name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Topology.WithGeneratedByTopology.isClosed_iff | Mathlib.Topology.Convenient.GeneratedBy | ∀ {ι : Type t} {X : ι → Type u} [inst : (i : ι) → TopologicalSpace (X i)] {Y : Type v} [tY : TopologicalSpace Y]
{U : Set (Topology.WithGeneratedByTopology X Y)},
IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (⇑f ⁻¹' ⇑Topology.WithGeneratedByTopology.equiv.symm ⁻¹' U) | null | true |
derivedSet | Mathlib.Topology.DerivedSet | {X : Type u_1} → [TopologicalSpace X] → Set X → Set X | The derived set of a set is the set of all accumulation points of it.
| true |
SSet.innerAnodyneExtensions.whiskerRight | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct | ∀ {X Y : SSet} {f : X ⟶ Y},
SSet.innerAnodyneExtensions f →
∀ (Z : SSet), SSet.innerAnodyneExtensions (CategoryTheory.MonoidalCategoryStruct.whiskerRight f Z) | null | true |
SupHom.rec | Mathlib.Order.Hom.Lattice | {α : Type u_6} →
{β : Type u_7} →
[inst : Max α] →
[inst_1 : Max β] →
{motive : SupHom α β → Sort u} →
((toFun : α → β) →
(map_sup' : ∀ (a b : α), toFun (a ⊔ b) = toFun a ⊔ toFun b) →
motive { toFun := toFun, map_sup' := map_sup' }) →
(t : SupHom α... | null | false |
RootPairing.IsIrreducible.mk' | Mathlib.LinearAlgebra.RootSystem.Irreducible | ∀ {ι : Type u_1} {M : Type u_3} {N : Type u_4} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] {K : Type u_5}
[inst_2 : Field K] [inst_3 : Module K M] [inst_4 : Module K N] [Nontrivial M] (P : RootPairing ι K M N),
(∀ (q : Submodule K M), (∀ (i : ι), q ∈ Module.End.invtSubmodule ↑(P.reflection i)) → q ≠ ⊥ → q = ⊤... | When the coefficients are a field, the coroot conditions for irreducibility follow from those
for the roots. | true |
_private.Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid.0.Affine.Simplex.eq_centroid_of_forall_mem_median._proof_1_5 | Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | ∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
[inst_3 : AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n),
(AffineIndependent k fun a => if h : a = 0 then ⋯ ▸ s.centroid else s.points a) →
AffineIndependent k fun x => if x = 0 then s.centr... | null | false |
BitVec.mk_zero | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {h : 0 < 2 ^ w}, { toFin := ⟨0, h⟩ } = 0#w | null | true |
BooleanSubalgebra.subtype_comp_inclusion | Mathlib.Order.BooleanSubalgebra | ∀ {α : Type u_2} [inst : BooleanAlgebra α] {L M : BooleanSubalgebra α} (h : L ≤ M),
M.subtype.comp (BooleanSubalgebra.inclusion h) = L.subtype | null | true |
FiniteIndexNormalAddSubgroup.rec | Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {G : Type u_1} →
[inst : AddGroup G] →
{motive : FiniteIndexNormalAddSubgroup G → Sort u} →
((toAddSubgroup : AddSubgroup G) →
(isNormal' : toAddSubgroup.Normal) →
(isFiniteIndex' : toAddSubgroup.FiniteIndex) →
motive { toAddSubgroup := toAddSubgroup, isNormal' := isNorma... | null | false |
_private.Mathlib.Topology.Order.0.TopologicalSpace.nhds_mkOfNhds_single._simp_1_2 | Mathlib.Topology.Order | ∀ {α : Type u_1} {a : α} {p : α → Prop}, (∀ᶠ (x : α) in pure a, p x) = p a | null | false |
continuousAt_gauge_zero | Mathlib.Analysis.Convex.Gauge | ∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} [inst_2 : TopologicalSpace E]
[ContinuousSMul ℝ E], s ∈ nhds 0 → ContinuousAt (gauge s) 0 | If `s` is a neighborhood of the origin, then `gauge s` is continuous at the origin.
See also `continuousAt_gauge`. | true |
Tropical.le_zero._simp_1 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u} [inst : LE R] [inst_1 : OrderTop R] (x : Tropical R), (x ≤ 0) = True | null | false |
CategoryTheory.SimplicialObject.IsCoskeletal.isRightKanExtension | Mathlib.AlgebraicTopology.SimplicialObject.Coskeletal | ∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {X : CategoryTheory.SimplicialObject C} {n : ℕ}
[self : X.IsCoskeletal n],
CategoryTheory.Functor.IsRightKanExtension X
(CategoryTheory.CategoryStruct.id ((SimplexCategory.Truncated.inclusion n).op.comp X)) | null | true |
Std.Tactic.BVDecide.LRAT.Internal.Assignment.addAssignment | Std.Tactic.BVDecide.LRAT.Internal.Assignment | Bool → Std.Tactic.BVDecide.LRAT.Internal.Assignment → Std.Tactic.BVDecide.LRAT.Internal.Assignment | null | true |
Lean.Meta.SimpAll.M | Lean.Meta.Tactic.Simp.SimpAll | Type → Type | null | true |
Differentiable.fun_add_iff_right | Mathlib.Analysis.Calculus.FDeriv.Add | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f g : E → F},
Differentiable 𝕜 f → ((Differentiable 𝕜 fun i => f i + g i) ↔ Differentiable 𝕜 g) | Eta-expanded form of `Differentiable.add_iff_right` | true |
List.untrop_prod | Mathlib.Algebra.Tropical.BigOperators | ∀ {R : Type u_1} [inst : AddMonoid R] (l : List (Tropical R)), Tropical.untrop l.prod = (List.map Tropical.untrop l).sum | null | true |
SSet.instIsStableUnderCoproductsMonomorphismsOfHasCoproductsType | Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms | ∀ [CategoryTheory.Limits.HasCoproducts (Type u)],
CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{v', u, u + 1}
(CategoryTheory.MorphismProperty.monomorphisms SSet) | null | true |
MeasureTheory.llr.eq_1 | Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | ∀ {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) (x : α),
MeasureTheory.llr μ ν x = Real.log (μ.rnDeriv ν x).toReal | null | true |
_private.Mathlib.Topology.Separation.Hausdorff.0.t2_iff_ultrafilter._simp_1_2 | Mathlib.Topology.Separation.Hausdorff | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
Ideal.fiberIsoOfBijectiveResidueField._proof_7 | Mathlib.RingTheory.Etale.QuasiFinite | ∀ {R' : Type u_1} [inst : CommRing R'] {q : Ideal R'} [inst_1 : q.IsPrime],
SMulCommClass R' q.ResidueField q.ResidueField | null | false |
_private.Mathlib.Analysis.Convex.Segment.0.mem_openSegment_translate._simp_1_2 | Mathlib.Analysis.Convex.Segment | ∀ (𝕜 : Type u_1) {E : Type u_2} {G : Type u_4} [inst : Ring 𝕜] [inst_1 : PartialOrder 𝕜] [AddRightMono 𝕜]
[inst_3 : AddCommGroup E] [inst_4 : AddCommGroup G] [inst_5 : Module 𝕜 E] [inst_6 : AddTorsor G E]
[VAddCommClass G E E] (a : G) (b c : E), openSegment 𝕜 (a +ᵥ b) (a +ᵥ c) = a +ᵥ openSegment 𝕜 b c | null | false |
ContinuousLinearEquiv.comp_contDiffWithinAt_iff | Mathlib.Analysis.Calculus.ContDiff.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
[inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {s : Set E} {f : E → F} {x : E} {n : ... | Composition by continuous linear equivs on the left respects higher differentiability at a
point in a domain. | true |
_private.Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace.0.Besicovitch.exists_goodδ._simp_1_8 | Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | ∀ {α : Type u_1} [inst : Fintype α] (x : α), (x ∈ Finset.univ) = True | null | false |
AlgebraicGeometry.Scheme.ProEt.forget | Mathlib.AlgebraicGeometry.Sites.Proetale | (S : AlgebraicGeometry.Scheme) → CategoryTheory.Functor S.ProEt (CategoryTheory.Over S) | The forgetful functor the pro-étale site of `S` to schemes over `S`. | true |
CategoryTheory.ObjectProperty.isThin_of_isSeparating_bot | Mathlib.CategoryTheory.Generator.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C], ⊥.IsSeparating → Quiver.IsThin C | null | true |
Vector.append_assoc._proof_2 | Init.Data.Vector.Lemmas | ∀ {n m k : ℕ}, n + (m + k) = n + m + k | null | false |
SemistandardYoungTableau.col_strict' | Mathlib.Combinatorics.Young.SemistandardTableau | ∀ {μ : YoungDiagram} (self : SemistandardYoungTableau μ) {i1 i2 j : ℕ},
i1 < i2 → (i2, j) ∈ μ → self.entry i1 j < self.entry i2 j | The entries in each column are strictly increasing (top to bottom). | true |
FirstOrder.Language.instUniqueStructureEmpty | Mathlib.ModelTheory.Basic | {M : Type w} → Unique (FirstOrder.Language.empty.Structure M) | null | true |
CategoryTheory.SmallObject.SuccStruct.mk | Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
C → (succ : C → C) → ((X : C) → X ⟶ succ X) → CategoryTheory.SmallObject.SuccStruct C | null | true |
_private.Mathlib.Algebra.BigOperators.Fin.0.Fin.sum_insertNth_go._f | Mathlib.Algebra.BigOperators.Fin | ∀ {M : Type u_2} [inst : AddCommMonoid M] (x : ℕ)
(f :
Nat.below (motive := fun x =>
∀ (x_1 : ℕ) (x_2 : x_1 < x + 1) (x_3 : M) (x_4 : Fin x → M),
∑ j, ⟨x_1, x_2⟩.insertNth x_3 x_4 j = x_3 + ∑ j, x_4 j)
x)
(x_1 : ℕ) (x_2 : x_1 < x + 1) (x_3 : M) (x_4 : Fin x → M), ∑ j, ⟨x_1, x_2⟩.insertNth x_... | null | false |
ContinuousMonoidHom.comp | Mathlib.Topology.Algebra.ContinuousMonoidHom | {A : Type u_2} →
{B : Type u_3} →
{C : Type u_4} →
[inst : Monoid A] →
[inst_1 : Monoid B] →
[inst_2 : Monoid C] →
[inst_3 : TopologicalSpace A] →
[inst_4 : TopologicalSpace B] → [inst_5 : TopologicalSpace C] → (B →ₜ* C) → (A →ₜ* B) → A →ₜ* C | Composition of two continuous homomorphisms. | true |
Aesop.RuleStatsTotals.empty | Aesop.Stats.Basic | Aesop.RuleStatsTotals | null | true |
Lean.IR.IRType.usize | Lean.Compiler.IR.Basic | Lean.IR.IRType | null | true |
KummerDedekind.quotMapEquivQuotQuotMap._proof_4 | Mathlib.NumberTheory.KummerDedekind | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S} {I : Ideal R}
[inst_3 : IsDomain R] [inst_4 : IsIntegrallyClosed R] [inst_5 : IsDedekindDomain S]
[inst_6 : Module.IsTorsionFree R S] (hx' : IsIntegral R x),
(Ideal.span {Polynomial.map (Ideal.Quotient.mk I) (... | null | false |
_private.Mathlib.Probability.Kernel.Representation.0.ProbabilityTheory.Kernel.exists_measurable_map_eq_unitInterval_aux._simp_1_12 | Mathlib.Probability.Kernel.Representation | ∀ {α : Type u_1} [inst : CompleteSemilatticeSup α] {s : Set α} {a : α}, (sSup s ≤ a) = ∀ b ∈ s, b ≤ a | null | false |
Lean.JsonRpc.Message.noConfusion | Lean.Data.JsonRpc | {P : Sort u} → {t t' : Lean.JsonRpc.Message} → t = t' → Lean.JsonRpc.Message.noConfusionType P t t' | null | false |
_private.Std.Data.DTreeMap.Internal.Model.0.Std.DTreeMap.Internal.Impl.minView.match_3.splitter | Std.Data.DTreeMap.Internal.Model | {α : Type u_1} →
{β : α → Type u_2} →
(r : Std.DTreeMap.Internal.Impl α β) →
(motive :
(l : Std.DTreeMap.Internal.Impl α β) →
l.Balanced → Std.DTreeMap.Internal.Impl.BalancedAtRoot l.size r.size → Sort u_3) →
(l : Std.DTreeMap.Internal.Impl α β) →
(hl : l.Balanced) →
... | null | true |
MvPolynomial.restrictSupportIdeal.congr_simp | Mathlib.RingTheory.MvPolynomial.Ideal | ∀ {σ : Type u} (R : Type v) [inst : CommSemiring R] (s s_1 : Set (σ →₀ ℕ)) (e_s : s = s_1) (hs : IsUpperSet s),
MvPolynomial.restrictSupportIdeal R s hs = MvPolynomial.restrictSupportIdeal R s_1 ⋯ | null | true |
CategoryTheory.Presieve.singleton_eq_iff_domain._simp_1 | Mathlib.CategoryTheory.Sites.Sieves | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f g : Y ⟶ X),
CategoryTheory.Presieve.singleton f g = (f = g) | null | false |
UniformSpace.Completion.mapRingEquiv_symm_apply | Mathlib.Topology.Algebra.UniformRing | ∀ {α : Type u_1} [inst : Ring α] [inst_1 : UniformSpace α] [inst_2 : IsTopologicalRing α] [inst_3 : IsUniformAddGroup α]
{β : Type u} [inst_4 : UniformSpace β] [inst_5 : Ring β] [inst_6 : IsUniformAddGroup β] [inst_7 : IsTopologicalRing β]
(f : α ≃+* β) (hf : Continuous ⇑f) (hf' : Continuous ⇑f.symm) (a : UniformSp... | null | true |
Lean.Grind.Bool.or_eq_of_eq_true_left | Init.Grind.Lemmas | ∀ {a b : Bool}, a = true → (a || b) = true | null | true |
_private.Mathlib.Topology.MetricSpace.GromovHausdorff.0.GromovHausdorff.instSecondCountableTopologyGHSpace._simp_7 | Mathlib.Topology.MetricSpace.GromovHausdorff | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
sInfHom.ext | Mathlib.Order.Hom.CompleteLattice | ∀ {α : Type u_2} {β : Type u_3} [inst : InfSet α] [inst_1 : InfSet β] {f g : sInfHom α β},
(∀ (a : α), f a = g a) → f = g | null | true |
IsClub.csSup_mem | Mathlib.SetTheory.Cardinal.Cofinality.Club | ∀ {α : Type u_1} [inst : ConditionallyCompleteLinearOrder α] {s t : Set α},
IsClub s → t ⊆ s → t.Nonempty → BddAbove t → sSup t ∈ s | null | true |
HilbertBasis.hasSum_repr_symm | Mathlib.Analysis.InnerProductSpace.l2Space | ∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (f : ↥(lp (fun x => 𝕜) 2)),
HasSum (fun i => ↑f i • b i) (b.repr.symm f) | null | true |
AdjoinRoot.instCommRing._proof_3 | Mathlib.RingTheory.AdjoinRoot | ∀ {R : Type u_1} [inst : CommRing R] (f : Polynomial R) (a b c : AdjoinRoot f), a + b + c = a + (b + c) | null | false |
Std.Format.MonadPrettyFormat.mk | Init.Data.Format.Basic | {m : Type → Type} →
(String → m Unit) → (ℕ → m Unit) → m ℕ → (ℕ → m Unit) → (ℕ → m Unit) → Std.Format.MonadPrettyFormat m | null | true |
Lean.ErrorExplanation.rec | Lean.ErrorExplanation | {motive : Lean.ErrorExplanation → Sort u} →
((doc : String) →
(metadata : Lean.ErrorExplanation.Metadata) →
(declLoc? : Option Lean.DeclarationLocation) →
motive { doc := doc, metadata := metadata, declLoc? := declLoc? }) →
(t : Lean.ErrorExplanation) → motive t | null | false |
Real.HolderTriple.pos' | Mathlib.Data.Real.ConjExponents | ∀ {p q r : ℝ}, p.HolderTriple q r → 0 < r | For `r`, instead of `p` | true |
Algebra.PreSubmersivePresentation.differential._proof_2 | Mathlib.RingTheory.Extension.Presentation.Submersive | ∀ {R : Type u_1} {S : Type u_4} {ι : Type u_2} {σ : Type u_3} [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : Algebra R S] (P : Algebra.PreSubmersivePresentation R S ι σ), SMulCommClass P.Ring P.Ring (σ → P.Ring) | null | false |
AddUnits.mk_val | Mathlib.Algebra.Group.Units.Defs | ∀ {α : Type u} [inst : AddMonoid α] (u : AddUnits α) (y : α) (h₁ : ↑u + y = 0) (h₂ : y + ↑u = 0),
{ val := ↑u, neg := y, val_neg := h₁, neg_val := h₂ } = u | null | true |
Lean.Lsp.CompletionList.casesOn | Lean.Data.Lsp.LanguageFeatures | {motive : Lean.Lsp.CompletionList → Sort u} →
(t : Lean.Lsp.CompletionList) →
((isIncomplete : Bool) →
(items : Array Lean.Lsp.CompletionItem) → motive { isIncomplete := isIncomplete, items := items }) →
motive t | null | false |
round.eq_1 | Mathlib.Algebra.Order.Round | ∀ {α : Type u_2} [inst : Ring α] [inst_1 : LinearOrder α] [inst_2 : FloorRing α] (x : α),
round x = if 2 * Int.fract x < 1 then ⌊x⌋ else ⌈x⌉ | null | true |
_private.Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence.0.Mathlib.Tactic.BicategoryLike.normalize.match_3 | Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence | (motive : Mathlib.Tactic.BicategoryLike.Mor₁ → Sort u_1) →
(f : Mathlib.Tactic.BicategoryLike.Mor₁) →
((e : Lean.Expr) → (a : Mathlib.Tactic.BicategoryLike.Obj) → motive (Mathlib.Tactic.BicategoryLike.Mor₁.id e a)) →
((e : Lean.Expr) →
(f g : Mathlib.Tactic.BicategoryLike.Mor₁) → motive (Mathlib.T... | null | false |
_private.Init.Data.SInt.Lemmas.0.Int16.toISize_ne_minValue._proof_1_2 | Init.Data.SInt.Lemmas | ∀ (a : Int16), -2 ^ 15 ≤ a.toInt → ISize.minValue.toInt ≤ -2 ^ 31 → a.toInt = ISize.minValue.toInt → False | null | false |
Nat.ToInt.of_eq | Init.Data.Int.OfNat | ∀ {a b : ℕ} {a' b' : ℤ}, ↑a = a' → ↑b = b' → a = b → a' = b' | null | true |
IsValuativeTopology.hasBasis_nhds_zero' | Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | ∀ (R : Type u_1) [inst : Ring R] [inst_1 : ValuativeRel R] [inst_2 : TopologicalSpace R] [IsValuativeTopology R],
(nhds 0).HasBasis (fun x => x ≠ 0) fun x => {x_1 | (ValuativeRel.valuation R) x_1 < x} | A variant of `hasBasis_nhds_zero` where `· ≠ 0` is unbundled. | true |
MeasureTheory.«_aux_Mathlib_MeasureTheory_OuterMeasure_AE___macroRules_MeasureTheory_term∀ᵐ_∂_,__1» | Mathlib.MeasureTheory.OuterMeasure.AE | Lean.Macro | null | false |
CategoryTheory.ShortComplex.SnakeInput.functorL₃._proof_2 | Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
{X Y Z : CategoryTheory.ShortComplex.SnakeInput C} (f : X ⟶ Y) (g : Y ⟶ Z),
(CategoryTheory.CategoryStruct.comp f g).f₃ = CategoryTheory.CategoryStruct.comp f.f₃ g.f₃ | null | false |
pinGroup.instStarMulSubtypeCliffordAlgebraMemSubmonoid._proof_1 | Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | ∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{Q : QuadraticForm R M} (x : ↥(pinGroup Q)), star (star x) = x | null | false |
_private.Mathlib.Tactic.Positivity.Core.0.Mathlib.Meta.Positivity.OrderRel.ne | Mathlib.Tactic.Positivity.Core | Mathlib.Meta.Positivity.OrderRel✝ | null | true |
Fin.val_mk | Init.Data.Fin.Lemmas | ∀ {m n : ℕ} (h : m < n), ↑⟨m, h⟩ = m | null | true |
_private.Mathlib.NumberTheory.Multiplicity.0.Nat.eight_dvd_sq_sub_one_of_odd._proof_1_1 | Mathlib.NumberTheory.Multiplicity | ∀ (m : ℕ), (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1)) | null | false |
Std.TreeMap.Raw.getElem_insertMany_list.match_1 | Std.Data.TreeMap.Raw.Lemmas | {β : Type u_1} →
(motive : Option β → Sort u_2) →
(x : Option β) → ((v : β) → x = some v → motive (some v)) → (x = none → motive none) → motive x | null | false |
MulOpposite.instAddCommSemigroup | Mathlib.Algebra.Group.Opposite | {α : Type u_1} → [AddCommSemigroup α] → AddCommSemigroup αᵐᵒᵖ | null | true |
_private.Mathlib.Analysis.InnerProductSpace.l2Space.0.HilbertBasis.dense_span._simp_1_1 | Mathlib.Analysis.InnerProductSpace.l2Space | ∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p) | null | false |
WithLp.ofLp_multisetSum | Mathlib.Analysis.Normed.Lp.WithLp | ∀ (p : ENNReal) (V : Type u_4) [inst : AddCommGroup V] (s : Multiset (WithLp p V)),
s.sum.ofLp = (Multiset.map WithLp.ofLp s).sum | null | true |
AlgebraicGeometry.Scheme.kerFunctor._proof_4 | Mathlib.AlgebraicGeometry.IdealSheaf.Basic | ∀ (Y : AlgebraicGeometry.Scheme) {X Y_1 Z : (CategoryTheory.Over Y)ᵒᵖ} (x : X ⟶ Y_1) (x : Y_1 ⟶ Z),
AlgebraicGeometry.Scheme.Hom.ker (Opposite.unop X).hom ≤ AlgebraicGeometry.Scheme.Hom.ker (Opposite.unop Z).hom | null | false |
_private.Mathlib.NumberTheory.ModularForms.Bounds.0.ModularGroup.exists_bound_of_invariant_of_isBigO._simp_1_4 | Mathlib.NumberTheory.ModularForms.Bounds | ∀ {M₀ : Type u_1} [inst : Mul M₀] [inst_1 : Zero M₀] [NoZeroDivisors M₀] {a b : M₀}, a ≠ 0 → b ≠ 0 → (a * b = 0) = False | null | false |
CategoryTheory.bifunctorComp₂₃FunctorObj_obj | Mathlib.CategoryTheory.Functor.Trifunctor | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6}
[inst : CategoryTheory.Category.{v_1, u_1} C₁] [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂]
[inst_2 : CategoryTheory.Category.{v_3, u_3} C₃] [inst_3 : CategoryTheory.Category.{v_4, u_4} C₄]
[inst_4 : CategoryTheory.Category.{v_... | null | true |
EuclideanDomain.toNontrivial | Mathlib.Algebra.EuclideanDomain.Defs | ∀ {R : Type u} [self : EuclideanDomain R], Nontrivial R | null | true |
Std.ExtTreeMap.minKeyD_le_minKeyD_erase | Std.Data.ExtTreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α},
t.erase k ≠ ∅ → ∀ {fallback : α}, (cmp (t.minKeyD fallback) ((t.erase k).minKeyD fallback)).isLE = true | null | true |
BoxIntegral.TaggedPrepartition.noConfusionType | Mathlib.Analysis.BoxIntegral.Partition.Tagged | Sort u →
{ι : Type u_1} →
{I : BoxIntegral.Box ι} →
BoxIntegral.TaggedPrepartition I →
{ι' : Type u_1} → {I' : BoxIntegral.Box ι'} → BoxIntegral.TaggedPrepartition I' → Sort u | null | false |
_private.Mathlib.NumberTheory.Chebyshev.0.Nat.lcmUpto_dvd_factorial._simp_1_2 | Mathlib.NumberTheory.Chebyshev | ∀ {m n : ℕ}, 0 < m → m ≤ n → (m ∣ n.factorial) = True | null | false |
_private.Mathlib.Order.OmegaCompletePartialOrder.0.OmegaCompletePartialOrder.Chain.pair.match_1.eq_2 | Mathlib.Order.OmegaCompletePartialOrder | ∀ (motive : ℕ → Sort u_1) (x : ℕ) (h_1 : Unit → motive 0) (h_2 : (x : ℕ) → motive x),
(x = 0 → False) →
(match x with
| 0 => h_1 ()
| x => h_2 x) =
h_2 x | null | true |
ContinuousMap.instCompactSpaceElemCoeCompacts | Mathlib.Topology.ContinuousMap.Compact | ∀ {X : Type u_4} [inst : TopologicalSpace X] (K : TopologicalSpace.Compacts X), CompactSpace ↑↑K | null | true |
Lean.Elab.formatDeprecatedArgMsg | Lean.Elab.DeprecatedArg | Lean.Elab.DeprecatedArgEntry → Lean.MessageData | Format the deprecation warning message for a deprecated argument. | true |
FloatArray.instEmptyCollection | Init.Data.FloatArray.Basic | EmptyCollection FloatArray | null | true |
_private.Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel.0.CategoryTheory.Limits.CokernelCofork.mapBifunctor._simp_2 | Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(self : CategoryTheory.Functor C D) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp (self.map f) (self.map g) = self.map (CategoryTheory.CategoryStruct.comp f g) | null | false |
_private.Batteries.Data.BinomialHeap.Basic.0.Batteries.BinomialHeap.Imp.Heap.WF.of_le.match_1_3 | Batteries.Data.BinomialHeap.Basic | ∀ {n' : ℕ} {α : Type u_1} {le : α → α → Bool}
(motive : (s : Batteries.BinomialHeap.Imp.Heap α) → Batteries.BinomialHeap.Imp.Heap.WF le n' s → Prop)
(s : Batteries.BinomialHeap.Imp.Heap α) (h : Batteries.BinomialHeap.Imp.Heap.WF le n' s),
(∀ (h : Batteries.BinomialHeap.Imp.Heap.WF le n' Batteries.BinomialHeap.Imp... | null | false |
LinearIsometryEquiv.toLinearEquiv_symm | Mathlib.Analysis.Normed.Operator.LinearIsometry | ∀ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]
{σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [inst_2 : RingHomInvPair σ₁₂ σ₂₁] [inst_3 : RingHomInvPair σ₂₁ σ₁₂]
[inst_4 : SeminormedAddCommGroup E] [inst_5 : SeminormedAddCommGroup E₂] [inst_6 : Module R E] [inst_7 : Mo... | null | true |
ProbabilityTheory.defaultRatCDF | Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | ℚ → ℝ | A function with the property `IsMeasurableRatCDF`.
Used in a piecewise construction to convert a function which only satisfies the properties
defining `IsMeasurableRatCDF` on some set into a true `IsMeasurableRatCDF`. | true |
Real.rpowIntegrand₀₁_eqOn_pow_div | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | ∀ {p x : ℝ},
p ∈ Set.Ioo 0 1 →
0 ≤ x → Set.EqOn (fun x_1 => p.rpowIntegrand₀₁ x_1 x) (fun t => t ^ (p - 1) * x / (t + x)) (Set.Ioi 0) | null | true |
Nat.toArray_ric_eq_cons | Init.Data.Range.Polymorphic.NatLemmas | ∀ {n : ℕ}, (*...=n).toArray = #[0] ++ (1...=n).toArray | null | true |
_private.Mathlib.Algebra.Module.LocalizedModule.Basic.0.IsLocalizedModule.instLiftOfLE._simp_3 | Mathlib.Algebra.Module.LocalizedModule.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] {S : Submonoid R} {M : Type u_2} {M' : Type u_3} [inst_1 : AddCommMonoid M]
[inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] (f : M →ₗ[R] M')
[inst_5 : IsLocalizedModule S f] {R₀ : Type u_6} [inst_6 : SMul R₀ R] [inst_7 : SMul R₀ M] [inst_8 : SMul R₀ ... | null | false |
SimpleGraph.Subgraph.IsPerfectMatching.exists_of_isClique_supp | Mathlib.Combinatorics.SimpleGraph.Tutte | ∀ {V : Type u_1} {G : SimpleGraph V} [Finite V],
Even (Nat.card V) →
¬G.IsTutteViolator G.universalVerts →
(∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp) →
∃ M, M.IsPerfectMatching | If the universal vertices of a graph `G` decompose `G` into cliques such that the Tutte isn't
violated, then `G` has a perfect matching. | true |
List.attach_filterMap | Init.Data.List.Attach | ∀ {α : Type u_1} {β : Type u_2} {l : List α} {f : α → Option β},
(List.filterMap f l).attach =
List.filterMap
(fun x =>
match x with
| ⟨x, h⟩ => (f x).pbind fun b m => some ⟨b, ⋯⟩)
l.attach | null | true |
UniformContinuousConstVAdd.uniformContinuous_const_vadd | Mathlib.Topology.Algebra.UniformMulAction | ∀ {M : Type v} {X : Type x} {inst : UniformSpace X} {inst_1 : VAdd M X} [self : UniformContinuousConstVAdd M X] (c : M),
UniformContinuous fun x => c +ᵥ x | null | true |
_private.Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital.0._auto_271 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | Lean.Syntax | null | false |
SimpleGraph.ediam_eq_top | Mathlib.Combinatorics.SimpleGraph.Diam | ∀ {α : Type u_1} {G : SimpleGraph α}, G.ediam = ⊤ ↔ ∀ b < ⊤, ∃ u v, b < G.edist u v | null | true |
List.Forall._unsafe_rec | Mathlib.Data.List.Defs | {α : Type u_1} → (α → Prop) → List α → Prop | null | false |
Std.DHashMap.Const.get?_empty | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {a : α}, Std.DHashMap.Const.get? ∅ a = none | null | true |
NormedAddGroup.toBornology._inherited_default | Mathlib.Analysis.Normed.Group.Defs | {E : Type u_8} →
(dist : E → E → ℝ) →
(∀ (x y : E), dist x y = dist y x) → (∀ (x y z : E), dist x z ≤ dist x y + dist y z) → Bornology E | null | false |
WithTop.mul_eq_top_iff | Mathlib.Algebra.Order.Ring.WithTop | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : MulZeroClass α] {a b : WithTop α},
a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0 | null | true |
CategoryTheory.Presieve.singleton.rec | Mathlib.CategoryTheory.Sites.Sieves | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{X Y : C} →
{f : Y ⟶ X} →
{motive : ⦃Y_1 : C⦄ → (a : Y_1 ⟶ X) → CategoryTheory.Presieve.singleton f a → Sort u} →
motive f ⋯ → ⦃Y_1 : C⦄ → {a : Y_1 ⟶ X} → (t : CategoryTheory.Presieve.singleton f a) → motive a t | null | false |
_private.Std.Data.DTreeMap.Internal.Operations.0.Std.DTreeMap.Internal.Impl.minView.match_3.splitter | Std.Data.DTreeMap.Internal.Operations | {α : Type u_1} →
{β : α → Type u_2} →
(r : Std.DTreeMap.Internal.Impl α β) →
(motive :
(l : Std.DTreeMap.Internal.Impl α β) →
l.Balanced → Std.DTreeMap.Internal.Impl.BalancedAtRoot l.size r.size → Sort u_3) →
(l : Std.DTreeMap.Internal.Impl α β) →
(hl : l.Balanced) →
... | null | true |
_private.Lean.Elab.ConfigEval.DeriveEvalTerm.0.Lean.Elab.ConfigEval.EvalTerm.resolveDottedAtomicNameForConstNamespace | Lean.Elab.ConfigEval.DeriveEvalTerm | Lean.Name → Lean.Ident → Lean.Elab.TermElabM String | Resolves `id` as if it were a dotted identifier for namespace `c`,
and returns `s` if the resolved name is of the form `Name.str c s`.
| true |
List.modify_eq_nil_iff | Init.Data.List.Nat.Modify | ∀ {α : Type u_1} {f : α → α} {i : ℕ} {l : List α}, l.modify i f = [] ↔ l = [] | null | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter.0.SimplexCategory.δ_σ₀Iter._proof_1_10 | Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | ∀ {n : ℕ} (j : ℕ) (k : Fin ({ len := n }.len + 1)), ↑k < j → ↑k.succ < j + 1 | null | false |
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