ) :
closed_ball x r = set.pi univ (λ b, closed_ball (x b) r) | (le_or_lt 0 r).elim (closed_ball_pi x) $ λ hr, by simp [closed_ball_eq_empty.2 hr] | lemma | closed_ball_pi' | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"closed_ball_pi",
"set.pi"
] | A closed ball in a product space is a product of closed balls. See also `metric.closed_ball_pi`
for a version assuming `0 ≤ r` instead of `nonempty β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sphere_pi (x : Πb, π b) {r : ℝ} (h : 0 < r ∨ nonempty β) :
sphere x r = (⋃ i : β, function.eval i ⁻¹' sphere (x i) r) ∩ closed_ball x r | begin
obtain hr | rfl | hr := lt_trichotomy r 0,
{ simp [hr], },
{ rw [closed_ball_eq_sphere_of_nonpos le_rfl, eq_comm, set.inter_eq_right_iff_subset],
letI := h.resolve_left (lt_irrefl _),
inhabit β,
refine subset_Union_of_subset default _,
intros x hx,
replace hx := hx.le,
rw [dist_pi_le... | lemma | sphere_pi | 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",
"dist_pi_eq_iff",
"dist_pi_le_iff",
"function.eval",
"le_rfl",
"set.inter_eq_right_iff_subset"
] | A sphere in a product space is a union of spheres on each component restricted to the closed
ball. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin.nndist_insert_nth_insert_nth {n : ℕ} {α : fin (n + 1) → Type*}
[Π i, pseudo_metric_space (α i)] (i : fin (n + 1)) (x y : α i) (f g : Π j, α (i.succ_above j)) :
nndist (i.insert_nth x f) (i.insert_nth y g) = max (nndist x y) (nndist f g) | eq_of_forall_ge_iff $ λ c, by simp [nndist_pi_le_iff, i.forall_iff_succ_above] | lemma | fin.nndist_insert_nth_insert_nth | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"eq_of_forall_ge_iff",
"nndist_pi_le_iff",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin.dist_insert_nth_insert_nth {n : ℕ} {α : fin (n + 1) → Type*}
[Π i, pseudo_metric_space (α i)] (i : fin (n + 1)) (x y : α i) (f g : Π j, α (i.succ_above j)) :
dist (i.insert_nth x f) (i.insert_nth y g) = max (dist x y) (dist f g) | by simp only [dist_nndist, fin.nndist_insert_nth_insert_nth, nnreal.coe_max] | lemma | fin.dist_insert_nth_insert_nth | 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",
"fin.nndist_insert_nth_insert_nth",
"nnreal.coe_max",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.dist_le_of_mem_pi_Icc {x y x' y' : β → ℝ} (hx : x ∈ Icc x' y') (hy : y ∈ Icc x' y') :
dist x y ≤ dist x' y' | begin
refine (dist_pi_le_iff dist_nonneg).2 (λ b, (real.dist_le_of_mem_uIcc _ _).trans
(dist_le_pi_dist _ _ b)); refine Icc_subset_uIcc _,
exacts [⟨hx.1 _, hx.2 _⟩, ⟨hy.1 _, hy.2 _⟩]
end | lemma | real.dist_le_of_mem_pi_Icc | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"dist_le_pi_dist",
"dist_nonneg",
"dist_pi_le_iff",
"real.dist_le_of_mem_uIcc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_cover_balls_of_compact {α : Type u} [pseudo_metric_space α] {s : set α}
(hs : is_compact s) {e : ℝ} (he : 0 < e) :
∃t ⊆ s, set.finite t ∧ s ⊆ ⋃x∈t, ball x e | begin
apply hs.elim_finite_subcover_image,
{ simp [is_open_ball] },
{ intros x xs,
simp,
exact ⟨x, ⟨xs, by simpa⟩⟩ }
end | lemma | finite_cover_balls_of_compact | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"pseudo_metric_space",
"set.finite"
] | Any compact set in a pseudometric space can be covered by finitely many balls of a given
positive radius | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proper_space (α : Type u) [pseudo_metric_space α] : Prop | (is_compact_closed_ball : ∀x:α, ∀r, is_compact (closed_ball x r)) | class | proper_space | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"pseudo_metric_space"
] | A pseudometric space is proper if all closed balls are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_sphere {α : Type*} [pseudo_metric_space α] [proper_space α] (x : α) (r : ℝ) :
is_compact (sphere x r) | is_compact_of_is_closed_subset (is_compact_closed_ball x r) is_closed_sphere
sphere_subset_closed_ball | lemma | is_compact_sphere | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"is_compact",
"is_compact_of_is_closed_subset",
"proper_space",
"pseudo_metric_space"
] | In a proper pseudometric space, all spheres are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_of_proper [proper_space α] :
second_countable_topology α | begin
-- We already have `sigma_compact_space_of_locally_compact_second_countable`, so we don't
-- add an instance for `sigma_compact_space`.
suffices : sigma_compact_space α, by exactI emetric.second_countable_of_sigma_compact α,
rcases em (nonempty α) with ⟨⟨x⟩⟩|hn,
{ exact ⟨⟨λ n, closed_ball x n, λ n, is_c... | instance | second_countable_of_proper | topology.metric_space | src/topology/metric_space/basic.lean | [
"tactic.positivity",
"topology.algebra.order.compact",
"topology.metric_space.emetric_space",
"topology.bornology.constructions"
] | [
"em",
"emetric.second_countable_of_sigma_compact",
"is_compact_empty",
"proper_space",
"sigma_compact_space"
] | A proper pseudo metric space is sigma compact, and therefore second countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_dist_right_cocompact_at_top [proper_space α] (x : α) :
tendsto (λ y, dist y x) (cocompact α) at_top | (has_basis_cocompact.tendsto_iff at_top_basis).2 $ λ r hr,
⟨closed_ball x r, is_compact_closed_ball x r, λ y hy, (not_le.1 $ mt mem_closed_ball.2 hy).le⟩ | lemma | tendsto_dist_right_cocompact_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"
] | [
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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