name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.MeasureTheory.Function.UniformIntegrable.0.MeasureTheory.MemLp.tendsto_eLpNorm_restrict_zero._simp_1_1 | Mathlib.MeasureTheory.Function.UniformIntegrable | ∀ {α : Type u_1} [inst : Preorder α] {b x : α}, (x ∈ Set.Iio b) = (x < b) | null | false |
Set.Icc.mul_le_right | Mathlib.Algebra.Order.Interval.Set.Instances | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R] {x y : ↑(Set.Icc 0 1)},
x * y ≤ y | null | true |
_private.Init.Omega.IntList.0.List.getElem?_map.match_1_1 | Init.Omega.IntList | ∀ {α : Type u_1} (motive : List α → ℕ → Prop) (x : List α) (x_1 : ℕ),
(∀ (x : ℕ), motive [] x) →
(∀ (head : α) (tail : List α), motive (head :: tail) 0) →
(∀ (head : α) (l : List α) (i : ℕ), motive (head :: l) i.succ) → motive x x_1 | null | false |
FunLike.ring | Mathlib.Algebra.Ring.FunLike | {F : Type u_1} →
{α : Type u_2} →
{β : Type u_3} →
[inst : FunLike F α β] →
[inst_1 : Zero F] →
[inst_2 : One F] →
[inst_3 : Add F] →
[inst_4 : Neg F] →
[inst_5 : Sub F] →
[inst_6 : Mul F] →
[inst_7 : SMul ℕ F]... | A `FunLike` type with `(f * g) x = f x * g x` is a `Ring` if `β` is a `Ring`. | true |
StarRingEquiv.trans | Mathlib.Algebra.Star.StarRingHom | {A : Type u_1} →
{B : Type u_2} →
{C : Type u_3} →
[inst : Add A] →
[inst_1 : Add B] →
[inst_2 : Mul A] →
[inst_3 : Mul B] →
[inst_4 : Star A] →
[inst_5 : Star B] →
[inst_6 : Add C] → [inst_7 : Mul C] → [inst_8 : Star C] → (A ≃⋆+*... | Transitivity of `StarRingEquiv`. | true |
_private.Mathlib.LinearAlgebra.BilinearForm.Orthogonal.0.LinearMap.BilinForm.ker_restrict_eq_of_codisjoint._simp_1_2 | Mathlib.LinearAlgebra.BilinearForm.Orthogonal | ∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂}
{x : M} {f : M →ₛₗ[σ₁₂] M₂} {p : Submodule R₂ M₂}, (x ∈ Submodule.comap f p) = (f x ∈ p) | null | false |
IsPRadical.injective_comp | Mathlib.FieldTheory.IsPerfectClosure | ∀ {K : Type u_1} {L : Type u_2} (M : Type u_3) [inst : CommRing K] [inst_1 : CommRing L] [inst_2 : CommRing M]
(i : K →+* L) (p : ℕ) [ExpChar M p] [IsPRadical i p] [IsReduced M], Function.Injective fun f => f.comp i | If `i : K →+* L` is `p`-radical, then for any reduced ring `M` of exponential characteristic
`p`, the map `(L →+* M) → (K →+* M)` induced by `i` is injective.
A special case of `IsPRadical.injective_comp_of_pNilradical_eq_bot`
and a generalization of `IsPurelyInseparable.injective_comp_algebraMap`. | true |
Algebra.FinitePresentation.mvPolynomial_of_finitePresentation | Mathlib.RingTheory.FinitePresentation | ∀ {R : Type w₁} {A : Type w₂} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
[Algebra.FinitePresentation R A] (ι : Type v) [Finite ι], Algebra.FinitePresentation R (MvPolynomial ι A) | If `A` is a finitely presented `R`-algebra, then `MvPolynomial (Fin n) A` is finitely presented
as `R`-algebra. | true |
_private.Lean.Meta.Tactic.Grind.Action.0.Lean.Meta.Grind.Action.loop.match_1 | Lean.Meta.Tactic.Grind.Action | (motive : ℕ → Sort u_1) → (n : ℕ) → (Unit → motive 0) → ((n : ℕ) → motive n.succ) → motive n | null | false |
isAddRegular_unop | Mathlib.Algebra.Regular.Opposite | ∀ {R : Type u_1} [inst : Add R] {a : Rᵃᵒᵖ}, IsAddRegular (AddOpposite.unop a) ↔ IsAddRegular a | null | true |
Polynomial.add_modByMonic | Mathlib.Algebra.Polynomial.Div | ∀ {R : Type u} [inst : CommRing R] {q : Polynomial R} (p₁ p₂ : Polynomial R), (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q | null | true |
Nat.maxPowDvdDiv.eq_1 | Mathlib.Data.Nat.MaxPowDiv | ∀ (p n : ℕ), p.maxPowDvdDiv n = if H : 1 < p ∧ n ≠ 0 then Nat.maxPowDvdDiv.go n p H else (0, n) | null | true |
_private.Init.Data.Nat.Gcd.0.Nat.gcd_pos_iff._simp_1_2 | Init.Data.Nat.Gcd | ∀ {p q : Prop} [d₁ : Decidable p] [d₂ : Decidable q], (¬(p ∧ q)) = (¬p ∨ ¬q) | null | false |
Function.OfArity.curry_two_eq_curry | Mathlib.Data.Fin.Tuple.Curry | ∀ {α β : Type u} (f : (Fin 2 → α) → β), Function.OfArity.curry f = Function.curry (f ∘ ⇑(finTwoArrowEquiv α).symm) | null | true |
Turing.TM2.stepAux.eq_1 | Mathlib.Computability.TuringMachine.StackTuringMachine | ∀ {K : Type u_1} {Γ : K → Type u_2} {Λ : Type u_3} {σ : Type u_4} [inst : DecidableEq K] (x : σ)
(x_1 : (k : K) → List (Γ k)) (k : K) (f : σ → Γ k) (q : Turing.TM2.Stmt Γ Λ σ),
Turing.TM2.stepAux (Turing.TM2.Stmt.push k f q) x x_1 = Turing.TM2.stepAux q x (Function.update x_1 k (f x :: x_1 k)) | null | true |
_private.Lean.Compiler.LCNF.Simp.DiscrM.0.Lean.Compiler.LCNF.Simp.withDiscrCtorImp.updateCtx.match_1 | Lean.Compiler.LCNF.Simp.DiscrM | (motive : Option (List Lean.Level × Array (Lean.Compiler.LCNF.Arg Lean.Compiler.LCNF.Purity.pure)) → Sort u_1) →
(__do_lift : Option (List Lean.Level × Array (Lean.Compiler.LCNF.Arg Lean.Compiler.LCNF.Purity.pure))) →
((us : List Lean.Level) →
(params : Array (Lean.Compiler.LCNF.Arg Lean.Compiler.LCNF.Pur... | null | false |
addMonoidHomOfMemClosureRangeCoe_apply | Mathlib.Topology.Algebra.Monoid | ∀ {M₁ : Type u_6} {M₂ : Type u_7} [inst : TopologicalSpace M₂] [inst_1 : T2Space M₂] [inst_2 : AddZeroClass M₁]
[inst_3 : AddZeroClass M₂] [inst_4 : ContinuousAdd M₂] {F : Type u_8} [inst_5 : FunLike F M₁ M₂]
[inst_6 : AddMonoidHomClass F M₁ M₂] (f : M₁ → M₂) (hf : f ∈ closure (Set.range fun f x => f x)),
⇑(addMo... | null | true |
ExpChar.zero | Mathlib.Algebra.CharP.Defs | ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [CharZero R], ExpChar R 1 | null | true |
UInt8.toUInt16_div | Init.Data.UInt.Lemmas | ∀ (a b : UInt8), (a / b).toUInt16 = a.toUInt16 / b.toUInt16 | null | true |
Rat.instSemilatticeInf | Mathlib.Algebra.Order.Ring.Unbundled.Rat | SemilatticeInf ℚ | null | true |
CategoryTheory.Subfunctor.range_eq_top | Mathlib.CategoryTheory.Subfunctor.Image | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {F F' : CategoryTheory.Functor C (Type w)} (p : F' ⟶ F)
[CategoryTheory.Epi p], CategoryTheory.Subfunctor.range p = ⊤ | null | true |
Associates.instCommMonoidWithZero._proof_1 | Mathlib.Algebra.GroupWithZero.Associated | ∀ {M : Type u_1} [inst : CommMonoidWithZero M] (a : M), 0 * Associates.mk a = 0 | null | false |
ISize.minValue_le_toInt | Init.Data.SInt.Lemmas | ∀ (x : ISize), ISize.minValue.toInt ≤ x.toInt | null | true |
ContinuousLinearMap.isPositive_self_comp_adjoint | Mathlib.Analysis.InnerProductSpace.Positive | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F]
[inst_5 : CompleteSpace E] [inst_6 : CompleteSpace F] (S : E →L[𝕜] F),
(S ∘SL ContinuousLinearMap.adjoint S).IsPos... | null | true |
Std.DTreeMap.Internal.Impl.Const.getKey!_insertManyIfNewUnit!_list_of_not_mem_of_mem | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α fun x => Unit} [Std.TransOrd α] [inst : Inhabited α],
t.WF →
∀ {l : List α} {k k' : α},
compare k k' = Ordering.eq →
k ∉ t →
List.Pairwise (fun a b => ¬compare a b = Ordering.eq) l →
k ∈ l → (↑(Std.DTreeMap.Inte... | null | true |
Aesop.Frontend.AttrConfig.noConfusionType | Aesop.Frontend.Attribute | Sort u → Aesop.Frontend.AttrConfig → Aesop.Frontend.AttrConfig → Sort u | null | false |
Std.TreeSet.le_max!_of_mem | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp] [inst : Inhabited α] {k : α},
k ∈ t → (cmp k t.max!).isLE = true | null | true |
CategoryTheory.Abelian.SpectralObject.homologyDataIdId_right_p | Mathlib.Algebra.Homology.SpectralObject.Page | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j : ι} (f : i ⟶ j) (n₀ n₁ n₂ : ℤ)
(hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObjec... | null | true |
AdicCompletion.sum._proof_1 | Mathlib.RingTheory.AdicCompletion.Functoriality | ∀ {R : Type u_1} [inst : CommRing R] {ι : Type u_2} (M : ι → Type u_3) [inst_1 : (i : ι) → AddCommGroup (M i)]
[inst_2 : (i : ι) → Module R (M i)], SMulCommClass R R (DirectSum ι fun j => M j) | null | false |
Std.ExtHashSet.ext_mem | Std.Data.ExtHashSet.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} [inst : LawfulBEq α] {m₁ m₂ : Std.ExtHashSet α},
(∀ (k : α), k ∈ m₁ ↔ k ∈ m₂) → m₁ = m₂ | null | true |
_private.Lean.Meta.Tactic.Grind.EMatchTheorem.0.Lean.Meta.Grind.EMatchTheoremKind.toAttributeCore.match_1 | Lean.Meta.Tactic.Grind.EMatchTheorem | (motive : Lean.Meta.Grind.EMatchTheoremKind → Sort u_1) →
(kind : Lean.Meta.Grind.EMatchTheoremKind) →
(Unit → motive (Lean.Meta.Grind.EMatchTheoremKind.eqLhs true)) →
(Unit → motive (Lean.Meta.Grind.EMatchTheoremKind.eqLhs false)) →
(Unit → motive (Lean.Meta.Grind.EMatchTheoremKind.eqRhs true)) →
... | null | false |
Sigma.forall | Mathlib.Data.Sigma.Basic | ∀ {α : Type u_1} {β : α → Type u_4} {p : (a : α) × β a → Prop},
(∀ (x : (a : α) × β a), p x) ↔ ∀ (a : α) (b : β a), p ⟨a, b⟩ | null | true |
Lean._aux_Init_Meta___macroRules_Lean_Parser_Syntax_addPrio_1 | Init.Meta | Lean.Macro | null | false |
SchwartzMap.instIsUniformAddGroup | Mathlib.Analysis.Distribution.SchwartzSpace.Basic | ∀ {E : Type u_5} {F : Type u_6} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℝ F], IsUniformAddGroup (SchwartzMap E F) | null | true |
ChainComplex.truncateAugment._proof_3 | Mathlib.Algebra.Homology.Augment | ∀ {V : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V]
(C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : CategoryTheory.CategoryStruct.comp (C.d 1 0) f = 0) (i j : ℕ),
(ComplexShape.down ℕ).Rel i j →
CategoryTheory.CategoryStruct.comp (CategoryTheory... | null | false |
NumberField.CMExtension.algebraMap_equivMaximalRealSubfield_symm_apply | Mathlib.NumberTheory.NumberField.CMField | ∀ (F : Type u_1) (K : Type u_2) [inst : Field F] [inst_1 : NumberField.IsTotallyReal F] [inst_2 : Field K]
[inst_3 : CharZero K] [inst_4 : Algebra.IsIntegral ℚ K] [inst_5 : NumberField.IsTotallyComplex K]
[inst_6 : Algebra F K] [inst_7 : Algebra.IsQuadraticExtension F K] (x : ↥(NumberField.maximalRealSubfield K)),
... | null | true |
IsLocalization.Away.awayToAwayRight_eq | Mathlib.RingTheory.Localization.Away.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] {S : Type u_2} [inst_1 : CommSemiring S] [inst_2 : Algebra R S] {P : Type u_3}
[inst_3 : CommSemiring P] (x : R) [inst_4 : IsLocalization.Away x S] (y : R) [inst_5 : Algebra R P]
[inst_6 : IsLocalization.Away (x * y) P] (a : R),
(IsLocalization.Away.awayToAwayRight x y) ((... | null | true |
ZetaAsymptotics.termSum | Mathlib.NumberTheory.Harmonic.ZetaAsymp | ℝ → ℕ → ℝ | Sum of finitely many `term`s. | true |
Int.mul_le_mul_left._simp_1 | Init.Data.Int.DivMod.Lemmas | ∀ {a b c : ℤ}, 0 < a → (a * b ≤ a * c) = (b ≤ c) | null | false |
EReal.Tendsto.const_mul | Mathlib.Topology.Instances.EReal.Lemmas | ∀ {α : Type u_2} {f : Filter α} {m : α → EReal} {a b : EReal},
Filter.Tendsto m f (nhds b) → a ≠ ⊥ ∨ b ≠ 0 → a ≠ ⊤ ∨ b ≠ 0 → Filter.Tendsto (fun b => a * m b) f (nhds (a * b)) | null | true |
Lean.Elab.Command.AssertExists._sizeOf_1 | Lean.Elab.AssertExists | Lean.Elab.Command.AssertExists → ℕ | null | false |
MvPowerSeries.isEmptyEquiv_symm_apply | Mathlib.RingTheory.MvPowerSeries.Equiv | ∀ (σ : Type u_1) (R : Type u_2) [inst : CommSemiring R] [inst_1 : IsEmpty σ] (a : R),
(MvPowerSeries.isEmptyEquiv σ R).symm a = MvPowerSeries.C a | null | true |
CategoryTheory.Discrete.monoidalFunctorMonoidal._proof_10 | Mathlib.CategoryTheory.Monoidal.Discrete | ∀ {M : Type u_1} [inst : Monoid M] {N : Type u_2} [inst_1 : Monoid N] (F : M →* N) (X : CategoryTheory.Discrete M),
(CategoryTheory.MonoidalCategoryStruct.rightUnitor ((CategoryTheory.Discrete.monoidalFunctor F).obj X)).hom =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerL... | null | false |
_private.Mathlib.Topology.Bornology.BoundedOperation.0.isBounded_pow._simp_1_2 | Mathlib.Topology.Bornology.BoundedOperation | ∀ {α : Type u} {β : Type v} (f : α → β) (s : Set α) (y : β), (y ∈ f '' s) = ∃ x ∈ s, f x = y | null | false |
Multiset.le_union_left | Mathlib.Data.Multiset.UnionInter | ∀ {α : Type u_1} [inst : DecidableEq α] {s t : Multiset α}, s ≤ s ∪ t | null | true |
SzemerediRegularity.coe_stepBound | Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | ∀ {α : Type u_1} [inst : Semiring α] (n : ℕ), ↑(SzemerediRegularity.stepBound n) = ↑n * 4 ^ n | null | true |
HomologicalComplex.Hom.ext_iff | 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}
{x y : A.Hom B}, x = y ↔ x.f = y.f | null | true |
BooleanSubalgebra.sSup_mem | Mathlib.Order.BooleanSubalgebra | ∀ {α : Type u_2} [inst : CompleteBooleanAlgebra α] {L : BooleanSubalgebra α} {s : Set α}, s.Finite → s ⊆ ↑L → sSup s ∈ L | null | true |
Concept.strictAnti_intent | Mathlib.Order.Concept | ∀ {α : Type u_2} {β : Type u_3} {r : α → β → Prop}, StrictAnti Concept.intent | null | true |
AddSubgroup.inf_addSubgroupOf_left | Mathlib.Algebra.Group.Subgroup.Map | ∀ {G : Type u_1} [inst : AddGroup G] (H K : AddSubgroup G), (K ⊓ H).addSubgroupOf K = H.addSubgroupOf K | null | true |
_private.Init.Data.Int.LemmasAux.0.Int.toNat_sub_of_le._proof_1_1 | Init.Data.Int.LemmasAux | ∀ {a b : ℤ}, b ≤ a → ¬↑(a - b).toNat = a - b → False | null | false |
ULift.nonUnitalNonAssocSemiring._proof_3 | Mathlib.Algebra.Ring.ULift | ∀ {R : Type u_2} [inst : NonUnitalNonAssocSemiring R] (a : ULift.{u_1, u_2} R), 0 * a = 0 | null | false |
Polynomial.content.eq_1 | Mathlib.RingTheory.Polynomial.Content | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : NormalizedGCDMonoid R] (p : Polynomial R),
p.content = p.support.gcd ⇑p.coeff | null | true |
CochainComplex.singleFunctors._proof_8 | Mathlib.Algebra.Homology.HomotopyCategory.SingleFunctors | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] (n m a a' a'' : ℤ) (ha' : n + a = a') (ha'' : m + a' = a''),
CategoryTheory.NatIso.ofComponents
(fun X => HomologicalComplex.Hom.isoOfComponents (fun i => Cat... | null | false |
CategoryTheory.Functor.sheafInducedTopologyEquivOfIsCoverDense._proof_1 | Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_2, u_4} D] (G : CategoryTheory.Functor C D)
(K : CategoryTheory.GrothendieckTopology D) [G.LocallyCoverDense K] [G.IsLocallyFull K] [G.IsLocallyFaithful K]
[G.IsCoverDense K], G.IsContinuous (G.induc... | null | false |
Semiquot.IsPure | Mathlib.Data.Semiquot | {α : Type u_1} → Semiquot α → Prop | Assert that a `Semiquot` contains only one possible value. | true |
CategoryTheory.Limits.CatCospanTransform.category | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | {A : Type u₁} →
{B : Type u₂} →
{C : Type u₃} →
{A' : Type u₄} →
{B' : Type u₅} →
{C' : Type u₆} →
[inst : CategoryTheory.Category.{v₁, u₁} A] →
[inst_1 : CategoryTheory.Category.{v₂, u₂} B] →
[inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
... | null | true |
Vector.map_zip_eq_zipWith | Init.Data.Vector.Zip | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {n : ℕ} {f : α × β → γ} {as : Vector α n} {bs : Vector β n},
Vector.map f (as.zip bs) = Vector.zipWith (Function.curry f) as bs | null | true |
Lean.Grind.AC.Seq.startsWithVar_k_cons | Init.Grind.AC | ∀ (y x : Lean.Grind.AC.Var) (s : Lean.Grind.AC.Seq), (Lean.Grind.AC.Seq.cons y s).startsWithVar_k x = (x == y) | null | true |
LieAlgebra.LoopAlgebra.residuePairing._proof_3 | Mathlib.Algebra.Lie.Loop | ∀ (R : Type u_1) (A : Type u_2) (L : Type u_3) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
[inst_3 : AddCommGroup A] [inst_4 : DistribSMul A R] (Φ : LinearMap.BilinForm R L)
(f x y : LieAlgebra.loopAlgebra R A L),
(((LieAlgebra.LoopAlgebra.toFinsupp R A L) (x + y)).sum fun a v =>
a • ... | null | false |
Lean.Omega.Int.add_le_zero_iff_le_neg | Init.Omega.Int | ∀ {a b : ℤ}, a + b ≤ 0 ↔ a ≤ -b | null | true |
_private.Mathlib.Analysis.Complex.Periodic.0.Function.Periodic.qParam_ne_zero._simp_1_2 | Mathlib.Analysis.Complex.Periodic | ∀ (x : ℂ), (Complex.exp x = 0) = False | null | false |
ENat.epow_zero | Mathlib.Data.ENat.Pow | ∀ {x : ℕ∞}, x ^ 0 = 1 | null | true |
Filter.indicator_const_eventuallyEq | Mathlib.Order.Filter.IndicatorFunction | ∀ {α : Type u_1} {β : Type u_2} [inst : Zero β] {l : Filter α} {c : β},
c ≠ 0 → ∀ {s t : Set α}, ((s.indicator fun x => c) =ᶠ[l] t.indicator fun x => c) ↔ s =ᶠ[l] t | null | true |
Cardinal.mul_natCast_lt_mul_natCast._simp_1 | Mathlib.SetTheory.Cardinal.Arithmetic | ∀ {n : ℕ} {a b : Cardinal.{u_1}}, n ≠ 0 → (a * ↑n < b * ↑n) = (a < b) | null | false |
ContinuousMapZero.instCanLift | 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], CanLift C(X, R) (ContinuousMapZero X R) toContinuousMap fun f => f 0 = 0 | null | true |
List.drop.eq_2 | Init.Data.Array.GetLit | ∀ {α : Type u} (n : ℕ), List.drop n.succ [] = [] | null | true |
Int32.or_comm | Init.Data.SInt.Bitwise | ∀ (a b : Int32), a ||| b = b ||| a | null | true |
_private.Batteries.Recycling.RBTree.Alter.0.RBTree.RBNode.Path.fill.match_1.eq_3 | Batteries.Recycling.RBTree.Alter | ∀ {α : Type u_1} (motive : RBTree.RBNode.Path α → RBTree.RBNode α → Sort u_2) (c : RBTree.RBColor) (a : RBTree.RBNode α)
(y : α) (parent : RBTree.RBNode.Path α) (b : RBTree.RBNode α)
(h_1 : (t : RBTree.RBNode α) → motive RBTree.RBNode.Path.root t)
(h_2 :
(c : RBTree.RBColor) →
(parent : RBTree.RBNode.Pa... | null | true |
Flow.isSemiconjugacy_id_iff_eq | Mathlib.Dynamics.Flow | ∀ {τ : Type u_1} {α : Type u_2} [inst : TopologicalSpace τ] [inst_1 : TopologicalSpace α] [inst_2 : AddMonoid τ]
(ϕ ψ : Flow τ α), Flow.IsSemiconjugacy id ϕ ψ ↔ ϕ = ψ | The identity is a semiconjugacy from `ϕ` to `ψ` if and only if `ϕ` and `ψ` are equal. | true |
ImportGraph.Shake.StanceM.run' | ImportGraph.Shake.DeclNeeds | {α : Type} →
ImportGraph.Shake.StanceM α → optParam (Std.HashMap Lean.Name (Option ImportGraph.Shake.Stance)) ∅ → Lean.CoreM α | null | true |
AlgebraicGeometry.Scheme.isLocallyArtinianScheme_Spec | Mathlib.AlgebraicGeometry.Artinian | ∀ {R : CommRingCat}, AlgebraicGeometry.IsLocallyArtinian (AlgebraicGeometry.Spec R) ↔ IsArtinianRing ↑R | A commutative ring `R` is Artinian if and only if `Spec R` is an Artinian scheme. | true |
Projectivization.lift_mk | Mathlib.LinearAlgebra.Projectivization.Basic | ∀ {K : Type u_1} {V : Type u_2} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {α : Type u_3}
(f : { v // v ≠ 0 } → α) (hf : ∀ (a b : { v // v ≠ 0 }) (t : K), ↑a = t • ↑b → f a = f b) (v : V) (hv : v ≠ 0),
Projectivization.lift f hf (Projectivization.mk K v hv) = f ⟨v, hv⟩ | null | true |
Nonneg.linearOrderedCommGroupWithZero._proof_5 | Mathlib.Algebra.Order.Nonneg.Field | ∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α], Nontrivial (Nonneg α) | null | false |
AlgCat.instMonoidalCategory | Mathlib.Algebra.Category.AlgCat.Monoidal | {R : Type u} → [inst : CommRing R] → CategoryTheory.MonoidalCategory (AlgCat R) | null | true |
Std.Rxo.Iterator.mk.noConfusion | Init.Data.Range.Polymorphic.RangeIterator | {α : Type u} →
{P : Sort u_1} →
{next : Option α} →
{upperBound : α} →
{next' : Option α} →
{upperBound' : α} →
{ next := next, upperBound := upperBound } = { next := next', upperBound := upperBound' } →
(next ≍ next' → upperBound ≍ upperBound' → P) → P | null | false |
_private.Lean.Meta.Sym.Simp.DiscrTree.0.Lean.Meta.Sym.resolveAssignedMVars | Lean.Meta.Sym.Simp.DiscrTree | Lean.MetavarContext → Lean.Expr → Lean.Expr | If its head is an assigned metavariable, replaces it with its value,
beta-reducing any applied arguments. Goal types may mention metavariables assigned after the
goal was created (e.g. by `Pattern.unify?` or by solving a sibling subgoal), so retrieval must
see through assignments; unassigned metavariables remain opaque... | true |
SimpleGraph.Walk.nil_nil._simp_1 | Mathlib.Combinatorics.SimpleGraph.Walk.Basic | ∀ {V : Type u} {G : SimpleGraph V} {u : V}, SimpleGraph.Walk.nil.Nil = True | null | false |
CategoryTheory.Adjunction.corepresentableBy._proof_1 | Mathlib.CategoryTheory.Adjunction.Basic | ∀ {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 : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C}
(adj : F ⊣ G) (X : C) {Y Y' : D} (g : Y ⟶ Y') (f : F.obj X ⟶ Y),
(adj.homEquiv X Y') (CategoryTheory.CategoryStruct.comp f... | null | false |
RingHom.instMonoid._proof_1 | Mathlib.Algebra.Ring.Hom.Defs | ∀ {α : Type u_1} {x : NonAssocSemiring α} (n : ℕ) (f : α →+* α), (⇑f)^[n] = ⇑(npowRec n f) | null | false |
Nat.shiftLeft'.eq_1 | Mathlib.Data.Nat.Bits | ∀ (b : Bool) (m : ℕ), Nat.shiftLeft' b m 0 = m | null | true |
_private.Mathlib.RingTheory.Grassmannian.0.Module.Grassmannian.map._proof_3 | Mathlib.RingTheory.Grassmannian | ∀ {R : Type u_3} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {k : ℕ}
{A : Type u_1} [inst_3 : CommRing A] [inst_4 : Algebra R A] {B : Type u_1} [inst_5 : CommRing B]
[inst_6 : Algebra R B] (f : A →ₐ[R] B) (N : Module.Grassmannian A (TensorProduct R A M) k),
Module.Finite B (... | null | false |
QuadraticForm.isometryEquivWeightedSumSquares._proof_3 | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | ∀ {K : Type u_1} [inst : Field K], SMulCommClass K K K | null | false |
CategoryTheory.GrothendieckTopology.Point.skyscraperSheafFunctor | Mathlib.CategoryTheory.Sites.Point.Skyscraper | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{J : CategoryTheory.GrothendieckTopology C} →
J.Point →
{A : Type u'} →
[inst_1 : CategoryTheory.Category.{v', u'} A] →
[CategoryTheory.Limits.HasProducts A] → CategoryTheory.Functor A (CategoryTheory.Sheaf J A) | Given a point `Φ` of a site `(C, J)`, this is the skyscraper sheaf functor
`A ⥤ Sheaf J A`. | true |
Std.Tactic.BVDecide.BVExpr.bitblast.OverflowInput.recOn | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Carry | {α : Type} →
[inst : Hashable α] →
[inst_1 : DecidableEq α] →
{aig : Std.Sat.AIG α} →
{motive : Std.Tactic.BVDecide.BVExpr.bitblast.OverflowInput aig → Sort u} →
(t : Std.Tactic.BVDecide.BVExpr.bitblast.OverflowInput aig) →
((w : ℕ) → (vec : aig.BinaryRefVec w) → (cin : aig.Ref... | null | false |
CategoryTheory.ObjectProperty.IsSeparating.isDetecting | Mathlib.CategoryTheory.Generator.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {P : CategoryTheory.ObjectProperty C}
[CategoryTheory.Balanced C], P.IsSeparating → P.IsDetecting | null | true |
_private.Lean.Parser.Extension.0.Lean.Parser.addParserCategoryCore | Lean.Parser.Extension | Lean.Parser.ParserCategories → Lean.Name → Lean.Parser.ParserCategory → Except String Lean.Parser.ParserCategories | null | true |
Submodule.topEquiv | Mathlib.Algebra.Module.Submodule.Lattice | {R : Type u_1} → {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → ↥⊤ ≃ₗ[R] M | The top submodule is linearly equivalent to the module.
This is the module version of `AddSubmonoid.topEquiv`. | true |
AddAction.block_stabilizerOrderIso.match_5 | Mathlib.GroupTheory.GroupAction.Blocks | ∀ (G : Type u_2) [inst : AddGroup G] {X : Type u_1} [inst_1 : AddAction G X] (a : X)
(motive : { B // a ∈ B ∧ AddAction.IsBlock G B } → Prop) (x : { B // a ∈ B ∧ AddAction.IsBlock G B }),
(∀ (val : Set X) (ha : a ∈ val) (hB : AddAction.IsBlock G val), motive ⟨val, ⋯⟩) → motive x | null | false |
CategoryTheory.Mod.forget._proof_2 | Mathlib.CategoryTheory.Monoidal.Mod | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] [inst_1 : CategoryTheory.MonoidalCategory C]
{D : Type u_4} [inst_2 : CategoryTheory.Category.{u_2, u_4} D]
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [inst_4 : CategoryTheory.MonObj A]
{X Y Z : CategoryTheory.Mod D A} (... | null | false |
_private.Lean.Meta.Tactic.Grind.Split.0.Lean.Meta.Grind.Action.isSorryAlt._sparseCasesOn_1 | Lean.Meta.Tactic.Grind.Split | {α : Type u} →
{motive : List α → Sort u_1} →
(t : List α) →
((head : α) → (tail : List α) → motive (head :: tail)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
IsOrderBornology.cobounded_le_atBot_sup_atTop | Mathlib.Topology.Order.Bornology | ∀ {α : Type u_1} [inst : Bornology α] [inst_1 : LinearOrder α] [IsOrderBornology α],
Bornology.cobounded α ≤ Filter.atBot ⊔ Filter.atTop | null | true |
Std.DTreeMap.Internal.Impl.minKey?.eq_def | Std.Data.DTreeMap.Internal.Queries | ∀ {α : Type u} {β : α → Type v} (x : Std.DTreeMap.Internal.Impl α β),
x.minKey? =
match x with
| Std.DTreeMap.Internal.Impl.leaf => none
| Std.DTreeMap.Internal.Impl.inner size k v Std.DTreeMap.Internal.Impl.leaf r => some k
| Std.DTreeMap.Internal.Impl.inner size k v (l@h:(Std.DTreeMap.Internal.Impl.... | null | true |
_private.Mathlib.MeasureTheory.Group.Action.0.MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_compact_ne_zero.match_1_1 | Mathlib.MeasureTheory.Group.Action | ∀ (G : Type u_1) {α : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] {K U : Set α}
(motive : (∃ I, K ⊆ ⋃ g ∈ I, g +ᵥ U) → Prop) (x : ∃ I, K ⊆ ⋃ g ∈ I, g +ᵥ U),
(∀ (t : Finset G) (ht : K ⊆ ⋃ g ∈ t, g +ᵥ U), motive ⋯) → motive x | null | false |
CategoryTheory.Limits.CofanTypes | Mathlib.CategoryTheory.Limits.Types.Coproducts | {C : Type u} → (C → Type v) → Type (max (max u v) (w + 1)) | Given a functor `F : Discrete C ⥤ Type v`, this is a "cofan" for `F`,
but we allow the point to be in `Type w` for an arbitrary universe `w`. | true |
InfHom.top_apply | Mathlib.Order.Hom.Lattice | ∀ {α : Type u_2} {β : Type u_3} [inst : Min α] [inst_1 : SemilatticeInf β] [inst_2 : Top β] (a : α), ⊤ a = ⊤ | null | true |
CategoryTheory.CostructuredArrow.costructuredArrowToOverEquivalence.functor._proof_3 | Mathlib.CategoryTheory.Comma.Over.Basic | ∀ {T : Type u_4} [inst : CategoryTheory.Category.{u_2, u_4} T] {D : Type u_3}
[inst_1 : CategoryTheory.Category.{u_1, u_3} D] (F : CategoryTheory.Functor D T) {X : T} (Y : CategoryTheory.Over X)
{X_1 Y_1 Z : CategoryTheory.CostructuredArrow (CategoryTheory.CostructuredArrow.toOver F X) Y} (f : X_1 ⟶ Y_1)
(g : Y_1... | null | false |
Asymptotics.«term_~[_]_» | Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | Lean.TrailingParserDescr | Two functions `u` and `v` are said to be asymptotically equivalent along a filter `l`
(denoted as `u ~[l] v` in the `Asymptotics` namespace)
when `u x - v x = o(v x)` as `x` converges along `l`. | true |
Ordinal.small_Icc | Mathlib.SetTheory.Ordinal.Basic | ∀ (a b : Ordinal.{u}), Small.{u, u + 1} ↑(Set.Icc a b) | null | true |
Rack.toEnvelGroup.map._proof_2 | Mathlib.Algebra.Quandle | ∀ {R : Type u_1} [inst : Rack R] {G : Type u_2} [inst_1 : Group G] (f : ShelfHom R (Quandle.Conj G))
(x y : Rack.EnvelGroup R),
Quotient.liftOn (x * y) (Rack.toEnvelGroup.mapAux f) ⋯ =
Quotient.liftOn x (Rack.toEnvelGroup.mapAux f) ⋯ * Quotient.liftOn y (Rack.toEnvelGroup.mapAux f) ⋯ | null | false |
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