Loogle!
Result
Found 12 definitions whose name contains "of_open".
- T0Space.of_open_cover Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X], (∀ (x : X), ∃ s, x ∈ s ∧ IsOpen s ∧ T0Space ↑s) → T0Space X - quasiSober_of_open_cover Mathlib.Topology.Sober
∀ {α : Type u_1} [inst : TopologicalSpace α] (S : Set (Set α)), (∀ (s : ↑S), IsOpen ↑s) → ∀ [hS' : ∀ (s : ↑S), QuasiSober ↑↑s], ⋃₀ S = ⊤ → QuasiSober α - StrictConvex.eq_of_openSegment_subset_frontier Mathlib.Analysis.Convex.Strict
∀ {𝕜 : Type u_1} {E : Type u_3} [inst : OrderedRing 𝕜] [inst_1 : TopologicalSpace E] [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜 E] {s : Set E} {x y : E} [inst_4 : Nontrivial 𝕜] [inst_5 : DenselyOrdered 𝕜], StrictConvex 𝕜 s → x ∈ s → y ∈ s → openSegment 𝕜 x y ⊆ frontier s → x = y - QuasiSeparatedSpace.of_openEmbedding Mathlib.Topology.QuasiSeparated
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β}, OpenEmbedding f → ∀ [inst_2 : QuasiSeparatedSpace β], QuasiSeparatedSpace α - mem_tangentCone_of_openSegment_subset Mathlib.Analysis.Calculus.TangentCone
∀ {G : Type u_4} [inst : NormedAddCommGroup G] [inst_1 : NormedSpace ℝ G] {s : Set G} {x y : G}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s x - TopCat.pullback_map_openEmbedding_of_open_embeddings Mathlib.Topology.Category.TopCat.Limits.Pullbacks
∀ {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z}, OpenEmbedding ⇑i₁ → OpenEmbedding ⇑i₂ → ∀ (i₃ : S ⟶ T) [H₃ : CategoryTheory.Mono i₃] (eq₁ : CategoryTheory.CategoryStruct.comp f₁ i₃ = CategoryTheory.CategoryStruct.comp i₁ g₁) (eq₂ : CategoryTheory.CategoryStruct.comp f₂ i₃ = CategoryTheory.CategoryStruct.comp i₂ g₂), OpenEmbedding ⇑(CategoryTheory.Limits.pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) - TopCat.Presheaf.isSheaf_of_openEmbedding Mathlib.Topology.Sheaves.SheafCondition.Sites
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : TopCat} {f : X ⟶ Y} {F : TopCat.Presheaf C Y} (h : OpenEmbedding ⇑f), TopCat.Presheaf.IsSheaf F → TopCat.Presheaf.IsSheaf (CategoryTheory.Functor.comp (IsOpenMap.functor ⋯).op F) - TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding Mathlib.Topology.Sheaves.Stalks
∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasColimits C] {X Y : TopCat} {f : X ⟶ Y}, OpenEmbedding ⇑f → ∀ (F : TopCat.Presheaf C X) (x : ↑X), CategoryTheory.IsIso (TopCat.Presheaf.stalkPushforward C f F x) - AlgebraicGeometry.PropertyIsLocalAtTarget.of_openCover Mathlib.AlgebraicGeometry.Morphisms.Basic
∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme}, AlgebraicGeometry.PropertyIsLocalAtTarget P → ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : AlgebraicGeometry.Scheme.OpenCover Y), (∀ (i : 𝒰.J), P CategoryTheory.Limits.pullback.snd) → P f - AddSubgroup.isOpen_of_openAddSubgroup Mathlib.Topology.Algebra.OpenSubgroup
∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : TopologicalSpace G] [inst_2 : ContinuousAdd G] (H : AddSubgroup G) {U : OpenAddSubgroup G}, ↑U ≤ H → IsOpen ↑H - Subgroup.isOpen_of_openSubgroup Mathlib.Topology.Algebra.OpenSubgroup
∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] [inst_2 : ContinuousMul G] (H : Subgroup G) {U : OpenSubgroup G}, ↑U ≤ H → IsOpen ↑H - isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis Mathlib.Topology.Compactness.Lindelof
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] (b : ι → Set X), TopologicalSpace.IsTopologicalBasis (Set.range b) → (∀ (i : ι), IsLindelof (b i)) → ∀ (U : Set X), IsLindelof U ∧ IsOpen U ↔ ∃ s, Set.Countable s ∧ U = ⋃ i ∈ s, b 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 34713b2
serving mathlib revision a547a94