name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
NonUnitalStarAlgebra.elemental | Mathlib.Topology.Algebra.NonUnitalStarAlgebra | (R : Type u_1) →
{A : Type u_2} →
[inst : CommSemiring R] →
[inst_1 : StarRing R] →
[inst_2 : NonUnitalSemiring A] →
[inst_3 : StarRing A] →
[inst_4 : Module R A] →
[IsScalarTower R A A] →
[SMulCommClass R A A] →
[StarModule R A] ... | The topological closure of the non-unital star subalgebra generated by a single element. | true |
Std.Async.TCP.Socket.Client._sizeOf_1 | Std.Async.TCP | Std.Async.TCP.Socket.Client → ℕ | null | false |
Left.nsmul_nonpos | Mathlib.Algebra.Order.Monoid.Unbundled.Pow | ∀ {M : Type u_3} [inst : AddMonoid M] [inst_1 : Preorder M] [AddLeftMono M] {a : M}, a ≤ 0 → ∀ (n : ℕ), n • a ≤ 0 | null | true |
inhomogeneousCochains.d._proof_3 | Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | ∀ {k G : Type u_1} [inst : CommRing k] [inst_1 : Monoid G] (A : Rep.{u_2, u_1, u_1} k G) (n : ℕ) (r : k)
(f : (Fin n → G) → ↑A),
(fun g =>
(A.ρ (g 0)) ((r • f) fun i => g i.succ) +
∑ j, (-1) ^ (↑j + 1) • (r • f) (j.contractNth (fun x1 x2 => x1 * x2) g)) =
(RingHom.id k) r • fun g =>
(A.ρ (g ... | null | false |
LinearRecurrence.mkSol._proof_3 | Mathlib.Algebra.LinearRecurrence | ∀ {R : Type u_1} [inst : CommSemiring R] (E : LinearRecurrence R) (a : ℕ) (k : Fin E.order),
a - E.order + ↑k < a → InvImage (fun x1 x2 => x1 < x2) (fun x => x) (a - E.order + ↑k) a | null | false |
Submodule.mul_le | Mathlib.Algebra.Algebra.Operations | ∀ {R : Type u} [inst : Semiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Module R A]
[inst_3 : IsScalarTower R A A] {M N P : Submodule R A}, M * N ≤ P ↔ ∀ m ∈ M, ∀ n ∈ N, m * n ∈ P | null | true |
Equiv.pSigmaAssoc._proof_2 | Mathlib.Logic.Equiv.Defs | ∀ {α : Sort u_1} {β : α → Sort u_2} (γ : (a : α) → β a → Sort u_3),
Function.RightInverse (fun x => ⟨⟨x.fst, x.snd.fst⟩, x.snd.snd⟩) fun x => ⟨x.fst.fst, ⟨x.fst.snd, x.snd⟩⟩ | null | false |
_private.Mathlib.Order.Monotone.Basic.0.Nat.stabilises_of_antitone._proof_1_3 | Mathlib.Order.Monotone.Basic | ∀ {f : ℕ → ℕ}, f 0 = f 0 | null | false |
OpenPartialHomeomorph.univUnitBall_apply | Mathlib.Analysis.Normed.Module.Ball.Homeomorph | ∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] (x : E),
↑OpenPartialHomeomorph.univUnitBall x = (√(1 + ‖x‖ ^ 2))⁻¹ • x | null | true |
Int32.toISize_le | Init.Data.SInt.Lemmas | ∀ {a b : Int32}, a.toISize ≤ b.toISize ↔ a ≤ b | null | true |
_private.Mathlib.LinearAlgebra.LinearIndependent.Defs.0.not_linearIndependent_iffₛ._simp_1_2 | Mathlib.LinearAlgebra.LinearIndependent.Defs | ∀ {α : Sort u_1} {p : α → Prop}, (¬∀ (x : α), p x) = ∃ x, ¬p x | null | false |
ModuleCat.MonoidalCategory.instBraidedSemimoduleCatFunctorEquivalenceSemimoduleCat | Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric | {R : Type u} → [inst : CommRing R] → ModuleCat.equivalenceSemimoduleCat.functor.Braided | null | true |
LSeries_one_mul_Lseries_moebius | Mathlib.NumberTheory.LSeries.Dirichlet | ∀ {s : ℂ}, 1 < s.re → LSeries 1 s * LSeries (fun n => ↑(ArithmeticFunction.moebius n)) s = 1 | The L-series of the constant sequence `1` and of the Möbius function are inverses. | true |
Std.DTreeMap.Internal.Impl.Const.toListModel_alter! | Std.Data.DTreeMap.Internal.WF.Lemmas | ∀ {α : Type u} {β : Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{t : Std.DTreeMap.Internal.Impl α fun x => β} {a : α} {f : Option β → Option β},
t.Balanced →
t.Ordered →
(Std.DTreeMap.Internal.Impl.Const.alter! a f t).toListModel.Perm
(Std.Internal.List.Const.alte... | null | true |
_private.Mathlib.Topology.Instances.ENNReal.Lemmas.0.ENNReal.continuous_pow._simp_1_1 | Mathlib.Topology.Instances.ENNReal.Lemmas | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {y : Y},
(Continuous fun x => y) = True | null | false |
PFun.graph' | Mathlib.Data.PFun | {α : Type u_1} → {β : Type u_2} → (α →. β) → SetRel α β | Graph of a partial function as a relation. `x` and `y` are related iff `f x` is defined and
"equals" `y`. | true |
AlgebraicGeometry.Scheme.isoOfEq_inv_ι_assoc | Mathlib.AlgebraicGeometry.Restrict | ∀ (X : AlgebraicGeometry.Scheme) {U V : X.Opens} (e : U = V) {Z : AlgebraicGeometry.Scheme} (h : X ⟶ Z),
CategoryTheory.CategoryStruct.comp (X.isoOfEq e).inv (CategoryTheory.CategoryStruct.comp U.ι h) =
CategoryTheory.CategoryStruct.comp V.ι h | null | true |
CategoryTheory.GrpObj.lift_inv_left_eq | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{A B : C} [inst_2 : CategoryTheory.GrpObj B] (f g h : A ⟶ B),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.CartesianMonoidalCategory.lift (CategoryTheory.CategoryStruct.comp f CategoryTheo... | null | true |
ModuleCat.HasColimit.reflectsColimit | Mathlib.Algebra.Category.ModuleCat.Colimits | ∀ {R : Type w} [inst : Ring R] {J : Type u} [inst_1 : CategoryTheory.Category.{v, u} J]
(F : CategoryTheory.Functor J (ModuleCat R))
[CategoryTheory.Limits.HasColimit (F.comp (CategoryTheory.forget₂ (ModuleCat R) AddCommGrpCat))],
CategoryTheory.Limits.ReflectsColimit F (CategoryTheory.forget₂ (ModuleCat R) AddCo... | null | true |
Std.TreeMap.getElem?_eq_some_getElem | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {a : α} (h : a ∈ t),
t[a]? = some t[a] | null | true |
Nat.compare_eq_ite_lt | Init.Data.Nat.Compare | ∀ (a b : ℕ), compare a b = if a < b then Ordering.lt else if b < a then Ordering.gt else Ordering.eq | null | true |
num_smul_one_lt_den_smul_add | Mathlib.Data.Real.Embedding | ∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [IsOrderedAddMonoid M] [inst_3 : One M] {u v : ℚ}
{x y : M}, u.num • 1 < u.den • x → v.num • 1 < v.den • y → (u + v).num • 1 < (u + v).den • (x + y) | For `u v : ℚ` and `x y : M`, one can informally write
`u < x → v < y → u + v < x + y`. We formalize this using smul. | true |
_private.Mathlib.Algebra.Homology.HomotopyCategory.MappingCone.0.CochainComplex.mappingCone.lift_f_snd_v._proof_1_1 | Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | ∀ (p q : ℤ), p + 0 = q → q = p | null | false |
Int.natAbs_one | Init.Data.Int.Order | Int.natAbs 1 = 1 | null | true |
Aesop.ForwardRuleMatch.mk.inj | Aesop.Forward.Match.Types | ∀ {rule : Aesop.ForwardRule} {«match» : Aesop.CompleteMatch} {rule_1 : Aesop.ForwardRule}
{match_1 : Aesop.CompleteMatch},
{ rule := rule, «match» := «match» } = { rule := rule_1, «match» := match_1 } → rule = rule_1 ∧ «match» = match_1 | null | true |
RingAut.toAddAut._proof_1 | Mathlib.Algebra.Ring.Aut | ∀ (R : Type u_1) [inst : Mul R] [inst_1 : Add R], RingEquiv.toAddEquiv 1 = RingEquiv.toAddEquiv 1 | null | false |
CategoryTheory.Limits.ChosenPullback.mk._flat_ctor | Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{X₁ X₂ S : C} →
{f₁ : X₁ ⟶ S} →
{f₂ : X₂ ⟶ S} →
(pullback : C) →
(p₁ : pullback ⟶ X₁) →
(p₂ : pullback ⟶ X₂) →
(condition : CategoryTheory.CategoryStruct.comp p₁ f₁ = CategoryTheory.Categor... | null | false |
Matrix.submatrixEquivInvertible._proof_1 | Mathlib.LinearAlgebra.Matrix.NonsingularInverse | ∀ {m : Type u_3} {n : Type u_1} {α : Type u_2} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing α]
[inst_3 : Fintype m] [inst_4 : DecidableEq m] (A : Matrix m m α) (e₁ e₂ : n ≃ m) [inst_5 : Invertible A],
A.submatrix ⇑e₁ ⇑e₂ * (⅟A).submatrix ⇑e₂ ⇑e₁ = 1 | null | false |
List.nodup_permutations_iff | Mathlib.Data.List.Permutation | ∀ {α : Type u_1} {s : List α}, s.permutations.Nodup ↔ s.Nodup | null | true |
Plausible.TotalFunction.zeroDefault.eq_1 | Mathlib.Testing.Plausible.Functions | ∀ {α : Type u} {β : Type v} [inst : Zero β] (A : List ((_ : α) × β)) (a : β),
(Plausible.TotalFunction.withDefault A a).zeroDefault = Plausible.TotalFunction.withDefault A 0 | null | true |
Associates.one_or_eq_of_le_of_prime | Mathlib.Algebra.GroupWithZero.Associated | ∀ {M : Type u_1} [inst : CommMonoidWithZero M] [IsCancelMulZero M] {p m : Associates M}, Prime p → m ≤ p → m = 1 ∨ m = p | null | true |
Set.infsep_of_fintype._proof_1 | Mathlib.Topology.MetricSpace.Infsep | ∀ {α : Type u_1} {s : Set α} [inst : Fintype ↑s], s.Nontrivial → s.offDiag.toFinset.Nonempty | null | false |
NormedAddGroupHom.inhabited | Mathlib.Analysis.Normed.Group.Hom | {V₁ : Type u_2} →
{V₂ : Type u_3} →
[inst : SeminormedAddCommGroup V₁] → [inst_1 : SeminormedAddCommGroup V₂] → Inhabited (NormedAddGroupHom V₁ V₂) | null | true |
MulArchimedeanClass.liftOrderHom.eq_1 | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : CommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedMonoid M] {α : Type u_2}
[inst_3 : PartialOrder α] (f : M → α)
(h : ∀ (a b : M), MulArchimedeanClass.mk a ≤ MulArchimedeanClass.mk b → f a ≤ f b),
MulArchimedeanClass.liftOrderHom f h = { toFun := MulArchimedeanClass.lift f ⋯, monot... | null | true |
LieAlgebra.Orthogonal.soIndefiniteEquiv._proof_1 | Mathlib.Algebra.Lie.Classical | ∀ (p : Type u_1) (q : Type u_2) (R : Type u_3) [inst : DecidableEq p] [inst_1 : DecidableEq q] [inst_2 : CommRing R]
[inst_3 : Fintype p] [inst_4 : Fintype q] {i : R},
i * i = -1 →
↑(skewAdjointMatricesLieSubalgebra
((LieAlgebra.Orthogonal.Pso p q R i).transpose * LieAlgebra.Orthogonal.indefiniteDiago... | null | false |
AbstractCompletion.extend_unique | Mathlib.Topology.UniformSpace.AbstractCompletion | ∀ {α : Type uα} [inst : UniformSpace α] (pkg : AbstractCompletion.{vα, uα} α) {β : Type uβ} [inst_1 : UniformSpace β]
{f : α → β} [CompleteSpace β] [T0Space β],
UniformContinuous f → ∀ {g : pkg.space → β}, UniformContinuous g → (∀ (a : α), f a = g (pkg.coe a)) → pkg.extend f = g | null | true |
partialOrderOfSO.match_5 | Mathlib.Order.RelClasses | ∀ {α : Type u_1} (r : α → α → Prop) (x : α) (motive : (y : α) → x = y ∨ r x y → y = x ∨ r y x → Prop) (y : α)
(h₁ : x = y ∨ r x y) (h₂ : y = x ∨ r y x),
(∀ (x_1 : x = x ∨ r x x), motive x ⋯ x_1) →
(∀ (x_1 : x = x ∨ r x x), motive x x_1 ⋯) →
(∀ (x_1 : α) (h₁ : r x x_1) (h₂ : r x_1 x), motive x_1 ⋯ ⋯) → mot... | null | false |
Set.mem_cIcc | Mathlib.Order.Circular | ∀ {α : Type u_1} [inst : CircularPreorder α] {a b x : α}, x ∈ Set.cIcc a b ↔ btw a x b | null | true |
Asymptotics.IsBigOTVS.add | Mathlib.Analysis.Asymptotics.TVS | ∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
[inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E] [inst_3 : Module 𝕜 E] [inst_4 : AddCommGroup F]
[inst_5 : TopologicalSpace F] [inst_6 : Module 𝕜 F] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {f₁ f₂ : α → E} {g ... | null | true |
Set.Finite.inter_of_left | Mathlib.Data.Set.Finite.Basic | ∀ {α : Type u} {s : Set α}, s.Finite → ∀ (t : Set α), (s ∩ t).Finite | null | true |
RingOfIntegers.exponent_eq_one_iff | Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | ∀ {K : Type u_1} [inst : Field K] {θ : NumberField.RingOfIntegers K}, RingOfIntegers.exponent θ = 1 ↔ ℤ[θ] = ⊤ | null | true |
OrderMonoidIso.unitsCongr | Mathlib.Algebra.Order.Hom.Units | {α : Type u_1} →
{β : Type u_2} →
[inst : Preorder α] → [inst_1 : Monoid α] → [inst_2 : Preorder β] → [inst_3 : Monoid β] → (α ≃*o β) → αˣ ≃*o βˣ | An isomorphism of ordered monoids descends to their units. | true |
thickenedIndicatorAux.eq_1 | Mathlib.Topology.MetricSpace.ThickenedIndicator | ∀ {α : Type u_1} [inst : PseudoEMetricSpace α] (δ : ℝ) (E : Set α) (x : α),
thickenedIndicatorAux δ E x = 1 - Metric.infEDist x E / ENNReal.ofReal δ | null | true |
ValuationSubring.primeSpectrumEquiv_apply | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] (A : ValuationSubring K) (P : PrimeSpectrum ↥A),
A.primeSpectrumEquiv P = ⟨A.ofPrime P.asIdeal, ⋯⟩ | null | true |
Std.DTreeMap.Raw.ordered_keys | Std.Data.DTreeMap.Raw.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap.Raw α β cmp} [Std.TransCmp cmp],
t.WF → List.Pairwise (fun a b => cmp a b = Ordering.lt) t.keys | null | true |
_private.Mathlib.CategoryTheory.Subfunctor.Basic.0.CategoryTheory.instCompleteLatticeSubfunctor._simp_5 | Mathlib.CategoryTheory.Subfunctor.Basic | ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i | null | false |
GradedTensorProduct.instRing._proof_17 | Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | ∀ {R : Type u_3} {ι : Type u_4} {A : Type u_1} {B : Type u_2} [inst : CommSemiring ι] [inst_1 : DecidableEq ι]
[inst_2 : CommRing R] [inst_3 : Ring A] [inst_4 : Ring B] [inst_5 : Algebra R A] [inst_6 : Algebra R B]
(𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [inst_7 : GradedAlgebra 𝒜] [inst_8 : GradedAlgebra ... | null | false |
Metric.sphere_subset_closedBall | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, Metric.sphere x ε ⊆ Metric.closedBall x ε | null | true |
CategoryTheory.FreeBicategory.normalizeIso_comp | Mathlib.CategoryTheory.Bicategory.Coherence | ∀ {B : Type u} [inst : Quiver B] {a : B} {b c d : CategoryTheory.FreeBicategory B} (p : Quiver.Path a b) (f : b ⟶ c)
(g : c ⟶ d),
CategoryTheory.FreeBicategory.normalizeIso p (CategoryTheory.CategoryStruct.comp f g) =
(CategoryTheory.Bicategory.associator ((CategoryTheory.FreeBicategory.preinclusion B).map { as... | null | true |
Lean.Meta.Grind.Arith.CommRing.MonadRing.recOn | Lean.Meta.Tactic.Grind.Arith.CommRing.MonadRing | {m : Type → Type} →
{motive : Lean.Meta.Grind.Arith.CommRing.MonadRing m → Sort u} →
(t : Lean.Meta.Grind.Arith.CommRing.MonadRing m) →
((getRing : m Lean.Meta.Grind.Arith.CommRing.Ring) →
(modifyRing : (Lean.Meta.Grind.Arith.CommRing.Ring → Lean.Meta.Grind.Arith.CommRing.Ring) → m Unit) →
... | null | false |
Complex.tsum_exp_neg_quadratic | Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | ∀ {a : ℂ},
0 < a.re →
∀ (b : ℂ),
∑' (n : ℤ), Complex.exp (-↑Real.pi * a * ↑n ^ 2 + 2 * ↑Real.pi * b * ↑n) =
1 / a ^ (1 / 2) * ∑' (n : ℤ), Complex.exp (-↑Real.pi / a * (↑n + Complex.I * b) ^ 2) | Jacobi's theta-function transformation formula for the sum of `exp -Q(x)`, where `Q` is a
negative definite quadratic form. | true |
_private.Mathlib.Algebra.Lie.Semisimple.Basic.0.LieAlgebra.IsSemisimple.isSimple_of_isAtom._simp_1_7 | Mathlib.Algebra.Lie.Semisimple.Basic | ∀ {α : Sort u_1} {p : α → Prop} {a' : α}, (∃ a, p a ∧ a = a') = p a' | null | false |
CauSeq.IsComplete.isComplete | Mathlib.Algebra.Order.CauSeq.Completion | ∀ {α : Type u_1} {inst : Field α} {inst_1 : LinearOrder α} {inst_2 : IsStrictOrderedRing α} {β : Type u_2}
{inst_3 : Ring β} {abv : β → α} {inst_4 : IsAbsoluteValue abv} [self : CauSeq.IsComplete β abv] (s : CauSeq β abv),
∃ b, s ≈ CauSeq.const abv b | Every Cauchy sequence has a limit. | true |
exists_isClopen_upper_or_lower_of_ne | Mathlib.Topology.Order.Priestley | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : PartialOrder α] [PriestleySpace α] {x y : α},
x ≠ y → ∃ U, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U | null | true |
SeparationQuotient.exists_out_continuousLinearMap | Mathlib.Topology.Algebra.SeparationQuotient.Section | ∀ (K : Type u_1) (E : Type u_2) [inst : DivisionRing K] [inst_1 : AddCommGroup E] [inst_2 : Module K E]
[inst_3 : TopologicalSpace E] [inst_4 : IsTopologicalAddGroup E] [inst_5 : ContinuousConstSMul K E],
∃ f, SeparationQuotient.mkCLM K E ∘SL f = ContinuousLinearMap.id K (SeparationQuotient E) | There exists a continuous `K`-linear map from `SeparationQuotient E` to `E`
such that `mk (outCLM x) = x` for all `x`.
Note that continuity of this map comes for free, because `mk` is a topology inducing map.
| true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate.0.SSet.ιNormalizedChainComplex._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate | IsRightCancelAdd ℕ | null | false |
CategoryTheory.CoreSmallCategoryOfSet.functor | Mathlib.CategoryTheory.SmallRepresentatives | {Ω : Type w} →
{C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(h : CategoryTheory.CoreSmallCategoryOfSet Ω C) → CategoryTheory.Functor (↑h.smallCategoryOfSet.obj) C | Given `h : CoreSmallCategoryOfSet Ω C`, this is the
obvious functor `h.smallCategoryOfSet.obj ⥤ C`. | true |
DFinsupp.coe_sup | Mathlib.Data.DFinsupp.Order | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Zero (α i)] [inst_1 : (i : ι) → SemilatticeSup (α i)]
(f g : Π₀ (i : ι), α i), ⇑(f ⊔ g) = ⇑f ⊔ ⇑g | null | true |
CategoryTheory.op_add | Mathlib.CategoryTheory.Preadditive.Opposite | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {X Y : C}
(f g : X ⟶ Y), (f + g).op = f.op + g.op | null | true |
Std.DHashMap.Internal.Raw₀.replaceₘ._proof_1 | Std.Data.DHashMap.Internal.Model | ∀ {α : Type u_1} {β : α → Type u_2} [inst : BEq α] [inst_1 : Hashable α] (m : Std.DHashMap.Internal.Raw₀ α β) (a : α)
(b : β a),
0 <
{ size := (↑m).size,
buckets :=
Std.DHashMap.Internal.updateBucket (↑m).buckets ⋯ a fun l =>
Std.DHashMap.Internal.AssocList.replace a b l }.bu... | null | false |
BoundedLatticeHom.dual._proof_4 | Mathlib.Order.Hom.BoundedLattice | ∀ {α : Type u_2} {β : Type u_1} [inst : Lattice α] [inst_1 : BoundedOrder α] [inst_2 : Lattice β]
[inst_3 : BoundedOrder β] (f : BoundedLatticeHom αᵒᵈ βᵒᵈ), f.toFun ⊤ = ⊤ | null | false |
CategoryTheory.Comonad.Coalgebra | Mathlib.CategoryTheory.Monad.Algebra | {C : Type u₁} → [inst : CategoryTheory.Category.{v₁, u₁} C] → CategoryTheory.Comonad C → Type (max u₁ v₁) | An Eilenberg-Moore coalgebra for a comonad `T`. | true |
CategoryTheory.PreOneHypercover.IsStronglySheafFor.map_amalgamate | Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X : C} {E : CategoryTheory.PreOneHypercover X}
{F : CategoryTheory.Functor Cᵒᵖ (Type u_2)} (h : E.IsStronglySheafFor F)
(x : (i : E.I₀) → F.obj (Opposite.op (E.X i)))
(hc :
∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),
(CategoryTheory.ConcreteCategory.hom... | null | true |
RCLike.continuous_im | Mathlib.Analysis.RCLike.Basic | ∀ {K : Type u_1} [inst : RCLike K], Continuous ⇑RCLike.im | null | true |
Std.Iterators.Types.Flatten.IsPlausibleStep.outerDone_flatMap | Init.Data.Iterators.Lemmas.Combinators.Monadic.FlatMap | ∀ {α β α₂ γ : Type w} {m : Type w → Type w'} [inst : Monad m] [LawfulMonad m] [inst_2 : Std.Iterator α m β]
[inst_3 : Std.Iterator α₂ m γ] {f : β → Std.IterM m γ} {it₁ : Std.IterM m β},
it₁.IsPlausibleStep Std.IterStep.done → (Std.IterM.flatMapAfter f it₁ none).IsPlausibleStep Std.IterStep.done | null | true |
Std.HashMap.erase_emptyWithCapacity | Std.Data.HashMap.Lemmas | ∀ {α : Type u} {β : Type v} {x : BEq α} {x_1 : Hashable α} {a : α} {c : ℕ},
(Std.HashMap.emptyWithCapacity c).erase a = Std.HashMap.emptyWithCapacity c | null | true |
_private.Mathlib.Dynamics.Ergodic.Extreme.0.Ergodic.of_mem_extremePoints._simp_1_1 | Mathlib.Dynamics.Ergodic.Extreme | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α},
MeasureTheory.IsProbabilityMeasure μ = (μ Set.univ = 1) | null | false |
PseudoMetric.IsUltra.sup | Mathlib.Topology.MetricSpace.BundledFun | ∀ {X : Type u_1} {R : Type u_2} [inst : AddZeroClass R] [inst_1 : SemilatticeSup R] [inst_2 : AddLeftMono R]
[inst_3 : AddRightMono R] {d d' : PseudoMetric X R} [d.IsUltra] [d'.IsUltra], (d ⊔ d').IsUltra | null | true |
Submodule.mem_span_image_finset_iff_exists_fun | Mathlib.LinearAlgebra.Finsupp.LinearCombination | ∀ {α : Type u_1} {M : Type u_2} (R : Type u_3) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{v : α → M} {x : M} {s : Finset α}, x ∈ Submodule.span R (v '' ↑s) ↔ ∃ c, ∑ i, c i • v ↑i = x | null | true |
instBooleanAlgebraAsBoolAlg._proof_6 | Mathlib.Algebra.Ring.BooleanRing | ∀ {α : Type u_1} [inst : BooleanRing α] (a b : AsBoolAlg α), b ≤ Add.add (a + b) (a * b) | null | false |
CategoryTheory.NatTrans.instCommShiftOppositeShiftHomFunctorNatIsoComp | Mathlib.CategoryTheory.Shift.Opposite | ∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] (A : Type u_3) [inst_2 : AddMonoid A]
[inst_3 : CategoryTheory.HasShift C A] [inst_4 : CategoryTheory.HasShift D A] (F : CategoryTheory.Functor C D)
{E : Type u_4} [inst_5 : CategoryTheory... | null | true |
_private.Mathlib.Topology.Compactness.LocallyCompact.0.IsCompact.nhdsSet_basis_isCompact._proof_1_1 | Mathlib.Topology.Compactness.LocallyCompact | ∀ {X : Type u_1} [inst : TopologicalSpace X] {K : Set X} ⦃s t : Set X⦄,
s ⊆ t → (∃ r ∈ nhdsSet K, IsCompact r ∧ r ⊆ s) → ∃ r ∈ nhdsSet K, IsCompact r ∧ r ⊆ t | null | false |
Std.MaxEqOr.max_eq_or | Init.Data.Order.Classes | ∀ {α : Type u} {inst : Max α} [self : Std.MaxEqOr α] (a b : α), a ⊔ b = a ∨ a ⊔ b = b | null | true |
Lean.Widget.instToJsonRpcEncodablePacket._@.Lean.Widget.InteractiveGoal.1056429149._hygCtx._hyg.45 | Lean.Widget.InteractiveGoal | Lean.ToJson Lean.Widget.RpcEncodablePacket✝ | null | false |
IntermediateField.coe_algebraMap_over_bot | Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | ∀ {F : Type u_1} [inst : Field F] {E : Type u_2} [inst_1 : Field E] [inst_2 : Algebra F E],
⇑(algebraMap (↥⊥) F) = ⇑(IntermediateField.botEquiv F E) | null | true |
_private.Mathlib.Algebra.Homology.Augment.0.ChainComplex.augment.match_3.splitter | Mathlib.Algebra.Homology.Augment | (motive : ℕ → ℕ → Sort u_1) →
(x x_1 : ℕ) →
(Unit → motive 1 0) →
((i j : ℕ) → motive i.succ j.succ) →
((x x_2 : ℕ) → (x = 1 → x_2 = 0 → False) → (∀ (i j : ℕ), x = i.succ → x_2 = j.succ → False) → motive x x_2) →
motive x x_1 | null | true |
MonoidAlgebra.liftMagma._proof_2 | Mathlib.Algebra.MonoidAlgebra.Basic | ∀ (R : Type u_3) {A : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : Mul M]
[inst_2 : NonUnitalNonAssocSemiring A] [inst_3 : Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (f : M →ₙ* A)
(a₁ a₂ : MonoidAlgebra R M),
(Finsupp.sum (a₁ * a₂) fun m t => t • f m) = (Finsupp.sum a₁ fun m t => t • f m) *... | null | false |
minimal_ge_iff._simp_2 | Mathlib.Order.Minimal | ∀ {α : Type u_2} {x y : α} [inst : PartialOrder α], Minimal (fun x => y ≤ x) x = (x = y) | null | false |
CategoryTheory.Limits.prod.symmetry' | Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P Q : C) [inst_1 : CategoryTheory.Limits.HasBinaryProduct P Q]
[inst_2 : CategoryTheory.Limits.HasBinaryProduct Q P],
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Limits.prod.lift CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)
... | null | true |
Matrix.empty_vecAlt1 | Mathlib.Data.Fin.VecNotation | ∀ (α : Type u_1) {h : 0 = 0 + 0}, Matrix.vecAlt1 h ![] = ![] | null | true |
_private.Std.Sat.AIG.RefVecOperator.Fold.0.Std.Sat.AIG.RefVec.denote_fold_and._proof_1_1 | Std.Sat.AIG.RefVecOperator.Fold | ∀ {len : ℕ}, ¬0 ≤ len → False | null | false |
CategoryTheory.NatTrans.leftOp._proof_1 | Mathlib.CategoryTheory.Opposites | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] {F G : CategoryTheory.Functor C Dᵒᵖ} (α : F ⟶ G) (X Y : Cᵒᵖ)
(f : X ⟶ Y),
(CategoryTheory.CategoryStruct.comp (G.leftOp.map f) (α.app (Opposite.unop Y)).unop).op =
(CategoryTheory.Cate... | null | false |
IsUnit.isUnit_iff_mulLeft_bijective | Mathlib.Algebra.Group.Units.Basic | ∀ {M : Type u_1} [inst : Monoid M] {a : M}, IsUnit a ↔ Function.Bijective fun x => a * x | null | true |
_private.Lean.Elab.Tactic.NormCast.0.Lean.Elab.Tactic.NormCast.splittingProcedure.match_1 | Lean.Elab.Tactic.NormCast | (motive : Option Lean.Meta.Simp.Result → Sort u_1) →
(__x : Option Lean.Meta.Simp.Result) →
((x_x2 : Lean.Meta.Simp.Result) → motive (some x_x2)) → ((x : Option Lean.Meta.Simp.Result) → motive x) → motive __x | null | false |
Functor.map_comp_map | Mathlib.Control.Functor | ∀ {F : Type u → Type v} {α β γ : Type u} [inst : Functor F] [LawfulFunctor F] (f : α → β) (g : β → γ),
((fun x => g <$> x) ∘ fun x => f <$> x) = fun x => (g ∘ f) <$> x | null | true |
TypeCat.EquivalenceRelation.ofEquivalence._proof_3 | Mathlib.CategoryTheory.EquivalenceRelation | ∀ {X : Type u_1} {φ : X → X → Prop} (hφ : Equivalence φ),
CategoryTheory.CategoryStruct.comp (TypeCat.TransitiveRelation.ofIsTrans ⋯).t (TypeCat.p₁OfRel φ) =
CategoryTheory.CategoryStruct.comp (TypeCat.TransitiveRelation.ofIsTrans ⋯).c.fst (TypeCat.p₁OfRel φ) | null | false |
UniformSpace.Completion.ring._proof_30 | Mathlib.Topology.Algebra.UniformRing | ∀ {α : Type u_1} [inst : Ring α] [inst_1 : UniformSpace α] [IsTopologicalRing α] [inst_3 : IsUniformAddGroup α]
(a b c : UniformSpace.Completion α), a * (b + c) = a * b + a * c | null | false |
TopologicalSpace.Opens.comap._proof_4 | Mathlib.Topology.Sets.Opens | ∀ {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : C(α, β))
(x x_1 : TopologicalSpace.Opens β),
{ carrier := ⇑f ⁻¹' ↑(x ⊓ x_1), is_open' := ⋯ } = { carrier := ⇑f ⁻¹' ↑(x ⊓ x_1), is_open' := ⋯ } | null | false |
Matrix.normedSpace | Mathlib.Analysis.Matrix.Normed | {R : Type u_1} →
{m : Type u_3} →
{n : Type u_4} →
{α : Type u_5} →
[inst : Fintype m] →
[inst_1 : Fintype n] →
[inst_2 : NormedField R] →
[inst_3 : SeminormedAddCommGroup α] → [NormedSpace R α] → NormedSpace R (Matrix m n α) | Normed space instance (using sup norm of sup norm) for matrices over a normed space. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. | true |
Std.ExtTreeMap.diff.congr_simp | Std.Data.ExtTreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} [inst : Std.TransCmp cmp] (t₁ t₁_1 : Std.ExtTreeMap α β cmp),
t₁ = t₁_1 → ∀ (t₂ t₂_1 : Std.ExtTreeMap α β cmp), t₂ = t₂_1 → t₁.diff t₂ = t₁_1.diff t₂_1 | null | true |
BoundedContinuousFunction.instFunLike | Mathlib.Topology.ContinuousMap.Bounded.Basic | {α : Type u} →
{β : Type v} →
[inst : TopologicalSpace α] → [inst_1 : PseudoMetricSpace β] → FunLike (BoundedContinuousFunction α β) α β | null | true |
CategoryTheory.Sheaf.isLocallySurjective_comp | Mathlib.CategoryTheory.Sites.LocallySurjective | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {A : Type u'}
[inst_1 : CategoryTheory.Category.{v', u'} A] {FA : A → A → Type u_1} {CA : A → Type w'}
[inst_2 : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [inst_3 : CategoryTheory.ConcreteCategory A FA]
{F₁ F₂ F... | null | true |
unitInterval.sigmoid._proof_1 | Mathlib.Analysis.SpecialFunctions.Sigmoid | ∀ (x : ℝ), 0 ≤ x.sigmoid ∧ x.sigmoid ≤ 1 | null | false |
NormedSpace.restrictScalars._proof_1 | Mathlib.Analysis.Normed.Module.Basic | ∀ (𝕜 : Type u_2) (𝕜' : Type u_1) (E : Type u_3) [inst : NormedField 𝕜] [inst_1 : NormedField 𝕜']
[inst_2 : NormedAlgebra 𝕜 𝕜'] [inst_3 : SeminormedAddCommGroup E] (c : 𝕜) (x : E),
‖(algebraMap 𝕜 𝕜') c‖ * ‖x‖ = ‖c‖ * ‖x‖ | null | false |
CategoryTheory.FreeBicategory.Hom₂.whisker_right | Mathlib.CategoryTheory.Bicategory.Free | {B : Type u} →
[inst : Quiver B] →
{a b c : CategoryTheory.FreeBicategory B} →
{f g : a ⟶ b} →
(h : b ⟶ c) →
CategoryTheory.FreeBicategory.Hom₂ f g →
CategoryTheory.FreeBicategory.Hom₂ (CategoryTheory.FreeBicategory.Hom.comp f h)
(CategoryTheory.FreeBicategory.Hom... | null | true |
Lean.Core.instMonadLogCoreM | Lean.CoreM | Lean.MonadLog Lean.CoreM | null | true |
CategoryTheory.ComposableArrows.threeδ₁Toδ₀_app_zero | Mathlib.CategoryTheory.ComposableArrows.Three | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {i j k l : C} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l)
(f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂),
(CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂).app 0 = f₁ | null | true |
CategoryTheory.Oplax.StrongTrans.Modification.ext_iff | Mathlib.CategoryTheory.Bicategory.Modification.Oplax | ∀ {B : Type u₁} {inst : CategoryTheory.Bicategory B} {C : Type u₂} {inst_1 : CategoryTheory.Bicategory C}
{F G : CategoryTheory.OplaxFunctor B C} {η θ : F ⟶ G} {x y : CategoryTheory.Oplax.StrongTrans.Modification η θ},
x = y ↔ x.app = y.app | null | true |
Polynomial.reflect_mul_induction | Mathlib.Algebra.Polynomial.Reverse | ∀ {R : Type u_1} [inst : Semiring R] (cf cg N O : ℕ) (f g : Polynomial R),
f.support.card ≤ cf.succ →
g.support.card ≤ cg.succ →
f.natDegree ≤ N →
g.natDegree ≤ O → Polynomial.reflect (N + O) (f * g) = Polynomial.reflect N f * Polynomial.reflect O g | null | true |
Algebra.trdeg | Mathlib.RingTheory.AlgebraicIndependent.Basic | (R : Type u_2) → (A : Type v) → [inst : CommRing R] → [inst_1 : CommRing A] → [Algebra R A] → Cardinal.{v} | The transcendence degree of a commutative algebra `A` over a commutative ring `R` is
defined to be the maximal cardinality of an `R`-algebraically independent set in `A`. | true |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.