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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sum.uniform_space : uniform_space (α ⊕ β) | { to_core := uniform_space.core.sum,
is_open_uniformity := λ s, ⟨uniformity_sum_of_open_aux, open_of_uniformity_sum_aux⟩ } | instance | sum.uniform_space | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_open_uniformity",
"uniform_space",
"uniform_space.core.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.uniformity : 𝓤 (α ⊕ β) =
map (λ p : α × α, (inl p.1, inl p.2)) (𝓤 α) ⊔
map (λ p : β × β, (inr p.1, inr p.2)) (𝓤 β) | rfl | lemma | sum.uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lebesgue_number_lemma {α : Type u} [uniform_space α] {s : set α} {ι} {c : ι → set α}
(hs : is_compact s) (hc₁ : ∀ i, is_open (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
∃ n ∈ 𝓤 α, ∀ x ∈ s, ∃ i, {y | (x, y) ∈ n} ⊆ c i | begin
let u := λ n, {x | ∃ i (m ∈ 𝓤 α), {y | (x, y) ∈ m ○ n} ⊆ c i},
have hu₁ : ∀ n ∈ 𝓤 α, is_open (u n),
{ refine λ n hn, is_open_uniformity.2 _,
rintro x ⟨i, m, hm, h⟩,
rcases comp_mem_uniformity_sets hm with ⟨m', hm', mm'⟩,
apply (𝓤 α).sets_of_superset hm',
rintros ⟨x, y⟩ hp rfl,
refine ... | lemma | lebesgue_number_lemma | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"comp_mem_uniformity_sets",
"comp_rel_assoc",
"is_compact",
"is_open",
"mem_comp_rel",
"monotone_const",
"prod_mk_mem_comp_rel",
"refl_mem_uniformity",
"uniform_space"
] | Let `c : ι → set α` be an open cover of a compact set `s`. Then there exists an entourage
`n` such that for each `x ∈ s` its `n`-neighborhood is contained in some `c i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lebesgue_number_lemma_sUnion {α : Type u} [uniform_space α] {s : set α} {c : set (set α)}
(hs : is_compact s) (hc₁ : ∀ t ∈ c, is_open t) (hc₂ : s ⊆ ⋃₀ c) :
∃ n ∈ 𝓤 α, ∀ x ∈ s, ∃ t ∈ c, ∀ y, (x, y) ∈ n → y ∈ t | by rw sUnion_eq_Union at hc₂;
simpa using lebesgue_number_lemma hs (by simpa) hc₂ | lemma | lebesgue_number_lemma_sUnion | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"is_compact",
"is_open",
"lebesgue_number_lemma",
"uniform_space"
] | Let `c : set (set α)` be an open cover of a compact set `s`. Then there exists an entourage
`n` such that for each `x ∈ s` its `n`-neighborhood is contained in some `t ∈ c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lebesgue_number_of_compact_open [uniform_space α]
{K U : set α} (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :
∃ V ∈ 𝓤 α, is_open V ∧ ∀ x ∈ K, uniform_space.ball x V ⊆ U | begin
let W : K → set (α × α) := λ k, classical.some $ is_open_iff_open_ball_subset.mp hU k.1 $ hKU k.2,
have hW : ∀ k, W k ∈ 𝓤 α ∧ is_open (W k) ∧ uniform_space.ball k.1 (W k) ⊆ U,
{ intros k,
obtain ⟨h₁, h₂, h₃⟩ := classical.some_spec (is_open_iff_open_ball_subset.mp hU k.1 (hKU k.2)),
exact ⟨h₁, h₂, h... | lemma | lebesgue_number_of_compact_open | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"ball_mono",
"interior_mem_uniformity",
"interior_subset",
"is_compact",
"is_open",
"is_open_interior",
"lebesgue_number_lemma",
"set_coe.exists",
"uniform_space",
"uniform_space.ball",
"uniform_space.is_open_ball",
"uniform_space.mem_ball_self"
] | A useful consequence of the Lebesgue number lemma: given any compact set `K` contained in an
open set `U`, we can find an (open) entourage `V` such that the ball of size `V` about any point of
`K` is contained in `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_right {f : filter β} {u : β → α} {a : α} :
tendsto u f (𝓝 a) ↔ tendsto (λ x, (a, u x)) f (𝓤 α) | by rw [nhds_eq_comap_uniformity, tendsto_comap_iff] | theorem | uniform.tendsto_nhds_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_left {f : filter β} {u : β → α} {a : α} :
tendsto u f (𝓝 a) ↔ tendsto (λ x, (u x, a)) f (𝓤 α) | by rw [nhds_eq_comap_uniformity', tendsto_comap_iff] | theorem | uniform.tendsto_nhds_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"nhds_eq_comap_uniformity'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff'_right [topological_space β] {f : β → α} {b : β} :
continuous_at f b ↔ tendsto (λ x, (f b, f x)) (𝓝 b) (𝓤 α) | by rw [continuous_at, tendsto_nhds_right] | theorem | uniform.continuous_at_iff'_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff'_left [topological_space β] {f : β → α} {b : β} :
continuous_at f b ↔ tendsto (λ x, (f x, f b)) (𝓝 b) (𝓤 α) | by rw [continuous_at, tendsto_nhds_left] | theorem | uniform.continuous_at_iff'_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_iff_prod [topological_space β] {f : β → α} {b : β} :
continuous_at f b ↔ tendsto (λ x : β × β, (f x.1, f x.2)) (𝓝 (b, b)) (𝓤 α) | ⟨λ H, le_trans (H.prod_map' H) (nhds_le_uniformity _),
λ H, continuous_at_iff'_left.2 $ H.comp $ tendsto_id.prod_mk_nhds tendsto_const_nhds⟩ | theorem | uniform.continuous_at_iff_prod | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_at",
"nhds_le_uniformity",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff'_right [topological_space β] {f : β → α} {b : β} {s : set β} :
continuous_within_at f s b ↔ tendsto (λ x, (f b, f x)) (𝓝[s] b) (𝓤 α) | by rw [continuous_within_at, tendsto_nhds_right] | theorem | uniform.continuous_within_at_iff'_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_within_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff'_left [topological_space β] {f : β → α} {b : β} {s : set β} :
continuous_within_at f s b ↔ tendsto (λ x, (f x, f b)) (𝓝[s] b) (𝓤 α) | by rw [continuous_within_at, tendsto_nhds_left] | theorem | uniform.continuous_within_at_iff'_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_within_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff'_right [topological_space β] {f : β → α} {s : set β} :
continuous_on f s ↔ ∀ b ∈ s, tendsto (λ x, (f b, f x)) (𝓝[s] b) (𝓤 α) | by simp [continuous_on, continuous_within_at_iff'_right] | theorem | uniform.continuous_on_iff'_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff'_left [topological_space β] {f : β → α} {s : set β} :
continuous_on f s ↔ ∀ b ∈ s, tendsto (λ x, (f x, f b)) (𝓝[s] b) (𝓤 α) | by simp [continuous_on, continuous_within_at_iff'_left] | theorem | uniform.continuous_on_iff'_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff'_right [topological_space β] {f : β → α} :
continuous f ↔ ∀ b, tendsto (λ x, (f b, f x)) (𝓝 b) (𝓤 α) | continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds_right | theorem | uniform.continuous_iff'_right | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff'_left [topological_space β] {f : β → α} :
continuous f ↔ ∀ b, tendsto (λ x, (f x, f b)) (𝓝 b) (𝓤 α) | continuous_iff_continuous_at.trans $ forall_congr $ λ b, tendsto_nhds_left | theorem | uniform.continuous_iff'_left | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.congr_uniformity {α β} [uniform_space β] {f g : α → β} {l : filter α} {b : β}
(hf : tendsto f l (𝓝 b)) (hg : tendsto (λ x, (f x, g x)) l (𝓤 β)) :
tendsto g l (𝓝 b) | uniform.tendsto_nhds_right.2 $ (uniform.tendsto_nhds_right.1 hf).uniformity_trans hg | lemma | filter.tendsto.congr_uniformity | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform.tendsto_congr {α β} [uniform_space β] {f g : α → β} {l : filter α} {b : β}
(hfg : tendsto (λ x, (f x, g x)) l (𝓤 β)) :
tendsto f l (𝓝 b) ↔ tendsto g l (𝓝 b) | ⟨λ h, h.congr_uniformity hfg, λ h, h.congr_uniformity hfg.uniformity_symm⟩ | lemma | uniform.tendsto_congr | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"filter",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy (f : filter α) | ne_bot f ∧ f ×ᶠ f ≤ (𝓤 α) | def | cauchy | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"filter"
] | A filter `f` is Cauchy if for every entourage `r`, there exists an
`s ∈ f` such that `s × s ⊆ r`. This is a generalization of Cauchy
sequences, because if `a : ℕ → α` then the filter of sets containing
cofinitely many of the `a n` is Cauchy iff `a` is a Cauchy sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_complete (s : set α) | ∀f, cauchy f → f ≤ 𝓟 s → ∃x∈s, f ≤ 𝓝 x | def | is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy"
] | A set `s` is called *complete*, if any Cauchy filter `f` such that `s ∈ f`
has a limit in `s` (formally, it satisfies `f ≤ 𝓝 x` for some `x ∈ s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.cauchy_iff {ι} {p : ι → Prop} {s : ι → set (α × α)} (h : (𝓤 α).has_basis p s)
{f : filter α} :
cauchy f ↔ (ne_bot f ∧ (∀ i, p i → ∃ t ∈ f, ∀ x y ∈ t, (x, y) ∈ s i)) | and_congr iff.rfl $ (f.basis_sets.prod_self.le_basis_iff h).trans $
by simp only [subset_def, prod.forall, mem_prod_eq, and_imp, id, ball_mem_comm] | lemma | filter.has_basis.cauchy_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"and_imp",
"ball_mem_comm",
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_iff' {f : filter α} :
cauchy f ↔ (ne_bot f ∧ (∀ s ∈ 𝓤 α, ∃t∈f, ∀ x y ∈ t, (x, y) ∈ s)) | (𝓤 α).basis_sets.cauchy_iff | lemma | cauchy_iff' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_iff {f : filter α} :
cauchy f ↔ (ne_bot f ∧ (∀ s ∈ 𝓤 α, ∃t∈f, t ×ˢ t ⊆ s)) | cauchy_iff'.trans $ by simp only [subset_def, prod.forall, mem_prod_eq, and_imp, id, ball_mem_comm] | lemma | cauchy_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"and_imp",
"ball_mem_comm",
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.ultrafilter_of {l : filter α} (h : cauchy l) :
cauchy (@ultrafilter.of _ l h.1 : filter α) | begin
haveI := h.1,
have := ultrafilter.of_le l,
exact ⟨ultrafilter.ne_bot _, (filter.prod_mono this this).trans h.2⟩
end | lemma | cauchy.ultrafilter_of | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"filter.prod_mono",
"ultrafilter.of",
"ultrafilter.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_map_iff {l : filter β} {f : β → α} :
cauchy (l.map f) ↔ (ne_bot l ∧ tendsto (λp:β×β, (f p.1, f p.2)) (l ×ᶠ l) (𝓤 α)) | by rw [cauchy, map_ne_bot_iff, prod_map_map_eq, tendsto] | lemma | cauchy_map_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_map_iff' {l : filter β} [hl : ne_bot l] {f : β → α} :
cauchy (l.map f) ↔ tendsto (λp:β×β, (f p.1, f p.2)) (l ×ᶠ l) (𝓤 α) | cauchy_map_iff.trans $ and_iff_right hl | lemma | cauchy_map_iff' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.mono {f g : filter α} [hg : ne_bot g] (h_c : cauchy f) (h_le : g ≤ f) : cauchy g | ⟨hg, le_trans (filter.prod_mono h_le h_le) h_c.right⟩ | lemma | cauchy.mono | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"filter.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.mono' {f g : filter α} (h_c : cauchy f) (hg : ne_bot g) (h_le : g ≤ f) : cauchy g | h_c.mono h_le | lemma | cauchy.mono' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_nhds {a : α} : cauchy (𝓝 a) | ⟨nhds_ne_bot, nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)⟩ | lemma | cauchy_nhds | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"nhds_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_pure {a : α} : cauchy (pure a) | cauchy_nhds.mono (pure_le_nhds a) | lemma | cauchy_pure | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cauchy_map {l : filter β} [ne_bot l] {f : β → α} {a : α}
(h : tendsto f l (𝓝 a)) :
cauchy (map f l) | cauchy_nhds.mono h | lemma | filter.tendsto.cauchy_map | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.prod [uniform_space β] {f : filter α} {g : filter β} (hf : cauchy f) (hg : cauchy g) :
cauchy (f ×ᶠ g) | begin
refine ⟨hf.1.prod hg.1, _⟩,
simp only [uniformity_prod, le_inf_iff, ← map_le_iff_le_comap, ← prod_map_map_eq],
exact ⟨le_trans (prod_mono tendsto_fst tendsto_fst) hf.2,
le_trans (prod_mono tendsto_snd tendsto_snd) hg.2⟩
end | lemma | cauchy.prod | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"le_inf_iff",
"uniform_space",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_of_cauchy_adhp_aux {f : filter α} {x : α}
(adhs : ∀ s ∈ 𝓤 α, ∃ t ∈ f, (t ×ˢ t ⊆ s) ∧ ∃ y, (x, y) ∈ s ∧ y ∈ t) :
f ≤ 𝓝 x | begin
-- Consider a neighborhood `s` of `x`
assume s hs,
-- Take an entourage twice smaller than `s`
rcases comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 hs) with ⟨U, U_mem, hU⟩,
-- Take a set `t ∈ f`, `t × t ⊆ U`, and a point `y ∈ t` such that `(x, y) ∈ U`
rcases adhs U U_mem with ⟨t, t_mem, ht... | lemma | le_nhds_of_cauchy_adhp_aux | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"comp_mem_uniformity_sets",
"filter",
"prod_mk_mem_comp_rel"
] | The common part of the proofs of `le_nhds_of_cauchy_adhp` and
`sequentially_complete.le_nhds_of_seq_tendsto_nhds`: if for any entourage `s`
one can choose a set `t ∈ f` of diameter `s` such that it contains a point `y`
with `(x, y) ∈ s`, then `f` converges to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_nhds_of_cauchy_adhp {f : filter α} {x : α} (hf : cauchy f)
(adhs : cluster_pt x f) : f ≤ 𝓝 x | le_nhds_of_cauchy_adhp_aux
begin
assume s hs,
obtain ⟨t, t_mem, ht⟩ : ∃ t ∈ f, t ×ˢ t ⊆ s,
from (cauchy_iff.1 hf).2 s hs,
use [t, t_mem, ht],
exact (forall_mem_nonempty_iff_ne_bot.2 adhs _
(inter_mem_inf (mem_nhds_left x hs) t_mem ))
end | lemma | le_nhds_of_cauchy_adhp | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"cluster_pt",
"filter",
"le_nhds_of_cauchy_adhp_aux",
"mem_nhds_left"
] | If `x` is an adherent (cluster) point for a Cauchy filter `f`, then it is a limit point
for `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_nhds_iff_adhp_of_cauchy {f : filter α} {x : α} (hf : cauchy f) :
f ≤ 𝓝 x ↔ cluster_pt x f | ⟨assume h, cluster_pt.of_le_nhds' h hf.1, le_nhds_of_cauchy_adhp hf⟩ | lemma | le_nhds_iff_adhp_of_cauchy | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"cluster_pt",
"cluster_pt.of_le_nhds'",
"filter",
"le_nhds_of_cauchy_adhp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.map [uniform_space β] {f : filter α} {m : α → β}
(hf : cauchy f) (hm : uniform_continuous m) : cauchy (map m f) | ⟨hf.1.map _,
calc map m f ×ᶠ map m f = map (λp:α×α, (m p.1, m p.2)) (f ×ᶠ f) : filter.prod_map_map_eq
... ≤ map (λp:α×α, (m p.1, m p.2)) (𝓤 α) : map_mono hf.right
... ≤ 𝓤 β : hm⟩ | lemma | cauchy.map | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"filter.prod_map_map_eq",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.comap [uniform_space β] {f : filter β} {m : α → β}
(hf : cauchy f) (hm : comap (λp:α×α, (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α)
[ne_bot (comap m f)] : cauchy (comap m f) | ⟨‹_›,
calc comap m f ×ᶠ comap m f = comap (λp:α×α, (m p.1, m p.2)) (f ×ᶠ f) : filter.prod_comap_comap_eq
... ≤ comap (λp:α×α, (m p.1, m p.2)) (𝓤 β) : comap_mono hf.right
... ≤ 𝓤 α : hm⟩ | lemma | cauchy.comap | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"filter.prod_comap_comap_eq",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy.comap' [uniform_space β] {f : filter β} {m : α → β}
(hf : cauchy f) (hm : comap (λp:α×α, (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α)
(hb : ne_bot (comap m f)) : cauchy (comap m f) | hf.comap hm | lemma | cauchy.comap' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq [semilattice_sup β] (u : β → α) | cauchy (at_top.map u) | def | cauchy_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"semilattice_sup"
] | Cauchy sequences. Usually defined on ℕ, but often it is also useful to say that a function
defined on ℝ is Cauchy at +∞ to deduce convergence. Therefore, we define it in a type class that
is general enough to cover both ℕ and ℝ, which are the main motivating examples. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq.tendsto_uniformity [semilattice_sup β] {u : β → α} (h : cauchy_seq u) :
tendsto (prod.map u u) at_top (𝓤 α) | by simpa only [tendsto, prod_map_map_eq', prod_at_top_at_top_eq] using h.right | lemma | cauchy_seq.tendsto_uniformity | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.nonempty [semilattice_sup β] {u : β → α} (hu : cauchy_seq u) : nonempty β | @nonempty_of_ne_bot _ _ $ (map_ne_bot_iff _).1 hu.1 | lemma | cauchy_seq.nonempty | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.mem_entourage {β : Type*} [semilattice_sup β] {u : β → α}
(h : cauchy_seq u) {V : set (α × α)} (hV : V ∈ 𝓤 α) :
∃ k₀, ∀ i j, k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V | begin
haveI := h.nonempty,
have := h.tendsto_uniformity, rw ← prod_at_top_at_top_eq at this,
simpa [maps_to] using at_top_basis.prod_self.tendsto_left_iff.1 this V hV
end | lemma | cauchy_seq.mem_entourage | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cauchy_seq [semilattice_sup β] [nonempty β] {f : β → α} {x}
(hx : tendsto f at_top (𝓝 x)) :
cauchy_seq f | hx.cauchy_map | lemma | filter.tendsto.cauchy_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_const [semilattice_sup β] [nonempty β] (x : α) : cauchy_seq (λ n : β, x) | tendsto_const_nhds.cauchy_seq | lemma | cauchy_seq_const | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_iff_tendsto [nonempty β] [semilattice_sup β] {u : β → α} :
cauchy_seq u ↔ tendsto (prod.map u u) at_top (𝓤 α) | cauchy_map_iff'.trans $ by simp only [prod_at_top_at_top_eq, prod.map_def] | lemma | cauchy_seq_iff_tendsto | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"prod.map_def",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.comp_tendsto {γ} [semilattice_sup β] [semilattice_sup γ] [nonempty γ]
{f : β → α} (hf : cauchy_seq f) {g : γ → β} (hg : tendsto g at_top at_top) :
cauchy_seq (f ∘ g) | cauchy_seq_iff_tendsto.2 $ hf.tendsto_uniformity.comp (hg.prod_at_top hg) | lemma | cauchy_seq.comp_tendsto | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.comp_injective [semilattice_sup β] [no_max_order β] [nonempty β]
{u : ℕ → α} (hu : cauchy_seq u) {f : β → ℕ} (hf : injective f) :
cauchy_seq (u ∘ f) | hu.comp_tendsto $ nat.cofinite_eq_at_top ▸ hf.tendsto_cofinite.mono_left at_top_le_cofinite | lemma | cauchy_seq.comp_injective | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"nat.cofinite_eq_at_top",
"no_max_order",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.bijective.cauchy_seq_comp_iff {f : ℕ → ℕ} (hf : bijective f) (u : ℕ → α) :
cauchy_seq (u ∘ f) ↔ cauchy_seq u | begin
refine ⟨λ H, _, λ H, H.comp_injective hf.injective⟩,
lift f to ℕ ≃ ℕ using hf,
simpa only [(∘), f.apply_symm_apply] using H.comp_injective f.symm.injective
end | lemma | function.bijective.cauchy_seq_comp_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.subseq_subseq_mem {V : ℕ → set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α)
{u : ℕ → α} (hu : cauchy_seq u)
{f g : ℕ → ℕ} (hf : tendsto f at_top at_top) (hg : tendsto g at_top at_top) :
∃ φ : ℕ → ℕ, strict_mono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n | begin
rw cauchy_seq_iff_tendsto at hu,
exact ((hu.comp $ hf.prod_at_top hg).comp tendsto_at_top_diagonal).subseq_mem hV,
end | lemma | cauchy_seq.subseq_subseq_mem | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_iff_tendsto",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_iff' {u : ℕ → α} :
cauchy_seq u ↔ ∀ V ∈ 𝓤 α, ∀ᶠ k in at_top, k ∈ (prod.map u u) ⁻¹' V | by simpa only [cauchy_seq_iff_tendsto] | lemma | cauchy_seq_iff' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_iff {u : ℕ → α} :
cauchy_seq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V | by simp [cauchy_seq_iff', filter.eventually_at_top_prod_self', prod_map] | lemma | cauchy_seq_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_iff'",
"filter.eventually_at_top_prod_self'",
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.prod_map {γ δ} [uniform_space β] [semilattice_sup γ] [semilattice_sup δ]
{u : γ → α} {v : δ → β}
(hu : cauchy_seq u) (hv : cauchy_seq v) : cauchy_seq (prod.map u v) | by simpa only [cauchy_seq, prod_map_map_eq', prod_at_top_at_top_eq] using hu.prod hv | lemma | cauchy_seq.prod_map | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.prod {γ} [uniform_space β] [semilattice_sup γ] {u : γ → α} {v : γ → β}
(hu : cauchy_seq u) (hv : cauchy_seq v) : cauchy_seq (λ x, (u x, v x)) | begin
haveI := hu.nonempty,
exact (hu.prod hv).mono (tendsto.prod_mk le_rfl le_rfl)
end | lemma | cauchy_seq.prod | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"le_rfl",
"semilattice_sup",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.eventually_eventually [semilattice_sup β] {u : β → α} (hu : cauchy_seq u)
{V : set (α × α)} (hV : V ∈ 𝓤 α) :
∀ᶠ k in at_top, ∀ᶠ l in at_top, (u k, u l) ∈ V | eventually_at_top_curry $ hu.tendsto_uniformity hV | lemma | cauchy_seq.eventually_eventually | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous.comp_cauchy_seq {γ} [uniform_space β] [semilattice_sup γ]
{f : α → β} (hf : uniform_continuous f) {u : γ → α} (hu : cauchy_seq u) :
cauchy_seq (f ∘ u) | hu.map hf | lemma | uniform_continuous.comp_cauchy_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"semilattice_sup",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.subseq_mem {V : ℕ → set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α)
{u : ℕ → α} (hu : cauchy_seq u) :
∃ φ : ℕ → ℕ, strict_mono φ ∧ ∀ n, (u $ φ (n + 1), u $ φ n) ∈ V n | begin
have : ∀ n, ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n,
{ intro n,
rw [cauchy_seq_iff] at hu,
rcases hu _ (hV n) with ⟨N, H⟩,
exact ⟨N, λ k hk l hl, H _ (le_trans hk hl) _ hk ⟩ },
obtain ⟨φ : ℕ → ℕ, φ_extr : strict_mono φ, hφ : ∀ n, ∀ l ≥ φ n, (u l, u $ φ n) ∈ V n⟩ :=
extraction_forall_of_event... | lemma | cauchy_seq.subseq_mem | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"cauchy_seq_iff",
"lt_add_one",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.subseq_mem_entourage {V : ℕ → set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α)
{u : ℕ → α} {a : α} (hu : tendsto u at_top (𝓝 a)) :
∃ φ : ℕ → ℕ, strict_mono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ n, (u $ φ (n + 1), u $ φ n) ∈ V (n + 1) | begin
rcases mem_at_top_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity $ hV 0))) with ⟨n, hn⟩,
rcases (hu.comp (tendsto_add_at_top_nat n)).cauchy_seq.subseq_mem (λ n, hV (n + 1))
with ⟨φ, φ_mono, hφV⟩,
exact ⟨λ k, φ k + n, φ_mono.add_const _, hn _ le_add_self, hφV⟩
end | lemma | filter.tendsto.subseq_mem_entourage | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq.subseq_mem",
"strict_mono",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_of_cauchy_seq_of_subseq
[semilattice_sup β] {u : β → α} (hu : cauchy_seq u)
{ι : Type*} {f : ι → β} {p : filter ι} [ne_bot p]
(hf : tendsto f p at_top) {a : α} (ha : tendsto (u ∘ f) p (𝓝 a)) :
tendsto u at_top (𝓝 a) | le_nhds_of_cauchy_adhp hu (map_cluster_pt_of_comp hf ha) | lemma | tendsto_nhds_of_cauchy_seq_of_subseq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"filter",
"le_nhds_of_cauchy_adhp",
"map_cluster_pt_of_comp",
"semilattice_sup"
] | If a Cauchy sequence has a convergent subsequence, then it converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.cauchy_seq_iff {γ} [nonempty β] [semilattice_sup β] {u : β → α}
{p : γ → Prop} {s : γ → set (α × α)} (h : (𝓤 α).has_basis p s) :
cauchy_seq u ↔ ∀ i, p i → ∃N, ∀m n≥N, (u m, u n) ∈ s i | begin
rw [cauchy_seq_iff_tendsto, ← prod_at_top_at_top_eq],
refine (at_top_basis.prod_self.tendsto_iff h).trans _,
simp only [exists_prop, true_and, maps_to, preimage, subset_def, prod.forall,
mem_prod_eq, mem_set_of_eq, mem_Ici, and_imp, prod.map, ge_iff_le, @forall_swap (_ ≤ _) β]
end | lemma | filter.has_basis.cauchy_seq_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"and_imp",
"cauchy_seq",
"cauchy_seq_iff_tendsto",
"exists_prop",
"forall_swap",
"ge_iff_le",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.cauchy_seq_iff' {γ} [nonempty β] [semilattice_sup β] {u : β → α}
{p : γ → Prop} {s : γ → set (α × α)} (H : (𝓤 α).has_basis p s) :
cauchy_seq u ↔ ∀ i, p i → ∃N, ∀n≥N, (u n, u N) ∈ s i | begin
refine H.cauchy_seq_iff.trans ⟨λ h i hi, _, λ h i hi, _⟩,
{ exact (h i hi).imp (λ N hN n hn, hN n hn N le_rfl) },
{ rcases comp_symm_of_uniformity (H.mem_of_mem hi) with ⟨t, ht, ht', hts⟩,
rcases H.mem_iff.1 ht with ⟨j, hj, hjt⟩,
refine (h j hj).imp (λ N hN m hm n hn, hts ⟨u N, hjt _, ht' $ hjt _⟩),... | lemma | filter.has_basis.cauchy_seq_iff' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"comp_symm_of_uniformity",
"le_rfl",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_of_controlled [semilattice_sup β] [nonempty β]
(U : β → set (α × α)) (hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s)
{f : β → α} (hf : ∀ {N m n : β}, N ≤ m → N ≤ n → (f m, f n) ∈ U N) :
cauchy_seq f | cauchy_seq_iff_tendsto.2
begin
assume s hs,
rw [mem_map, mem_at_top_sets],
cases hU s hs with N hN,
refine ⟨(N, N), λ mn hmn, _⟩,
cases mn with m n,
exact hN (hf hmn.1 hmn.2)
end | lemma | cauchy_seq_of_controlled | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"mem_map",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_iff_cluster_pt {s : set α} :
is_complete s ↔ ∀ l, cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, cluster_pt x l | forall₃_congr $ λ l hl hls, exists₂_congr $ λ x hx, le_nhds_iff_adhp_of_cauchy hl | lemma | is_complete_iff_cluster_pt | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"cluster_pt",
"exists₂_congr",
"forall₃_congr",
"is_complete",
"le_nhds_iff_adhp_of_cauchy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_iff_ultrafilter {s : set α} :
is_complete s ↔ ∀ l : ultrafilter α, cauchy (l : filter α) → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x | begin
refine ⟨λ h l, h l, λ H, is_complete_iff_cluster_pt.2 $ λ l hl hls, _⟩,
haveI := hl.1,
rcases H (ultrafilter.of l) hl.ultrafilter_of ((ultrafilter.of_le l).trans hls)
with ⟨x, hxs, hxl⟩,
exact ⟨x, hxs, (cluster_pt.of_le_nhds hxl).mono (ultrafilter.of_le l)⟩
end | lemma | is_complete_iff_ultrafilter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"cluster_pt.of_le_nhds",
"filter",
"is_complete",
"ultrafilter",
"ultrafilter.of",
"ultrafilter.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_iff_ultrafilter' {s : set α} :
is_complete s ↔ ∀ l : ultrafilter α, cauchy (l : filter α) → s ∈ l → ∃ x ∈ s, ↑l ≤ 𝓝 x | is_complete_iff_ultrafilter.trans $ by simp only [le_principal_iff, ultrafilter.mem_coe] | lemma | is_complete_iff_ultrafilter' | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"is_complete",
"ultrafilter",
"ultrafilter.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete.union {s t : set α} (hs : is_complete s) (ht : is_complete t) :
is_complete (s ∪ t) | begin
simp only [is_complete_iff_ultrafilter', ultrafilter.union_mem_iff, or_imp_distrib] at *,
exact λ l hl, ⟨λ hsl, (hs l hl hsl).imp $ λ x hx, ⟨or.inl hx.fst, hx.snd⟩,
λ htl, (ht l hl htl).imp $ λ x hx, ⟨or.inr hx.fst, hx.snd⟩⟩
end | lemma | is_complete.union | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_complete",
"is_complete_iff_ultrafilter'",
"or_imp_distrib",
"ultrafilter.union_mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_Union_separated {ι : Sort*} {s : ι → set α} (hs : ∀ i, is_complete (s i))
{U : set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι) (x ∈ s i) (y ∈ s j), (x, y) ∈ U → i = j) :
is_complete (⋃ i, s i) | begin
set S := ⋃ i, s i,
intros l hl hls,
rw le_principal_iff at hls,
casesI cauchy_iff.1 hl with hl_ne hl',
obtain ⟨t, htS, htl, htU⟩ : ∃ t ⊆ S, t ∈ l ∧ t ×ˢ t ⊆ U,
{ rcases hl' U hU with ⟨t, htl, htU⟩,
exact ⟨t ∩ S, inter_subset_right _ _, inter_mem htl hls,
(set.prod_mono (inter_subset_left _ _... | lemma | is_complete_Union_separated | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"filter.nonempty_of_mem",
"is_complete",
"set.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space (α : Type u) [uniform_space α] : Prop | (complete : ∀{f:filter α}, cauchy f → ∃x, f ≤ 𝓝 x) | class | complete_space | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"uniform_space"
] | A complete space is defined here using uniformities. A uniform space
is complete if every Cauchy filter converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_univ {α : Type u} [uniform_space α] [complete_space α] :
is_complete (univ : set α) | begin
assume f hf _,
rcases complete_space.complete hf with ⟨x, hx⟩,
exact ⟨x, mem_univ x, hx⟩
end | lemma | complete_univ | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"is_complete",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space.prod [uniform_space β] [complete_space α] [complete_space β] :
complete_space (α × β) | { complete := λ f hf,
let ⟨x1, hx1⟩ := complete_space.complete $ hf.map uniform_continuous_fst in
let ⟨x2, hx2⟩ := complete_space.complete $ hf.map uniform_continuous_snd in
⟨(x1, x2), by rw [nhds_prod_eq, filter.prod_def];
from filter.le_lift.2 (λ s hs, filter.le_lift'.2 $ λ t ht,
inter_mem (... | instance | complete_space.prod | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"filter.prod_def",
"nhds_prod_eq",
"uniform_continuous_fst",
"uniform_continuous_snd",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space.mul_opposite [complete_space α] : complete_space αᵐᵒᵖ | { complete := λ f hf, mul_opposite.op_surjective.exists.mpr $
let ⟨x, hx⟩ := complete_space.complete (hf.map mul_opposite.uniform_continuous_unop) in
⟨x, (map_le_iff_le_comap.mp hx).trans_eq $ mul_opposite.comap_unop_nhds _⟩} | instance | complete_space.mul_opposite | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"mul_opposite.comap_unop_nhds",
"mul_opposite.uniform_continuous_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_of_is_complete_univ (h : is_complete (univ : set α)) : complete_space α | ⟨λ f hf, let ⟨x, _, hx⟩ := h f hf ((@principal_univ α).symm ▸ le_top) in ⟨x, hx⟩⟩ | lemma | complete_space_of_is_complete_univ | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"is_complete",
"le_top"
] | If `univ` is complete, the space is a complete space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_iff_is_complete_univ :
complete_space α ↔ is_complete (univ : set α) | ⟨@complete_univ α _, complete_space_of_is_complete_univ⟩ | lemma | complete_space_iff_is_complete_univ | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"complete_univ",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_iff_ultrafilter :
complete_space α ↔ ∀ l : ultrafilter α, cauchy (l : filter α) → ∃ x : α, ↑l ≤ 𝓝 x | by simp [complete_space_iff_is_complete_univ, is_complete_iff_ultrafilter] | lemma | complete_space_iff_ultrafilter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"complete_space",
"complete_space_iff_is_complete_univ",
"filter",
"is_complete_iff_ultrafilter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_iff_exists_le_nhds [complete_space α] {l : filter α} [ne_bot l] :
cauchy l ↔ (∃x, l ≤ 𝓝 x) | ⟨complete_space.complete, assume ⟨x, hx⟩, cauchy_nhds.mono hx⟩ | lemma | cauchy_iff_exists_le_nhds | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"complete_space",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_map_iff_exists_tendsto [complete_space α] {l : filter β} {f : β → α} [ne_bot l] :
cauchy (l.map f) ↔ (∃x, tendsto f l (𝓝 x)) | cauchy_iff_exists_le_nhds | lemma | cauchy_map_iff_exists_tendsto | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"cauchy_iff_exists_le_nhds",
"complete_space",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_tendsto_of_complete [semilattice_sup β] [complete_space α]
{u : β → α} (H : cauchy_seq u) : ∃x, tendsto u at_top (𝓝 x) | complete_space.complete H | theorem | cauchy_seq_tendsto_of_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"complete_space",
"semilattice_sup"
] | A Cauchy sequence in a complete space converges | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_tendsto_of_is_complete [semilattice_sup β] {K : set α} (h₁ : is_complete K)
{u : β → α} (h₂ : ∀ n, u n ∈ K) (h₃ : cauchy_seq u) : ∃ v ∈ K, tendsto u at_top (𝓝 v) | h₁ _ h₃ $ le_principal_iff.2 $ mem_map_iff_exists_image.2 ⟨univ, univ_mem,
by { simp only [image_univ], rintros _ ⟨n, rfl⟩, exact h₂ n }⟩ | lemma | cauchy_seq_tendsto_of_is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"is_complete",
"semilattice_sup"
] | If `K` is a complete subset, then any cauchy sequence in `K` converges to a point in `K` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy.le_nhds_Lim [complete_space α] [nonempty α] {f : filter α} (hf : cauchy f) :
f ≤ 𝓝 (Lim f) | le_nhds_Lim (complete_space.complete hf) | theorem | cauchy.le_nhds_Lim | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"Lim",
"cauchy",
"complete_space",
"filter",
"le_nhds_Lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.tendsto_lim [semilattice_sup β] [complete_space α] [nonempty α] {u : β → α}
(h : cauchy_seq u) :
tendsto u at_top (𝓝 $ lim at_top u) | h.le_nhds_Lim | theorem | cauchy_seq.tendsto_lim | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"complete_space",
"lim",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.is_complete [complete_space α] {s : set α}
(h : is_closed s) : is_complete s | λ f cf fs, let ⟨x, hx⟩ := complete_space.complete cf in
⟨x, is_closed_iff_cluster_pt.mp h x (cf.left.mono (le_inf hx fs)), hx⟩ | lemma | is_closed.is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"is_closed",
"is_complete",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded (s : set α) : Prop | ∀d ∈ 𝓤 α, ∃t : set α, t.finite ∧ s ⊆ (⋃ y ∈ t, {x | (x, y) ∈ d}) | def | totally_bounded | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | A set `s` is totally bounded if for every entourage `d` there is a finite
set of points `t` such that every element of `s` is `d`-near to some element of `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded.exists_subset {s : set α} (hs : totally_bounded s) {U : set (α × α)}
(hU : U ∈ 𝓤 α) :
∃ t ⊆ s, set.finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U} | begin
rcases comp_symm_of_uniformity hU with ⟨r, hr, rs, rU⟩,
rcases hs r hr with ⟨k, fk, ks⟩,
let u := k ∩ {y | ∃ x ∈ s, (x, y) ∈ r},
choose hk f hfs hfr using λ x : u, x.coe_prop,
refine ⟨range f, _, _, _⟩,
{ exact range_subset_iff.2 hfs },
{ haveI : fintype u := (fk.inter_of_left _).fintype,
exact ... | theorem | totally_bounded.exists_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"comp_symm_of_uniformity",
"fintype",
"set.finite",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_iff_subset {s : set α} : totally_bounded s ↔
∀d ∈ 𝓤 α, ∃t ⊆ s, set.finite t ∧ s ⊆ (⋃y∈t, {x | (x,y) ∈ d}) | ⟨λ H d hd, H.exists_subset hd, λ H d hd, let ⟨t, _, ht⟩ := H d hd in ⟨t, ht⟩⟩ | theorem | totally_bounded_iff_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.finite",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.totally_bounded_iff {ι} {p : ι → Prop} {U : ι → set (α × α)}
(H : (𝓤 α).has_basis p U) {s : set α} :
totally_bounded s ↔ ∀ i, p i → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U i} | H.forall_iff $ λ U V hUV h, h.imp $ λ t ht, ⟨ht.1, ht.2.trans $ Union₂_mono $ λ x hx y hy, hUV hy⟩ | lemma | filter.has_basis.totally_bounded_iff | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.finite",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_of_forall_symm {s : set α}
(h : ∀ V ∈ 𝓤 α, symmetric_rel V → ∃ t : set α, set.finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
totally_bounded s | uniform_space.has_basis_symmetric.totally_bounded_iff.2 $ λ V hV,
by simpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2 | lemma | totally_bounded_of_forall_symm | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"ball_eq_of_symmetry",
"set.finite",
"symmetric_rel",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_subset {s₁ s₂ : set α} (hs : s₁ ⊆ s₂)
(h : totally_bounded s₂) : totally_bounded s₁ | assume d hd, let ⟨t, ht₁, ht₂⟩ := h d hd in ⟨t, ht₁, subset.trans hs ht₂⟩ | lemma | totally_bounded_subset | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_empty : totally_bounded (∅ : set α) | λ d hd, ⟨∅, finite_empty, empty_subset _⟩ | lemma | totally_bounded_empty | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded.closure {s : set α} (h : totally_bounded s) :
totally_bounded (closure s) | uniformity_has_basis_closed.totally_bounded_iff.2 $ λ V hV, let ⟨t, htf, hst⟩ := h V hV.1
in ⟨t, htf, closure_minimal hst $ is_closed_bUnion htf $
λ y hy, hV.2.preimage (continuous_id.prod_mk continuous_const)⟩ | lemma | totally_bounded.closure | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"closure",
"closure_minimal",
"continuous_const",
"is_closed_bUnion",
"totally_bounded"
] | The closure of a totally bounded set is totally bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_bounded.image [uniform_space β] {f : α → β} {s : set α}
(hs : totally_bounded s) (hf : uniform_continuous f) : totally_bounded (f '' s) | assume t ht,
have {p:α×α | (f p.1, f p.2) ∈ t} ∈ 𝓤 α,
from hf ht,
let ⟨c, hfc, hct⟩ := hs _ this in
⟨f '' c, hfc.image f,
begin
simp [image_subset_iff],
simp [subset_def] at hct,
intros x hx, simp,
exact hct x hx
end⟩ | lemma | totally_bounded.image | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"totally_bounded",
"uniform_continuous",
"uniform_space"
] | The image of a totally bounded set under a uniformly continuous map is totally bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter.cauchy_of_totally_bounded {s : set α} (f : ultrafilter α)
(hs : totally_bounded s) (h : ↑f ≤ 𝓟 s) : cauchy (f : filter α) | ⟨f.ne_bot', assume t ht,
let ⟨t', ht'₁, ht'_symm, ht'_t⟩ := comp_symm_of_uniformity ht in
let ⟨i, hi, hs_union⟩ := hs t' ht'₁ in
have (⋃y∈i, {x | (x,y) ∈ t'}) ∈ f,
from mem_of_superset (le_principal_iff.mp h) hs_union,
have ∃y∈i, {x | (x,y) ∈ t'} ∈ f,
from (ultrafilter.finite_bUnion_mem_iff hi).1 this,
... | lemma | ultrafilter.cauchy_of_totally_bounded | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"comp_rel",
"comp_symm_of_uniformity",
"filter",
"totally_bounded",
"ultrafilter",
"ultrafilter.finite_bUnion_mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_iff_filter {s : set α} :
totally_bounded s ↔ (∀f, ne_bot f → f ≤ 𝓟 s → ∃c ≤ f, cauchy c) | begin
split,
{ introsI H f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f,
(ultrafilter.of f).cauchy_of_totally_bounded H ((ultrafilter.of_le f).trans hfs)⟩ },
{ intros H d hd,
contrapose! H with hd_cover,
set f := ⨅ t : finset α, 𝓟 (s \ ⋃ y ∈ t, {x | (x, y) ∈ d}),
have : ne_bot f,
... | lemma | totally_bounded_iff_filter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"directed_of_sup",
"finset",
"infi_le_of_le",
"le_rfl",
"totally_bounded",
"ultrafilter.of",
"ultrafilter.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_iff_ultrafilter {s : set α} :
totally_bounded s ↔ (∀f : ultrafilter α, ↑f ≤ 𝓟 s → cauchy (f : filter α)) | begin
refine ⟨λ hs f, f.cauchy_of_totally_bounded hs, λ H, totally_bounded_iff_filter.2 _⟩,
introsI f hf hfs,
exact ⟨ultrafilter.of f, ultrafilter.of_le f, H _ ((ultrafilter.of_le f).trans hfs)⟩
end | lemma | totally_bounded_iff_ultrafilter | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy",
"filter",
"totally_bounded",
"ultrafilter",
"ultrafilter.of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_iff_totally_bounded_is_complete {s : set α} :
is_compact s ↔ totally_bounded s ∧ is_complete s | ⟨λ hs, ⟨totally_bounded_iff_ultrafilter.2 (λ f hf,
let ⟨x, xs, fx⟩ := is_compact_iff_ultrafilter_le_nhds.1 hs f hf in cauchy_nhds.mono fx),
λ f fc fs,
let ⟨a, as, fa⟩ := @hs f fc.1 fs in
⟨a, as, le_nhds_of_cauchy_adhp fc fa⟩⟩,
λ ⟨ht, hc⟩, is_compact_iff_ultrafilter_le_nhds.2
(λf hf, hc _ (totally_boun... | lemma | is_compact_iff_totally_bounded_is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"is_complete",
"le_nhds_of_cauchy_adhp",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.totally_bounded {s : set α} (h : is_compact s) : totally_bounded s | (is_compact_iff_totally_bounded_is_complete.1 h).1 | lemma | is_compact.totally_bounded | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_complete {s : set α} (h : is_compact s) : is_complete s | (is_compact_iff_totally_bounded_is_complete.1 h).2 | lemma | is_compact.is_complete | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"is_compact",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_of_compact {α : Type u} [uniform_space α] [compact_space α] : complete_space α | ⟨λf hf, by simpa using (is_compact_iff_totally_bounded_is_complete.1 is_compact_univ).2 f hf⟩ | instance | complete_of_compact | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"compact_space",
"complete_space",
"is_compact_univ",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_of_totally_bounded_is_closed [complete_space α] {s : set α}
(ht : totally_bounded s) (hc : is_closed s) : is_compact s | (@is_compact_iff_totally_bounded_is_complete α _ s).2 ⟨ht, hc.is_complete⟩ | lemma | is_compact_of_totally_bounded_is_closed | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"complete_space",
"is_closed",
"is_compact",
"is_compact_iff_totally_bounded_is_complete",
"totally_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq.totally_bounded_range {s : ℕ → α} (hs : cauchy_seq s) :
totally_bounded (range s) | begin
refine totally_bounded_iff_subset.2 (λ a ha, _),
cases cauchy_seq_iff.1 hs a ha with n hn,
refine ⟨s '' {k | k ≤ n}, image_subset_range _ _, (finite_le_nat _).image _, _⟩,
rw [range_subset_iff, bUnion_image],
intro m,
rw [mem_Union₂],
cases le_total m n with hm hm,
exacts [⟨m, hm, refl_mem_uniform... | lemma | cauchy_seq.totally_bounded_range | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"cauchy_seq",
"refl_mem_uniformity",
"totally_bounded"
] | Every Cauchy sequence over `ℕ` is totally bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq_aux (n : ℕ) : {s : set α // ∃ (_ : s ∈ f), s ×ˢ s ⊆ U n } | indefinite_description _ $ (cauchy_iff.1 hf).2 (U n) (U_mem n) | def | sequentially_complete.set_seq_aux | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [] | An auxiliary sequence of sets approximating a Cauchy filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_seq (n : ℕ) : set α | ⋂ m ∈ set.Iic n, (set_seq_aux hf U_mem m).val | def | sequentially_complete.set_seq | topology.uniform_space | src/topology/uniform_space/cauchy.lean | [
"topology.algebra.constructions",
"topology.bases",
"topology.uniform_space.basic"
] | [
"set.Iic"
] | Given a Cauchy filter `f` and a sequence `U` of entourages, `set_seq` provides
an antitone sequence of sets `s n ∈ f` such that `s n ×ˢ s n ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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