name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
HomologicalComplex.acyclic_iff | Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c), K.Acyclic ↔ ∀ (i : ι), K.ExactAt i | null | true |
Lean.Name.getNumParts._unsafe_rec | Lean.Data.Name | Lean.Name → ℕ | null | false |
_private.Mathlib.Algebra.Homology.DerivedCategory.Ext.TStructure.0.CategoryTheory.HasExt.hasSmallLocalizedShiftedHom_of_isLE_of_isGE._simp_1_2 | Mathlib.Algebra.Homology.DerivedCategory.Ext.TStructure | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : Y ⟶ X) [CategoryTheory.Mono f] {g h : Z ⟶ Y},
(CategoryTheory.CategoryStruct.comp g f = CategoryTheory.CategoryStruct.comp h f) = (g = h) | null | false |
Lean.Grind.ToInt.of_le | Init.Grind.ToIntLemmas | ∀ {α : Type u_1} {i : Lean.Grind.IntInterval} [inst : Lean.Grind.ToInt α i] [inst_1 : LE α] [Lean.Grind.ToInt.LE α i]
{a b : α} {a' b' : ℤ}, ↑a = a' → ↑b = b' → a ≤ b → a' ≤ b' | null | true |
_private.Mathlib.GroupTheory.SpecificGroups.Alternating.0.Equiv.Perm.closure_three_cycles_eq_alternating._proof_1_3 | Mathlib.GroupTheory.SpecificGroups.Alternating | ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (n : ℕ),
(∀ (l : List (Equiv.Perm α)),
(∀ g ∈ l, g.IsSwap) → l.length = 2 * n → l.prod ∈ Subgroup.closure {σ | σ.IsThreeCycle}) →
∀ (a b : Equiv.Perm α) (l : List (Equiv.Perm α)),
(∀ g ∈ a :: b :: l, g.IsSwap) →
(b :: l).length = 2 *... | null | false |
Filter.HasBasis.cauchySeq_iff' | Mathlib.Topology.UniformSpace.Cauchy | ∀ {α : Type u} {β : Type v} [uniformSpace : UniformSpace α] {γ : Sort u_1} [Nonempty β] [inst : SemilatticeSup β]
{u : β → α} {p : γ → Prop} {s : γ → SetRel α α},
(uniformity α).HasBasis p s → (CauchySeq u ↔ ∀ (i : γ), p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i) | null | true |
RingPreordering | Mathlib.Algebra.Order.Ring.Ordering.Defs | (R : Type u_1) → [CommRing R] → Type u_1 | A preordering on a ring `R` is a subsemiring of `R` containing all squares,
but not containing `-1`. | true |
Real.fourierIntegral_convergent_iff._simp_1 | Mathlib.Analysis.Fourier.FourierTransform | ∀ {V : Type u_1} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : NormedAddCommGroup V]
[inst_3 : InnerProductSpace ℝ V] [inst_4 : MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {f : V → E}
(w : V), MeasureTheory.Integrable (fun v => Real.fourierChar (-inner ℝ v w) ... | null | false |
SSet.prodStdSimplex.orderHomOfSimplex_coe | Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex | ∀ {p q n : ℕ}
(x :
(CategoryTheory.MonoidalCategoryStruct.tensorObj (SSet.stdSimplex.obj { len := p })
(SSet.stdSimplex.obj { len := q })).obj
(Opposite.op { len := n }))
{m : ℕ} (hm : p + q = m) (i : Fin (n + 1)), (SSet.prodStdSimplex.orderHomOfSimplex x hm) i = ⟨↑(x.1 i) + ↑(x.2 i), ⋯⟩ | null | true |
AddMonoid.Coprod.clift._proof_1 | Mathlib.GroupTheory.Coprod.Basic | ∀ {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : AddZeroClass P],
AddHomClass (FreeAddMonoid (M ⊕ N) →+ P) (FreeAddMonoid (M ⊕ N)) P | null | false |
MeasureTheory.lintegral_rnDeriv_mul | Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | ∀ {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν],
μ.AbsolutelyContinuous ν →
∀ {f : α → ENNReal}, AEMeasurable f ν → ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂μ | null | true |
_private.Mathlib.Combinatorics.SetFamily.Shadow.0.Set.Sized.shadow_iterate._simp_1_2 | Mathlib.Combinatorics.SetFamily.Shadow | ∀ {α : Type u_1} [inst : DecidableEq α] {𝒜 : Finset (Finset α)} {t : Finset α} {k : ℕ},
(t ∈ Finset.shadow^[k] 𝒜) = ∃ s ∈ 𝒜, t ⊆ s ∧ (s \ t).card = k | null | false |
PiLp.basis_toMatrix_basisFun_mul | Mathlib.Analysis.Normed.Lp.PiLp | ∀ (p : ENNReal) {ι : Type u_2} [inst : Fintype ι] {𝕜 : Type u_5} [inst_1 : SeminormedCommRing 𝕜]
(b : Module.Basis ι 𝕜 (PiLp p fun x => 𝕜)) (A : Matrix ι ι 𝕜),
b.toMatrix ⇑(PiLp.basisFun p 𝕜 ι) * A = Matrix.of fun i j => (b.repr (WithLp.toLp p (A.transpose j))) i | null | true |
Polynomial.ringHom_ext' | Mathlib.Algebra.Polynomial.Monomial | ∀ {R : Type u} [inst : Semiring R] {S : Type u_1} [inst_1 : Semiring S] {f g : Polynomial R →+* S},
f.comp Polynomial.C = g.comp Polynomial.C → f Polynomial.X = g Polynomial.X → f = g | null | true |
AlgHom.Finite.comp | Mathlib.RingTheory.Finiteness.Basic | ∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [inst : CommRing R] [inst_1 : CommRing A]
[inst_2 : CommRing B] [inst_3 : CommRing C] [inst_4 : Algebra R A] [inst_5 : Algebra R B] [inst_6 : Algebra R C]
{g : B →ₐ[R] C} {f : A →ₐ[R] B}, g.Finite → f.Finite → (g.comp f).Finite | null | true |
ContinuousMap.partialOrder._proof_3 | Mathlib.Topology.ContinuousMap.Ordered | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : PartialOrder β]
(a : C(α, β)), a ≤ a | null | false |
_private.Mathlib.LinearAlgebra.Transvection.Basic.0.LinearEquiv.symm_mem_dilatransvections_iff._simp_1_2 | Mathlib.LinearAlgebra.Transvection.Basic | ∀ {R : Type u_1} {S : Type u_6} {M : Type u_7} {M₂ : Type u_9} [inst : Semiring R] [inst_1 : Semiring S]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] {module_M : Module R M} {module_S_M₂ : Module S M₂}
{σ : R →+* S} {σ' : S →+* R} {re₁ : RingHomInvPair σ σ'} {re₂ : RingHomInvPair σ' σ} (e : M ≃ₛₗ[σ] M₂) {... | null | false |
ContinuousMultilinearMap.analyticOn_uncurry_compContinuousLinearMap | Mathlib.Analysis.Analytic.CPolynomial | ∀ {𝕜 : Type u_1} {G : Type u_4} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup G]
[inst_2 : NormedSpace 𝕜 G] {ι : Type u_5} {Em : ι → Type u_6} {Fm : ι → Type u_7}
[inst_3 : (i : ι) → NormedAddCommGroup (Em i)] [inst_4 : (i : ι) → NormedSpace 𝕜 (Em i)]
[inst_5 : (i : ι) → NormedAddCommGroup (... | null | true |
Vector.neg_zero | Init.Data.Vector.Algebra | ∀ {α : Type u_1} {n : ℕ} [inst : Zero α] [inst_1 : Neg α], -0 = 0 → -0 = 0 | null | true |
AdjoinRoot.mul_div_root_cancel | Mathlib.RingTheory.AdjoinRoot | ∀ {K : Type u_5} [inst : Field K] (f : Polynomial K) [inst_1 : Fact (Irreducible f)],
(Polynomial.X - Polynomial.C (AdjoinRoot.root f)) *
(Polynomial.map (AdjoinRoot.of f) f / (Polynomial.X - Polynomial.C (AdjoinRoot.root f))) =
Polynomial.map (AdjoinRoot.of f) f | null | true |
Std.TreeMap.equiv_iff_keysArray_unit_perm | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap α Unit cmp}, t₁.Equiv t₂ ↔ t₁.keysArray.Perm t₂.keysArray | null | true |
Finset.map_subset_iff_subset_preimage | Mathlib.Data.Finset.Preimage | ∀ {α : Type u} {β : Type v} {f : α ↪ β} {s : Finset α} {t : Finset β}, Finset.map f s ⊆ t ↔ s ⊆ t.preimage ⇑f ⋯ | null | true |
_private.Mathlib.Computability.Primrec.List.0.Primrec.vector_head.match_1_1 | Mathlib.Computability.Primrec.List | ∀ {α : Type u_1} {n : ℕ} (motive : List.Vector α n.succ → Prop) (x : List.Vector α n.succ),
(∀ (head : α) (tail : List α) (property : (head :: tail).length = n.succ), motive ⟨head :: tail, property⟩) → motive x | null | false |
StandardEtalePair.equivAwayAdjoinRoot | Mathlib.RingTheory.Etale.StandardEtale | {R : Type u_1} →
[inst : CommRing R] → (P : StandardEtalePair R) → P.Ring ≃ₐ[R] Localization.Away ((AdjoinRoot.mk P.f) P.g) | `R[X][Y]/⟨f, Yg-1⟩ ≃ (R[X]/f)[1/g]` | true |
Algebra.coe_top | Mathlib.Algebra.Algebra.Subalgebra.Lattice | ∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A], ↑⊤ = Set.univ | null | true |
USize.toBitVec_ofNat | Init.Data.UInt.Lemmas | ∀ (n : ℕ), (OfNat.ofNat n).toBitVec = BitVec.ofNat System.Platform.numBits n | null | true |
Finset.Iic_toDual | Mathlib.Order.Interval.Finset.Defs | ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : LocallyFiniteOrderTop α] (a : α),
Finset.Iic (OrderDual.toDual a) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ici a) | null | true |
fderivWithin_comp_add_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 : E → F}
{x : E} {s : Set E} (a : E), fderivWithin 𝕜 (fun x => f (x + a)) s x = fderivWithin 𝕜 f (a +ᵥ s) (... | null | true |
AddCommGrpCat.instCoeSortType.eq_1 | Mathlib.Algebra.Category.Grp.Basic | AddCommGrpCat.instCoeSortType = { coe := AddCommGrpCat.carrier } | null | true |
_private.Mathlib.Analysis.Convex.MetricSpace.0.Convexity.continuous_convexCombPair_of_isBounded._simp_1_5 | Mathlib.Analysis.Convex.MetricSpace | ∀ {α : Type u} {β : Type v} {f : α → β} {s : Set β} {a : α}, (a ∈ f ⁻¹' s) = (f a ∈ s) | null | false |
Hypergraph.Adj.symm | Mathlib.Combinatorics.Hypergraph.Basic | ∀ {α : Type u_1} {x y : α} {H : Hypergraph α}, H.Adj x y → H.Adj y x | null | true |
AddActionHom.End.addOppositeEquiv_symm_apply_apply | Mathlib.GroupTheory.GroupAction.Hom | ∀ {M : Type u_2} [inst : AddMonoid M] (m x : M), (AddActionHom.End.addOppositeEquiv.symm m) x = m + x | null | true |
Equiv.Perm.sumCongr_refl | Mathlib.Logic.Equiv.Sum | ∀ {α : Type u_9} {β : Type u_10}, Equiv.Perm.sumCongr (Equiv.refl α) (Equiv.refl β) = Equiv.refl (α ⊕ β) | null | true |
AlgebraicGeometry.Scheme.exists_isQuasiAffine_of_isLimit | Mathlib.AlgebraicGeometry.AffineTransitionLimit | ∀ {I : Type u} [inst : CategoryTheory.Category.{u, u} I] (D : CategoryTheory.Functor I AlgebraicGeometry.Scheme)
(c : CategoryTheory.Limits.Cone D) (hc : CategoryTheory.Limits.IsLimit c) [CategoryTheory.IsCofiltered I]
[∀ {i j : I} (f : i ⟶ j), AlgebraicGeometry.IsAffineHom (D.map f)] [∀ (i : I), CompactSpace ↥(D.o... | Suppose `{ Xᵢ }` is an inverse system of qcqs schemes with affine transition maps.
If `lim Xᵢ` is quasi-affine, then some `Xᵢ` is quasi-affine. | true |
CategoryTheory.Functor.isoWhiskerLeft_hom | Mathlib.CategoryTheory.Whiskering | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D)
{G H : CategoryTheory.Functor D E} (α : G ≅ H), (F.isoWhiskerLeft α).hom = F.whiskerLeft α.hom | null | true |
_private.Mathlib.Data.Num.ZNum.0.Num.toZNumNeg_succ.match_1_1 | Mathlib.Data.Num.ZNum | ∀ (motive : Num → Prop) (x : Num), (∀ (_ : Unit), motive Num.zero) → (∀ (_n : PosNum), motive (Num.pos _n)) → motive x | null | false |
_private.Mathlib.NumberTheory.Bernoulli.0.Bernoulli.pIntegral_bernoulli_even_term._proof_1_9 | Mathlib.NumberTheory.Bernoulli | ∀ {k m : ℕ}, m < k → 2 ≤ 2 * k - 2 * m | null | false |
IsUpperSet.closure | Mathlib.Topology.Algebra.Order.UpperLower | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : Preorder α] [HasUpperLowerClosure α] {s : Set α},
IsUpperSet s → IsUpperSet (closure s) | null | true |
AlgebraicGeometry.Scheme.GlueData.oneHypercover_I₀ | Mathlib.AlgebraicGeometry.GluingOneHypercover | ∀ (D : AlgebraicGeometry.Scheme.GlueData), D.oneHypercover.I₀ = D.J | null | true |
RelIso.trans | Mathlib.Order.RelIso.Basic | {α : Type u_1} →
{β : Type u_2} →
{γ : Type u_3} → {r : α → α → Prop} → {s : β → β → Prop} → {t : γ → γ → Prop} → r ≃r s → s ≃r t → r ≃r t | Composition of two relation isomorphisms is a relation isomorphism. | true |
Filter.Germ.instField._proof_10 | Mathlib.Order.Filter.FilterProduct | ∀ {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [inst : Field β] (q : ℚ), ↑q = ↑q.num / ↑q.den | null | false |
Std.Time.instOrdDuration | Std.Time.Duration | Ord Std.Time.Duration | null | true |
IsConnected.iUnion_of_chain | Mathlib.Topology.Connected.Basic | ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : LinearOrder β] [inst_2 : SuccOrder β]
[IsSuccArchimedean β] [Nonempty β] {s : β → Set α},
(∀ (n : β), IsConnected (s n)) → (∀ (n : β), (s n ∩ s (Order.succ n)).Nonempty) → IsConnected (⋃ n, s n) | The iUnion of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is connected. | true |
_private.Mathlib.NumberTheory.RamificationInertia.Basic.0.Ideal.quotientToQuotientRangePowQuotSuccAux._simp_1 | Mathlib.NumberTheory.RamificationInertia.Basic | ∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
[inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {x : M₂}, (x ∈ f.range) = ∃ y, f y = x | null | false |
Std.Internal.Do.Spec.Iter.foldM_map | Std.Internal.Do.Triple.SpecLemmas | ∀ {α β γ δ : Type w} {n : Type w → Type w'} {Pred EPred : Type w} [inst : Std.Iterator α Id β]
[Std.Iterators.Finite α Id] [inst_2 : Monad n] [LawfulMonad n] [inst_4 : Std.Internal.Do.Assertion Pred]
[inst_5 : Std.Internal.Do.Assertion EPred] [inst_6 : Std.Internal.Do.WPMonad n Pred EPred]
[inst_7 : Std.IteratorL... | null | true |
MulAction.IsPretransitive.of_compHom | Mathlib.Algebra.Group.Action.Hom | ∀ {M : Type u_4} {N : Type u_5} {α : Type u_6} [inst : Monoid M] [inst_1 : Monoid N] [inst_2 : MulAction N α]
(f : M →* N) [h : MulAction.IsPretransitive M α], MulAction.IsPretransitive N α | null | true |
AlgebraicTopology.DoldKan.Γ₀'_obj | Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasFiniteCoproducts C] (K : ChainComplex C ℕ),
AlgebraicTopology.DoldKan.Γ₀'.obj K =
CategoryTheory.SimplicialObject.Split.mk' (AlgebraicTopology.DoldKan.Γ₀.splitting K) | null | true |
convex_sInter | Mathlib.Analysis.Convex.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : SMul 𝕜 E] {S : Set (Set E)}, (∀ s ∈ S, Convex 𝕜 s) → Convex 𝕜 (⋂₀ S) | null | true |
_private.Mathlib.Algebra.Homology.Opposite.0.HomologicalComplex.quasiIso_opFunctor_map_iff._simp_1_1 | Mathlib.Algebra.Homology.Opposite | ∀ {ι : Type u_1} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L)
[inst_2 : ∀ (i : ι), K.HasHomology i] [inst_3 : ∀ (i : ι), L.HasHomology i], QuasiIso f = ∀ (i : ι), QuasiIsoAt f i | null | false |
Asymptotics.instTransIsBigOTVSIsThetaTVS | Mathlib.Analysis.Asymptotics.TVS | {α : Type u_1} →
{𝕜 : Type u_3} →
{E : Type u_4} →
{F : Type u_5} →
{G : Type u_6} →
[inst : NontriviallyNormedField 𝕜] →
[inst_1 : AddCommGroup E] →
[inst_2 : TopologicalSpace E] →
[inst_3 : Module 𝕜 E] →
[inst_4 : AddCommGrou... | null | true |
TrivSqZeroExt.commSemiring._proof_2 | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : Module Rᵐᵒᵖ M] [inst_4 : IsCentralScalar R M] (a b : TrivSqZeroExt R M), a * b = b * a | null | false |
CochainComplex.liftCycles_shift_homologyπ._proof_2 | Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence | ∀ {n i : ℤ} (i' : ℤ), n + i = i' → i' = i + n | null | false |
TopHom.dual_id | Mathlib.Order.Hom.Bounded | ∀ {α : Type u_2} [inst : LE α] [inst_1 : OrderTop α], TopHom.dual (TopHom.id α) = BotHom.id αᵒᵈ | null | true |
CategoryTheory.CartesianMonoidalCategory.ofChosenFiniteProducts.leftUnitor_naturality | Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
(𝒯 : CategoryTheory.Limits.LimitCone (CategoryTheory.Functor.empty C))
(ℬ : (X Y : C) → CategoryTheory.Limits.LimitCone (CategoryTheory.Limits.pair X Y)) {X₁ X₂ : C} (f : X₁ ⟶ X₂),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.CartesianMonoidalC... | null | true |
Units.mulLeftLinearEquiv_apply | Mathlib.Algebra.Module.Equiv.Basic | ∀ (R : Type u_9) {A : Type u_10} [inst : Semiring R] [inst_1 : Semiring A] [inst_2 : Module R A]
[inst_3 : SMulCommClass R A A] (a : Aˣ) (x : A), ((Units.mulLeftLinearEquiv R A) a) x = ↑a * x | null | true |
_private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.processConstructor._unary.eq_def | Lean.Elab.Tactic.RCases | ∀ (ref : Lean.Syntax) (info : Array Lean.Meta.ParamInfo) (explicit : Bool)
(_x : (_ : ℕ) ×' List Lean.Elab.Tactic.RCases.RCasesPatt),
Lean.Elab.Tactic.RCases.processConstructor._unary✝ ref info explicit _x =
PSigma.casesOn _x fun idx ps =>
if x : idx < info.size then
if (!explicit && info[idx].bin... | null | false |
CategoryTheory.instCategoryInd._proof_7 | Mathlib.CategoryTheory.Limits.Indization.Category | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C],
autoParam
(∀ {X Y : CategoryTheory.Ind C} (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id X) f = f)
CategoryTheory.Category.id_comp._autoParam | null | false |
Ctop.Realizer.mk.sizeOf_spec | Mathlib.Data.Analysis.Topology | ∀ {α : Type u_6} [T : TopologicalSpace α] [inst : SizeOf α] (σ : Type u_5) (F : Ctop α σ) (eq : F.toTopsp = T),
sizeOf { σ := σ, F := F, eq := eq } = 1 + sizeOf σ + sizeOf F + sizeOf eq | null | true |
ProbabilityTheory.iCondIndepFun.indepFun_sub_left | Mathlib.Probability.Independence.Conditional | ∀ {Ω : Type u_1} {ι : Type u_2} {β : Type u_3} {m' mΩ : MeasurableSpace Ω} [inst : StandardBorelSpace Ω] {hm' : m' ≤ mΩ}
{μ : MeasureTheory.Measure Ω} [inst_1 : MeasureTheory.IsFiniteMeasure μ] {m : MeasurableSpace β} [inst_2 : Sub β]
[MeasurableSub₂ β] {f : ι → Ω → β},
ProbabilityTheory.iCondIndepFun m' hm' f μ ... | null | true |
continuousPregroupoid | Mathlib.Geometry.Manifold.StructureGroupoid | (H : Type u_2) → [inst : TopologicalSpace H] → Pregroupoid H | The pregroupoid of all partial maps on a topological space `H`. | true |
CategoryTheory.Subgroupoid.disconnect_normal | Mathlib.CategoryTheory.Groupoid.Subgroupoid | ∀ {C : Type u} [inst : CategoryTheory.Groupoid C] (S : CategoryTheory.Subgroupoid C), S.IsNormal → S.disconnect.IsNormal | null | true |
LinearEquiv.domMulActCongrRight._proof_1 | Mathlib.Algebra.Module.Equiv.Basic | ∀ {S : Type u_1} [inst : Semiring S], RingHomInvPair (RingHom.id Sᵈᵐᵃ) (RingHom.id Sᵈᵐᵃ) | null | false |
Std.Http.Protocol.H1.Reader.State.needHeader.sizeOf_spec | Std.Http.Protocol.H1.Reader | ∀ {dir : Std.Http.Protocol.H1.Direction} (a : ℕ), sizeOf (Std.Http.Protocol.H1.Reader.State.needHeader a) = 1 + sizeOf a | null | true |
isQuotientMap_projIcc | Mathlib.Topology.Order.ProjIcc | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α} {h : a ≤ b} [inst_1 : TopologicalSpace α] [OrderTopology α],
Topology.IsQuotientMap (Set.projIcc a b h) | null | true |
CategoryTheory.Under.isoMk_hom_right | Mathlib.CategoryTheory.Comma.Over.Basic | ∀ {T : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} T] {X : T} {f g : CategoryTheory.Under X}
(hr : f.right ≅ g.right)
(hw : autoParam (CategoryTheory.CategoryStruct.comp f.hom hr.hom = g.hom) CategoryTheory.Under.isoMk._auto_1),
(CategoryTheory.Under.isoMk hr hw).hom.right = hr.hom | null | true |
_private.Lean.Elab.DefView.0.Lean.Elab.DefKind.isTheorem._sparseCasesOn_1 | Lean.Elab.DefView | {motive : Lean.Elab.DefKind → Sort u} →
(t : Lean.Elab.DefKind) → motive Lean.Elab.DefKind.theorem → (Nat.hasNotBit 4 t.ctorIdx → motive t) → motive t | null | false |
_private.Init.GrindInstances.ToInt.0.Lean.Grind.instOfNatInt8SintOfNatNat._proof_2 | Init.GrindInstances.ToInt | ∀ (x : ℕ), 128 ≤ ↑x % 256 → ¬↑x % 256 - ↑256 = (↑x + 128) % 256 + -128 → False | null | false |
RootPairing.Base.casesOn | Mathlib.LinearAlgebra.RootSystem.Base | {ι : Type u_1} →
{R : Type u_2} →
{M : Type u_3} →
{N : Type u_4} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
[inst_3 : AddCommGroup N] →
[inst_4 : Module R N] →
{P : RootPairing ι R M N} →
... | null | false |
Int.fib_natCast | Mathlib.Data.Int.Fib.Basic | ∀ (n : ℕ), Int.fib ↑n = ↑(Nat.fib n) | null | true |
PartialEquiv.noConfusion | Mathlib.Logic.Equiv.PartialEquiv | {P : Sort u} →
{α : Type u_5} →
{β : Type u_6} →
{t : PartialEquiv α β} →
{α' : Type u_5} →
{β' : Type u_6} → {t' : PartialEquiv α' β'} → α = α' → β = β' → t ≍ t' → PartialEquiv.noConfusionType P t t' | null | false |
Lex.instAddCommSemigroup | Mathlib.Algebra.Order.Group.Synonym | {α : Type u_1} → [AddCommSemigroup α] → AddCommSemigroup (Lex α) | null | true |
IsIsotypicOfType.linearEquiv_fun | Mathlib.RingTheory.SimpleModule.Isotypic | ∀ {R : Type u_2} {M : Type u} {S : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup S]
[inst_3 : Module R M] [inst_4 : Module R S] [IsSemisimpleModule R M] [Module.Finite R M],
IsIsotypicOfType R M S → ∃ n, Nonempty (M ≃ₗ[R] Fin n → S) | null | true |
_private.Mathlib.Combinatorics.Additive.FreimanHom.0.isAddFreimanHom_antitone.match_1_1 | Mathlib.Combinatorics.Additive.FreimanHom | ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : AddCancelCommMonoid β] {A : Set α} {B : Set β}
{f : α → β} (s t : Multiset α) (motive : (n : ℕ) → IsAddFreimanHom (n + 1) A B f → s.card = n → t.card = n → Prop)
(n : ℕ) (hf : IsAddFreimanHom (n + 1) A B f) (hs : s.card = n) (x : t.card = n),
(∀ (... | null | false |
CategoryTheory.Abelian.SpectralObject.sequenceΨ_exact | Mathlib.Algebra.Homology.SpectralObject.Differentials | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k)
(h₁₂ : CategoryTheory.CategoryStruct.... | null | true |
CategoryTheory.MonoidalOpposite.mopMopEquivalence_inverse_obj_unmop_unmop | Mathlib.CategoryTheory.Monoidal.Opposite | ∀ (C : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} C] (X : C),
((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).inverse.obj X).unmop.unmop = X | null | true |
_private.Mathlib.MeasureTheory.Function.Holder.0.MeasureTheory.Lp.smul_assoc._simp_1_1 | Mathlib.MeasureTheory.Function.Holder | ∀ {α : Type u_1} {E : Type u_4} {m : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α}
[inst : NormedAddCommGroup E] {𝕜 : Type u_6} [inst_1 : NormedRing 𝕜] [inst_2 : Module 𝕜 E]
[inst_3 : IsBoundedSMul 𝕜 E] {f : α → E} (c : 𝕜) (hf : MeasureTheory.MemLp f p μ),
c • MeasureTheory.MemLp.toLp f hf =... | null | false |
uniformity_eq_comap_nhds_zero' | Mathlib.Topology.Algebra.IsUniformGroup.Defs | ∀ (G : Type u_1) [inst : AddGroup G] [inst_1 : TopologicalSpace G] [inst_2 : IsTopologicalAddGroup G],
uniformity G = Filter.comap (fun p => p.2 + -p.1) (nhds 0) | null | true |
CircleDeg1Lift.translate._proof_1 | Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | ∀ (x : Multiplicative ℝ), Monotone fun x_1 => Multiplicative.toAdd x + x_1 | null | false |
Lean.Lsp.WorkspaceEditClientCapabilities._sizeOf_inst | Lean.Data.Lsp.Capabilities | SizeOf Lean.Lsp.WorkspaceEditClientCapabilities | null | false |
CategoryTheory.Pseudofunctor.StrongTrans.id._proof_6 | Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo | ∀ {B : Type u_2} [inst : CategoryTheory.Bicategory B] {C : Type u_6} [inst_1 : CategoryTheory.Bicategory C]
(F : CategoryTheory.Pseudofunctor B C) {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Bicategory.rightUnitor (F.map (CategoryTheory.CategoryStruct.comp f g)) ≪≫
... | null | false |
Std.Tactic.BVDecide.BVExpr.bitblast.blastCpopLayer.go._unary._proof_1 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.Cpop | ∀ {α : Type} [inst : Hashable α] [inst_1 : DecidableEq α] {w len : ℕ} (aig : Std.Sat.AIG α) (iterNum : ℕ)
(oldLayer : aig.RefVec (len * w)),
aig.decls.size ≤
(Std.Tactic.BVDecide.BVExpr.bitblast.blastExtract aig
{ w := len * w, vec := oldLayer, start := 2 * iterNum * w }).aig.decls.size | null | false |
MonoidHom.map_mulIndicator | Mathlib.Algebra.Group.Indicator | ∀ {α : Type u_1} {M : Type u_6} {N : Type u_7} {F : Type u_8} [inst : One M] [inst_1 : One N] [inst_2 : FunLike F M N]
[OneHomClass F M N] (f : F) (s : Set α) (g : α → M) (x : α), f (s.mulIndicator g x) = s.mulIndicator (⇑f ∘ g) x | **Alias** of `map_mulIndicator`. | true |
Lean.ScopedEnvExtension.Entry._sizeOf_1 | Lean.ScopedEnvExtension | {α : Type} → [SizeOf α] → Lean.ScopedEnvExtension.Entry α → ℕ | null | false |
DyckWord.instUniqueAddUnits._proof_1 | Mathlib.Combinatorics.Enumerative.DyckWord | ∀ (p : AddUnits DyckWord), p = default | null | false |
_private.Mathlib.Data.Finset.Insert.0.Finset.singleton_subset_set_iff._proof_1_1 | Mathlib.Data.Finset.Insert | ∀ {α : Type u_1} {s : Set α} {a : α}, ↑{a} ⊆ s ↔ a ∈ s | null | false |
mem_asymptoticCone_iff | Mathlib.Topology.Algebra.AsymptoticCone | ∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Field k] [inst_1 : LinearOrder k] [inst_2 : AddCommGroup V]
[inst_3 : Module k V] [inst_4 : AddTorsor V P] [inst_5 : TopologicalSpace V] {v : V} {s : Set P},
v ∈ asymptoticCone k s ↔ ∃ᶠ (p : P) in AffineSpace.asymptoticNhds k P v, p ∈ s | null | true |
indiscreteTopology_iff_forall_norm_eq_zero' | Mathlib.Analysis.Normed.Group.Basic | ∀ {E : Type u_5} [inst : SeminormedGroup E], IndiscreteTopology E ↔ ∀ (x : E), ‖x‖ = 0 | null | true |
SimpleGraph.Subgraph.deleteEdges_coe_eq | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (s : Set (Sym2 ↑G'.verts)),
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map Subtype.val '' s)).coe | null | true |
ContinuousAffineEquiv.trans_apply | Mathlib.Topology.Algebra.ContinuousAffineEquiv | ∀ {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8}
[inst : Ring k] [inst_1 : AddCommGroup V₁] [inst_2 : Module k V₁] [inst_3 : AddTorsor V₁ P₁]
[inst_4 : TopologicalSpace P₁] [inst_5 : AddCommGroup V₂] [inst_6 : Module k V₂] [inst_7 : AddTorsor V₂ P₂]
... | null | true |
CliffordAlgebra.EvenHom.rec | Mathlib.LinearAlgebra.CliffordAlgebra.Even | {R : Type u_1} →
{M : Type u_2} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
{Q : QuadraticForm R M} →
{A : Type u_3} →
[inst_3 : Ring A] →
[inst_4 : Algebra R A] →
{motive : CliffordAlgebra.EvenHom ... | null | false |
MeasureTheory.SignedMeasure.toMeasureOfZeroLE'.eq_1 | Mathlib.MeasureTheory.VectorMeasure.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (i : Set α)
(hi : MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i) (j : Set α)
(hj : MeasurableSet j), s.toMeasureOfZeroLE' i hi j hj = ↑(NNReal.mk ((MeasureTheory.VectorMeasure.restrict s i) j) ⋯) | null | true |
CategoryTheory.Limits.hasLimit_of_equalizer_and_product | Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} [inst_1 : CategoryTheory.SmallCategory J]
(F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasLimit (CategoryTheory.Discrete.functor F.obj)]
[CategoryTheory.Limits.HasLimit (CategoryTheory.Discrete.functor fun f => F.obj f.fst.2)]
[Categ... | Given the existence of the appropriate (possibly finite) products and equalizers, we know a limit of
`F` exists.
(This assumes the existence of all equalizers, which is technically stronger than needed.)
| true |
_private.Mathlib.Lean.Meta.RefinedDiscrTree.Encode.0.Lean.Meta.RefinedDiscrTree.encodingStepAux._sparseCasesOn_5 | Mathlib.Lean.Meta.RefinedDiscrTree.Encode | {motive : Lean.Expr → Sort u} →
(t : Lean.Expr) →
((declName : Lean.Name) → (us : List Lean.Level) → motive (Lean.Expr.const declName us)) →
((typeName : Lean.Name) → (idx : ℕ) → (struct : Lean.Expr) → motive (Lean.Expr.proj typeName idx struct)) →
((fvarId : Lean.FVarId) → motive (Lean.Expr.fvar fv... | null | false |
_private.Mathlib.Tactic.SplitIfs.0.Mathlib.Tactic.SplitPosition.target.elim | Mathlib.Tactic.SplitIfs | {motive : Mathlib.Tactic.SplitPosition✝ → Sort u} →
(t : Mathlib.Tactic.SplitPosition✝) →
Mathlib.Tactic.SplitPosition.ctorIdx✝ t = 0 → motive Mathlib.Tactic.SplitPosition.target✝ → motive t | null | false |
LipschitzOnWith.absolutelyContinuousOnInterval | Mathlib.MeasureTheory.Function.AbsolutelyContinuous | ∀ {X : Type u_1} [inst : PseudoMetricSpace X] {a b : ℝ} {f : ℝ → X} {K : NNReal},
LipschitzOnWith K f (Set.uIcc a b) → AbsolutelyContinuousOnInterval f a b | If `f` is Lipschitz on `uIcc a b`, then `f` is absolutely continuous on `uIcc a b`. | true |
_private.Std.Time.Internal.Bounded.0.Std.Time.Internal.Bounded.LE.succ._proof_3 | Std.Time.Internal.Bounded | ∀ {lo hi : ℤ} (bounded : Std.Time.Internal.Bounded.LE lo hi), ↑bounded < hi → ¬↑bounded + 1 ≤ hi → False | null | false |
Metric.isBounded_iff_nndist | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α},
Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C | null | true |
Lean.Order.CompleteLattice.mk.noConfusion | Init.Internal.Order.Basic | {α : Sort u} →
{P : Sort u_1} →
{toPartialOrder : Lean.Order.PartialOrder α} →
{has_sup : ∀ (c : α → Prop), Exists (Lean.Order.is_sup c)} →
{toPartialOrder' : Lean.Order.PartialOrder α} →
{has_sup' : ∀ (c : α → Prop), Exists (Lean.Order.is_sup c)} →
{ toPartialOrder := toPartia... | null | false |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap.0.WeierstrassCurve.addSubMapCoeff._proof_15 | Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap | (27 + 1).AtLeastTwo | null | false |
Qq.Impl.PatVarDecl.mk._flat_ctor | Qq.MatchImpl | Option Q(Lean.Expr) → Lean.FVarId → Lean.Name → Qq.Impl.PatVarDecl | null | false |
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