name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ModuleCat.Hom.rec | Mathlib.Algebra.Category.ModuleCat.Basic | {R : Type u} →
[inst : Ring R] →
{M N : ModuleCat R} →
{motive : M.Hom N → Sort u_1} → ((hom' : ↑M →ₗ[R] ↑N) → motive { hom' := hom' }) → (t : M.Hom N) → motive t | null | false |
Lean.Meta.Grind.Arith.Linear.RingIneqCnstr.mk._flat_ctor | Lean.Meta.Tactic.Grind.Arith.Linear.Types | Lean.Grind.CommRing.Poly →
Bool → Lean.Meta.Grind.Arith.Linear.RingIneqCnstrProof → Lean.Meta.Grind.Arith.Linear.RingIneqCnstr | null | false |
QuadraticMap.associatedHom._proof_2 | Mathlib.LinearAlgebra.QuadraticForm.Basic | ∀ (S : Type u_4) {R : Type u_3} {M : Type u_2} {N : Type u_1} [inst : CommRing R] [inst_1 : AddCommGroup M]
[inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] [inst_5 : CommSemiring S] [inst_6 : Algebra S R]
[inst_7 : Module S N] [inst_8 : IsScalarTower S R N] [inst_9 : Invertible 2] (x : S) (x_1... | null | false |
PowerSeries.trunc._proof_3 | Mathlib.RingTheory.PowerSeries.Trunc | ∀ {R : Type u_1} [inst : Semiring R] (n : ℕ) (t : R) (φ : PowerSeries R),
PowerSeries.truncAux✝ n (t • φ) = (RingHom.id R) t • PowerSeries.truncAux✝ n φ | null | false |
MonotoneOn.integral_le_sum_Ico | Mathlib.Analysis.SumIntegralComparisons | ∀ {a b : ℕ} {f : ℝ → ℝ},
a ≤ b → MonotoneOn f (Set.Icc ↑a ↑b) → ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ i ∈ Finset.Ico a b, f ↑(i + 1) | null | true |
Lean.LabelExtensionMap | Lean.LabelAttribute | Type | The collection of all current `LabelExtension`s, indexed by name. | true |
Submodule.isTorsion'_iff_torsion'_eq_top | Mathlib.Algebra.Module.Torsion.Basic | ∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : Type u_3)
[inst_3 : CommMonoid S] [inst_4 : DistribMulAction S M] [inst_5 : SMulCommClass S R M],
Module.IsTorsion' M S ↔ Submodule.torsion' R M S = ⊤ | An `S`-torsion module is a module whose `S`-torsion submodule is the full space. | true |
_private.Mathlib.RingTheory.Extension.Cotangent.BaseChange.0.Algebra.tensorH1CotangentOfFlat._proof_4 | Mathlib.RingTheory.Extension.Cotangent.BaseChange | ∀ (R : Type u_3) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (T : Type u_1)
[inst_3 : CommRing T] [inst_4 : Algebra R T],
IsScalarTower T (TensorProduct R T S) (Algebra.Generators.self R S).toExtension.baseChange.Cotangent | null | false |
Lean.Lsp.RpcRef.ctorIdx | Lean.Server.Rpc.Basic | Lean.Lsp.RpcRef → ℕ | null | false |
_private.Batteries.Recycling.MonadSatisfying.Basic.0.SatisfiesM_ExceptT_eq.match_1_7 | Batteries.Recycling.MonadSatisfying.Basic | ∀ {α ρ : Type u_1} (e : ρ) (motive : (a : α) → Except.error e = Except.ok a → Prop) (a : α)
(a_1 : Except.error e = Except.ok a), motive a a_1 | null | false |
DivisionCommMonoid.npow._inherited_default | Mathlib.Algebra.Group.DivInvMonoid | {G : Type u_2} → G → (mul : G → G → G) → (∀ (a b c : G), a * b * c = a * (b * c)) → ℕ → G → G | null | false |
CategoryTheory.Idempotents.karoubiUniversal₁._proof_1 | Mathlib.CategoryTheory.Idempotents.FunctorExtension | ∀ (C : Type u_1) (D : Type u_4) [inst : CategoryTheory.Category.{u_2, u_1} C]
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (F : CategoryTheory.Functor C (CategoryTheory.Idempotents.Karoubi D)),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Idempotents.functorExtension₁ C D).map
((CategoryTheo... | null | false |
FormalMultilinearSeries.continuousOn | Mathlib.Analysis.Analytic.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
{p : FormalMultilinearSeries 𝕜 E F} [CompleteSpace F], ContinuousOn p.sum (Metric.eball 0 p.radius) | The sum of a converging power series is continuous in its disk of convergence. | true |
Std.Tactic.BVDecide.BVExpr.extract.sizeOf_spec | Std.Tactic.BVDecide.Bitblast.BVExpr.Basic | ∀ {w : ℕ} (start len : ℕ) (expr : Std.Tactic.BVDecide.BVExpr w),
sizeOf (Std.Tactic.BVDecide.BVExpr.extract start len expr) = 1 + sizeOf w + sizeOf start + sizeOf len + sizeOf expr | null | true |
_private.Init.Data.Int.DivMod.Lemmas.0.Int.tmod_zero.match_1_1 | Init.Data.Int.DivMod.Lemmas | ∀ (motive : ℤ → Prop) (x : ℤ), (∀ (a : ℕ), motive (Int.ofNat a)) → (∀ (a : ℕ), motive (Int.negSucc a)) → motive x | null | false |
WeierstrassCurve.Jacobian.dblXYZ_X | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | ∀ {R : Type r} [inst : CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R), W'.dblXYZ P 0 = W'.dblX P | null | true |
_private.Mathlib.Tactic.FunProp.FunctionData.0.Mathlib.Meta.FunProp.FunctionData.unfoldHeadFVar?.match_1 | Mathlib.Tactic.FunProp.FunctionData | (motive : Option Lean.Expr → Sort u_1) →
(__x : Option Lean.Expr) → ((val : Lean.Expr) → motive (some val)) → ((x : Option Lean.Expr) → motive x) → motive __x | null | false |
Function.eq_update_self_iff._simp_1 | Mathlib.Logic.Function.Basic | ∀ {α : Sort u} {β : α → Sort v} [inst : DecidableEq α] {f : (a : α) → β a} {a : α} {b : β a},
(f = Function.update f a b) = (f a = b) | null | false |
Submonoid.mem_mulSupport | Mathlib.Algebra.Group.Submonoid.Support | ∀ {G : Type u_1} [inst : Group G] {M : Submonoid G} {x : G}, x ∈ M.mulSupport ↔ x ∈ M ∧ x⁻¹ ∈ M | null | true |
Std.ExtDTreeMap.getKey!_alter_self | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.ExtDTreeMap α β cmp} [inst : Std.TransCmp cmp]
[inst_1 : Std.LawfulEqCmp cmp] [inst_2 : Inhabited α] {k : α} {f : Option (β k) → Option (β k)},
(t.alter k f).getKey! k = if (f (t.get? k)).isSome = true then k else default | null | true |
Combinatorics.Line.toSubspace._proof_3 | Mathlib.Combinatorics.HalesJewett | ∀ {η : Type u_1} {α : Type u_2} {ι : Type u_3} (l : Combinatorics.Line (η → α) ι) (e : η) (i : ι),
l.idxFun i = none → ((l.idxFun (i, e).1).elim (Sum.inr (i, e).2) fun f => Sum.inl (f (i, e).2)) = Sum.inr e | null | false |
_private.Mathlib.Analysis.Normed.Group.Ultra.0.IsUltrametricDist.nnnorm_prod_eq_sup_of_pairwise_ne._proof_1_3 | Mathlib.Analysis.Normed.Group.Ultra | ∀ {ι : Type u_1} (a : ι) (s : Finset ι), a ∉ s → ∀ j ∈ s, ¬a = j | null | false |
Std.DTreeMap.Internal.Zipper.cons.sizeOf_spec | Std.Data.DTreeMap.Internal.Zipper | ∀ {α : Type u} {β : α → Type v} [inst : SizeOf α] [inst_1 : (a : α) → SizeOf (β a)] (k : α) (v : β k)
(tree : Std.DTreeMap.Internal.Impl α β) (next : Std.DTreeMap.Internal.Zipper α β),
sizeOf (Std.DTreeMap.Internal.Zipper.cons k v tree next) = 1 + sizeOf k + sizeOf v + sizeOf tree + sizeOf next | null | true |
Filter.cofinite_eq_bot | Mathlib.Order.Filter.Cofinite | ∀ {α : Type u_2} [Finite α], Filter.cofinite = ⊥ | null | true |
Lean.Elab.InlayHintLinkLocation.casesOn | Lean.Elab.InfoTree.InlayHints | {motive : Lean.Elab.InlayHintLinkLocation → Sort u} →
(t : Lean.Elab.InlayHintLinkLocation) →
((module : Lean.Name) → (range : Lean.Syntax.Range) → motive { module := module, range := range }) → motive t | null | false |
_private.Mathlib.RingTheory.Extension.Cotangent.Basis.0.Algebra.Generators.PresentationOfFreeCotangent.Aux.presLeft._proof_3 | Mathlib.RingTheory.Extension.Cotangent.Basis | ∀ {R : Type u_2} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {ι : Type u_1}
{P : Algebra.Generators R S ι} {σ : Type u_3} {b : Module.Basis σ S P.toExtension.Cotangent}
(D : Algebra.Generators.PresentationOfFreeCotangent.Aux✝ P b),
(Ideal.span (Set.range (Subtype.val ∘ Algebra.... | null | false |
Mathlib.Tactic.MkIff.Shape.noConfusionType | Mathlib.Tactic.MkIffOfInductiveProp | Sort u → Mathlib.Tactic.MkIff.Shape → Mathlib.Tactic.MkIff.Shape → Sort u | null | false |
Std.DTreeMap.Internal.Impl.getKey!_inter | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {m₁ m₂ : Std.DTreeMap.Internal.Impl α β} [inst : Inhabited α]
[Std.TransOrd α] (h₁ : m₁.WF),
m₂.WF →
∀ {k : α},
(m₁.inter m₂ ⋯).getKey! k = if Std.DTreeMap.Internal.Impl.contains k m₂ = true then m₁.getKey! k else default | null | true |
ImportGraph.instToMessageDataError.match_1 | ImportGraph.WorkspaceModel.Model | (motive : ImportGraph.WorkspaceModel.Error → Sort u_1) →
(x : ImportGraph.WorkspaceModel.Error) →
((mod : Lean.Name) → motive (ImportGraph.WorkspaceModel.Error.noLibOfModule mod)) →
((mod : Lean.Name) →
(path : System.FilePath) →
(ioError : IO.Error) → motive (ImportGraph.WorkspaceMode... | null | false |
_private.Lean.Meta.Tactic.Simp.Arith.Int.Simp.0.Lean.Meta.Simp.Arith.Int.simpRel?.match_3 | Lean.Meta.Tactic.Simp.Arith.Int.Simp | (motive : Option Lean.Expr → Sort u_1) →
(eNew? : Option Lean.Expr) →
((eNew : Lean.Expr) → motive (some eNew)) → ((x : Option Lean.Expr) → motive x) → motive eNew? | null | false |
_private.Batteries.Tactic.PrintDependents.0.Batteries.Tactic.«_aux_Batteries_Tactic_PrintDependents___elabRules_Batteries_Tactic_command#printDependents____1».match_10 | Batteries.Tactic.PrintDependents | (motive : Lean.Name × ℕ → Sort u_1) → (x : Lean.Name × ℕ) → ((c : Lean.Name) → (snd : ℕ) → motive (c, snd)) → motive x | null | false |
ContinuousMapZero.toContinuousMapCLM._proof_2 | Mathlib.Topology.ContinuousMap.ContinuousMapZero | ∀ {X : Type u_1} {R : Type u_2} [inst : Zero X] [inst_1 : TopologicalSpace X] [inst_2 : TopologicalSpace R]
[inst_3 : CommSemiring R] [inst_4 : IsTopologicalSemiring R] (x x_1 : ContinuousMapZero X R), ↑(x + x_1) = ↑(x + x_1) | null | false |
DirectSum.Gmodule.mk._flat_ctor | Mathlib.Algebra.Module.GradedModule | {ιA : Type u_1} →
{ιB : Type u_2} →
{A : ιA → Type u_3} →
{M : ιB → Type u_4} →
[inst : AddMonoid ιA] →
[inst_1 : VAdd ιA ιB] →
[inst_2 : (i : ιA) → AddMonoid (A i)] →
[inst_3 : (i : ιB) → AddMonoid (M i)] →
[inst_4 : GradedMonoid.GMonoid A] →
... | null | false |
_private.Mathlib.RingTheory.DualNumber.0.DualNumber.fst_eq_zero_iff_eps_dvd._simp_1_4 | Mathlib.RingTheory.DualNumber | ∀ {p q : Prop}, (p ↔ p ∧ q) = (p → q) | null | false |
_private.Mathlib.Topology.MetricSpace.Infsep.0.Set.infsep_eq_iInf._simp_1_5 | Mathlib.Topology.MetricSpace.Infsep | ∀ {α : Type u_1} {β : Type u_2} {p : α × β → Prop}, (∀ (x : α × β), p x) = ∀ (a : α) (b : β), p (a, b) | null | false |
GradedMonoid.GMulAction.ctorIdx | Mathlib.Algebra.GradedMulAction | {ιA : Type u_1} →
{ιM : Type u_3} →
{A : ιA → Type u_4} →
{M : ιM → Type u_5} →
{inst : AddMonoid ιA} →
{inst_1 : VAdd ιA ιM} → {inst_2 : GradedMonoid.GMonoid A} → GradedMonoid.GMulAction A M → ℕ | null | false |
Ideal.mapHom._proof_2 | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u_1} [inst : CommSemiring R], IsScalarTower R R R | null | false |
_private.Mathlib.GroupTheory.FreeGroup.Reduce.0.FreeAddGroup.toWord_add_mk_eq._proof_1_15 | Mathlib.GroupTheory.FreeGroup.Reduce | ∀ {α : Type u_1} [inst : DecidableEq α] (x : FreeAddGroup α) (i : ℕ) (a_1 : α × Bool),
((List.map ((fun g => (g.1, !g.2)) ∘ fun g => (g.1, !g.2)) x.toWord).dropLast[i]? = some a_1) =
¬x.toWord.dropLast[i]? = some a_1 →
¬(List.map ((fun g => (g.1, !g.2)) ∘ fun g => (g.1, !g.2)) x.toWord).dropLast[i]? = some ... | null | false |
LinearEquiv.toLinearEquiv_toContinuousLinearEquiv | Mathlib.Topology.Algebra.Module.FiniteDimension | ∀ {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E]
[inst_2 : TopologicalSpace E] [inst_3 : IsTopologicalAddGroup E] [inst_4 : ContinuousSMul 𝕜 E] {F : Type w}
[inst_5 : AddCommGroup F] [inst_6 : Module 𝕜 F] [inst_7 : TopologicalSpace F] [inst_8 : IsTop... | null | true |
CategoryTheory.ComposableArrows.isoMk₁_hom_app | Mathlib.CategoryTheory.ComposableArrows.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {F G : CategoryTheory.ComposableArrows C 1}
(left :
F.obj' 0 CategoryTheory.ComposableArrows.homMk₁._proof_4 ≅ G.obj' 0 CategoryTheory.ComposableArrows.homMk₁._proof_4)
(right :
F.obj' 1 CategoryTheory.ComposableArrows.homMk₁._proof_5 ≅ G.obj' 1... | null | true |
MonomialOrder.degree_ne_zero_of_sub_leadingTerm_ne_zero | Mathlib.RingTheory.MvPolynomial.MonomialOrder | ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] {f : MvPolynomial σ R},
f - m.leadingTerm f ≠ 0 → m.degree f ≠ 0 | null | true |
DividedPowers.subDPIdeal_inf_of_quot._proof_1 | Mathlib.RingTheory.DividedPowers.SubDPIdeal | ∀ {A : Type u_1} [inst : CommRing A] {I : Ideal A} {hI : DividedPowers I} {J : Ideal A}
{hJ : DividedPowers (Ideal.map (Ideal.Quotient.mk J) I)} {φ : hI.DPMorphism hJ},
φ.toRingHom = Ideal.Quotient.mk J → ∀ (x : ℕ), x ≠ 0 → ∀ a ∈ J ⊓ I, hI.dpow x a ∈ J ⊓ I | null | false |
Submodule.submodule_torsionBy_orderIso._proof_9 | Mathlib.Algebra.Module.Torsion.Basic | ∀ {R : Type u_2} {M : Type u_1} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (a : R)
(p : Submodule R ↥(Submodule.torsionBy R M a)), 0 ∈ p | null | false |
min_eq_one | Mathlib.Algebra.Order.IsBotOne | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : One α] [IsBotOneClass α] {a b : α}, min a b = 1 ↔ a = 1 ∨ b = 1 | null | true |
LightCondMod.LocallyConstant.functorIsoDiscreteAux₂._proof_2 | Mathlib.Condensed.Discrete.Module | ∀ (R : Type u_1) [inst : Ring R],
CategoryTheory.HasSheafify (CategoryTheory.coherentTopology LightProfinite) (ModuleCat R) | null | false |
TrivSqZeroExt.isCentralScalar | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ {S : Type u_2} {R : Type u} {M : Type v} [inst : SMul S R] [inst_1 : SMul S M] [inst_2 : SMul Sᵐᵒᵖ R]
[inst_3 : SMul Sᵐᵒᵖ M] [IsCentralScalar S R] [IsCentralScalar S M], IsCentralScalar S (TrivSqZeroExt R M) | null | true |
IsDedekindDomain.HeightOneSpectrum.ext_iff | Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | ∀ {R : Type u_1} {inst : CommRing R} {x y : IsDedekindDomain.HeightOneSpectrum R}, x = y ↔ x.asIdeal = y.asIdeal | null | true |
ArithmeticFunction.instCommSemiring | Mathlib.NumberTheory.ArithmeticFunction.Defs | {R : Type u_1} → [inst : CommSemiring R] → CommSemiring (ArithmeticFunction R) | null | true |
_private.Mathlib.Order.Preorder.Chain.0.IsMaxChain.nonempty_iff._simp_1_1 | Mathlib.Order.Preorder.Chain | ∀ {α : Type u} {s : Set α}, (¬s.Nonempty) = (s = ∅) | null | false |
QuotientGroup.comapMk'OrderIso._proof_6 | Mathlib.GroupTheory.QuotientGroup.Basic | ∀ {G : Type u_1} [inst : Group G] (N : Subgroup G) [hn : N.Normal] {a b : Subgroup (G ⧸ N)},
Subgroup.comap (QuotientGroup.mk' N) a ≤ Subgroup.comap (QuotientGroup.mk' N) b ↔ a ≤ b | null | false |
CategoryTheory.Functor.leftKanExtensionCompIsoOfPreserves_hom_fac_app_assoc | Mathlib.CategoryTheory.Functor.KanExtension.Preserves | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} A]
[inst_1 : CategoryTheory.Category.{v_2, u_2} B] [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
[inst_3 : CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B... | null | true |
_private.Mathlib.Data.Multiset.Antidiagonal.0.Multiset.map_swap_antidiagonal._simp_1_1 | Mathlib.Data.Multiset.Antidiagonal | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β) (g : γ → δ),
Prod.swap ∘ Prod.map g f = Prod.map f g ∘ Prod.swap | null | false |
Lean.Grind.AddCommGroup.casesOn | Init.Grind.Module.Basic | {M : Type u} →
{motive : Lean.Grind.AddCommGroup M → Sort u_1} →
(t : Lean.Grind.AddCommGroup M) →
([toAddCommMonoid : Lean.Grind.AddCommMonoid M] →
[toNeg : Neg M] →
[toSub : Sub M] →
(neg_add_cancel : ∀ (a : M), -a + a = 0) →
(sub_eq_add_neg : ∀ (a b : M... | null | false |
ModelWithCorners.MfldCat.Hom.noConfusion | Mathlib.Geometry.Manifold.Category.MfldCat.OfModel | {P : Sort u_4} →
{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} →
... | null | false |
_private.Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity.0.ChevalleyThm.PolynomialC.InductionObj.ext.match_1 | Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | ∀ {R : Type u_1} {inst : CommRing R} {n : ℕ} (motive : ChevalleyThm.PolynomialC.InductionObj✝ R n → Prop)
(h : ChevalleyThm.PolynomialC.InductionObj✝ R n), (∀ (val : Fin n → Polynomial R), motive { val := val }) → motive h | null | false |
Nat.pow_eq | Init.Data.Nat.Lemmas | ∀ {m n : ℕ}, m.pow n = m ^ n | null | true |
Real.deriv_tan | Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv | ∀ (x : ℝ), deriv Real.tan x = 1 / Real.cos x ^ 2 | null | true |
SimpleGraph.nonuniformWitness_subset | Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | ∀ {α : Type u_1} {𝕜 : Type u_2} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [inst_2 : IsStrictOrderedRing 𝕜]
(G : SimpleGraph α) [inst_3 : DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α},
¬G.IsUniform ε s t → G.nonuniformWitness ε s t ⊆ s | null | true |
Lean.Meta.Grind.Goal.noConfusionType | Lean.Meta.Tactic.Grind.Types | Sort u → Lean.Meta.Grind.Goal → Lean.Meta.Grind.Goal → Sort u | null | false |
inv_mul_lt_one_iff_lt | Mathlib.Algebra.Order.Group.Unbundled.Basic | ∀ {α : Type u} [inst : Group α] [inst_1 : LT α] [MulRightStrictMono α] {a b : α}, a * b⁻¹ < 1 ↔ a < b | null | true |
Lean.Parser.ParserContextCore.currNamespace._inherited_default | Lean.Parser.Types | Lean.Name | null | false |
Std.Internal.List.getValue?.match_1 | Std.Data.Internal.List.Associative | {α : Type u_2} →
{β : Type u_1} →
(motive : List ((_ : α) × β) → Sort u_3) →
(x : List ((_ : α) × β)) →
(Unit → motive []) → ((k : α) → (v : β) → (l : List ((_ : α) × β)) → motive (⟨k, v⟩ :: l)) → motive x | null | false |
_private.Lean.Exception.0.Lean.ofExceptKernelException.match_1 | Lean.Exception | {α : Type} →
(motive : Except Lean.Kernel.Exception α → Sort u_1) →
(x : Except Lean.Kernel.Exception α) →
((a : α) → motive (Except.ok a)) → ((e : Lean.Kernel.Exception) → motive (Except.error e)) → motive x | null | false |
Lean.FuzzyMatching.instInhabitedCharRole.default | Lean.Data.FuzzyMatching | Lean.FuzzyMatching.CharRole | null | true |
Array.le_snd_of_mem_zipIdx | Init.Data.Array.Range | ∀ {α : Type u_1} {x : α × ℕ} {k : ℕ} {xs : Array α}, x ∈ xs.zipIdx k → k ≤ x.2 | null | true |
HomologicalComplex.Hom.casesOn | Mathlib.Algebra.Homology.HomologicalComplex | {ι : Type u_1} →
{V : Type u} →
[inst : CategoryTheory.Category.{v, u} V] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] →
{c : ComplexShape ι} →
{A B : HomologicalComplex V c} →
{motive : A.Hom B → Sort u_2} →
(t : A.Hom B) →
((f : (i : ι) ... | null | false |
FirstOrder.Language.Formula.exClosure | Mathlib.ModelTheory.Syntax | {L : FirstOrder.Language} → {α : Type u'} → [DecidableEq α] → L.Formula α → L.Sentence | `exClosure φ` is the sentence asserting that there exist values for all free variables of `φ`
such that `φ` holds. | true |
CantorBendixson.iteratedDerivedSet_mem_fixedPoints | Mathlib.Topology.CantorBendixson | ∀ {X : Type u} [inst : TopologicalSpace X] (s : Set X),
∃ a, CantorBendixson.iteratedDerivedSet s a ∈ Function.fixedPoints ⇑relDerivedSet | null | true |
MixedCharZero.recOn | Mathlib.Algebra.CharP.MixedCharZero | {R : Type u_1} →
[inst : CommRing R] →
{p : ℕ} →
{motive : MixedCharZero R p → Sort u} →
(t : MixedCharZero R p) →
([toCharZero : CharZero R] → (charP_quotient : ∃ I, I ≠ ⊤ ∧ CharP (R ⧸ I) p) → motive ⋯) → motive t | null | false |
Std.TreeMap.getElem?_insert | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {a k : α} {v : β},
(t.insert k v)[a]? = if cmp k a = Ordering.eq then some v else t[a]? | null | true |
AddSubgroup.addCommutator_top_right_le_iff | Mathlib.GroupTheory.Commutator.Basic | ∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G}, ⁅H, ⊤⁆ ≤ H ↔ H.Normal | null | true |
Fin.map_succEmb_Iic | Mathlib.Order.Interval.Finset.Fin | ∀ {n : ℕ} (i : Fin n), Finset.map (Fin.succEmb n) (Finset.Iic i) = Finset.Ioc 0 i.succ | null | true |
_private.Mathlib.Algebra.Lie.Basis.Prod.0.LieAlgebra.Basis.prodCartan | Mathlib.Algebra.Lie.Basis.Prod | {ι₁ : Type u_1} →
{ι₂ : Type u_2} →
{L₁ : Type u_3} →
{L₂ : Type u_4} →
[inst : Finite ι₁] →
[inst_1 : Finite ι₂] →
(eι : ι₁ ≃ ι₂) →
[inst_2 : LieRing L₁] →
[inst_3 : LieRing L₂] →
{R : Type u_5} →
[inst_4 : Co... | A distinguished subalgebra of the Lie algebra defined by `LieAlgebra.Basis.prod`.
Subject to the assumtions demanded by `LieAlgebra.Basis.isCartanSubalgebra` this will be a Cartan
subalgebra of `LieAlgebra.Basis.prod`. | true |
Lean.Grind.Nat.lo_ro_1 | Init.Grind.Offset | ∀ (u w v k₁ k₂ : ℕ), Lean.Grind.isLt k₂ k₁ = true → u + k₁ ≤ w → w ≤ v + k₂ → u + (k₁ - k₂) ≤ v | null | true |
Finset.HasAntidiagonal.mk._flat_ctor | Mathlib.Algebra.Order.Antidiag.Prod | {A : Type u_1} →
[inst : Add A] →
(antidiagonal : A → Finset (A × A)) →
(∀ {n : A} {a : A × A}, a ∈ antidiagonal n ↔ a.1 + a.2 = n) → Finset.HasAntidiagonal A | null | false |
ImportGraph.Shake.DeclNeed._sizeOf_1 | ImportGraph.Shake.DeclNeeds | ImportGraph.Shake.DeclNeed → ℕ | null | false |
ZeroAtInftyContinuousMap | Mathlib.Topology.ContinuousMap.ZeroAtInfty | (α : Type u) → (β : Type v) → [TopologicalSpace α] → [Zero β] → [TopologicalSpace β] → Type (max u v) | `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
topological space to a metric space with a zero element.
When possible, instead of parametrizing results over `(f : C₀(α, β))`,
you should parametrize over `(F : Type*) [ZeroAtInftyContinuousMapClass F α β] (f : F)`.
When you exten... | true |
Lean.logNamedErrorAt | Lean.Log | {m : Type → Type} →
[Monad m] →
[Lean.MonadLog m] →
[Lean.AddMessageContext m] → [Lean.MonadOptions m] → Lean.Syntax → Lean.Name → Lean.MessageData → m Unit | Log a named error message using the given message data. The position is provided by `ref`.
Note: Use the macro `logNamedErrorAt`, which validates error names, instead of calling this function
directly.
| true |
Nat.cast_npow | Mathlib.Algebra.Group.NatPowAssoc | ∀ (R : Type u_2) [inst : NonAssocSemiring R] [inst_1 : Pow R ℕ] [NatPowAssoc R] (n m : ℕ), ↑(n ^ m) = ↑n ^ m | null | true |
Std.ExtDTreeMap.minEntry._proof_3 | Std.Data.ExtDTreeMap.Basic | ∀ {α : Type u_1} {β : α → Type u_2} {cmp : α → α → Ordering} [inst : Std.TransCmp cmp] (t : Std.ExtDTreeMap α β cmp)
(h : t ≠ ∅) (x x_1 : Std.DTreeMap α β cmp) (x_2 : t = Std.ExtDTreeMap.mk x),
t = Std.ExtDTreeMap.mk x_1 → ∀ (h_1 : x.Equiv x_1), x.minEntry ⋯ = x_1.minEntry ⋯ | null | false |
_private.Mathlib.GroupTheory.FreeGroup.Basic.0.FreeGroup.instGroup.match_10 | Mathlib.GroupTheory.FreeGroup.Basic | ∀ {α : Type u_1} (motive : α × Bool → Prop) (x : α × Bool), (∀ (x : α) (b : Bool), motive (x, b)) → motive x | null | false |
Fin.isValue | Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin | Lean.Meta.Simp.DSimproc | Simplification procedure for ensuring `Fin n` literals are normalized. | true |
CategoryTheory.Pseudofunctor.ObjectProperty.IsClosedUnderMapObj | Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty | {B : Type u} →
[inst : CategoryTheory.Bicategory B] →
{F : CategoryTheory.Pseudofunctor B CategoryTheory.Cat} → F.ObjectProperty → Prop | If `P` is a property of objects for a pseudofunctor `F` to `Cat`,
this is the condition that `P` is preserved by the application of the functors `F.map`. | true |
WeakDual.instAddCommGroup._proof_10 | Mathlib.Topology.Algebra.Module.Spaces.WeakDual | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : CommRing 𝕜] [inst_1 : TopologicalSpace 𝕜] [inst_2 : IsTopologicalAddGroup 𝕜]
[inst_3 : ContinuousConstSMul 𝕜 𝕜] [inst_4 : AddCommGroup E] [inst_5 : Module 𝕜 E] [inst_6 : TopologicalSpace E]
(a b : WeakDual 𝕜 E), a + b = b + a | null | false |
Std.Time.Number.mk.injEq | Std.Time.Format.Modifier | ∀ (padding padding_1 : ℕ), ({ padding := padding } = { padding := padding_1 }) = (padding = padding_1) | null | true |
AlgebraicGeometry.pullback_of_geometrically' | Mathlib.AlgebraicGeometry.Geometrically.Basic | ∀ {P : CategoryTheory.ObjectProperty AlgebraicGeometry.Scheme} {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},
AlgebraicGeometry.geometrically P f →
∀ (K : Type u) [inst : Field K] (y : AlgebraicGeometry.Spec ↧K ⟶ Y), P (CategoryTheory.Limits.pullback y f) | null | true |
CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopyEquiv._proof_1 | Mathlib.AlgebraicTopology.ExtraDegeneracy | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] {X : CategoryTheory.SimplicialObject.Augmented C}
(ed : X.ExtraDegeneracy) (i : ℕ),
(CategoryTheory.CategoryStruct.comp (AlgebraicTopology.AlternatingFaceMapCompl... | null | false |
MeasureTheory.Measure.measurePreserving_zsmul | Mathlib.MeasureTheory.Measure.Haar.Unique | ∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : TopologicalSpace G] [IsTopologicalAddGroup G]
[inst_3 : MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [CompactSpace G]
[DivisibleBy G ℤ] {n : ℤ}, n ≠ 0 → MeasureTheory.MeasurePreserving (fun g => n • g) μ μ | null | true |
PartialEquiv.single._proof_2 | Mathlib.Logic.Equiv.PartialEquiv | ∀ {α : Type u_2} {β : Type u_1} (a : α) (b b' : β), b' ∈ {b} → Function.const α b (Function.const β a b') = b' | null | false |
_private.Mathlib.Algebra.Star.CHSH.0.CHSH_id._proof_1_2 | Mathlib.Algebra.Star.CHSH | (3 + 1).AtLeastTwo | null | false |
SimpleGraph.Subgraph.incidenceSet_subset | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u} {G : SimpleGraph V} (G' : G.Subgraph) (v : V), G'.incidenceSet v ⊆ G'.edgeSet | null | true |
_private.Init.Data.String.Lemmas.Pattern.Char.0.String.Slice.Pattern.Model.Char.revMatchesAt_iff_revMatchesAt_beq._simp_1_2 | Init.Data.String.Lemmas.Pattern.Char | ∀ {c : Char} {s : String.Slice} {pos pos' : s.Pos},
String.Slice.Pattern.Model.IsLongestRevMatchAt c pos pos' =
String.Slice.Pattern.Model.IsLongestRevMatchAt (fun x => x == c) pos pos' | null | false |
Condensed.instAB4StarCondensedMod | Mathlib.Condensed.AB | ∀ (R : Type (u + 1)) [inst : Ring R], CategoryTheory.AB4Star (CondensedMod R) | null | true |
CategoryTheory.AddModObj.ext | Mathlib.CategoryTheory.Monoidal.Mod | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C] {M : C}
[inst_2 : CategoryTheory.AddMonObj M] {X : C} (h₁ h₂ : CategoryTheory.AddModObj M X),
CategoryTheory.AddModObj.vadd = CategoryTheory.AddModObj.vadd → h₁ = h₂ | null | true |
Polynomial.prod_max_one_norm_roots_le_mahlerMeasure_of_one_le_leadingCoeff | Mathlib.Analysis.Polynomial.MahlerMeasure | ∀ {p : Polynomial ℂ}, 1 ≤ ‖p.leadingCoeff‖ → (Multiset.map (fun a => max 1 ‖a‖) p.roots).prod ≤ p.mahlerMeasure | null | true |
CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv | 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 : S.HasRightHomology],
CategoryTheory.CategoryStruct.comp S.fromOpcycles.op S.cyclesOpIso.inv = S.op.toCycles | null | true |
Aesop.GoalStats._sizeOf_inst | Aesop.Stats.Basic | SizeOf Aesop.GoalStats | null | false |
Convex.nhdsWithin_diff_eq_nhdsLT | Mathlib.Analysis.Convex.Topology | ∀ {𝕜 : Type u_3} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [inst_3 : TopologicalSpace 𝕜]
[OrderTopology 𝕜] {s : Set 𝕜},
Convex 𝕜 s →
∀ {a : 𝕜},
a ∈ closure s → (s ∩ Set.Iio a).Nonempty → s ∩ Set.Ioi a = ∅ → nhdsWithin a (s \ {a}) = nhdsWithin a (Set.Iio a) | **Alias** of `Convex.nhdsWithin_sdiff_eq_nhdsLT`. | true |
Perfection.lift.congr_simp | Mathlib.RingTheory.Perfection | ∀ (p : ℕ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [inst : CommSemiring R] [inst_1 : CharP R p]
[inst_2 : PerfectRing R p] (S : Type u₂) [inst_3 : CommSemiring S] [inst_4 : CharP S p],
Perfection.lift p R S = Perfection.lift p R S | null | true |
MonoidAlgebra.ofCoeff_eq_zero | Mathlib.Algebra.MonoidAlgebra.Defs | ∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] {x : M →₀ R}, MonoidAlgebra.ofCoeff x = 0 ↔ x = 0 | null | true |
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