Loogle!
Result
Found 48 definitions whose name contains "of_compact".
- CompleteLattice.Iic_coatomic_of_compact_element Mathlib.Order.CompactlyGenerated.Basic
∀ {α : Type u_2} [inst : CompleteLattice α] {k : α}, CompleteLattice.IsCompactElement k → IsCoatomic ↑(Set.Iic k) - finite_of_compact_of_discrete Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] [inst : DiscreteTopology X], Finite X - cluster_point_of_compact Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] (f : Filter X) [inst_2 : Filter.NeBot f], ∃ x, ClusterPt x f - exists_clusterPt_of_compactSpace Mathlib.Topology.Compactness.Compact
∀ {X : Type u} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] (f : Filter X) [inst_2 : Filter.NeBot f], ∃ x, ClusterPt x f - isClosedMap_fst_of_compactSpace Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : CompactSpace Y], IsClosedMap Prod.fst - isClosedMap_snd_of_compactSpace Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : CompactSpace X], IsClosedMap Prod.snd - LocallyFinite.finite_nonempty_of_compact Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] {f : ι → Set X}, LocallyFinite f → Set.Finite {i | Set.Nonempty (f i)} - LocallyFinite.finite_of_compact Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] {f : ι → Set X}, LocallyFinite f → (∀ (i : ι), Set.Nonempty (f i)) → Set.Finite Set.univ - exists_subset_nhds_of_compactSpace Mathlib.Topology.Compactness.Compact
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] [inst_2 : Nonempty ι] {V : ι → Set X}, Directed (fun x x_1 => x ⊇ x_1) V → (∀ (i : ι), IsClosed (V i)) → ∀ {U : Set X}, (∀ x ∈ ⋂ i, V i, U ∈ nhds x) → ∃ i, V i ⊆ U - locally_compact_of_compact Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : T2Space X] [inst_2 : CompactSpace X], LocallyCompactSpace X - NormalSpace.of_compactSpace_r1Space Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] [inst_2 : R1Space X], NormalSpace X - T4Space.of_compactSpace_t2Space Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : CompactSpace X] [inst_2 : T2Space X], T4Space X - lebesgue_number_of_compact_open Mathlib.Topology.UniformSpace.Basic
∀ {α : Type ua} [inst : UniformSpace α] {K U : Set α}, IsCompact K → IsOpen U → K ⊆ U → ∃ V ∈ uniformity α, IsOpen V ∧ ∀ x ∈ K, UniformSpace.ball x V ⊆ U - complete_of_compact Mathlib.Topology.UniformSpace.Cauchy
∀ {α : Type u} [inst : UniformSpace α] [inst_1 : CompactSpace α], CompleteSpace α - tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace Mathlib.Topology.UniformSpace.UniformConvergence
∀ {α : Type u} {β : Type v} {ι : Type x} [inst : UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} [inst_1 : TopologicalSpace α] [inst_2 : CompactSpace α], TendstoLocallyUniformly F f p ↔ TendstoUniformly F f p - tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact Mathlib.Topology.UniformSpace.UniformConvergence
∀ {α : Type u} {β : Type v} {ι : Type x} [inst : UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} [inst_1 : TopologicalSpace α], IsCompact s → (TendstoLocallyUniformlyOn F f p s ↔ TendstoUniformlyOn F f p s) - unique_uniformity_of_compact Mathlib.Topology.UniformSpace.Compact
∀ {γ : Type u_3} [t : TopologicalSpace γ] [inst : CompactSpace γ] {u u' : UniformSpace γ}, UniformSpace.toTopologicalSpace = t → UniformSpace.toTopologicalSpace = t → u = u' - topologicalAddGroup_is_uniform_of_compactSpace Mathlib.Topology.Algebra.UniformGroup
∀ (G : Type u_1) [inst : AddGroup G] [inst_1 : TopologicalSpace G] [inst_2 : TopologicalAddGroup G] [inst_3 : CompactSpace G], UniformAddGroup G - topologicalGroup_is_uniform_of_compactSpace Mathlib.Topology.Algebra.UniformGroup
∀ (G : Type u_1) [inst : Group G] [inst_1 : TopologicalSpace G] [inst_2 : TopologicalGroup G] [inst_3 : CompactSpace G], UniformGroup G - EMetric.subset_countable_closure_of_compact Mathlib.Topology.EMetricSpace.Basic
∀ {α : Type u} [inst : PseudoEMetricSpace α] {s : Set α}, IsCompact s → ∃ t ⊆ s, Set.Countable t ∧ s ⊆ closure t - EMetric.countable_closure_of_compact Mathlib.Topology.EMetricSpace.Basic
∀ {γ : Type w} [inst : EMetricSpace γ] {s : Set γ}, IsCompact s → ∃ t ⊆ s, Set.Countable t ∧ s = closure t - finite_cover_balls_of_compact Mathlib.Topology.MetricSpace.PseudoMetric
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α}, IsCompact s → ∀ {e : ℝ}, 0 < e → ∃ t ⊆ s, Set.Finite t ∧ s ⊆ ⋃ x ∈ t, Metric.ball x e - proper_of_compact Mathlib.Topology.MetricSpace.ProperSpace
∀ {α : Type u} [inst : PseudoMetricSpace α] [inst_1 : CompactSpace α], ProperSpace α - properSpace_of_compact_closedBall_of_le Mathlib.Topology.MetricSpace.ProperSpace
∀ {α : Type u} [inst : PseudoMetricSpace α] (R : ℝ), (∀ (x : α) (r : ℝ), R ≤ r → IsCompact (Metric.closedBall x r)) → ProperSpace α - Metric.isBounded_of_compactSpace Mathlib.Topology.MetricSpace.Bounded
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} [inst_1 : CompactSpace α], Bornology.IsBounded s - isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace Mathlib.MeasureTheory.Measure.Typeclasses
∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_2 : CompactSpace α], MeasureTheory.IsFiniteMeasure μ ↔ MeasureTheory.IsFiniteMeasureOnCompacts μ - FirstCountableTopology.seq_compact_of_compact Mathlib.Topology.Sequences
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : FirstCountableTopology X] [inst_2 : CompactSpace X], SeqCompactSpace X - Continuous.bounded_above_of_compact_support Mathlib.Analysis.Normed.Group.Basic
∀ {α : Type u_3} {E : Type u_6} [inst : NormedAddGroup E] [inst_1 : TopologicalSpace α] {f : α → E}, Continuous f → HasCompactSupport f → ∃ C, ∀ (x : α), ‖f x‖ ≤ C - MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero Mathlib.MeasureTheory.Group.Action
∀ (G : Type u) {α : Type w} {m : MeasurableSpace α} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : MeasurableSpace G] [inst_3 : MeasurableSMul G α] {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.SMulInvariantMeasure G α μ] [inst_5 : TopologicalSpace α] [inst_6 : ContinuousConstSMul G α] [inst : MulAction.IsMinimal G α] {K U : Set α}, IsCompact K → ↑↑μ K ≠ 0 → IsOpen U → Set.Nonempty U → 0 < ↑↑μ U - MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_compact_ne_zero Mathlib.MeasureTheory.Group.Action
∀ (G : Type u) {α : Type w} {m : MeasurableSpace α} [inst : AddGroup G] [inst_1 : AddAction G α] [inst_2 : MeasurableSpace G] [inst_3 : MeasurableVAdd G α] {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.VAddInvariantMeasure G α μ] [inst_5 : TopologicalSpace α] [inst_6 : ContinuousConstVAdd G α] [inst : AddAction.IsMinimal G α] {K U : Set α}, IsCompact K → ↑↑μ K ≠ 0 → IsOpen U → Set.Nonempty U → 0 < ↑↑μ U - ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact Mathlib.Topology.UniformSpace.CompactConvergence
∀ {α : Type u₁} {β : Type u₂} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] [inst_2 : CompactSpace α], Filter.HasBasis (uniformity C(α, β)) (fun V => V ∈ uniformity β) fun V => {fg | ∀ (x : α), (fg.1 x, fg.2 x) ∈ V} - BoundedContinuousFunction.norm_lt_iff_of_compact Mathlib.Topology.ContinuousFunction.Bounded
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : SeminormedAddCommGroup β] [inst_2 : CompactSpace α] {f : BoundedContinuousFunction α β} {M : ℝ}, 0 < M → (‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M) - BoundedContinuousFunction.dist_lt_iff_of_compact Mathlib.Topology.ContinuousFunction.Bounded
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} [inst_2 : CompactSpace α], 0 < C → (dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C) - ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv Mathlib.Topology.ContinuousFunction.Compact
∀ {α : Type u_1} {E : Type u_3} [inst : TopologicalSpace α] [inst_1 : CompactSpace α] [inst_2 : NormedAddCommGroup E] {𝕜 : Type u_4} [inst_3 : NormedField 𝕜] [inst_4 : NormedSpace 𝕜 E], LinearEquiv.toEquiv (ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv = ContinuousMap.equivBoundedOfCompact α E - continuousOn_integral_of_compact_support Mathlib.MeasureTheory.Integral.SetIntegral
∀ {X : Type u_1} {Y : Type u_2} {E : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : MeasurableSpace Y] [inst_3 : OpensMeasurableSpace Y] {μ : MeasureTheory.Measure Y} [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace ℝ E] {f : X → Y → E} {s : Set X} {k : Set Y} [inst_6 : MeasureTheory.IsFiniteMeasureOnCompacts μ], IsCompact k → ContinuousOn (Function.uncurry f) (s ×ˢ Set.univ) → (∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0) → ContinuousOn (fun x => ∫ (y : Y), f x y ∂μ) s - continuousOn_integral_bilinear_of_locally_integrable_of_compact_support Mathlib.MeasureTheory.Integral.SetIntegral
∀ {X : Type u_1} {Y : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {𝕜 : Type u_6} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [inst_2 : MeasurableSpace Y] [inst_3 : OpensMeasurableSpace Y] {μ : MeasureTheory.Measure Y} [inst_4 : NontriviallyNormedField 𝕜] [inst_5 : NormedAddCommGroup E] [inst_6 : NormedSpace ℝ E] [inst_7 : NormedAddCommGroup F] [inst_8 : NormedSpace 𝕜 F] [inst_9 : NormedAddCommGroup G] [inst_10 : NormedSpace 𝕜 G] [inst_11 : NormedSpace 𝕜 E] (L : F →L[𝕜] G →L[𝕜] E) {f : X → Y → G} {s : Set X} {k : Set Y} {g : Y → F}, IsCompact k → ContinuousOn (Function.uncurry f) (s ×ˢ Set.univ) → (∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0) → MeasureTheory.IntegrableOn g k μ → ContinuousOn (fun x => ∫ (y : Y), (L (g y)) (f x y) ∂μ) s - MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_compact Mathlib.MeasureTheory.Group.Measure
∀ {G : Type u_2} [inst : MeasurableSpace G] [inst_1 : TopologicalSpace G] [inst_2 : BorelSpace G] {μ : MeasureTheory.Measure G} [inst_3 : AddGroup G] [inst_4 : TopologicalAddGroup G] [inst_5 : MeasureTheory.Measure.IsAddLeftInvariant μ] (K : Set G), IsCompact K → ↑↑μ K ≠ 0 → MeasureTheory.Measure.IsOpenPosMeasure μ - MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_compact Mathlib.MeasureTheory.Group.Measure
∀ {G : Type u_2} [inst : MeasurableSpace G] [inst_1 : TopologicalSpace G] [inst_2 : BorelSpace G] {μ : MeasureTheory.Measure G} [inst_3 : Group G] [inst_4 : TopologicalGroup G] [inst_5 : MeasureTheory.Measure.IsMulLeftInvariant μ] (K : Set G), IsCompact K → ↑↑μ K ≠ 0 → MeasureTheory.Measure.IsOpenPosMeasure μ - MeasureTheory.Measure.isAddInvariant_eq_smul_of_compactSpace Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : TopologicalAddGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [inst_5 : CompactSpace G] (μ' μ : MeasureTheory.Measure G) [inst_6 : MeasureTheory.Measure.IsAddHaarMeasure μ] [inst_7 : MeasureTheory.Measure.IsAddLeftInvariant μ'] [inst_8 : MeasureTheory.IsFiniteMeasureOnCompacts μ'], μ' = MeasureTheory.Measure.addHaarScalarFactor μ' μ • μ - MeasureTheory.Measure.isMulInvariant_eq_smul_of_compactSpace Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : TopologicalGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [inst_5 : CompactSpace G] (μ' μ : MeasureTheory.Measure G) [inst_6 : MeasureTheory.Measure.IsHaarMeasure μ] [inst_7 : MeasureTheory.Measure.IsMulLeftInvariant μ'] [inst_8 : MeasureTheory.IsFiniteMeasureOnCompacts μ'], μ' = MeasureTheory.Measure.haarScalarFactor μ' μ • μ - paracompact_of_compact Mathlib.Topology.Compactness.Paracompact
∀ {X : Type v} [inst : TopologicalSpace X] [inst_1 : CompactSpace X], ParacompactSpace X - RCLike.uniqueContinuousFunctionalCalculus_of_compactSpace_spectrum Mathlib.Topology.ContinuousFunction.UniqueCFC
∀ {𝕜 : Type u_1} {A : Type u_2} [inst : RCLike 𝕜] [inst_1 : TopologicalSpace A] [inst_2 : T2Space A] [inst_3 : Ring A] [inst_4 : StarRing A] [inst_5 : Algebra 𝕜 A] [h : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)], UniqueContinuousFunctionalCalculus 𝕜 A - TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system Mathlib.Topology.Category.TopCat.Limits.Konig
∀ {J : Type u} [inst : CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCatMax) [inst_1 : CategoryTheory.IsCofilteredOrEmpty J] [inst_2 : ∀ (j : J), Nonempty ↑(F.obj j)] [inst_3 : ∀ (j : J), CompactSpace ↑(F.obj j)] [inst_4 : ∀ (j : J), T2Space ↑(F.obj j)], Nonempty ↑(TopCat.limitCone F).pt - exists_idempotent_of_compact_t2_of_continuous_add_left Mathlib.Topology.Algebra.Semigroup
∀ {M : Type u_1} [inst : Nonempty M] [inst : AddSemigroup M] [inst_1 : TopologicalSpace M] [inst_2 : CompactSpace M] [inst_3 : T2Space M], (∀ (r : M), Continuous fun x => x + r) → ∃ m, m + m = m - exists_idempotent_of_compact_t2_of_continuous_mul_left Mathlib.Topology.Algebra.Semigroup
∀ {M : Type u_1} [inst : Nonempty M] [inst : Semigroup M] [inst_1 : TopologicalSpace M] [inst_2 : CompactSpace M] [inst_3 : T2Space M], (∀ (r : M), Continuous fun x => x * r) → ∃ m, m * m = m - MDifferentiable.apply_eq_of_compactSpace Mathlib.Geometry.Manifold.Complex
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type u_2} [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℂ F] {H : Type u_3} [inst_4 : TopologicalSpace H] {I : ModelWithCorners ℂ E H} [inst_5 : ModelWithCorners.Boundaryless I] {M : Type u_4} [inst_6 : TopologicalSpace M] [inst_7 : CompactSpace M] [inst_8 : ChartedSpace H M] [inst_9 : SmoothManifoldWithCorners I M] [inst_10 : PreconnectedSpace M] {f : M → F}, MDifferentiable I (modelWithCornersSelf ℂ F) f → ∀ (a b : M), f a = f b - MDifferentiable.exists_eq_const_of_compactSpace Mathlib.Geometry.Manifold.Complex
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type u_2} [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℂ F] {H : Type u_3} [inst_4 : TopologicalSpace H] {I : ModelWithCorners ℂ E H} [inst_5 : ModelWithCorners.Boundaryless I] {M : Type u_4} [inst_6 : TopologicalSpace M] [inst_7 : CompactSpace M] [inst_8 : ChartedSpace H M] [inst_9 : SmoothManifoldWithCorners I M] [inst_10 : PreconnectedSpace M] {f : M → F}, MDifferentiable I (modelWithCornersSelf ℂ F) f → ∃ v, f = Function.const M v - exists_embedding_euclidean_of_compact Mathlib.Geometry.Manifold.WhitneyEmbedding
∀ {E : Type uE} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] {H : Type uH} [inst_3 : TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [inst_6 : SmoothManifoldWithCorners I M] [inst_7 : T2Space M] [inst_8 : CompactSpace M], ∃ n e, Smooth I (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) e ∧ ClosedEmbedding e ∧ ∀ (x : M), Function.Injective ⇑(mfderiv I (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) e x)
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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 c38e122