name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
List.head_cons | Init.Data.List.Basic | ∀ {α : Type u} {a : α} {l : List α} {h : a :: l ≠ []}, (a :: l).head h = a | null | true |
_private.Mathlib.MeasureTheory.MeasurableSpace.Constructions.0.measurable_from_prod_countable_left'._simp_1_1 | Mathlib.MeasureTheory.MeasurableSpace.Constructions | ∀ {α : Type u} {β : Type v} {f : α → β} {s : Set β} {a : α}, (a ∈ f ⁻¹' s) = (f a ∈ s) | null | false |
Std.DHashMap.instForInSigmaOfMonad | Std.Data.DHashMap.Basic | {α : Type u} →
{β : α → Type v} →
{m : Type w → Type w'} →
[Monad m] → [inst : BEq α] → [inst_1 : Hashable α] → ForIn m (Std.DHashMap α β) ((a : α) × β a) | null | true |
ThreeGPFree.of_image | Mathlib.Combinatorics.Additive.AP.Three.Defs | ∀ {α : Type u_2} {β : Type u_3} [inst : CommMonoid α] [inst_1 : CommMonoid β] {s A : Set α} {t : Set β} {f : α → β},
IsMulFreimanHom 2 s t f → Set.InjOn f s → A ⊆ s → ThreeGPFree (f '' A) → ThreeGPFree A | Geometric progressions of length three are reflected under `2`-Freiman homomorphisms. | true |
Std.Tactic.BVDecide.BVExpr.bitblast.FullAdderOutput.aig | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.Add | {α : Type} →
[inst : Hashable α] →
[inst_1 : DecidableEq α] →
{old : Std.Sat.AIG α} → Std.Tactic.BVDecide.BVExpr.bitblast.FullAdderOutput old → Std.Sat.AIG α | null | true |
_private.Lean.Parser.Module.0.Lean.Parser.parseCommand.match_1 | Lean.Parser.Module | (motive : Option Lean.Parser.Error → Sort u_1) →
(x : Option Lean.Parser.Error) →
(Unit → motive none) → ((errorMsg : Lean.Parser.Error) → motive (some errorMsg)) → motive x | null | false |
CoalgCat.concreteCategory._proof_1 | Mathlib.Algebra.Category.CoalgCat.Basic | ∀ {R : Type u_2} [inst : CommRing R] {X Y : CoalgCat R} (f : ↑X.toModuleCat →ₗc[R] ↑Y.toModuleCat),
{ toCoalgHom' := f }.toCoalgHom' = f | null | false |
Submonoid.mem_map_iff_mem | Mathlib.Algebra.Group.Submonoid.Operations | ∀ {M : Type u_1} {N : Type u_2} [inst : MulOneClass M] [inst_1 : MulOneClass N] {F : Type u_4} [inst_2 : FunLike F M N]
[mc : MonoidHomClass F M N] {f : F},
Function.Injective ⇑f → ∀ {S : Submonoid M} {x : M}, f x ∈ Submonoid.map f S ↔ x ∈ S | null | true |
CategoryTheory.Limits.Sigma.reindex | Mathlib.CategoryTheory.Limits.Shapes.Products | {β : Type w} →
{C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{γ : Type w'} →
(ε : β ≃ γ) →
(f : γ → C) →
[inst_1 : CategoryTheory.Limits.HasCoproduct f] →
[inst_2 : CategoryTheory.Limits.HasCoproduct (f ∘ ⇑ε)] → ∐ f ∘ ⇑ε ≅ ∐ f | Reindex a categorical coproduct via an equivalence of the index types. | true |
Lean.Meta.Grind.InjectiveInfo.mk.noConfusion | Lean.Meta.Tactic.Grind.Types | {P : Sort u} →
{us : List Lean.Level} →
{α β h : Lean.Expr} →
{inv? : Option (Lean.Expr × Lean.Expr)} →
{us' : List Lean.Level} →
{α' β' h' : Lean.Expr} →
{inv?' : Option (Lean.Expr × Lean.Expr)} →
{ us := us, α := α, β := β, h := h, inv? := inv? } =
... | null | false |
Units.mulLECancellable_val | Mathlib.Algebra.Order.Monoid.Unbundled.Units | ∀ {M : Type u_1} [inst : Monoid M] [inst_1 : LE M] [MulLeftMono M] (u : Mˣ), MulLECancellable ↑u | null | true |
CategoryTheory.Subobject.map_mk | Mathlib.CategoryTheory.Subobject.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {A X Y : C} (i : A ⟶ X) [inst_1 : CategoryTheory.Mono i]
(f : X ⟶ Y) [inst_2 : CategoryTheory.Mono f],
(CategoryTheory.Subobject.map f).obj (CategoryTheory.Subobject.mk i) =
CategoryTheory.Subobject.mk (CategoryTheory.CategoryStruct.comp i f) | null | true |
Std.Time.Duration.subHours | Std.Time.Duration | Std.Time.Duration → Std.Time.Hour.Offset → Std.Time.Duration | Subtracts an `Hour.Offset` from a `Duration`
| true |
Std.DHashMap.Internal.Raw₀.get!_alter | Std.Data.DHashMap.Internal.RawLemmas | ∀ {α : Type u} {β : α → Type v} (m : Std.DHashMap.Internal.Raw₀ α β) [inst : BEq α] [inst_1 : Hashable α]
[inst_2 : LawfulBEq α] {k k' : α},
(↑m).WF →
∀ [inst_3 : Inhabited (β k')] {f : Option (β k) → Option (β k)},
(m.alter k f).get! k' = if heq : (k == k') = true then (Option.map (cast ⋯) (f (m.get? k))... | null | true |
Lean.Meta.MVarRenaming.noConfusionType | Lean.Meta.Match.MVarRenaming | Sort u → Lean.Meta.MVarRenaming → Lean.Meta.MVarRenaming → Sort u | null | false |
Absorbs.sInter | Mathlib.Topology.Bornology.Absorbs | ∀ {G₀ : Type u_1} {α : Type u_2} [inst : GroupWithZero G₀] [inst_1 : Bornology G₀] [inst_2 : MulAction G₀ α] {t : Set α}
{S : Set (Set α)}, S.Finite → (∀ s ∈ S, Absorbs G₀ s t) → Absorbs G₀ (⋂₀ S) t | **Alias** of the reverse direction of `Set.Finite.absorbs_sInter`. | true |
_private.Mathlib.CategoryTheory.Preadditive.Injective.InjectiveObject.0.CategoryTheory.InjectiveObject.instContainsZeroIsInjectiveOfHasZeroMorphismsOfHasZeroObject._simp_1 | Mathlib.CategoryTheory.Preadditive.Injective.InjectiveObject | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] (X : C),
CategoryTheory.Limits.IsZero X = (CategoryTheory.CategoryStruct.id X = 0) | null | false |
CategoryTheory.MonoOver.imageForgetAdj._proof_1 | Mathlib.CategoryTheory.Subobject.MonoOver | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {X : C} [inst_1 : CategoryTheory.Limits.HasImages C]
(f : CategoryTheory.Over X) (g : CategoryTheory.MonoOver X) (k : f ⟶ (CategoryTheory.MonoOver.forget X).obj g),
(fun k =>
CategoryTheory.Over.homMk
(CategoryTheory.CategoryStruct.com... | null | false |
AddActionHom.prodMap._proof_2 | Mathlib.GroupTheory.GroupAction.Hom | ∀ {M : Type u_5} {N : Type u_6} {α : Type u_1} {β : Type u_2} {γ : Type u_4} {δ : Type u_3} [inst : VAdd M α]
[inst_1 : VAdd M β] [inst_2 : VAdd N γ] [inst_3 : VAdd N δ] {σ : M → N} (f : α →ₑ[σ] γ) (g : β →ₑ[σ] δ) (m : M)
(x : α × β),
((f.comp (AddActionHom.fst M α β)).prod (g.comp (AddActionHom.snd M α β))).toFu... | null | false |
Lean.DefinitionVal.getSafetyEx | Lean.Declaration | Lean.DefinitionVal → Lean.DefinitionSafety | null | true |
CategoryTheory.Functor.WellOrderInductionData.ofExists._proof_4 | Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | ∀ {J : Type u_1} [inst : LinearOrder J] [inst_1 : SuccOrder J] {F : CategoryTheory.Functor Jᵒᵖ (Type u_2)}
(h₁ :
∀ (j : J),
¬IsMax j → Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.homOfLE ⋯).op)))
(j : J) (hj : ¬IsMax j) (x : F.obj (Opposite.op j)),
(CategoryTheory.Co... | null | false |
_private.Mathlib.AlgebraicTopology.ExtraDegeneracy.0.CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_13 | Mathlib.AlgebraicTopology.ExtraDegeneracy | ∀ {n : ℕ} (j : Fin (n + 1)) (k : Fin (n + 1 + 1)), j.succ.rev = k → ↑k + ↑j = n | null | false |
Aesop.EqualUpToIdsM.run' | Aesop.Util.EqualUpToIds | {α : Type} →
Aesop.EqualUpToIdsM α →
Option Lean.MetavarContext →
Lean.MetavarContext → Lean.MetavarContext → Bool → Lean.MetaM (α × Aesop.EqualUpToIdsM.State) | null | true |
Lean.isIdRestAscii | Init.Meta.Defs | UInt8 → Bool | null | true |
_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Ici_erase._simp_1_4 | Mathlib.Order.Interval.Finset.Basic | ∀ {α : Type u_2} [inst : PartialOrder α] {a b : α}, (a < b) = (a ≤ b ∧ a ≠ b) | null | false |
ContinuousLinearMap.IsFredholm.of_codRestrict | Mathlib.Analysis.Normed.Operator.Fredholm.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E]
[inst_2 : AddCommGroup F] [inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 F] [inst_5 : TopologicalSpace E]
[inst_6 : TopologicalSpace F] [CompleteSpace 𝕜] [IsTopologicalAddGroup E] [IsTopologicalAddGroup F]
[... | **Alias** of the forward direction of `ContinuousLinearMap.isFredholm_codRestrict_iff`. | true |
ImportGraph.Shake.Stance.casesOn | ImportGraph.Shake.DeclNeeds | {motive : ImportGraph.Shake.Stance → Sort u} →
(t : ImportGraph.Shake.Stance) →
((isPublic : Bool) →
(isExposed : Option Bool) →
(isPublic_if_isExposed : (isExposed != some true || isPublic) = true) →
(isMeta : Option Bool) →
(isExact : Bool) →
motive
... | null | false |
Lean.Grind.CommRing.Poly.combine.go.eq_5 | Init.Grind.Ring.CommSolver | ∀ (fuel_2 : ℕ) (k₁ : ℤ) (m₁ : Lean.Grind.CommRing.Mon) (p₁_2 : Lean.Grind.CommRing.Poly) (k₂ : ℤ)
(m₂ : Lean.Grind.CommRing.Mon) (p₂_2 : Lean.Grind.CommRing.Poly),
Lean.Grind.CommRing.Poly.combine.go fuel_2.succ (Lean.Grind.CommRing.Poly.add k₁ m₁ p₁_2)
(Lean.Grind.CommRing.Poly.add k₂ m₂ p₂_2) =
match m₁... | null | true |
Int32.maxValue_le_iff._simp_1 | Init.Data.SInt.Lemmas | ∀ {a : Int32}, (Int32.maxValue ≤ a) = (a = Int32.maxValue) | null | false |
Filter.Germ.instAddAction._proof_2 | Mathlib.Order.Filter.Germ.Basic | ∀ {α : Type u_1} {β : Type u_2} {l : Filter α} {M : Type u_3} [inst : AddMonoid M] [inst_1 : AddAction M β] (f : α → β),
0 +ᵥ ↑f = ↑f | null | false |
RingHom.FinitePresentation.comp_surjective | Mathlib.RingTheory.FinitePresentation | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : CommRing C]
{f : A →+* B} {g : B →+* C},
f.FinitePresentation → Function.Surjective ⇑g → (RingHom.ker g).FG → (g.comp f).FinitePresentation | null | true |
CategoryTheory.WithTerminal.coneEquiv._proof_5 | Mathlib.CategoryTheory.WithTerminal.Cone | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_2, u_4} C] {J : Type u_3}
[inst_1 : CategoryTheory.Category.{u_1, u_3} J] {X : C} {K : CategoryTheory.Functor J (CategoryTheory.Over X)}
{X_1 Y : CategoryTheory.Limits.Cone (CategoryTheory.WithTerminal.liftFromOver.obj K)} (f : X_1 ⟶ Y),
CategoryTheory.CategoryS... | null | false |
SubgroupClass.inclusion_self | Mathlib.Algebra.Group.Subgroup.Defs | ∀ {G : Type u_1} [inst : Group G] {S : Type u_4} {H : S} [inst_1 : SetLike S G] [inst_2 : SubgroupClass S G]
[inst_3 : Preorder S] [inst_4 : IsConcreteLE S G] (x : ↥H), (SubgroupClass.inclusion ⋯) x = x | null | true |
Congruent.index_equiv._simp_1 | Mathlib.Topology.MetricSpace.Congruence | ∀ {ι : Type u_1} {ι' : Type u_2} {P₁ : Type u_3} {P₂ : Type u_4} [inst : PseudoEMetricSpace P₁]
[inst_1 : PseudoEMetricSpace P₂] {E : Type u_7} [inst_2 : EquivLike E ι' ι] (f : E) (v₁ : ι → P₁) (v₂ : ι → P₂),
Congruent (v₁ ∘ ⇑f) (v₂ ∘ ⇑f) = Congruent v₁ v₂ | null | false |
diagonal_dotProduct | Mathlib.Data.Matrix.Mul | ∀ {m : Type u_2} {α : Type v} [inst : Fintype m] [inst_1 : DecidableEq m] [inst_2 : NonUnitalNonAssocSemiring α]
(v w : m → α) (i : m), Matrix.diagonal v i ⬝ᵥ w = v i * w i | null | true |
CategoryTheory.ShortComplex.descRightHomology | 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) →
{A : C} →
(k : S.X₂ ⟶ A) →
CategoryTheory.CategoryStruct.comp S.f k = 0 → [inst_2 : S.HasRightHomology] → S.rightHomology... | The morphism `S.rightHomology ⟶ A` obtained from a morphism `k : S.X₂ ⟶ A`
such that `S.f ≫ k = 0.` | true |
CategoryTheory.Limits.Types.limitConeIsLimit._proof_1 | Mathlib.CategoryTheory.Limits.Types.Limits | ∀ {J : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} J] (F : CategoryTheory.Functor J (Type (max u_3 u_2)))
(s : CategoryTheory.Limits.Cone F) (j : J),
CategoryTheory.CategoryStruct.comp
(TypeCat.ofHom fun v => ⟨fun j => (CategoryTheory.ConcreteCategory.hom (s.π.app j)) v, ⋯⟩)
((CategoryTheory.Li... | null | false |
_private.Std.Data.DHashMap.Lemmas.0.Std.DHashMap.size_union_of_not_mem._simp_1_1 | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} {k : α},
(k ∉ m) = (m.contains k = false) | null | false |
List.range'_eq_singleton_iff._simp_1 | Init.Data.List.Nat.Range | ∀ {s n a : ℕ}, (List.range' s n = [a]) = (s = a ∧ n = 1) | null | false |
CommBialgCat.isoEquivBialgEquiv._proof_1 | Mathlib.Algebra.Category.CommBialgCat | ∀ {R : Type u_2} [inst : CommRing R] {X Y : Type u_1} [inst_1 : CommRing X] [inst_2 : Bialgebra R X]
[inst_3 : CommRing Y] [inst_4 : Bialgebra R Y] (x : ↧X ≅ ↧Y),
CommBialgCat.isoMk (CommBialgCat.bialgEquivOfIso x) = CommBialgCat.isoMk (CommBialgCat.bialgEquivOfIso x) | null | false |
AddEquiv.ext_int_iff | Mathlib.Data.Int.Cast.Lemmas | ∀ {A : Type u_4} [inst : AddMonoid A] {f g : ℤ ≃+ A}, f = g ↔ f 1 = g 1 | null | true |
logDeriv.eq_1 | Mathlib.Analysis.Calculus.LogDeriv | ∀ {𝕜 : Type u_1} {𝕜' : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜']
[inst_2 : NormedAlgebra 𝕜 𝕜'] (f : 𝕜 → 𝕜'), logDeriv f = deriv f / f | null | true |
_private.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.0.definition._simp_5._@.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.1878421158._hygCtx._hyg.2 | Mathlib.MeasureTheory.Measure.Typeclasses.Finite | ∀ {α : Type u} (x : α), (x ∈ Set.univ) = True | null | false |
closureAddCommutatorRepresentatives.eq_1 | Mathlib.GroupTheory.Commutator.Basic | ∀ (G : Type u_1) [inst : AddGroup G],
closureAddCommutatorRepresentatives G =
AddSubgroup.closure (Prod.fst '' addCommutatorRepresentatives G ∪ Prod.snd '' addCommutatorRepresentatives G) | null | true |
_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule.0.trapezoidal_error_le_of_lt._simp_1_4 | Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 4] [NeZero 4], (4 = 0) = False | null | false |
ContinuousLinearMap.instMeasurableSpace._proof_1 | Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap | ∀ {F : Type u_1} [inst : NormedAddCommGroup F], IsTopologicalAddGroup F | null | false |
Path.trans_apply | Mathlib.Topology.Path | ∀ {X : Type u_1} [inst : TopologicalSpace X] {x y z : X} (γ : Path x y) (γ' : Path y z) (t : ↑unitInterval),
(γ.trans γ') t = if h : ↑t ≤ 1 / 2 then γ ⟨2 * ↑t, ⋯⟩ else γ' ⟨2 * ↑t - 1, ⋯⟩ | null | true |
mfderivWithin_prodMap | Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_5} [inst_6 : NormedAddComm... | null | true |
RelSeries.head_map | Mathlib.Order.RelSeries | ∀ {α : Type u_1} {r : SetRel α α} {β : Type u_2} {s : SetRel β β} (p : RelSeries r) (f : r.Hom s),
(p.map f).head = f p.head | null | true |
Lean.Server.Completion.HoverInfo.noConfusionType | Lean.Server.Completion.CompletionUtils | Sort u → Lean.Server.Completion.HoverInfo → Lean.Server.Completion.HoverInfo → Sort u | null | false |
_private.Mathlib.Basic.IsEmpty.Basic.0.isEmpty_sum._simp_1_2 | Mathlib.Basic.IsEmpty.Basic | ∀ {α : Type u_1} {β : Type u_2}, Nonempty (α ⊕ β) = (Nonempty α ∨ Nonempty β) | null | false |
IsAddUnit.addUnit_add | Mathlib.Algebra.Group.Units.Defs | ∀ {M : Type u_1} [inst : AddMonoid M] {a b : M} (ha : IsAddUnit a) (hb : IsAddUnit b),
⋯.addUnit = ha.addUnit + hb.addUnit | null | true |
MonoidHom.finsuppProd_apply | Mathlib.Algebra.BigOperators.Finsupp.Basic | ∀ {α : Type u_1} {β : Type u_7} {N : Type u_10} {P : Type u_11} [inst : Zero β] [inst_1 : MulOneClass N]
[inst_2 : CommMonoid P] (f : α →₀ β) (g : α → β → N →* P) (x : N), (f.prod g) x = f.prod fun i fi => (g i fi) x | null | true |
CategoryTheory.Limits.Types.Small.productLimitCone | Mathlib.CategoryTheory.Limits.Types.Products | {J : Type v} → (F : J → Type u) → [Small.{u, v} J] → CategoryTheory.Limits.LimitCone (CategoryTheory.Discrete.functor F) | A variant of `productLimitCone` using a `Small` hypothesis rather than a function to `Type`.
| true |
Std.Do.SPred.entails.refl._simp_1 | Std.Do.SPred.Laws | ∀ {σs : List (Type u)} (P : Std.Do.SPred σs), (P ⊢ₛ P) = True | null | false |
abs_pow_eq_one | Mathlib.Algebra.Order.Ring.Abs | ∀ {α : Type u_1} [inst : Ring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {n : ℕ} (a : α),
n ≠ 0 → (|a ^ n| = 1 ↔ |a| = 1) | null | true |
Pi.uniformSpace.eq_1 | Mathlib.Topology.UniformSpace.UniformConvergenceTopology | ∀ {ι : Type u_1} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)],
Pi.uniformSpace α =
UniformSpace.ofCoreEq (⨅ i, UniformSpace.comap (Function.eval i) (U i)).toCore Pi.topologicalSpace ⋯ | null | true |
Aesop.RappId.instLT | Aesop.Tree.Data | LT Aesop.RappId | null | true |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftLeft.match_1 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.ShiftLeft | {α : Type} →
[inst : Hashable α] →
[inst_1 : DecidableEq α] →
{w : ℕ} →
(aig : Std.Sat.AIG α) →
(motive : aig.ArbitraryShiftTarget w → Sort u_1) →
(target : aig.ArbitraryShiftTarget w) →
((n : ℕ) →
(input : aig.RefVec w) →
(di... | null | false |
mem_absConvexHull_iff | Mathlib.Analysis.LocallyConvex.AbsConvex | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : SeminormedRing 𝕜] [inst_1 : SMul 𝕜 E] [inst_2 : AddCommMonoid E]
[inst_3 : PartialOrder 𝕜] {s : Set E} {x : E}, x ∈ (absConvexHull 𝕜) s ↔ ∀ (t : Set E), s ⊆ t → AbsConvex 𝕜 t → x ∈ t | null | true |
_private.Lean.Meta.Basic.0.Lean.Meta.getConstTemp?._sparseCasesOn_1 | Lean.Meta.Basic | {motive : Lean.ConstantInfo → Sort u} →
(t : Lean.ConstantInfo) →
((val : Lean.TheoremVal) → motive (Lean.ConstantInfo.thmInfo val)) →
((val : Lean.DefinitionVal) → motive (Lean.ConstantInfo.defnInfo val)) →
(Nat.hasNotBit 6 t.ctorIdx → motive t) → motive t | null | false |
Polynomial.eval₂_finset_sum | Mathlib.Algebra.Polynomial.Eval.Defs | ∀ {R : Type u} {S : Type v} {ι : Type y} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) (s : Finset ι)
(g : ι → Polynomial R) (x : S), Polynomial.eval₂ f x (∑ i ∈ s, g i) = ∑ i ∈ s, Polynomial.eval₂ f x (g i) | **Alias** of `Polynomial.eval₂_finsetSum`. | true |
Aesop.ForwardHypData | Aesop.RuleTac.Forward.Basic | Type | null | true |
CoeT.mk.noConfusion | Init.Coe | {α : Sort u} →
{x : α} → {β : Sort v} → {P : Sort u_1} → {coe coe' : β} → { coe := coe } = { coe := coe' } → (coe ≍ coe' → P) → P | null | false |
Lean.MetavarContext.MkBinding.Context.noConfusionType | Lean.MetavarContext | Sort u → Lean.MetavarContext.MkBinding.Context → Lean.MetavarContext.MkBinding.Context → Sort u | null | false |
WeierstrassCurve.Jacobian.addZ_self | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | ∀ {R : Type r} [inst : CommRing R] (P : Fin 3 → R), WeierstrassCurve.Jacobian.addZ P P = 0 | null | true |
_private.Init.Data.List.Sort.Basic.0.List.mergeSort.match_1.congr_eq_2 | Init.Data.Array.Sort.Lemmas | ∀ {α : Type u_1} (motive : List α → (α → α → Bool) → Sort u_2) (x : List α) (x_1 : α → α → Bool)
(h_1 : (x : α → α → Bool) → motive [] x) (h_2 : (a : α) → (x : α → α → Bool) → motive [a] x)
(h_3 : (a b : α) → (xs : List α) → (le : α → α → Bool) → motive (a :: b :: xs) le) (a : α) (x_2 : α → α → Bool),
x = [a] →
... | null | true |
Nat.getD_toList_roc_eq_fallback | Init.Data.Range.Polymorphic.NatLemmas | ∀ {m n i fallback : ℕ}, n ≤ i + m → (m<...=n).toList.getD i fallback = fallback | null | true |
_private.Mathlib.Algebra.TrivSqZeroExt.Basic.0.TrivSqZeroExt.isUnit_inv_iff._simp_1_3 | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] {a : G₀}, (a⁻¹ = 0) = (a = 0) | null | false |
_private.Mathlib.CategoryTheory.WithTerminal.Basic.0.CategoryTheory.WithTerminal.map.match_1.eq_3 | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C]
(motive : (X Y : CategoryTheory.WithTerminal C) → (X ⟶ Y) → Sort u_3)
(x : CategoryTheory.WithTerminal.star ⟶ CategoryTheory.WithTerminal.star)
(h_1 :
(a a_1 : C) →
(f : CategoryTheory.WithTerminal.of a ⟶ CategoryTheory.WithTerminal.of a_1) ... | null | true |
Lean.Parser.Term.optSemicolon.parenthesizer | Lean.Parser.Term | Lean.PrettyPrinter.Parenthesizer → Lean.PrettyPrinter.Parenthesizer | null | true |
Finset.biUnion_congr | Mathlib.Data.Finset.Union | ∀ {α : Type u_1} {β : Type u_2} {s₁ s₂ : Finset α} {t₁ t₂ : α → Finset β} [inst : DecidableEq β],
s₁ = s₂ → (∀ a ∈ s₁, t₁ a = t₂ a) → s₁.biUnion t₁ = s₂.biUnion t₂ | null | true |
_private.Mathlib.GroupTheory.Perm.Centralizer.0.Equiv.Perm.OnCycleFactors.nat_card_range_toPermHom._simp_1_7 | Mathlib.GroupTheory.Perm.Centralizer | ∀ {α : Type u_1} {β : Type v} {f : α → β} {b : β} {s : Multiset α}, (b ∈ Multiset.map f s) = ∃ a ∈ s, f a = b | null | false |
Std.TreeSet.forIn_eq_forIn | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} {δ : Type w} {m : Type w → Type w'} [inst : Monad m]
[LawfulMonad m] {f : α → δ → m (ForInStep δ)} {init : δ}, Std.TreeSet.forIn f init t = forIn t init f | null | true |
Rep.standardComplex.hint._@.Mathlib.RepresentationTheory.Homological.Resolution.927499924._hygCtx._hyg.13 | Mathlib.RepresentationTheory.Homological.Resolution | (G : Type u) → {n : ℕ} → [Monoid G] → Prop | null | false |
Matrix.submatrix_gram | Mathlib.Analysis.InnerProductSpace.GramMatrix | ∀ {E : Type u_1} {n : Type u_2} {𝕜 : Type u_4} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] (v : n → E) {m : Set n} (f : ↑m → n),
(Matrix.gram 𝕜 v).submatrix f f = Matrix.gram 𝕜 (v ∘ f) | null | true |
Subalgebra.frontier_spectrum | Mathlib.Analysis.Normed.Algebra.Spectrum | ∀ {𝕜 : Type u_3} {A : Type u_4} {SA : Type u_5} [inst : NormedRing A] [CompleteSpace A] [inst_2 : SetLike SA A]
[inst_3 : SubringClass SA A] [inst_4 : NormedField 𝕜] [inst_5 : NormedAlgebra 𝕜 A] [instSMulMem : SMulMemClass SA 𝕜 A]
(S : SA) [hS : IsClosed ↑S] (x : ↥S), frontier (spectrum 𝕜 x) ⊆ spectrum 𝕜 ↑x | If `S : Subalgebra 𝕜 A` is a closed subalgebra of a Banach algebra `A`, then for any
`x : S`, the boundary of the spectrum of `x` relative to `S` is a subset of the spectrum of
`↑x : A` relative to `A`. | true |
HahnSeries.SummableFamily.hasFiniteSupport_smul | Mathlib.RingTheory.HahnSeries.Summable | ∀ {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} {α : Type u_5} {β : Type u_6} [inst : PartialOrder Γ]
[inst_1 : PartialOrder Γ'] [inst_2 : AddCommMonoid V] [inst_3 : AddCommMonoid R] [inst_4 : SMulWithZero R V]
(s : HahnSeries.SummableFamily Γ R α) (t : HahnSeries.SummableFamily Γ' V β) (gh : Γ × Γ')... | null | true |
ProbabilityTheory.IsFiniteKernel.exists_univ_le | Mathlib.Probability.Kernel.Defs | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
[self : ProbabilityTheory.IsFiniteKernel κ], ∃ C < ⊤, ∀ (a : α), (κ a) Set.univ ≤ C | null | true |
Lean.Compiler.LCNF.SimpleGroundExpr.reference.noConfusion | Lean.Compiler.LCNF.SimpleGroundExpr | {P : Sort u} →
{n n' : Lean.Name} →
Lean.Compiler.LCNF.SimpleGroundExpr.reference n = Lean.Compiler.LCNF.SimpleGroundExpr.reference n' →
(n = n' → P) → P | null | false |
CompletelyPositiveMap.map_cstarMatrix_nonneg' | Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | ∀ {A₁ : Type u_1} {A₂ : Type u_2} [inst : NonUnitalCStarAlgebra A₁] [inst_1 : NonUnitalCStarAlgebra A₂]
[inst_2 : PartialOrder A₁] [inst_3 : PartialOrder A₂] [inst_4 : StarOrderedRing A₁] [inst_5 : StarOrderedRing A₂]
(self : CompletelyPositiveMap A₁ A₂) (k : ℕ) (M : CStarMatrix (Fin k) (Fin k) A₁), 0 ≤ M → 0 ≤ M.m... | null | true |
CategoryTheory.Functor.IsLocallyDirected.casesOn | Mathlib.CategoryTheory.LocallyDirected | {J : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} J] →
{F : CategoryTheory.Functor J (Type u_2)} →
{motive : F.IsLocallyDirected → Sort u} →
(t : F.IsLocallyDirected) →
((cond :
∀ {i j k : J} (fi : i ⟶ k) (fj : j ⟶ k) (xi : F.obj i) (xj : F.obj j),
... | null | false |
contDiffOn_clm_apply | Mathlib.Analysis.Calculus.ContDiff.FiniteDimension | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {D : Type uD} [inst_1 : NormedAddCommGroup D]
[inst_2 : NormedSpace 𝕜 D] {E : Type uE} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {F : Type uF}
[inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace 𝕜 F] {n : WithTop ℕ∞} [CompleteSpace 𝕜] {f :... | A family of continuous linear maps is `C^n` on `s` if and only if all its applications are. | true |
CategoryTheory.Functor.obj.ζ_def | Mathlib.CategoryTheory.Monoidal.Mon | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C] {D : Type u₂}
[inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.MonoidalCategory D]
{F : CategoryTheory.Functor C D} [inst_4 : F.LaxMonoidal] (X : C) [inst_5 : CategoryTheory.AddMonObj X],
... | null | true |
Lean.Meta.hasAssignableMVar | Lean.Meta.HasAssignableMVar | Lean.Expr → Lean.MetaM Bool | Return `true` iff expression contains a metavariable that can be assigned. | true |
ZMod.intCast_eq_zero_iff_even | Mathlib.Data.ZMod.Basic | ∀ {n : ℤ}, ↑n = 0 ↔ Even n | null | true |
_private.Mathlib.Computability.StateTransition.0.StateTransition.tr_eval'.match_1_1 | Mathlib.Computability.StateTransition | ∀ {σ₁ σ₂ : Type u_1} (f₁ : σ₁ → Option σ₁) (tr : σ₁ → σ₂) (a₁ : σ₁) (b₂ : σ₂)
(motive : (∃ b₁, tr b₁ = b₂ ∧ b₁ ∈ StateTransition.eval f₁ a₁) → Prop)
(x : ∃ b₁, tr b₁ = b₂ ∧ b₁ ∈ StateTransition.eval f₁ a₁),
(∀ (b₁ : σ₁) (bb : tr b₁ = b₂) (hb : b₁ ∈ StateTransition.eval f₁ a₁), motive ⋯) → motive x | null | false |
Matroid.Indep.closure_eq_setOfPred_isBasis_insert | Mathlib.Combinatorics.Matroid.Closure | ∀ {α : Type u_2} {M : Matroid α} {I : Set α}, M.Indep I → M.closure I = {x | M.IsBasis I (insert x I)} | null | true |
ContinuousLinearMap.mulLeftRight._proof_3 | Mathlib.Analysis.Normed.Operator.Mul | ∀ (𝕜 : Type u_2) [inst : NontriviallyNormedField 𝕜] (R : Type u_1) [inst_1 : NonUnitalSeminormedRing R]
[inst_2 : NormedSpace 𝕜 R], IsTopologicalAddGroup (R →L[𝕜] R) | null | false |
Std.HashMap.getKey!_filterMap | Std.Data.HashMap.Lemmas | ∀ {α : Type u} {β : Type v} {γ : Type w} {x : BEq α} {x_1 : Hashable α} {m : Std.HashMap α β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] [inst_2 : Inhabited α] {f : α → β → Option γ} {k : α},
(Std.HashMap.filterMap f m).getKey! k = ((m.getKey? k).pfilter fun x_2 h' => (f x_2 m[x_2]).isSome).get! | null | true |
Bifunctor.mapEquiv_refl_refl | Mathlib.Logic.Equiv.Functor | ∀ {α : Type u} {α' : Type v} (F : Type u → Type v → Type w) [inst : Bifunctor F] [inst_1 : LawfulBifunctor F],
Bifunctor.mapEquiv F (Equiv.refl α) (Equiv.refl α') = Equiv.refl (F α α') | null | true |
Std.TreeMap.Raw.getKey?_insert_self | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap.Raw α β cmp} [Std.TransCmp cmp],
t.WF → ∀ {k : α} {v : β}, (t.insert k v).getKey? k = some k | null | true |
DistLat.mk.injEq | Mathlib.Order.Category.DistLat | ∀ (carrier : Type u_1) [str : DistribLattice carrier] (carrier_1 : Type u_1) (str_1 : DistribLattice carrier_1),
({ carrier := carrier, str := str } = { carrier := carrier_1, str := str_1 }) = (carrier = carrier_1 ∧ str ≍ str_1) | null | true |
DiffContOnCl.ball_subset_image_closedBall | Mathlib.Analysis.Complex.OpenMapping | ∀ {f : ℂ → ℂ} {z₀ : ℂ} {ε r : ℝ},
DiffContOnCl ℂ f (Metric.ball z₀ r) →
0 < r →
(∀ z ∈ Metric.sphere z₀ r, ε ≤ ‖f z - f z₀‖) →
(∃ᶠ (z : ℂ) in nhds z₀, f z ≠ f z₀) → Metric.ball (f z₀) (ε / 2) ⊆ f '' Metric.closedBall z₀ r | If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. | true |
PartialEquiv.disjointUnion._proof_5 | Mathlib.Logic.Equiv.PartialEquiv | ∀ {α : Type u_2} {β : Type u_1} (e e' : PartialEquiv α β), e.target.ite e.target e'.target = e.target ∪ e'.target | null | false |
NoetherNormalization.T1._proof_1 | Mathlib.RingTheory.NoetherNormalization | ∀ {n : ℕ}, NeZero (n + 1) | null | false |
_private.Mathlib.Probability.ProbabilityMassFunction.Monad.0.PMF.bindOnSupport_eq_zero_iff._simp_1_1 | Mathlib.Probability.ProbabilityMassFunction.Monad | ∀ {α : Type u_1} {f : α → ENNReal}, (∑' (i : α), f i = 0) = ∀ (i : α), f i = 0 | null | false |
Homotopy.mkCoinductiveAux₂._proof_1 | Mathlib.Algebra.Homology.Homotopy | ∀ {V : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} V] [inst_1 : CategoryTheory.Preadditive V]
{P Q : CochainComplex V ℕ} (e : P ⟶ Q) (zero : P.X 1 ⟶ Q.X 0),
e.f 0 = CategoryTheory.CategoryStruct.comp (P.d 0 1) zero →
e.f 0 =
CategoryTheory.CategoryStruct.comp 0 (HomologicalComplex.dTo Q 0) +
... | null | false |
BitVec.toNat_intMin | Init.Data.BitVec.Lemmas | ∀ {w : ℕ}, (BitVec.intMin w).toNat = 2 ^ (w - 1) % 2 ^ w | The RHS is zero in case `w = 0` which is modeled by wrapping the expression in `... % 2 ^ w`.
| true |
HMul.hMul | Init.Prelude | {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ | `a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent.
Conventions for notations in identifiers:
* The recommended spelling of `*` in identifiers is `mul`. | true |
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