name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Finset.noncommProd_mulSingle | Mathlib.Data.Finset.NoncommProd | ∀ {ι : Type u_2} {M : ι → Type u_6} [inst : (i : ι) → Monoid (M i)] [inst_1 : Fintype ι] [inst_2 : DecidableEq ι]
(x : (i : ι) → M i), Finset.univ.noncommProd (fun i => Pi.mulSingle i (x i)) ⋯ = x | null | true |
instLinearOrderPNat._proof_24 | Mathlib.Data.PNat.Defs | autoParam (∀ (a b : ℕ+), compare a b = compareOfLessAndEq a b) LinearOrder.compare_eq_compareOfLessAndEq._autoParam | null | false |
Topology.IsQuotientMap.trivializationOfSMulDisjoint.match_7 | Mathlib.Topology.Covering.Quotient | ∀ {E : Type u_2} {X : Type u_1} {f : E → X} (U : Set E) (e : E) (motive : f e ∈ f '' U → Prop) (x : f e ∈ f '' U),
(∀ (e' : E) (he' : e' ∈ U) (hfe : f e' = f e), motive ⋯) → motive x | null | false |
CategoryTheory.pi.coconeOfCoconeCompEval | Mathlib.CategoryTheory.Limits.Pi | {I : Type v₁} →
{C : I → Type u₁} →
[inst : (i : I) → CategoryTheory.Category.{v₁, u₁} (C i)] →
{J : Type v₁} →
[inst_1 : CategoryTheory.SmallCategory J] →
{F : CategoryTheory.Functor J ((i : I) → C i)} →
((i : I) → CategoryTheory.Limits.Cocone (F.comp (CategoryTheory.Pi.eval C... | Given a family of cocones over the `F ⋙ Pi.eval C i`,
we can assemble these together as a `Cocone F`.
| true |
CategoryTheory.ShortComplex.ShortExact.fIsKernel._proof_1 | Mathlib.Algebra.Homology.ShortComplex.ShortExact | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
{S : CategoryTheory.ShortComplex C}, S.ShortExact → CategoryTheory.Mono S.f | null | false |
Mathlib.Linter.OverlappingInstances.linter.overlappingInstances | Mathlib.Tactic.Linter.OverlappingInstances | Lean.Option Bool | Lints against overlaps between instances in the local contexts of declarations. | true |
Nat.getElem?_toArray_rio | Init.Data.Range.Polymorphic.NatLemmas | ∀ {n i : ℕ}, (*...n).toArray[i]? = if i < n then some i else none | null | true |
Batteries.PairingHeapImp.Heap.foldTreeM._f | Batteries.Data.PairingHeap | {m : Type u_1 → Type u_2} →
{β : Type u_1} →
{α : Type u_3} →
[Monad m] →
β → (α → β → β → m β) → (x : Batteries.PairingHeapImp.Heap α) → Batteries.PairingHeapImp.Heap.below x → m β | null | false |
ENat.card_prod | Mathlib.SetTheory.Cardinal.Finite | ∀ (α : Type u_3) (β : Type u_4), ENat.card (α × β) = ENat.card α * ENat.card β | null | true |
ContIntertwiningMap.mapInvariants_apply | Mathlib.RepresentationTheory.Continuous.Basic | ∀ {R : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [inst : Monoid G] [inst_1 : Ring R]
[inst_2 : AddCommGroup V] [inst_3 : TopologicalSpace V] [inst_4 : IsTopologicalAddGroup V] [inst_5 : Module R V]
[inst_6 : AddCommGroup W] [inst_7 : TopologicalSpace W] [inst_8 : IsTopologicalAddGroup W] [inst_9 : Modu... | null | true |
_private.Mathlib.Data.Set.Image.0.Set.disjoint_image_left._simp_1_1 | Mathlib.Data.Set.Image | ∀ {α : Type u} {s t : Set α}, Disjoint s t = (s ∩ t = ∅) | null | false |
lp.mapCLM | Mathlib.Analysis.Normed.Lp.lpHolder | {α : Type u_1} →
{𝕜 : Type u_2} →
{E : α → Type u_3} →
{F : α → Type u_4} →
[inst : NontriviallyNormedField 𝕜] →
[inst_1 : (i : α) → NormedAddCommGroup (E i)] →
[inst_2 : (i : α) → NormedSpace 𝕜 (E i)] →
[inst_3 : (i : α) → NormedAddCommGroup (F i)] →
... | A uniformly bounded family of continuous linear maps, as a continuous linear map
on the `lp` space. | true |
_private.Mathlib.Tactic.GCongr.Core.0.Lean.MVarId.gcongr.match_1 | Mathlib.Tactic.GCongr.Core | (motive : Option Bool → Sort u_1) →
(mdataLhs? : Option Bool) →
((mdataLhs : Bool) → motive (some mdataLhs)) → ((x : Option Bool) → motive x) → motive mdataLhs? | null | false |
ModuleCat.forget₂AddCommGroup_reflectsLimitOfSize | Mathlib.Algebra.Category.ModuleCat.Limits | ∀ {R : Type u} [inst : Ring R],
CategoryTheory.Limits.ReflectsLimitsOfSize.{t, v, w, w, max u (w + 1), w + 1}
(CategoryTheory.forget₂ (ModuleCat R) AddCommGrpCat) | null | true |
ENNReal.instCommSemiring._proof_16 | Mathlib.Basic.ENNReal.Basic | ∀ (a b : ENNReal), a * b = b * a | null | false |
smul_left_cancel_iff._simp_2 | Mathlib.Algebra.Group.Action.Basic | ∀ {α : Type u_5} {β : Type u_6} [inst : Group α] [inst_1 : MulAction α β] (g : α) {x y : β}, (g • x = g • y) = (x = y) | null | false |
Matrix.vecCons_inj._simp_1 | Mathlib.Data.Fin.VecNotation | ∀ {α : Type u} {n : ℕ} {x y : α} {u v : Fin n → α}, (Matrix.vecCons x u = Matrix.vecCons y v) = (x = y ∧ u = v) | null | false |
LinearMap.ofAEval._proof_6 | Mathlib.Algebra.Polynomial.Module.AEval | ∀ {R : Type u_3} {A : Type u_4} {M : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] (a : A)
[inst_2 : Algebra R A] [inst_3 : AddCommMonoid M] [inst_4 : Module A M] [inst_5 : Module R M]
[inst_6 : IsScalarTower R A M] {N : Type u_1} [inst_7 : AddCommMonoid N] [inst_8 : Module R N]
[inst_9 : Module (Polyno... | null | false |
Rat.sub | Init.Data.Rat.Basic | ℚ → ℚ → ℚ | Subtraction of rational numbers. (This definition is `@[irreducible]` because you don't want to
unfold it. Use `Rat.sub_def` instead.)
| true |
HopfAlgCat.MonoidalCategory.inducingFunctorData._proof_6 | Mathlib.Algebra.Category.HopfAlgCat.Monoidal | ∀ (R : Type u_1) [inst : CommRing R] (x : HopfAlgCat R),
(TensorProduct.mk R x.carrier (CategoryTheory.MonoidalCategoryStruct.tensorUnit (HopfAlgCat R)).carrier).compr₂ₛₗ
((fun x_1 => ↑x_1)
((CategoryTheory.forget₂ (HopfAlgCat R) (BialgCat R)).map
(CategoryTheory.MonoidalCategoryStruct.right... | null | false |
CategoryTheory.Limits.natTransIntoForgetCompFiberwiseColimit._proof_2 | Mathlib.CategoryTheory.Limits.Shapes.Grothendieck | ∀ {C : Type u_5} [inst : CategoryTheory.Category.{u_3, u_5} C] {F : CategoryTheory.Functor C CategoryTheory.Cat}
{H : Type u_2} [inst_1 : CategoryTheory.Category.{u_1, u_2} H]
(G : CategoryTheory.Functor (CategoryTheory.Grothendieck F) H)
[inst_2 :
∀ {X Y : C} (f : X ⟶ Y),
CategoryTheory.Limits.HasColim... | null | false |
ContinuousLinearMap.lTensor_smul | Mathlib.Analysis.InnerProductSpace.TensorProduct | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} (G : Type u_4) [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F]
[inst_5 : NormedAddCommGroup G] [inst_6 : InnerProductSpace 𝕜 G] (r : 𝕜) (f : E →L[𝕜] F),
Continu... | null | true |
CategoryTheory.BraidedCategory.Hexagon.functor₁₃₂ | Mathlib.CategoryTheory.Monoidal.Braided.Multifunctor | (C : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[CategoryTheory.MonoidalCategory C] →
CategoryTheory.Functor C (CategoryTheory.Functor C (CategoryTheory.Functor C C)) | The trifunctor `X₁ X₂ X₃ ↦ (X₁ ⊗ X₃) ⊗ X₂` | true |
jacobiTheta₂.eq_1 | Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | ∀ (z τ : ℂ), jacobiTheta₂ z τ = ∑' (n : ℤ), jacobiTheta₂_term n z τ | null | true |
PeriodPair.mem_lattice | Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | ∀ {L : PeriodPair} {x : ℂ}, x ∈ L.lattice ↔ ∃ m n, ↑m * L.ω₁ + ↑n * L.ω₂ = x | null | true |
Std.Do._aux_Std_Do_PostCond___unexpand_Std_Do_PostCond_entails_1 | Std.Do.PostCond | Lean.PrettyPrinter.Unexpander | null | false |
DFinsupp.Lex.acc_zero | Mathlib.Data.DFinsupp.WellFounded | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Zero (α i)] {r : ι → ι → Prop} {s : (i : ι) → α i → α i → Prop},
(∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0) → Acc (DFinsupp.Lex r s) 0 | null | true |
Prod.instTorsor.eq_1 | Mathlib.Algebra.Torsor.Basic | ∀ {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [inst : Group G] [inst_1 : Group G']
[inst_2 : Torsor G P] [inst_3 : Torsor G' P'],
Prod.instTorsor =
{ smul := fun v p => (v.1 • p.1, v.2 • p.2), mul_smul := ⋯, one_smul := ⋯,
sdiv := fun p₁ p₂ => (p₁.1 /ₛ p₂.1, p₁.2 /ₛ p₂.2), nonempty := ⋯,... | null | true |
Algebra.FormallySmooth.liftOfSurjective.eq_1 | Mathlib.RingTheory.Smooth.Basic | ∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {B : Type u_1}
[inst_3 : CommRing B] [inst_4 : Algebra R B] {C : Type u_4} [inst_5 : CommRing C] [inst_6 : Algebra R C]
[inst_7 : Algebra.FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C) (hg : Function.Surjective ⇑g)
(... | null | true |
Lean.Omega.IntList.dvd_gcd | Init.Omega.IntList | ∀ (xs : Lean.Omega.IntList) (c : ℕ), (∀ {a : ℤ}, a ∈ xs → ↑c ∣ a) → c ∣ xs.gcd | null | true |
_private.Mathlib.MeasureTheory.Group.FundamentalDomain.0.MeasureTheory.termπ_3 | Mathlib.MeasureTheory.Group.FundamentalDomain | Lean.ParserDescr | null | true |
_private.Mathlib.Data.EReal.BigOperators.0.EReal.sum_mul_of_nonneg._proof_1_1 | Mathlib.Data.EReal.BigOperators | ∀ {ι : Type u_1} {f : ι → EReal} (i : ι) (s : Finset ι), (∀ i_1 ∈ insert i s, 0 ≤ f i_1) → ∀ i_1 ∈ s, 0 ≤ f i_1 | null | false |
_private.Std.Data.DTreeMap.Internal.Balancing.0.Std.DTreeMap.Internal.Impl.balance!_eq_balanceₘ._proof_1_22 | Std.Data.DTreeMap.Internal.Balancing | ∀ {α : Type u_1} {β : α → Type u_2} (ls size : ℕ) (l r : Std.DTreeMap.Internal.Impl α β),
(l.Balanced ∧
r.Balanced ∧ (l.size + r.size ≤ 1 ∨ l.size ≤ 3 * r.size ∧ r.size ≤ 3 * l.size) ∧ size = l.size + 1 + r.size) ∧
(size + 0 ≤ 1 ∨ size ≤ 3 * 0 ∧ 0 ≤ 3 * size) ∧ ls = size + 1 + 0 →
¬l.size + 1 + r.si... | null | false |
_private.Lean.Elab.Tactic.Conv.Basic.0.Lean.Elab.Tactic.Conv.evalClear._regBuiltin.Lean.Elab.Tactic.Conv.evalClear_1 | Lean.Elab.Tactic.Conv.Basic | IO Unit | null | false |
NNReal.instLinearOrderedCommGroupWithZero._proof_1 | Mathlib.Basic.NNReal.Defs | ∀ (h : NNReal), 0 ≤ ↑h | null | false |
RingHom.Etale.toAlgebra | Mathlib.RingTheory.RingHom.Etale | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] {f : R →+* S}, f.Etale → Algebra.Etale R S | Helper lemma for the `algebraize` tactic | true |
Finset.prod_insert_of_eq_one_if_notMem | Mathlib.Algebra.BigOperators.Group.Finset.Basic | ∀ {ι : Type u_1} {M : Type u_4} {s : Finset ι} {a : ι} [inst : CommMonoid M] {f : ι → M} [inst_1 : DecidableEq ι],
(a ∉ s → f a = 1) → ∏ x ∈ insert a s, f x = ∏ x ∈ s, f x | The product of `f` over `insert a s` is the same as
the product over `s`, as long as `a` is in `s` or `f a = 1`. | true |
BooleanSubalgebra.copy._proof_3 | Mathlib.Order.BooleanSubalgebra | ∀ {α : Type u_1} [inst : BooleanAlgebra α] (L : BooleanSubalgebra α) (s : Set α) (hs : s = ↑L), ⊥ ∈ (L.copy s ⋯).carrier | null | false |
codisjoint_bot._simp_1 | Mathlib.Order.Disjoint | ∀ {α : Type u_1} [inst : PartialOrder α] [inst_1 : BoundedOrder α] {a : α}, Codisjoint a ⊥ = (a = ⊤) | null | false |
RCLike.inv_re | Mathlib.Analysis.RCLike.Basic | ∀ {K : Type u_1} [inst : RCLike K] (z : K), RCLike.re z⁻¹ = RCLike.re z / RCLike.normSq z | null | true |
MeasureTheory.Measure.AbsolutelyContinuous.add_left_iff | Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | ∀ {α : Type u_1} {mα : MeasurableSpace α} {μ₁ μ₂ ν : MeasureTheory.Measure α},
(μ₁ + μ₂).AbsolutelyContinuous ν ↔ μ₁.AbsolutelyContinuous ν ∧ μ₂.AbsolutelyContinuous ν | null | true |
Matrix.comp_diagonal_diagonal | Mathlib.Data.Matrix.Composition | ∀ {I : Type u_1} {J : Type u_2} {R : Type u_5} [inst : DecidableEq I] [inst_1 : DecidableEq J] [inst_2 : Zero R]
(d : I → J → R),
(Matrix.comp I I J J R) (Matrix.diagonal fun i => Matrix.diagonal fun j => d i j) =
Matrix.diagonal fun ij => d ij.1 ij.2 | null | true |
iteratedDerivWithin_fun_id | Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : ℕ} {x : 𝕜} {s : Set 𝕜},
x ∈ s → UniqueDiffOn 𝕜 s → iteratedDerivWithin n (fun x => x) s x = if n = 0 then x else if n = 1 then 1 else 0 | Eta-expanded form of `iteratedDerivWithin_id` | true |
MeasurableSpace.CountablySeparated.rec | Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {α : Type u_3} →
[inst : MeasurableSpace α] →
{motive : MeasurableSpace.CountablySeparated α → Sort u} →
((countably_separated : HasCountableSeparatingOn α MeasurableSet Set.univ) → motive ⋯) →
(t : MeasurableSpace.CountablySeparated α) → motive t | null | false |
PresheafOfModules.colimitPresheafOfModules_map | Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {J : Type u₂}
[inst_1 : CategoryTheory.Category.{v₂, u₂} J] (F : CategoryTheory.Functor J (PresheafOfModules R))
[inst_2 :
∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),
CategoryTheory.Limits.PreservesColimit (F.comp (Preshea... | null | true |
CategoryTheory.ProjectiveResolution.extAddEquivCohomologyClass._proof_1 | Mathlib.CategoryTheory.Abelian.Projective.Ext | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : CategoryTheory.HasExt C] {X Y : C} (R : CategoryTheory.ProjectiveResolution X) {n : ℕ}
(x y : CochainComplex.HomComplex.CohomologyClass R.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) ↑n),
R.ex... | null | false |
CategoryTheory.Functor.Elements.coconeπOpCompShrinkYonedaObj_pt | Mathlib.CategoryTheory.Limits.Presheaf | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.LocallySmall.{w, v₁, u₁} C]
(F : CategoryTheory.Functor C (Type w)) (X : C),
(CategoryTheory.Functor.Elements.coconeπOpCompShrinkYonedaObj F X).pt = F.obj X | null | true |
_private.Std.Data.Iterators.Lemmas.Consumers.Monadic.Set.0.Std.IterM.toExtTreeSet_eq_ofList._simp_1_3 | Std.Data.Iterators.Lemmas.Consumers.Monadic.Set | ∀ {α : Type u} {cmp : α → α → Ordering} [inst : Std.TransCmp cmp] {l : List α},
List.foldl (fun acc a => acc.insert a) ∅ l = Std.ExtTreeSet.ofList l cmp | null | false |
GetElem.getElem.congr_simp | Init.GetElem | ∀ {coll : Type u} {idx : Type v} {elem : Type w} {valid : coll → idx → Prop} [self : GetElem coll idx elem valid]
(xs xs_1 : coll) (e_xs : xs = xs_1) (i i_1 : idx) (e_i : i = i_1) (h : valid xs i), xs[i] = xs_1[i_1] | null | true |
Aesop.Frontend.evalSaturate? | Aesop.Frontend.Saturate | Lean.Elab.Tactic.Tactic | null | true |
Lean.Elab.CommandContextInfo.fileMap | Lean.Elab.InfoTree.Types | Lean.Elab.CommandContextInfo → Lean.FileMap | null | true |
zero_le_four | Mathlib.Algebra.Order.Monoid.NatCast | ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_1 : Preorder α] [ZeroLEOneClass α] [AddLeftMono α], 0 ≤ 4 | null | true |
Aesop.NormM.Context.mk | Aesop.Search.Expansion.Norm | Aesop.Options' → Aesop.LocalRuleSet → Aesop.NormSimpContext → Aesop.NormM.Context | null | true |
_private.Mathlib.RingTheory.Localization.Module.0.IsLocalizedModule.linearIndependent_lift._simp_1_1 | Mathlib.RingTheory.Localization.Module | ∀ {M' : Type u_1} {α : Type u_2} [inst : MulOneClass M'] [inst_1 : SMul M' α] {S : Submonoid M'} (g : ↥S) (a : α),
↑g • a = g • a | null | false |
_private.Lean.PrettyPrinter.Parenthesizer.0.Lean.PrettyPrinter.categoryParenthesizerAttribute._regBuiltin.Lean.PrettyPrinter.categoryParenthesizerAttribute.declRange_3 | Lean.PrettyPrinter.Parenthesizer | IO Unit | null | false |
_private.Lean.Elab.Binders.0.Lean.Elab.Term.expandFun._regBuiltin.Lean.Elab.Term.expandFun_1 | Lean.Elab.Binders | IO Unit | null | false |
ENNReal.hasSum_coe._simp_1 | Mathlib.Topology.Algebra.InfiniteSum.ENNReal | ∀ {α : Type u_1} {f : α → NNReal} {r : NNReal}, HasSum (fun a => ↑(f a)) ↑r = HasSum f r | null | false |
Nat.mod_pow_succ | Init.Data.Nat.Mod | ∀ {x b k : ℕ}, x % b ^ (k + 1) = x % b ^ k + b ^ k * (x / b ^ k % b) | null | true |
FractionalIdeal.map_div | Mathlib.RingTheory.FractionalIdeal.Operations | ∀ {R₁ : Type u_3} [inst : CommRing R₁] {K : Type u_4} [inst_1 : Field K] [inst_2 : Algebra R₁ K]
[inst_3 : IsFractionRing R₁ K] [inst_4 : IsDomain R₁] {K' : Type u_5} [inst_5 : Field K'] [inst_6 : Algebra R₁ K']
[inst_7 : IsFractionRing R₁ K'] (I J : FractionalIdeal (nonZeroDivisors R₁) K) (h : K ≃ₐ[R₁] K'),
Frac... | null | true |
Int8.ofInt_neg | Init.Data.SInt.Lemmas | ∀ (a : ℤ), Int8.ofInt (-a) = -Int8.ofInt a | null | true |
ProbabilityTheory.integral_tilted_mul_eq_mgf | Mathlib.Probability.Moments.Tilted | ∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ} {t : ℝ} {E : Type u_2}
[inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] (g : Ω → E),
(∫ (ω : Ω), g ω ∂μ.tilted fun x => t * X x) = ∫ (ω : Ω), (Real.exp (t * X ω) / ProbabilityTheory.mgf X μ t) • g ω ∂μ | null | true |
ImportGraph.Shake.ImportNeedsKind.privNoMeta._proof_2 | ImportGraph.Shake.DeclNeeds | (!ImportGraph.Shake.NeedsKind.priv.isMeta || false) = true | null | false |
MeasureTheory.SimpleFunc.approxOn.congr_simp | Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : PseudoEMetricSpace α]
[inst_2 : OpensMeasurableSpace α] [inst_3 : MeasurableSpace β] (f f_1 : β → α) (e_f : f = f_1) (hf : Measurable f)
(s s_1 : Set α) (e_s : s = s_1) (y₀ y₀_1 : α) (e_y₀ : y₀ = y₀_1) (h₀ : y₀ ∈ s)
[inst_4 : TopologicalSpace.Se... | null | true |
LieSubmodule.restr_toSubmodule | Mathlib.Algebra.Lie.Submodule | ∀ {R : Type u} {L : Type v} {M : Type w} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : AddCommGroup M]
[inst_3 : Module R M] [inst_4 : LieRingModule L M] [inst_5 : LieAlgebra R L] (N : LieSubmodule R L M)
(H : LieSubalgebra R L), ↑(N.restr H) = ↑N | null | true |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.minKey?_alterKey_eq_self._simp_1_3 | Std.Data.Internal.List.Associative | ∀ {α : Type u_1} {o : Option α}, (o.isNone = true) = (o = none) | null | false |
Turing.PartrecToTM2.unrev | Mathlib.Computability.TuringMachine.ToPartrec | Turing.PartrecToTM2.Λ' → Turing.PartrecToTM2.Λ' | Move everything from the `rev` stack to the `main` stack (reversed). | true |
CategoryTheory.Limits.Sigma.isoColimit.eq_1 | Mathlib.CategoryTheory.Limits.Shapes.Products | ∀ {α : Type w₂} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
(X : CategoryTheory.Functor (CategoryTheory.Discrete α) C)
[inst_1 : CategoryTheory.Limits.HasCoproduct fun j => X.obj { as := j }]
[inst_2 : CategoryTheory.Limits.HasColimit X],
CategoryTheory.Limits.Sigma.isoColimit X =
(CategoryTheory... | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction.0.SSet.Truncated.liftOfStrictSegal_app_1._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | 1 ≤ 2 | null | false |
CategoryTheory.Abelian.LeftResolution.chainComplexMap_f_succ_succ._proof_1 | Mathlib.Algebra.Homology.LeftResolution.Basic | ∀ {A : Type u_2} {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C]
[inst_1 : CategoryTheory.Category.{u_1, u_2} A] {ι : CategoryTheory.Functor C A}
(Λ : CategoryTheory.Abelian.LeftResolution ι) {X Y : A} (f : X ⟶ Y) [inst_2 : ι.Full] [inst_3 : ι.Faithful]
[inst_4 : CategoryTheory.Limits.HasZeroMorphism... | null | false |
WeierstrassCurve.Projective.Point.toAffine | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {F : Type u} → [inst : Field F] → (W : WeierstrassCurve.Projective F) → (Fin 3 → F) → W.toAffine.Point | The natural map from a nonsingular projective point representative on a Weierstrass curve to its
corresponding nonsingular point in affine coordinates. | true |
_private.Mathlib.SetTheory.Ordinal.Notation.0.ONote.nf_repr_split'._simp_1_3 | Mathlib.SetTheory.Ordinal.Notation | ONote.NF 0 = True | null | false |
MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge | Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α)
[μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → NNReal),
Measurable f →
∀ {ε : ENNReal},
ε ≠ 0 → ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ ... | Given a measurable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
Formulation in terms of `lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. | true |
Continuous.cfcₙ_fun._auto_3 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | Lean.Syntax | null | false |
Topology.IsLocallyConstructible.isConstructible_of_subset_of_isCompact | Mathlib.Topology.Constructible | ∀ {X : Type u_2} [inst : TopologicalSpace X] {s t : Set X} [PrespectralSpace X] [QuasiSeparatedSpace X],
Topology.IsLocallyConstructible s → s ⊆ t → IsCompact t → Topology.IsConstructible s | null | true |
List.splitLengths_nil | Mathlib.Data.List.SplitLengths | ∀ {α : Type u_1} (l : List α), [].splitLengths l = [] | null | true |
Ultrafilter.mk.inj | Mathlib.Order.Filter.Ultrafilter.Defs | ∀ {α : Type u_2} {toFilter : Filter α} {neBot' : toFilter.NeBot}
{le_of_le : ∀ (g : Filter α), g.NeBot → g ≤ toFilter → toFilter ≤ g} {toFilter_1 : Filter α}
{neBot'_1 : toFilter_1.NeBot} {le_of_le_1 : ∀ (g : Filter α), g.NeBot → g ≤ toFilter_1 → toFilter_1 ≤ g},
{ toFilter := toFilter, neBot' := neBot', le_of_le... | null | true |
Valued.toUniformSpace | Mathlib.Topology.Algebra.Valued.ValuationTopology | {R : Type u} →
{inst : Ring R} →
{Γ₀ : outParam (Type v)} → {inst_1 : LinearOrderedCommGroupWithZero Γ₀} → [self : Valued R Γ₀] → UniformSpace R | null | true |
Nat.Partition.UniquePartitionZero._proof_2 | Mathlib.Combinatorics.Enumerative.Partition.Basic | ∀ (x : Nat.Partition 0), x = default | null | false |
Set.notMem_Ioc_of_le | Mathlib.Order.Interval.Set.Basic | ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, c ≤ a → c ∉ Set.Ioc a b | null | true |
Order.Coframe.recOn | Mathlib.Order.CompleteBooleanAlgebra | {α : Type u_1} →
{motive : Order.Coframe α → Sort u} →
(t : Order.Coframe α) →
([toCompleteLattice : CompleteLattice α] →
[toSDiff : SDiff α] →
(sdiff_le_iff : ∀ (a b c : α), a \ b ≤ c ↔ a ≤ b ⊔ c) →
[toHNot : HNot α] →
(top_sdiff : ∀ (a : α), ⊤ \ a = ￢a) ... | null | false |
CategoryTheory.SingleObj.inv_as_inv | Mathlib.CategoryTheory.SingleObj | ∀ (G : Type u) [inst : Group G] {x y : CategoryTheory.SingleObj G} (f : x ⟶ y), CategoryTheory.inv f = f⁻¹ | null | true |
iSupIndep.subtype_ne_bot_le_finrank | Mathlib.LinearAlgebra.Dimension.Finite | ∀ {ι : Type w} {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsDomain R]
[Module.IsTorsionFree R M] [Module.Finite R M] [StrongRankCondition R] {p : ι → Submodule R M},
iSupIndep p → ∀ [inst_7 : Fintype { i // p i ≠ ⊥ }], Fintype.card { i // p i ≠ ⊥ } ≤ Module.finrank R ... | If `p` is an independent family of submodules of an `R`-finite module `M`, then the
number of nontrivial subspaces in the family `p` is bounded above by the dimension of `M`.
Note that the `Fintype` hypothesis required here can be provided by
`iSupIndep.fintypeNeBotOfFiniteDimensional`. | true |
LieDerivation.SMulBracketCommClass.rec | Mathlib.Algebra.Lie.Derivation.Basic | {S : Type u_4} →
{L : Type u_5} →
{α : Type u_6} →
[inst : SMul S α] →
[inst_1 : LieRing L] →
[inst_2 : AddCommGroup α] →
[inst_3 : LieRingModule L α] →
{motive : LieDerivation.SMulBracketCommClass S L α → Sort u} →
((smul_bracket_comm : ∀ (s : S) ... | null | false |
Lean.Grind.IntModule.OfNatModule.instLTQOfOrderedAdd | Init.Grind.Module.Envelope | {α : Type u} →
[inst : Lean.Grind.NatModule α] →
[inst_1 : LE α] →
[inst_2 : Std.IsPreorder α] → [Lean.Grind.OrderedAdd α] → LT (Lean.Grind.IntModule.OfNatModule.Q α) | null | true |
_private.Mathlib.FieldTheory.Extension.0.IntermediateField.Lifts.nonempty_algHom_of_exist_lifts_finset._simp_1_5 | Mathlib.FieldTheory.Extension | ∀ {α : Type u} [inst : SemilatticeSup α] {a b : α}, (a < a ⊔ b) = ¬b ≤ a | null | false |
HahnSeries.instDistribMulAction._proof_4 | Mathlib.RingTheory.HahnSeries.Addition | ∀ {Γ : Type u_1} {R : Type u_3} [inst : PartialOrder Γ] {V : Type u_2} [inst_1 : Monoid R] [inst_2 : AddMonoid V]
[inst_3 : DistribMulAction R V] (x : R) (x_1 x_2 : HahnSeries Γ V), x • (x_1 + x_2) = x • x_1 + x • x_2 | null | false |
PerfectClosure.instCommRing._proof_8 | Mathlib.FieldTheory.PerfectClosure | ∀ (K : Type u_1) [inst : CommRing K] (p : ℕ) [inst_1 : Fact (Nat.Prime p)] [inst_2 : CharP K p]
(a b c : PerfectClosure K p), a * b * c = a * (b * c) | null | false |
CategoryTheory.ShortComplex.RightHomologyData.ofHasKernel_p | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : CategoryTheory.Limits.HasKernel S.g] (hf : S.f = 0),
(CategoryTheory.ShortComplex.RightHomologyData.ofHasKernel S hf).p = CategoryTheory.CategoryStruct.id ... | null | true |
_private.Mathlib.RingTheory.Smooth.Kaehler.0.derivationOfSectionOfKerSqZero._simp_6 | Mathlib.RingTheory.Smooth.Kaehler | ∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)
{x : M} (h : x ∈ p), (⟨x, h⟩ = 0) = (x = 0) | null | false |
_private.Lean.Elab.MutualDef.0.Lean.Elab.Term.ExprWithHoles.mk.inj | Lean.Elab.MutualDef | ∀ {ref : Lean.Syntax} {expr : Lean.Expr} {ref_1 : Lean.Syntax} {expr_1 : Lean.Expr},
{ ref := ref, expr := expr } = { ref := ref_1, expr := expr_1 } → ref = ref_1 ∧ expr = expr_1 | null | true |
CategoryTheory.ObjectProperty.IsClosedUnderQuotients.mk | Mathlib.CategoryTheory.ObjectProperty.EpiMono | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {P : CategoryTheory.ObjectProperty C},
(∀ {X Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f], P X → P Y) → P.IsClosedUnderQuotients | null | true |
SimpleGraph.Subgraph.symm | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u} {G : SimpleGraph V} (self : G.Subgraph), Std.Symm self.Adj | null | true |
Complex.instNormedField._proof_1 | Mathlib.Analysis.Complex.Basic | ∀ (x x_1 : ℂ), dist x x_1 = dist x x_1 | null | false |
_private.Init.Data.Nat.Lemmas.0.Nat.anyLTTR_loop_eq_true._proof_1_5 | Init.Data.Nat.Lemmas | ∀ {n : ℕ} (m : ℕ), ¬n - (m + 1) ≤ n - (m + 1) → False | null | false |
Dyadic.instHShiftRightInt | Init.Data.Dyadic.Basic | HShiftRight Dyadic ℤ Dyadic | null | true |
Mathlib.Tactic.Choose.choose1WithInfo | Mathlib.Tactic.Choose | Lean.MVarId →
Bool →
Option Lean.Expr →
Mathlib.Tactic.Choose.ChooseArg → Lean.Elab.TermElabM (Mathlib.Tactic.Choose.ElimStatus × Lean.MVarId) | A wrapper around `choose1` that parses identifiers, adds variable info to new variables,
and optionally checks the type annotation. | true |
Std.DHashMap.containsThenInsertIfNew | Std.Data.DHashMap.Basic | {α : Type u} →
{β : α → Type v} → {x : BEq α} → {x_1 : Hashable α} → Std.DHashMap α β → (a : α) → β a → Bool × Std.DHashMap α β | Checks whether a key is present in a map and inserts a value for the key if it was not found.
If the returned `Bool` is `true`, then the returned map is unaltered. If the `Bool` is `false`, then
the returned map has a new value inserted.
Equivalent to (but potentially faster than) calling `contains` followed by `inse... | true |
Complex.sin_add_int_mul_two_pi | Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | ∀ (x : ℂ) (n : ℤ), Complex.sin (x + ↑n * (2 * ↑Real.pi)) = Complex.sin x | null | true |
Option.forM_some | Init.Data.Option.Monadic | ∀ {m : Type u_1 → Type u_2} {α : Type u_3} [inst : Monad m] (f : α → m PUnit.{u_1 + 1}) (a : α), forM (some a) f = f a | null | true |
_private.Init.Data.String.Lemmas.Pattern.Find.String.0.String.Slice.isInfix_toList_iff | Init.Data.String.Lemmas.Pattern.Find.String | ∀ {t s : String}, t.toList <:+: s.toList ↔ ∃ s₁ s₂, s = s₁ ++ t ++ s₂ | null | true |
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