Loogle!
Result
Found 15 definitions mentioning Exists, IsOpen, Set.iUnion, IsCompact and Set. Of these, 15 match your pattern(s).
- IsCompact.elim_directed_cover Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X} {ι : Type v} [hι : Nonempty ι], IsCompact s → ∀ (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → Directed (fun x x_1 => x ⊆ x_1) U → ∃ i, s ⊆ U i - isCompact_of_finite_subcover Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, (∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i) → IsCompact s - IsCompact.elim_finite_subcover Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X} {ι : Type v}, IsCompact s → ∀ (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i - isCompact_iff_finite_subcover Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsCompact s ↔ ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → s ⊆ ⋃ i, U i → ∃ t, s ⊆ ⋃ i ∈ t, U i - isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] (b : ι → Set X), TopologicalSpace.IsTopologicalBasis (Set.range b) → (∀ (i : ι), IsCompact (b i)) → ∀ (U : Set X), IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i - IsCompact.elim_finite_subcover_image Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] {s : Set X} {b : Set ι} {c : ι → Set X}, IsCompact s → (∀ i ∈ b, IsOpen (c i)) → s ⊆ ⋃ i ∈ b, c i → ∃ b' ⊆ b, b'.Finite ∧ s ⊆ ⋃ i ∈ b', c i - IsCompact.finite_compact_cover Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : R1Space X] {s : Set X}, IsCompact s → ∀ {ι : Type u_3} (t : Finset ι) (U : ι → Set X), (∀ i ∈ t, IsOpen (U i)) → s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i - Filter.HasBasis.lebesgue_number_lemma Mathlib.Topology.UniformSpace.Basic
∀ {α : Type ua} [inst : UniformSpace α] {K : Set α} {ι' : Sort u_2} {ι : Sort u_3} {p : ι' → Prop} {V : ι' → Set (α × α)} {U : ι → Set α}, (uniformity α).HasBasis p V → IsCompact K → (∀ (j : ι), IsOpen (U j)) → K ⊆ ⋃ j, U j → ∃ i, p i ∧ ∀ x ∈ K, ∃ j, UniformSpace.ball x (V i) ⊆ U j - lebesgue_number_lemma Mathlib.Topology.UniformSpace.Basic
∀ {α : Type ua} [inst : UniformSpace α] {K : Set α} {ι : Sort u_2} {U : ι → Set α}, IsCompact K → (∀ (i : ι), IsOpen (U i)) → K ⊆ ⋃ i, U i → ∃ V ∈ uniformity α, ∀ x ∈ K, ∃ i, UniformSpace.ball x V ⊆ U i - lebesgue_number_lemma_of_metric Mathlib.Topology.MetricSpace.PseudoMetric
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} {ι : Sort u_3} {c : ι → Set α}, IsCompact s → (∀ (i : ι), IsOpen (c i)) → s ⊆ ⋃ i, c i → ∃ δ > 0, ∀ x ∈ s, ∃ i, Metric.ball x δ ⊆ c i - TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion Mathlib.Topology.Sets.Opens
∀ {α : Type u_2} [inst : TopologicalSpace α] {ι : Type u_5} (b : ι → TopologicalSpace.Opens α), TopologicalSpace.Opens.IsBasis (Set.range b) → (∀ (i : ι), IsCompact ↑(b i)) → ∀ (U : Set α), IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, ↑(b i) - IsCompact.exists_finite_cover_smul Mathlib.Dynamics.Minimal
∀ (G : Type u_2) {α : Type u_3} [inst : Group G] [inst_1 : TopologicalSpace α] [inst_2 : MulAction G α] [inst_3 : MulAction.IsMinimal G α] [inst_4 : ContinuousConstSMul G α] {K U : Set α}, IsCompact K → IsOpen U → U.Nonempty → ∃ I, K ⊆ ⋃ g ∈ I, g • U - IsCompact.exists_finite_cover_vadd Mathlib.Dynamics.Minimal
∀ (G : Type u_2) {α : Type u_3} [inst : AddGroup G] [inst_1 : TopologicalSpace α] [inst_2 : AddAction G α] [inst_3 : AddAction.IsMinimal G α] [inst_4 : ContinuousConstVAdd G α] {K U : Set α}, IsCompact K → IsOpen U → U.Nonempty → ∃ I, K ⊆ ⋃ g ∈ I, g +ᵥ U - AlgebraicGeometry.isCompact_open_iff_eq_finset_affine_union Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
∀ {X : AlgebraicGeometry.Scheme} (U : Set ↑↑X.toPresheafedSpace), IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, ↑↑i - AlgebraicGeometry.isCompact_open_iff_eq_basicOpen_union Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
∀ {X : AlgebraicGeometry.Scheme} [inst : AlgebraicGeometry.IsAffine X] (U : Set ↑↑X.toPresheafedSpace), IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, ↑(X.basicOpen i)
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision d874bdf