name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
PartialEquiv.symm_image_target_eq_source | Mathlib.Logic.Equiv.PartialEquiv | ∀ {α : Type u_1} {β : Type u_2} (e : PartialEquiv α β), ↑e.symm '' e.target = e.source | null | true |
Std.TreeMap.mem_union_iff | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap α β cmp} [Std.TransCmp cmp] {k : α},
k ∈ t₁ ∪ t₂ ↔ k ∈ t₁ ∨ k ∈ t₂ | null | true |
_private.Mathlib.Topology.AlexandrovDiscrete.0.alexandrovDiscrete_iff_nhds._simp_1_2 | Mathlib.Topology.AlexandrovDiscrete | ∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsClosed s = ∀ (a : X), ClusterPt a (Filter.principal s) → a ∈ s | null | false |
Std.Do.PredTrans.throw | Std.Do.PredTrans | {ps : Std.Do.PostShape} → {α ε : Type u} → ε → Std.Do.PredTrans (Std.Do.PostShape.except ε ps) α | The predicate transformer that asserts the first exception condition.
| true |
CategoryTheory.Functor.mapSquare_obj | Mathlib.CategoryTheory.Square | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
(F : CategoryTheory.Functor C D) (sq : CategoryTheory.Square C), F.mapSquare.obj sq = sq.map F | null | true |
RCLike.add_conj | Mathlib.Analysis.RCLike.Basic | ∀ {K : Type u_1} [inst : RCLike K] (z : K), z + (starRingEnd K) z = 2 * ↑(RCLike.re z) | null | true |
Matrix.SpecialLinearGroup.isTopologicalGroup | Mathlib.Topology.Algebra.Group.Matrix | ∀ {n : Type u_1} {R : Type u_2} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing R]
[inst_3 : TopologicalSpace R] [IsTopologicalRing R], IsTopologicalGroup (Matrix.SpecialLinearGroup n R) | The special linear group over a topological ring is a topological group. | true |
ModuleCat.shortComplexOfConj_exact | Mathlib.Algebra.Homology.ShortComplex.ModuleCat | ∀ {R : Type u} [inst : Ring R] {M : Type v} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type v}
[inst_3 : AddCommGroup N] [inst_4 : Module R N] {L : Type v} [inst_5 : AddCommGroup L] [inst_6 : Module R L]
{M' : Type u_1} {N' : Type u_2} {L' : Type u_3} [inst_7 : AddCommGroup M'] [inst_8 : AddCommGroup N']
... | null | true |
Lean.Meta.Simp.Arith.Nat.toLinearExpr | Lean.Meta.Tactic.Simp.Arith.Nat.Basic | Lean.Expr → Lean.MetaM (Lean.Meta.Simp.Arith.Nat.LinearExpr × Array Lean.Expr) | null | true |
IsLocalization.adjoin_inv | Mathlib.RingTheory.Localization.Away.AdjoinRoot | ∀ {R : Type u_1} [inst : CommRing R] (r : R), IsLocalization.Away r (AdjoinRoot (Polynomial.C r * Polynomial.X - 1)) | null | true |
WeierstrassCurve.Projective.baseChange_polynomialY | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | ∀ {R : Type r} [inst : CommRing R] (W' : WeierstrassCurve.Projective R) {S : Type s} [inst_1 : CommRing S] {A : Type u}
[inst_2 : CommRing A] {B : Type v} [inst_3 : CommRing B] [inst_4 : Algebra R S] [inst_5 : Algebra R A]
[inst_6 : Algebra S A] [IsScalarTower R S A] [inst_8 : Algebra R B] [inst_9 : Algebra S B] [I... | null | true |
Lean.Linter.MissingDocs.lintNamed | Lean.Linter.MissingDocs | Lean.Syntax → String → Lean.Elab.Command.CommandElabM Unit | null | true |
_private.Lean.Elab.Tactic.Omega.Frontend.0.Lean.Elab.Tactic.Omega.asLinearComboImpl.handleNatCast.match_1 | Lean.Elab.Tactic.Omega.Frontend | (motive : Option Lean.Expr → Sort u_1) →
(__do_lift : Option Lean.Expr) →
((v : Lean.Expr) → motive (some v)) → ((x : Option Lean.Expr) → motive x) → motive __do_lift | null | false |
isExtrFilter_dual_iff | Mathlib.Order.Filter.Extr | ∀ {α : Type u} {β : Type v} [inst : Preorder β] {f : α → β} {l : Filter α} {a : α},
IsExtrFilter (⇑OrderDual.toDual ∘ f) l a ↔ IsExtrFilter f l a | null | true |
BddLat.hasForgetToBddOrd._proof_1 | Mathlib.Order.Category.BddLat | ∀ (X : BddLat),
BddOrd.ofHom (BddLat.Hom.hom (CategoryTheory.CategoryStruct.id X)).toBoundedOrderHom =
CategoryTheory.CategoryStruct.id ↧↑X.toLat | null | false |
AddHom.instAdd.eq_1 | Mathlib.Algebra.Group.Hom.Basic | ∀ {M : Type u_2} {N : Type u_3} [inst : Add M] [inst_1 : AddCommSemigroup N],
AddHom.instAdd = { add := fun f g => { toFun := fun m => f m + g m, map_add' := ⋯ } } | null | true |
Float32.Model.instMax | Init.Data.Float.Model.Float32 | Max Float32.Model | null | true |
Ring.KrullDimLE.minimalPrimes_eq_setOfPred_isMaximal | Mathlib.RingTheory.KrullDimension.Zero | ∀ (R : Type u_1) [inst : CommSemiring R] [Ring.KrullDimLE 0 R], minimalPrimes R = {I | I.IsMaximal} | null | true |
CoalgCat.mk.noConfusion | Mathlib.Algebra.Category.CoalgCat.Basic | {R : Type u} →
{inst : CommRing R} →
{P : Sort u_1} →
{toModuleCat : ModuleCat R} →
{instCoalgebra : Coalgebra R ↑toModuleCat} →
{toModuleCat' : ModuleCat R} →
{instCoalgebra' : Coalgebra R ↑toModuleCat'} →
{ toModuleCat := toModuleCat, instCoalgebra := instCoalge... | null | false |
Prod.seminormedRing._proof_12 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {α : Type u_2} {β : Type u_1} [inst : SeminormedRing α] [inst_1 : SeminormedRing β] (n : ℕ), ↑(n + 1) = ↑n + 1 | null | false |
Nat.div_ne_zero_iff_of_dvd | Init.Data.Nat.Lemmas | ∀ {b a : ℕ}, b ∣ a → (a / b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0) | null | true |
Lean.Widget.instEmptyCollectionInteractiveGoals | Lean.Widget.InteractiveGoal | EmptyCollection Lean.Widget.InteractiveGoals | null | true |
CategoryTheory.Functor.Faithful.div_comp | Mathlib.CategoryTheory.Functor.FullyFaithful | ∀ {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 E) [F.Faithful]
(G : CategoryTheory.Functor D E) [inst_4 : G.Faithful] (obj : C → D) (h_obj : ∀ (X : C), G... | null | true |
ModuleCat.Hom._sizeOf_1 | Mathlib.Algebra.Category.ModuleCat.Basic | {R : Type u} → {inst : Ring R} → {M N : ModuleCat R} → [SizeOf R] → M.Hom N → ℕ | null | false |
PseudoEpimorphismClass.recOn | Mathlib.Topology.Order.Hom.Esakia | {F : Type u_6} →
{α : Type u_7} →
{β : Type u_8} →
[inst : Preorder α] →
[inst_1 : Preorder β] →
[inst_2 : FunLike F α β] →
{motive : PseudoEpimorphismClass F α β → Sort u} →
(t : PseudoEpimorphismClass F α β) →
([toRelHomClass : RelHomClass F (fun... | null | false |
CategoryTheory.Subgroupoid.obj_surjective_of_im_eq_top | Mathlib.CategoryTheory.Groupoid.Subgroupoid | ∀ {C : Type u} [inst : CategoryTheory.Groupoid C] {D : Type u_1} [inst_1 : CategoryTheory.Groupoid D]
(φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj),
CategoryTheory.Subgroupoid.im φ hφ = ⊤ → Function.Surjective φ.obj | null | true |
NNRat.coe_pow._simp_1 | Mathlib.Data.NNRat.Defs | ∀ (q : ℚ≥0) (n : ℕ), ↑q ^ n = ↑(q ^ n) | null | false |
_private.Mathlib.LinearAlgebra.Matrix.ToLin.0.LinearMap.toMatrix_basis_equiv._simp_1_1 | Mathlib.LinearAlgebra.Matrix.ToLin | ∀ {α : Sort u_1} {a b : α}, (a = b) = (b = a) | null | false |
List.IsChain.backwards_cons_induction | Mathlib.Data.List.Chain | ∀ {α : Type u_1} {r : α → α → Prop} {a b : α} (p : α → Prop) (l : List α),
List.IsChain r (a :: l) → (a :: l).getLast ⋯ = b → (∀ ⦃x y : α⦄, r x y → p y → p x) → p b → ∀ i ∈ a :: l, p i | null | true |
NonUnitalAlgHom.Lmul._proof_4 | Mathlib.Analysis.Normed.Operator.Mul | ∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] (R : Type u_2) [inst_1 : NonUnitalSeminormedRing R]
[inst_2 : NormedSpace 𝕜 R], ContinuousConstSMul 𝕜 R | null | false |
CategoryTheory.Pretopology.trivial._proof_2 | Mathlib.CategoryTheory.Sites.Pretopology | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_1, u_2} C] ⦃X : C⦄ (S : CategoryTheory.Presieve X)
(Ti : ⦃Y : C⦄ → (f : Y ⟶ X) → S f → CategoryTheory.Presieve Y),
S ∈ {S | ∃ Y f, ∃ (_ : CategoryTheory.IsIso f), S = CategoryTheory.Presieve.singleton f} →
(∀ ⦃Y : C⦄ (f : Y ⟶ X) (H : S f),
Ti f H ∈ {S ... | null | false |
_private.Lean.Meta.Tactic.Grind.Types.0.Lean.Meta.Grind.Methods.toMethodsRef.unsafe_impl_2 | Lean.Meta.Tactic.Grind.Types | Lean.Meta.Grind.Methods → Lean.Meta.Grind.MethodsRef | null | true |
Module.Relations.solutionFinsupp._proof_1 | Mathlib.Algebra.Module.Presentation.Free | ∀ {A : Type u_1} [inst : Ring A] (relations : Module.Relations A) [IsEmpty relations.R] (r : relations.R),
(Finsupp.linearCombination A fun g => fun₀ | g => 1) (relations.relation r) = 0 | null | false |
CategoryTheory.MorphismProperty.epimorphisms.iff._simp_1 | Mathlib.CategoryTheory.MorphismProperty.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y),
CategoryTheory.MorphismProperty.epimorphisms C f = CategoryTheory.Epi f | null | false |
OrderRingHom.apply_eq_self | Mathlib.Algebra.Order.Archimedean.Hom | ∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] [Archimedean α] (f : α →+*o α)
(x : α), f x = x | null | true |
SheafOfModules.evaluationPreservesLimitsOfSize | Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C}
(R : CategoryTheory.Sheaf J RingCat) (X : Cᵒᵖ),
CategoryTheory.Limits.PreservesLimitsOfSize.{v₂, v, max u₁ v, v, max (max (max u u₁) (v + 1)) v₁, max u (v + 1)}
(SheafOfModules.evaluation R X) | null | true |
RelIso.emptySumLex_apply | Mathlib.Order.RelIso.Basic | ∀ {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [inst : IsEmpty α] (a : α ⊕ β),
(RelIso.emptySumLex r s) a = (Equiv.sumEmpty β α) a.swap | null | true |
_private.Mathlib.Probability.Moments.Variance.0.ProbabilityTheory.evariance_eq_zero_iff._simp_1_2 | Mathlib.Probability.Moments.Variance | ∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a - b = 0) = (a = b) | null | false |
cfcₙ_smul._auto_1 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | Lean.Syntax | null | false |
NonUnitalCommCStarAlgebra.recOn | Mathlib.Analysis.CStarAlgebra.Classes | {A : Type u_1} →
{motive : NonUnitalCommCStarAlgebra A → Sort u} →
(t : NonUnitalCommCStarAlgebra A) →
([toNonUnitalNormedCommRing : NonUnitalNormedCommRing A] →
[toStarRing : StarRing A] →
[toCompleteSpace : CompleteSpace A] →
[toCStarRing : CStarRing A] →
... | null | false |
Mathlib.Tactic.Basify.ElimEntry.rec | Mathlib.Tactic.Basify.Attr | {motive : Mathlib.Tactic.Basify.ElimEntry → Sort u} →
((elimName : Lean.Name) →
(altBinders : Array (Array (Option Lean.Name))) → motive { elimName := elimName, altBinders := altBinders }) →
(t : Mathlib.Tactic.Basify.ElimEntry) → motive t | null | false |
IsDedekindDomain.HeightOneSpectrum.adicAbv._proof_1 | Mathlib.RingTheory.DedekindDomain.AdicValuation | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] {K : Type u_2} [inst_2 : Field K]
[inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal}
(hb : 1 < b) (x : K), 0 ≤ ↑(v.adicAbvDef hb x) | null | false |
Nat.dvd_of_mem_primeFactorsList | Mathlib.Data.Nat.Factors | ∀ {n p : ℕ}, p ∈ n.primeFactorsList → p ∣ n | null | true |
_private.Mathlib.LinearAlgebra.Finsupp.Span.0.Finsupp.iInf_ker_lapply_le_bot._simp_1_2 | Mathlib.LinearAlgebra.Finsupp.Span | ∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_4}
(p : ι → Submodule R M) {x : M}, (x ∈ ⨅ i, p i) = ∀ (i : ι), x ∈ p i | null | false |
Std.LinearPreorderPackage.le_total | Init.Data.Order.PackageFactories | ∀ {α : Type u} [self : Std.LinearPreorderPackage α] (a b : α), a ≤ b ∨ b ≤ a | null | true |
_private.Mathlib.Algebra.Lie.Semisimple.Basic.0.LieAlgebra.IsSemisimple.isSimple_of_isAtom._simp_1_15 | Mathlib.Algebra.Lie.Semisimple.Basic | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
_private.Lean.Compiler.LCNF.LiveVars.0.Lean.Compiler.LCNF.State.rec | Lean.Compiler.LCNF.LiveVars | {motive : Lean.Compiler.LCNF.State✝ → Sort u} →
((visitedJps : Lean.FVarIdHashSet) → motive { visitedJps := visitedJps }) → (t : Lean.Compiler.LCNF.State✝) → motive t | null | false |
Complex.norm_log_one_add_sub_self_le | Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | ∀ {z : ℂ}, ‖z‖ < 1 → ‖Complex.log (1 + z) - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 | The difference `log (1+z) - z` is bounded by `‖z‖^2/(2*(1-‖z‖))` when `‖z‖ < 1`. | true |
AddCommute.addUnits_neg_right | Mathlib.Algebra.Group.Commute.Units | ∀ {M : Type u_1} [inst : AddMonoid M] {a : M} {u : AddUnits M}, AddCommute a ↑u → AddCommute a ↑(-u) | null | true |
_private.Lean.Elab.BuiltinCommand.0.Lean.Elab.Command.elabSection._regBuiltin.Lean.Elab.Command.elabSection.declRange_3 | Lean.Elab.BuiltinCommand | IO Unit | null | false |
_private.Std.Data.DTreeMap.Internal.WF.Lemmas.0.Std.DTreeMap.Internal.Cell.Const.alter.match_1.eq_2 | Std.Data.DTreeMap.Internal.WF.Lemmas | ∀ {α : Type u_2} {β : Type u_1} (motive : Option ((_ : α) × β) → Sort u_3) (fst : α) (v' : β) (h_1 : Unit → motive none)
(h_2 : (fst : α) → (v' : β) → motive (some ⟨fst, v'⟩)),
(match some ⟨fst, v'⟩ with
| none => h_1 ()
| some ⟨fst, v'⟩ => h_2 fst v') =
h_2 fst v' | null | true |
_private.Mathlib.Tactic.Linter.UnusedTactic.0.Mathlib.Linter.UnusedTactic.eraseUsedTactics | Mathlib.Tactic.Linter.UnusedTactic | Std.HashSet Lean.SyntaxNodeKind → Lean.PersistentArray Lean.Elab.InfoTree → Mathlib.Linter.UnusedTactic.M✝ Unit | Search for tactic executions in the info tree and remove the syntax of the tactics that
changed something. | true |
Bornology.IsBounded.snd_of_prod | Mathlib.Topology.Bornology.Constructions | ∀ {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst_1 : Bornology β] {s : Set α} {t : Set β},
Bornology.IsBounded (s ×ˢ t) → s.Nonempty → Bornology.IsBounded t | null | true |
continuous_sigmaMk | Mathlib.Topology.Constructions | ∀ {ι : Type u_2} {σ : ι → Type u_4} [inst : (i : ι) → TopologicalSpace (σ i)] {i : ι}, Continuous (Sigma.mk i) | null | true |
PadicInt.toZMod._proof_2 | Mathlib.NumberTheory.Padics.RingHoms | ∀ {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p]) (a b : ℕ),
x - ↑a ∈ Ideal.span {↑p} → x - ↑b ∈ Ideal.span {↑p} → ↑a = ↑b | null | false |
_private.Mathlib.RingTheory.Algebraic.Defs.0.Algebra.isAlgebraic_iff._simp_1_3 | Mathlib.RingTheory.Algebraic.Defs | ∀ {p : Prop} {q : p → Prop} (h : p), (∀ (h' : p), q h') = q h | null | false |
Vector.swap._auto_3 | Init.Data.Vector.Basic | Lean.Syntax | null | false |
WithVal.instNumberField | Mathlib.Topology.Algebra.Valued.WithVal | ∀ {R : Type u_1} {Γ₀ : Type u_2} [inst : LinearOrderedCommGroupWithZero Γ₀] [inst_1 : Field R] (v : Valuation R Γ₀)
[NumberField R], NumberField (WithVal v) | null | true |
Topology.IsEmbedding.comp | Mathlib.Topology.Maps.Basic | ∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [inst : TopologicalSpace X]
[inst_1 : TopologicalSpace Y] [inst_2 : TopologicalSpace Z],
Topology.IsEmbedding g → Topology.IsEmbedding f → Topology.IsEmbedding (g ∘ f) | null | true |
IsOpen.stronglyLocallyContractibleSpace | Mathlib.Topology.Homotopy.LocallyContractible | ∀ {X : Type u_1} [inst : TopologicalSpace X] [StronglyLocallyContractibleSpace X] {U : Set X},
IsOpen U → StronglyLocallyContractibleSpace ↑U | Open subsets of strongly locally contractible spaces are strongly locally contractible. | true |
ModuleCat.CoextendScalars.map' | Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {R : Type u₁} →
{S : Type u₂} →
[inst : Ring R] →
[inst_1 : Ring S] →
(f : R →+* S) →
{M M' : ModuleCat R} → (M ⟶ M') → (ModuleCat.CoextendScalars.obj' f M ⟶ ModuleCat.CoextendScalars.obj' f M') | If `M, M'` are `R`-modules, then any `R`-linear map `g : M ⟶ M'` induces an `S`-linear map
`(S →ₗ[R] M) ⟶ (S →ₗ[R] M')` defined by `h ↦ g ∘ h` | true |
Equiv.prodEmbeddingDisjointEquivSigmaEmbeddingRestricted._proof_1 | Mathlib.Logic.Equiv.Embedding | ∀ {α : Type u_3} {β : Type u_1} {γ : Type u_2} (a : α ↪ γ),
(fun b => Disjoint (Set.range ⇑a) (Set.range ⇑b)) = fun f => ∀ (a_1 : β), f a_1 ∈ (Set.range ⇑a)ᶜ | null | false |
_private.Mathlib.Logic.Equiv.Prod.0.Equiv.piEquivPiSubtypeProd._proof_7 | Mathlib.Logic.Equiv.Prod | ∀ {α : Type u_2} (p : α → Prop) (β : α → Type u_1) [inst : DecidablePred p] (f : (i : { x // p x }) → β ↑i)
(g : (i : { x // ¬p x }) → β ↑i),
(fun x => if h : p ↑x then (f, g).1 ⟨↑x, h⟩ else (f, g).2 ⟨↑x, h⟩, fun x =>
if h : p ↑x then (f, g).1 ⟨↑x, h⟩ else (f, g).2 ⟨↑x, h⟩).1 =
(f, g).1 | null | false |
Eq.trans_ssubset | Mathlib.Order.RelClasses | ∀ {α : Type u_1} [UsesSetNotationForOrder α] {a b c : α} [inst : LT α], a = b → b ⊂ c → a ⊂ c | Set notation form of `Eq.trans_lt` | true |
Std.DTreeMap.isEmpty | Std.Data.DTreeMap.Basic | {α : Type u} → {β : α → Type v} → {cmp : α → α → Ordering} → Std.DTreeMap α β cmp → Bool | Returns `true` if the tree map contains no mappings. | true |
CochainComplex.HomComplex.Cochain.toSingleEquiv_smul | Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] {X : C} {K : CochainComplex C ℤ} {R : Type u_1} [inst_3 : Ring R]
[inst_4 : CategoryTheory.Linear R C] (r : R) {p q n : ℤ} (h : p + n = q)
(f : CochainComplex.HomComple... | null | true |
CategoryTheory.PreOneHypercover.Hom.ext'_iff | Mathlib.CategoryTheory.Sites.Hypercover.One | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {S : C} {E : CategoryTheory.PreOneHypercover S}
{F : CategoryTheory.PreOneHypercover S} {f g : E.Hom F},
f = g ↔
∃ (hs₀ : f.s₀ = g.s₀) (_ :
∀ (i : E.I₀), f.h₀ i = CategoryTheory.CategoryStruct.comp (g.h₀ i) (CategoryTheory.eqToHom ⋯)) (hs₁ :
∀... | null | true |
Set.projIcc_of_le_left | Mathlib.Order.Interval.Set.ProjIcc | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α} (h : a ≤ b) {x : α}, x ≤ a → Set.projIcc a b h x = ⟨a, ⋯⟩ | null | true |
Mathlib.Tactic.ClickSuggestions.Choice | Mathlib.Tactic.ClickSuggestions.FindPremises | Type | A choice of which discrimination trees to build. | true |
CategoryTheory.Category.comp_id._autoParam | Mathlib.CategoryTheory.Category.Basic | Lean.Syntax | null | false |
Std.DHashMap.Internal.Raw₀.getEntry?_eq_getEntry? | Std.Data.DHashMap.Internal.WF | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] [PartialEquivBEq α] [LawfulHashable α]
{m : Std.DHashMap.Internal.Raw₀ α β},
Std.DHashMap.Internal.Raw.WFImp ↑m →
∀ {a : α}, m.getEntry? a = Std.Internal.List.getEntry? a (Std.DHashMap.Internal.toListModel (↑m).buckets) | null | true |
Std.Format.bracket | Init.Data.Format.Basic | String → Std.Format → String → Std.Format | Creates a format `l ++ f ++ r` with a flattening group, nesting the contents by the length of `l`.
The group's `FlattenBehavior` is `allOrNone`; for `fill` use `Std.Format.bracketFill`.
| true |
Lean.Meta.Grind.Context.cheapCases._default | Lean.Meta.Tactic.Grind.Types | Bool | null | false |
linearIndependent_fin_cons | Mathlib.LinearAlgebra.LinearIndependent.Lemmas | ∀ {K : Type u_3} {V : Type u} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {x : V} {n : ℕ}
{v : Fin n → V}, LinearIndependent K (Fin.cons x v) ↔ LinearIndependent K v ∧ x ∉ Submodule.span K (Set.range v) | **Alias** of `linearIndependent_finCons`. | true |
Std.WP.WPMonad.wp_orElse_EStateM_apply_eq | Std.WP.Monad.Lemmas | ∀ {ε σ α : Type} {post : α → σ → Prop} {epost : ε → σ → Prop} (x : EStateM ε σ α) (h : Unit → EStateM ε σ α),
Std.WP.wp (OrElse.orElse x h) post epost = fun s => Std.WP.wp x post (fun x s' => Std.WP.wp (h ()) post epost s') s | null | true |
WeierstrassCurve.Projective.PointClass | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | (R : Type r) → [CommRing R] → Type r | The equivalence class of a projective point representative on a Weierstrass curve. | true |
eq_zero_or_one_of_sq_eq_self | Mathlib.Algebra.GroupWithZero.Defs | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] [IsRightCancelMulZero M₀] {x : M₀}, x ^ 2 = x → x = 0 ∨ x = 1 | null | true |
Std.Time.Database.TZdb.getLocalZoneRules | Std.Time.Zoned.Database.TZdb | Std.Time.Database.TZdb → IO Std.Time.TimeZone.ZoneRules | Retrieves the timezone rules for the local timezone, re-reading the `TZ` and `TZDIR`
environment variables on each call so that runtime changes are reflected immediately.
| true |
IsOpen.add_closure | Mathlib.Topology.Algebra.Group.Pointwise | ∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [IsTopologicalAddGroup G] {s : Set G},
IsOpen s → ∀ (t : Set G), s + closure t = s + t | null | true |
Set.IsWF.min_add | Mathlib.Data.Finset.MulAntidiagonal | ∀ {α : Type u_1} {s t : Set α} [inst : AddCommMonoid α] [inst_1 : LinearOrder α] [inst_2 : IsOrderedCancelAddMonoid α]
(hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty), ⋯.min ⋯ = hs.min hsn + ht.min htn | null | true |
Order.not_isSuccLimit_add_one | Mathlib.Algebra.Order.SuccPred | ∀ {α : Type u_1} [inst : PartialOrder α] (a : α) [inst_1 : Add α] [inst_2 : One α] [SuccAddOrder α] [NoMaxOrder α],
¬Order.IsSuccLimit (a + 1) | null | true |
UInt16.pow | Init.Data.UInt.Basic | UInt16 → ℕ → UInt16 | The power operation, raising a 16-bit unsigned integer to a natural number power,
wrapping around on overflow. Usually accessed via the `^` operator.
This function is currently *not* overridden at runtime with an efficient implementation,
and should be used with caution. See https://github.com/leanprover/lean4/issues/... | true |
PartialHomeomorph.mk.noConfusion | Mathlib.Topology.PartialHomeomorph.Defs | {X : Type u_5} →
{Y : Type u_6} →
{inst : TopologicalSpace X} →
{inst_1 : TopologicalSpace Y} →
{P : Sort u} →
{toPartialEquiv : PartialEquiv X Y} →
{continuousOn_toFun : ContinuousOn (↑toPartialEquiv) toPartialEquiv.source} →
{continuousOn_invFun : ContinuousOn t... | null | false |
ModuleCat.smulNatTrans._proof_8 | Mathlib.Algebra.Category.ModuleCat.Basic | ∀ (R : Type u_1) [inst : Ring R] (x x_1 : R),
{ asHom := { app := fun M => (M.smul (x + x_1)).asHom, naturality := ⋯ } } =
{ asHom := { app := fun M => (M.smul x).asHom, naturality := ⋯ } } +
{ asHom := { app := fun M => (M.smul x_1).asHom, naturality := ⋯ } } | null | false |
Function.Surjective.smulCommClass | Mathlib.Algebra.Group.Action.Defs | ∀ {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [inst : SMul M α] [inst_1 : SMul N α] [inst_2 : SMul M β]
[inst_3 : SMul N β] [SMulCommClass M N α] {f : α → β},
Function.Surjective f →
(∀ (c : M) (x : α), f (c • x) = c • f x) → (∀ (c : N) (x : α), f (c • x) = c • f x) → SMulCommClass M N β | null | true |
CompHausLike.LocallyConstant.sigmaIso._proof_3 | Mathlib.Condensed.Discrete.LocallyConstant | ∀ {P : TopCat → Prop} [inst : ∀ (S : CompHausLike P) (p : ↑S.toTop → Prop), CompHausLike.HasProp P (Subtype p)]
{Q : CompHausLike P} {Z : Type (max u_1 u_2)} (r : LocallyConstant (↑Q.toTop) Z),
CompactSpace ((i : Function.Fiber ⇑r) × ↑(CompHausLike.LocallyConstant.fiber r i).toTop) | null | false |
extChartAt_self_apply | Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | ∀ (𝕜 : Type u_1) {E : Type u_2} {H : Type u_4} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {x y : H},
↑(extChartAt I x) y = ↑I y | null | true |
CategoryTheory.InjectiveResolution.definition._@.Mathlib.CategoryTheory.Abelian.Injective.Resolution.4211954440._hygCtx._hyg.8 | Mathlib.CategoryTheory.Abelian.Injective.Resolution | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.Abelian C] →
[CategoryTheory.EnoughInjectives C] → (Z : C) → CategoryTheory.InjectiveResolution Z | null | false |
Lean.Elab.Info.ofChoiceInfo.elim | Lean.Elab.InfoTree.Types | {motive : Lean.Elab.Info → Sort u} →
(t : Lean.Elab.Info) →
t.ctorIdx = 14 → ((i : Lean.Elab.ChoiceInfo) → motive (Lean.Elab.Info.ofChoiceInfo i)) → motive t | null | false |
one_half_le_sum_primes_ge_one_div | Mathlib.NumberTheory.SumPrimeReciprocals | ∀ (k : ℕ), 1 / 2 ≤ ∑ p ∈ (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \ k.primesBelow, 1 / ↑p | The sum over primes `k ≤ p ≤ 4^(π(k-1)+1)` over `1/p` (as a real number) is at least `1/2`. | true |
List.isEmpty_eq_false_iff._simp_1 | Init.Data.List.Lemmas | ∀ {α : Type u_1} {l : List α}, (l.isEmpty = false) = (l ≠ []) | null | false |
_private.Mathlib.Order.Interval.Set.Fin.0.Fin.preimage_rev_Ico._simp_1_2 | Mathlib.Order.Interval.Set.Fin | ∀ {n : ℕ} {i j : Fin n}, (i.rev < j) = (j.rev < i) | null | false |
Stream'.Seq.update._proof_2 | Mathlib.Data.Seq.Defs | ∀ {α : Type u_1} (s : Stream'.Seq α) (n : ℕ) (f : α → α), Stream'.IsSeq (Function.update (↑s) n (Option.map f (↑s n))) | null | false |
Complex.continuousWithinAt_log_of_re_neg_of_im_zero | Mathlib.Analysis.SpecialFunctions.Complex.Log | ∀ {z : ℂ}, z.re < 0 → z.im = 0 → ContinuousWithinAt Complex.log {z | 0 ≤ z.im} z | null | true |
FreeGroup.Red.Step.cons_not | Mathlib.GroupTheory.FreeGroup.Basic | ∀ {α : Type u} {L : List (α × Bool)} {x : α} {b : Bool}, FreeGroup.Red.Step ((x, b) :: (x, !b) :: L) L | null | true |
_private.Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform.0.SimpleGraph.unreduced_edges_subset._simp_1_4 | Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a ≤ b) = (b < a) | null | false |
_private.Mathlib.Order.CompleteLattice.PiLex.0.Pi.Colex.instInfSetColexForall._proof_1 | Mathlib.Order.CompleteLattice.PiLex | ∀ {ι : Type u_1} [inst : LinearOrder ι] [WellFoundedGT ι], WellFounded fun x1 x2 => x1 < x2 | null | false |
StrictConcaveOn.lt_map_sum | Mathlib.Analysis.Convex.Jensen | ∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} {ι : Type u_4} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜]
[IsStrictOrderedRing 𝕜] [inst_3 : AddCommGroup E] [inst_4 : AddCommGroup β] [inst_5 : PartialOrder β]
[IsOrderedAddMonoid β] [inst_7 : Module 𝕜 E] [inst_8 : Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set... | Concave **strict Jensen inequality**.
If the function is strictly concave, the weights are strictly positive and the indexed family of
points is non-constant, then Jensen's inequality is strict.
See also `StrictConcaveOn.map_sum_eq_iff`. | true |
Lean.PrettyPrinter.Formatter.pushToken | Lean.PrettyPrinter.Formatter | Lean.SourceInfo → String → Bool → Lean.PrettyPrinter.FormatterM Unit | null | true |
Lean.Meta.instReduceEvalLiteral_qq | Qq.ForLean.ReduceEval | Lean.Meta.ReduceEval Lean.Literal | null | true |
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