name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.Topology.MetricSpace.HausdorffDistance.0.Metric.infEDist_eq_top_iff._simp_1_1 | Mathlib.Topology.MetricSpace.HausdorffDistance | ∀ {α : Type u} {s : Set α}, s.Nonempty → (s = ∅) = False | null | false |
Std.TreeSet.Raw.toList_iter | Std.Data.TreeSet.Raw.Iterator | ∀ {α : Type u_1} {cmp : α → α → Ordering} (m : Std.TreeSet.Raw α cmp), m.iter.toList = m.toList | null | true |
Lean.Language.Lean.HeaderProcessedSnapshot.infoTree?._inherited_default | Lean.Language.Lean.Types | Option Lean.Elab.InfoTree | null | false |
add_tsub_cancel_left | Mathlib.Algebra.Order.Sub.Defs | ∀ {α : Type u_1} [inst : PartialOrder α] [inst_1 : AddCommSemigroup α] [inst_2 : Sub α] [OrderedSub α]
[AddLeftReflectLE α] (a b : α), a + b - a = b | null | true |
CategoryTheory.Lax.OplaxTrans.LaxFunctor.bicategory._proof_6 | Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax | ∀ (B : Type u_5) [inst : CategoryTheory.Bicategory B] (C : Type u_6) [inst_1 : CategoryTheory.Bicategory C]
{a b : CategoryTheory.LaxFunctor B C} {f g : a ⟶ b} (η : f ⟶ g),
CategoryTheory.Lax.OplaxTrans.whiskerLeft (CategoryTheory.CategoryStruct.id a) η =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Lax.O... | null | false |
Submodule.sndEquiv_symm_apply_coe | Mathlib.LinearAlgebra.Prod | ∀ (R : Type u) (M : Type v) (M₂ : Type w) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]
[inst_3 : Module R M] [inst_4 : Module R M₂] (n : M₂), ↑((Submodule.sndEquiv R M M₂).symm n) = (0, n) | null | true |
MeasureTheory.diracProbaEquiv | Mathlib.MeasureTheory.Measure.DiracProba | {X : Type u_1} →
[inst : MeasurableSpace X] →
[inst_1 : TopologicalSpace X] → [OpensMeasurableSpace X] → [T0Space X] → X ≃ ↑(Set.range MeasureTheory.diracProba) | In a T0 topological space `X`, the assignment `x ↦ diracProba x` is a bijection to its
range in `ProbabilityMeasure X`. | true |
CategoryTheory.Pretriangulated.Triangle.instSMulHom | Mathlib.CategoryTheory.Triangulated.Basic | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.HasShift C ℤ] →
{T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C} →
[inst_2 : CategoryTheory.Preadditive C] →
{R : Type u_1} →
[inst_3 : Semiring R] →
[inst_4 : CategoryTheory.Li... | null | true |
Set.card_empty | Mathlib.Data.Set.Finite.Basic | ∀ {α : Type u}, Fintype.card ↑∅ = 0 | null | true |
eventually_mabs_div_lt | Mathlib.Topology.Order.LeftRightNhds | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : CommGroup α] [inst_2 : LinearOrder α] [IsOrderedMonoid α]
[OrderTopology α] (a : α) {ε : α}, 1 < ε → ∀ᶠ (x : α) in nhds a, |x / a|ₘ < ε | null | true |
AddUnits.instAddZeroClass | Mathlib.Algebra.Group.Units.Defs | {α : Type u} → [inst : AddMonoid α] → AddZeroClass (AddUnits α) | Additive units of an additive monoid have an addition and an additive identity. | true |
Std.Rcc.count_iter | Std.Data.Iterators.Lemmas.Producers.Range | ∀ {α : Type u_1} [inst : LE α] [inst_1 : DecidableLE α] [inst_2 : Std.PRange.UpwardEnumerable α]
[inst_3 : Std.PRange.LawfulUpwardEnumerableLE α] [Std.Rxc.IsAlwaysFinite α]
[inst_5 : Std.PRange.LawfulUpwardEnumerable α] [inst_6 : Std.Rxc.HasSize α] [Std.Rxc.LawfulHasSize α] {r : Std.Rcc α},
r.iter.length = r.size | null | true |
Lean.Meta.Context.localInstances._default | Lean.Meta.Basic | Array Lean.LocalInstance | null | false |
Lean.Name.eraseMacroScopes | Init.Prelude | Lean.Name → Lean.Name | Remove the macro scopes from the name. | true |
_private.Lean.Meta.Tactic.Grind.Arith.Linear.Proof.0.Lean.Meta.Grind.Arith.Linear.RingEqCnstr.toExprProof.match_1 | Lean.Meta.Tactic.Grind.Arith.Linear.Proof | (motive : Lean.Meta.Grind.Arith.Linear.RingEqCnstrProof → Sort u_1) →
(x : Lean.Meta.Grind.Arith.Linear.RingEqCnstrProof) →
((a b : Lean.Expr) →
(la lb : Lean.Grind.CommRing.Expr) → motive (Lean.Meta.Grind.Arith.Linear.RingEqCnstrProof.core a b la lb)) →
((c : Lean.Meta.Grind.Arith.Linear.RingEqCnst... | null | false |
MeromorphicAt.meromorphicTrailingCoeffAt_of_order_eq_top | Mathlib.Analysis.Meromorphic.TrailingCoefficient | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜}, meromorphicOrderAt f x = ⊤ → meromorphicTrailingCoeffAt f x = 0 | If `f` is meromorphic of infinite order at `x`, the trailing coefficient is zero by definition.
| true |
_private.Mathlib.Algebra.Polynomial.Monic.0.Polynomial.Monic.natDegree_mul'._simp_1_1 | Mathlib.Algebra.Polynomial.Monic | ∀ {R : Type u} [inst : Semiring R] {p : Polynomial R}, (p.leadingCoeff ≠ 0) = (p ≠ 0) | null | false |
Subsemigroup.isClosed_topologicalClosure | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : Semigroup M] [inst_2 : SeparatelyContinuousMul M]
(s : Subsemigroup M), IsClosed ↑s.topologicalClosure | null | true |
_private.Mathlib.Algebra.Module.LocalizedModule.Basic.0.LocalizedModule.add_smul_aux | Mathlib.Algebra.Module.LocalizedModule.Basic | ∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{T : Type u_1} [inst_3 : CommSemiring T] [inst_4 : Algebra R T] [inst_5 : IsLocalization S T] (x y : T)
(p : LocalizedModule S M), (x + y) • p = x • p + y • p | null | true |
Std.HashMap.Raw.mem_modify | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} [inst : BEq α] [inst_1 : Hashable α] {m : Std.HashMap.Raw α β} [inst_2 : EquivBEq α]
[LawfulHashable α] {k k' : α} {f : β → β}, m.WF → (k' ∈ m.modify k f ↔ k' ∈ m) | null | true |
LieModule.toEnd | Mathlib.Algebra.Lie.OfAssociative | (R : Type u) →
(L : Type v) →
(M : Type w) →
[inst : CommRing R] →
[inst_1 : LieRing L] →
[inst_2 : LieAlgebra R L] →
[inst_3 : AddCommGroup M] →
[inst_4 : Module R M] → [inst_5 : LieRingModule L M] → [LieModule R L M] → L →ₗ⁅R⁆ Module.End R M | A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `LieModule.toModuleHom`. | true |
_private.Std.Http.Data.Status.0.Std.Http.CustomStatus.ofCodeAndPhrase?._proof_1 | Std.Http.Data.Status | ∀ (code : UInt16) (phrase : String),
Std.Http.IsValidReasonPhrase phrase ∧ (100 ≤ code ∧ code ≤ 999) ∧ ¬Std.Http.isKnownStatusCode code = true →
Std.Http.IsValidReasonPhrase phrase | null | false |
EReal.toReal_zero | Mathlib.Data.EReal.Basic | EReal.toReal 0 = 0 | null | true |
Ordinal.aleph0_le_cof_iff._simp_1 | Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | ∀ {o : Ordinal.{u_1}}, (Cardinal.aleph0 ≤ o.cof) = (1 < o.cof) | null | false |
Partrec.sumCasesOn_right | Mathlib.Computability.Partrec | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [inst : Primcodable α] [inst_1 : Primcodable β]
[inst_2 : Primcodable γ] [inst_3 : Primcodable σ] {f : α → β ⊕ γ} {g : α → β → σ} {h : α → γ →. σ},
Computable f → Computable₂ g → Partrec₂ h → Partrec fun a => Sum.casesOn (f a) (fun b => Part.some (g a b)... | null | true |
HomologicalComplex₂.ιTotalOrZero | Mathlib.Algebra.Homology.TotalComplex | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
{I₁ : Type u_2} →
{I₂ : Type u_3} →
{I₁₂ : Type u_4} →
{c₁ : ComplexShape I₁} →
{c₂ : ComplexShape I₂} →
(K : HomologicalComplex₂ C c₁ c₂) →
... | The inclusion of a summand in the total complex, or zero if the degrees do not match. | true |
_private.Mathlib.Computability.TuringMachine.ToPartrec.0.Turing.PartrecToTM2.trPosNum.match_1.eq_2 | Mathlib.Computability.TuringMachine.ToPartrec | ∀ (motive : PosNum → Sort u_1) (n : PosNum) (h_1 : Unit → motive PosNum.one) (h_2 : (n : PosNum) → motive n.bit0)
(h_3 : (n : PosNum) → motive n.bit1),
(match n.bit0 with
| PosNum.one => h_1 ()
| n.bit0 => h_2 n
| n.bit1 => h_3 n) =
h_2 n | null | true |
Std.Roi.mem_succ_iff | Init.Data.Range.Polymorphic.Lemmas | ∀ {α : Type u} [inst : LT α] [inst_1 : Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α]
[Std.PRange.LawfulUpwardEnumerableLT α] [inst_4 : Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxi.IsAlwaysFinite α]
[Std.PRange.LawfulUpwardEnumerable α] {lo a : α},
a ∈ (Std.PRange.succ lo)<...* ↔ ∃ a'... | null | true |
CategoryTheory.ChosenPullbacksAlong.Over.tensorHom_left_fst | Mathlib.CategoryTheory.LocallyCartesianClosed.Over | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.ChosenPullbacks C] {X : C}
{R S T U : CategoryTheory.Over X} (f : R ⟶ S) (g : T ⟶ U),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Over.Hom.left (CategoryTheory.MonoidalCategoryStruct.tensorHom f g))
(CategoryTheo... | null | true |
_private.Mathlib.RingTheory.Spectrum.Prime.Topology.0.PrimeSpectrum.isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton.match_1_2 | Mathlib.RingTheory.Spectrum.Prime.Topology | ∀ {R : Type u_1} [inst : CommSemiring R] {f : R} (r : R)
(motive : (∃ p, p.IsPrime ∧ Ideal.span {r} ≤ p ∧ Disjoint ↑p ↑(Submonoid.powers f)) → Prop)
(x : ∃ p, p.IsPrime ∧ Ideal.span {r} ≤ p ∧ Disjoint ↑p ↑(Submonoid.powers f)),
(∀ (q : Ideal R) (prime : q.IsPrime) (le : Ideal.span {r} ≤ q) (disj : Disjoint ↑q ↑(S... | null | false |
Nat.ModEq.listSum_zero | Mathlib.Algebra.BigOperators.ModEq | ∀ {n : ℕ} {l : List ℕ}, (∀ x ∈ l, x ≡ 0 [MOD n]) → l.sum ≡ 0 [MOD n] | null | true |
Std.DTreeMap.Internal.Cell.getEntry?.match_1 | Std.Data.DTreeMap.Internal.Cell | {α : Type u_2} →
{β : α → Type u_1} →
(motive : Option ((a : α) × β a) → Sort u_3) →
(x : Option ((a : α) × β a)) → (Unit → motive none) → ((p : (a : α) × β a) → motive (some p)) → motive x | null | false |
CategoryTheory.Join.instCategory._proof_15 | Mathlib.CategoryTheory.Join.Basic | ∀ {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] {x y : CategoryTheory.Join C D} (f : x.Hom y),
CategoryTheory.Join.comp f y.id = f | null | false |
ball_add_ball | Mathlib.Analysis.Normed.Module.Ball.Pointwise | ∀ {E : Type u_2} [inst : SeminormedAddCommGroup E] [NormedSpace ℝ E] {δ ε : ℝ},
0 < ε → 0 < δ → ∀ (a b : E), Metric.ball a ε + Metric.ball b δ = Metric.ball (a + b) (ε + δ) | null | true |
_private.Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality.0.groupHomology.mapShortComplexH1._simp_1 | Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_1 : Group G] [inst_2 : DivisionMonoid H]
[MonoidHomClass F G H] (f : F) (a : G), (f a)⁻¹ = f a⁻¹ | null | false |
ProbabilityTheory.Kernel.lintegral_piecewise | Mathlib.Probability.Kernel.Basic | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : ProbabilityTheory.Kernel α β}
{s : Set α} {hs : MeasurableSet s} [inst : DecidablePred fun x => x ∈ s] (a : α) (g : β → ENNReal),
∫⁻ (b : β), g b ∂(ProbabilityTheory.Kernel.piecewise hs κ η) a =
if a ∈ s then ∫⁻ (b : β), g ... | null | true |
Lean.Meta.Grind.CanonArgKey.i | Lean.Meta.Tactic.Grind.Types | Lean.Meta.Grind.CanonArgKey → ℕ | null | true |
CategoryTheory.Comonad.id_δ_app | Mathlib.CategoryTheory.Monad.Basic | ∀ (C : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} C] (X : C),
(CategoryTheory.Comonad.id C).δ.app X = CategoryTheory.CategoryStruct.id X | null | true |
CategoryTheory.Triangulated.TStructure.noConfusion | Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {P : Sort u} →
{C : Type u_1} →
{inst : CategoryTheory.Category.{v_1, u_1} C} →
{inst_1 : CategoryTheory.Preadditive C} →
{inst_2 : CategoryTheory.Limits.HasZeroObject C} →
{inst_3 : CategoryTheory.HasShift C ℤ} →
{inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive}... | null | false |
SuccOrder.limitRecOn_succ_of_not_isMax | Mathlib.Order.SuccPred.Limit | ∀ {α : Type u_1} {b : α} {motive : α → Sort u_2} [inst : LinearOrder α] [inst_1 : SuccOrder α]
[inst_2 : WellFoundedLT α] (isMin : (a : α) → IsMin a → motive a)
(succ : (a : α) → ¬IsMax a → motive a → motive (Order.succ a))
(isSuccLimit : (a : α) → Order.IsSuccLimit a → ((b : α) → b < a → motive b) → motive a) (h... | null | true |
OrderRingHom.coe_orderAddMonoidHom_id | Mathlib.Algebra.Order.Hom.Ring | ∀ {α : Type u_2} [inst : NonAssocSemiring α] [inst_1 : Preorder α], ↑(OrderRingHom.id α) = OrderAddMonoidHom.id α | null | true |
Std.HashMap.Raw.getD_emptyWithCapacity | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} [inst : BEq α] [inst_1 : Hashable α] {a : α} {fallback : β} {c : ℕ},
(Std.HashMap.Raw.emptyWithCapacity c).getD a fallback = fallback | null | true |
InnerProductSpace.ringOfCoalgebra._proof_21 | Mathlib.Analysis.InnerProductSpace.Coalgebra | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] (a : E), -a + a = 0 | null | false |
NFA.evalFrom.eq_1 | Mathlib.Computability.NFA | ∀ {α : Type u} {σ : Type v} (M : NFA α σ) (S : Set σ), M.evalFrom S = List.foldl M.stepSet S | null | true |
Finset.prod_congr_of_eq_on_inter | Mathlib.Algebra.BigOperators.Group.Finset.Basic | ∀ {ι : Type u_5} {M : Type u_6} {s₁ s₂ : Finset ι} {f g : ι → M} [inst : CommMonoid M],
(∀ a ∈ s₁, a ∉ s₂ → f a = 1) →
(∀ a ∈ s₂, a ∉ s₁ → g a = 1) → (∀ a ∈ s₁, a ∈ s₂ → f a = g a) → ∏ a ∈ s₁, f a = ∏ a ∈ s₂, g a | The products of two functions `f g : ι → M` over finite sets `s₁ s₂ : Finset ι`
are equal if the functions agree on `s₁ ∩ s₂`, `f = 1` and `g = 1` on the respective
set differences. | true |
NonUnitalStarAlgebra.adjoin_induction | Mathlib.Algebra.Star.NonUnitalSubalgebra | ∀ (R : Type u) {A : Type v} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
[inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : IsScalarTower R A A] [inst_6 : SMulCommClass R A A]
[inst_7 : StarModule R A] {s : Set A} {p : (x : A) → x ∈ NonUnitalStarAlgebra.adjoin R s → Prop},
(∀ (x... | null | true |
FirstOrder.Language.closedUnder_univ | Mathlib.ModelTheory.Substructures | ∀ (L : FirstOrder.Language) {M : Type w} [inst : L.Structure M] {n : ℕ} (f : L.Functions n),
FirstOrder.Language.ClosedUnder f Set.univ | null | true |
Lean.Meta.Sym.Arith.getAddFn | Lean.Meta.Sym.Arith.Functions | {m : Type → Type} →
[MonadLiftT Lean.MetaM m] →
[Lean.MonadError m] →
[Monad m] → [Lean.Meta.Sym.Arith.MonadCanon m] → [Lean.Meta.Sym.Arith.MonadRing m] → m Lean.Expr | null | true |
Std.DTreeMap.Internal.Impl.isEmpty_diff_iff | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {m₁ m₂ : Std.DTreeMap.Internal.Impl α β} [Std.TransOrd α]
(h₁ : m₁.WF),
m₂.WF →
((m₁.diff m₂ ⋯).isEmpty = true ↔
∀ (k : α), Std.DTreeMap.Internal.Impl.contains k m₁ = true → Std.DTreeMap.Internal.Impl.contains k m₂ = true) | null | true |
gcd_zero_right | Mathlib.Algebra.GCDMonoid.Basic | ∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] (a : α), gcd a 0 = normalize a | null | true |
_private.Mathlib.Geometry.Manifold.VectorBundle.Riemannian.0.«term⟪_,_⟫» | Mathlib.Geometry.Manifold.VectorBundle.Riemannian | Lean.ParserDescr | null | true |
DilationEquiv.self_trans_symm | Mathlib.Topology.MetricSpace.DilationEquiv | ∀ {X : Type u_1} {Y : Type u_2} [inst : PseudoEMetricSpace X] [inst_1 : PseudoEMetricSpace Y] (e : X ≃ᵈ Y),
e.trans e.symm = DilationEquiv.refl X | null | true |
Rat.instNormedField._proof_1 | Mathlib.Analysis.Normed.Field.Lemmas | ∀ (a b : ℚ), ‖a * b‖ = ‖a‖ * ‖b‖ | null | false |
CategoryTheory.LaxBraidedFunctor.instCategory._proof_8 | Mathlib.CategoryTheory.Monoidal.Braided.Basic | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] {D : Type u_4} [inst_3 : CategoryTheory.Category.{u_2, u_4} D]
[inst_4 : CategoryTheory.MonoidalCategory D] [inst_5 : CategoryTheory.BraidedCategory D],
autoParam... | null | false |
CStarMatrix.mul_zero | Mathlib.Analysis.CStarAlgebra.CStarMatrix | ∀ {m : Type u_1} {n : Type u_2} {A : Type u_5} {o : Type u_7} [inst : Fintype n] [inst_1 : NonUnitalNonAssocSemiring A]
(M : CStarMatrix m n A), M * 0 = 0 | null | true |
hahnEmbedding_isOrderedAddMonoid | Mathlib.RingTheory.HahnSeries.HahnEmbedding | ∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M],
∃ f,
Function.Injective ⇑f ∧
∀ (a : M), ArchimedeanClass.mk a = (FiniteArchimedeanClass.withTopOrderIso M) (ofLex (f a)).orderTop | **Hahn embedding theorem**
For a linearly ordered additive group `M`, there exists an injective `OrderAddMonoidHom` from `M` to
`Lex ℝ⟦FiniteArchimedeanClass M⟧` that sends each `a : M` to an element of the `a`-Archimedean class
of the Hahn series.
| true |
Lean.Doc.instOrdMathMode.ord | Lean.DocString.Types | Lean.Doc.MathMode → Lean.Doc.MathMode → Ordering | null | true |
Lean.Doc.Parser.UnorderedListType.asterisk.elim | Lean.DocString.Parser | {motive : Lean.Doc.Parser.UnorderedListType → Sort u} →
(t : Lean.Doc.Parser.UnorderedListType) → t.ctorIdx = 0 → motive Lean.Doc.Parser.UnorderedListType.asterisk → motive t | null | false |
HasSummableGeomSeries.mk | Mathlib.Analysis.SpecificLimits.Normed | ∀ {K : Type u_4} [inst : NormedRing K], (∀ (ξ : K), ‖ξ‖ < 1 → Summable fun n => ξ ^ n) → HasSummableGeomSeries K | null | true |
MvPowerSeries.gaussNorm | Mathlib.RingTheory.MvPowerSeries.GaussNorm | {R : Type u_1} → {σ : Type u_2} → [Semiring R] → (R → ℝ) → (σ → ℝ) → MvPowerSeries σ R → ℝ | Given a multivariate power series `f` in, a function `v : R → ℝ` and a tuple `c` of real
numbers, the Gauss norm is defined as the supremum of the set of all values of
`v (coeff t f) * ∏ i : t.support, c i` for all `t : σ →₀ ℕ`. | true |
Lean.Elab.Term.GeneralizeResult.mk.sizeOf_spec | Lean.Elab.Match | ∀ (discrs : Array Lean.Elab.Term.Discr) (toClear : Array Lean.FVarId) (matchType : Lean.Expr)
(altViews : Array Lean.Elab.Term.TermMatchAltView) (refined : Bool),
sizeOf { discrs := discrs, toClear := toClear, matchType := matchType, altViews := altViews, refined := refined } =
1 + sizeOf discrs + sizeOf toClea... | null | true |
_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic.0.CFC.exp_log._proof_1_1 | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | ∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : NormedAlgebra ℝ A] [inst_2 : PartialOrder A] [NonnegSpectrumClass ℝ A]
(a : A), IsStrictlyPositive a → ∀ x ∈ spectrum ℝ a, ¬x = 0 | null | false |
Mathlib.Tactic.ITauto.Proof.hyp.injEq | Mathlib.Tactic.ITauto | ∀ (n n_1 : Lean.Name), (Mathlib.Tactic.ITauto.Proof.hyp n = Mathlib.Tactic.ITauto.Proof.hyp n_1) = (n = n_1) | null | true |
ShrinkingLemma.PartialRefinement.chainSup._proof_1 | Mathlib.Topology.ShrinkingLemma | ∀ {ι : Type u_2} {X : Type u_1} [inst : TopologicalSpace X] {u : ι → Set X} {s : Set X} {p : Set X → Prop}
(c : Set (ShrinkingLemma.PartialRefinement u s p)) (ne : c.Nonempty) (i : ι),
IsOpen ((ShrinkingLemma.PartialRefinement.find c ne i).toFun i) | null | false |
_private.Mathlib.CategoryTheory.GradedObject.Monoidal.0.CategoryTheory.GradedObject.instFiniteElemProdNatPreimageHAddFstSndSingletonSet._proof_3 | Mathlib.CategoryTheory.GradedObject.Monoidal | ∀ (n i₁ i₂ : ℕ), i₁ + i₂ = n → i₁ < n + 1 | null | false |
Matrix.linftyOpNormSMulClass | 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 : SeminormedRing R] [inst_3 : SeminormedAddCommGroup α] [inst_4 : Module R α] [NormSMulClass R α],
NormSMulClass R (Matrix m n α) | This applies to the sup norm of L1 norm. | true |
MeasureTheory.VectorMeasure.trim._proof_2 | Mathlib.MeasureTheory.VectorMeasure.Basic | ∀ {α : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : TopologicalSpace M] {m n : MeasurableSpace α}
(v : MeasureTheory.VectorMeasure α M) (i : Set α), ¬MeasurableSet i → (if MeasurableSet i then ↑v i else 0) = 0 | null | false |
CategoryTheory.MorphismProperty.IsStableUnderColimitsOfShape.rec | Mathlib.CategoryTheory.MorphismProperty.Limits | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{W : CategoryTheory.MorphismProperty C} →
{J : Type u_1} →
[inst_1 : CategoryTheory.Category.{v_1, u_1} J] →
{motive : W.IsStableUnderColimitsOfShape J → Sort u_2} →
((condition :
∀ (X₁ X₂ : CategoryTh... | null | false |
TensorPower.multilinearMapToDual._proof_3 | Mathlib.LinearAlgebra.TensorPower.Pairing | ∀ (R : Type u_1) (M : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (n : ℕ),
(∀ (x : DecidableEq (Fin n)) (f : Fin n → Module.Dual R M) (φ : Module.Dual R M) (i j : Fin n) (v : Fin n → M),
(Function.update f i φ j) (v j) = Function.update (fun j => (f j) (v j)) i (φ (v i)) j... | null | false |
_private.BatteriesRecycling.RBTree.Lemmas.0.RBTree.RBSet.lowerBoundP?_exists._simp_1_3 | BatteriesRecycling.RBTree.Lemmas | ∀ {α : Type u_1} {cmp : α → α → Ordering} {x : α} {t : RBTree.RBSet α cmp},
(x ∈ t) = ∃ y ∈ t.toList, cmp x y = Ordering.eq | null | false |
orderOf_eq_zero_iff_eq_zero | Mathlib.GroupTheory.OrderOfElement | ∀ {G₀ : Type u_6} [inst : GroupWithZero G₀] [Finite G₀] {a : G₀}, orderOf a = 0 ↔ a = 0 | null | true |
_private.Init.Data.Range.Polymorphic.Internal.SignedBitVec.0.BitVec.Signed.sle_iff_rotate_le_rotate._proof_1_24 | Init.Data.Range.Polymorphic.Internal.SignedBitVec | ∀ (n : ℕ) (x y : BitVec (n + 1)), ¬x.toNat < 2 ^ n → ¬x.toNat - 2 ^ n ≤ y.toNat + 2 ^ n → False | null | false |
Option.elim'.eq_2 | Mathlib.MeasureTheory.OuterMeasure.Induced | ∀ {α : Type u_1} {β : Type u_2} (b : β) (f : α → β), Option.elim' b f none = b | null | true |
MonoidHom.inverse._proof_4 | Mathlib.Algebra.Group.Hom.Defs | ∀ {A : Type u_1} {B : Type u_2} [inst : Monoid A] [inst_1 : Monoid B] (f : A →* B) (g : B → A)
(h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (x y : B),
((↑f).inverse g h₁ h₂).toFun (x * y) = ((↑f).inverse g h₁ h₂).toFun x * ((↑f).inverse g h₁ h₂).toFun y | null | false |
CategoryTheory.ShortComplex.cyclesMapIso_hom | Mathlib.Algebra.Homology.ShortComplex.LeftHomology | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ : CategoryTheory.ShortComplex C} (e : S₁ ≅ S₂) [inst_2 : S₁.HasLeftHomology] [inst_3 : S₂.HasLeftHomology],
(CategoryTheory.ShortComplex.cyclesMapIso e).hom = CategoryTheory.ShortComplex.cyclesM... | null | true |
AlgebraicGeometry.Scheme.OpenCover.pullbackCoverAffineRefinementObjIso._proof_7 | Mathlib.AlgebraicGeometry.Cover.Open | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : Y.OpenCover)
(i : (CategoryTheory.Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).I₀),
CategoryTheory.Limits.HasPullback f (𝒰.affineRefinement.openCover.f i) | null | false |
Turing.TM2ComputableInTime.casesOn | Mathlib.Computability.TuringMachine.Computable | {α β αΓ βΓ : Type} →
{ea : α → List αΓ} →
{eb : β → List βΓ} →
{f : α → β} →
{motive : Turing.TM2ComputableInTime ea eb f → Sort u} →
(t : Turing.TM2ComputableInTime ea eb f) →
((toTM2ComputableAux : Turing.TM2ComputableAux αΓ βΓ) →
(time : ℕ → ℕ) →
... | null | false |
Lean.Elab.Term.Arg._sizeOf_inst | Lean.Elab.Arg | SizeOf Lean.Elab.Term.Arg | null | false |
_private.Aesop.Search.Expansion.Basic.0.Aesop.runRuleTac.match_6 | Aesop.Search.Expansion.Basic | (motive : Except Lean.Exception Aesop.RuleTacOutput → Sort u_1) →
(result : Except Lean.Exception Aesop.RuleTacOutput) →
((ruleOutput : Aesop.RuleTacOutput) → motive (Except.ok ruleOutput)) →
((x : Except Lean.Exception Aesop.RuleTacOutput) → motive x) → motive result | null | false |
Lean.Expr.getForallBinderNames._unsafe_rec | Lean.Expr | Lean.Expr → List Lean.Name | null | false |
Lean.Lsp.LeanPrepareModuleHierarchyParams.mk.injEq | Lean.Data.Lsp.Extra | ∀ (textDocument textDocument_1 : Lean.Lsp.TextDocumentIdentifier),
({ textDocument := textDocument } = { textDocument := textDocument_1 }) = (textDocument = textDocument_1) | null | true |
LinearMap.fst_surjective | Mathlib.LinearAlgebra.Prod | ∀ {R : Type u} {M : Type v} {M₂ : Type w} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]
[inst_3 : Module R M] [inst_4 : Module R M₂], Function.Surjective ⇑(LinearMap.fst R M M₂) | null | true |
BoxIntegral.Prepartition.IsPartition.eq_1 | Mathlib.Analysis.BoxIntegral.Partition.Basic | ∀ {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I), π.IsPartition = ∀ x ∈ I, ∃ J ∈ π, x ∈ J | null | true |
Set.not_infinite | Mathlib.Data.Finite.Defs | ∀ {α : Type u} {s : Set α}, ¬s.Infinite ↔ s.Finite | null | true |
ENNReal.add_sub_add_eq_sub_right | Mathlib.Data.ENNReal.Operations | ∀ {a b c : ENNReal}, autoParam (c ≠ ⊤) ENNReal.add_sub_add_eq_sub_right._auto_1 → a + c - (b + c) = a - b | null | true |
HurwitzKernelBounds.isBigO_atTop_F_int_one | Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | ∀ (a : UnitAddCircle), ∃ p, 0 < p ∧ HurwitzKernelBounds.F_int 1 a =O[Filter.atTop] fun t => Real.exp (-p * t) | null | true |
_private.Mathlib.Probability.Process.Filtration.0.MeasureTheory.Filtration.instInfSet._simp_5 | Mathlib.Probability.Process.Filtration | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
ContinuousMap.nnnorm_sum_eq_sup | Mathlib.Topology.ContinuousMap.Compact | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : CompactSpace α] {R : Type u_4}
[inst_2 : NonUnitalSeminormedRing R] [IsCancelMulZero R] {ι : Type u_5} {f : ι → C(α, R)} (s : Finset ι),
Pairwise (Function.onFun (fun x1 x2 => x1 * x2 = 0) f) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x => ‖f x‖₊ | If the pairwise products of continuous functions on a compact space are all zero, then the norm
of their sum is the maximum of their norms. | true |
_private.Mathlib.LinearAlgebra.Dual.Lemmas.0.Module.finite_dual_iff.match_1_1 | Mathlib.LinearAlgebra.Dual.Lemmas | ∀ (K : Type u_2) {V : Type u_1} [inst : CommSemiring K] [inst_1 : AddCommMonoid V] [inst_2 : Module K V]
(motive : Nonempty ((I : Type u_1) × Module.Basis I K V) → Prop) (x : Nonempty ((I : Type u_1) × Module.Basis I K V)),
(∀ (ι : Type u_1) (b : Module.Basis ι K V), motive ⋯) → motive x | null | false |
Matrix.replicateCol_smul | Mathlib.LinearAlgebra.Matrix.RowCol | ∀ {m : Type u_2} {R : Type u_5} {α : Type v} {ι : Type u_6} [inst : SMul R α] (x : R) (v : m → α),
Matrix.replicateCol ι (x • v) = x • Matrix.replicateCol ι v | null | true |
Algebra.coe_inf | Mathlib.Algebra.Algebra.Subalgebra.Lattice | ∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (S T : Subalgebra R A),
↑(S ⊓ T) = ↑S ∩ ↑T | null | true |
CategoryTheory.MorphismProperty.RightFraction.ofInv.congr_simp | Mathlib.CategoryTheory.Localization.CalculusOfFractions | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C}
(s s_1 : Y ⟶ X) (e_s : s = s_1) (hs : W s),
CategoryTheory.MorphismProperty.RightFraction.ofInv s hs = CategoryTheory.MorphismProperty.RightFraction.ofInv s_1 ⋯ | null | true |
smul_one_strictMono | Mathlib.Algebra.Order.Module.Defs | ∀ {α : Type u_1} (β : Type u_2) [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : PartialOrder β] [inst_3 : Zero β]
[inst_4 : One β] [ZeroLEOneClass β] [NeZero 1] [SMulPosStrictMono α β], StrictMono fun x => x • 1 | null | true |
InfTopHomClass.toInfHomClass | Mathlib.Order.Hom.BoundedLattice | ∀ {F : Type u_6} {α : Type u_7} {β : Type u_8} {inst : Min α} {inst_1 : Min β} {inst_2 : Top α} {inst_3 : Top β}
{inst_4 : FunLike F α β} [self : InfTopHomClass F α β], InfHomClass F α β | null | true |
_private.Mathlib.Topology.MetricSpace.Pseudo.Defs.0.Metric.continuousOn_iff._simp_1_1 | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u} {β : Type v} [inst : PseudoMetricSpace α] [inst_1 : PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α},
ContinuousWithinAt f s a = ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) (f a) < ε | null | false |
AddEquiv.module._proof_4 | Mathlib.Algebra.Module.TransferInstance | ∀ {α : Type u_1} {β : Type u_3} (A : Type u_2) [inst : Semiring A] [inst_1 : AddCommMonoid α] [inst_2 : AddCommMonoid β]
[inst_3 : Module A β] (e : α ≃+ β) (a : A) (x y : α), a • (x + y) = a • x + a • y | null | false |
_private.Mathlib.AlgebraicTopology.ModelCategory.Basic.0.HomotopicalAlgebra.ModelCategory.mk'.cm3a_aux._simp_1_4 | Mathlib.AlgebraicTopology.ModelCategory.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C],
HomotopicalAlgebra.WeakEquivalence f = HomotopicalAlgebra.weakEquivalences C f | null | false |
DifferentiableAt.finsetProd | Mathlib.Analysis.Calculus.Deriv.Mul | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_2} {𝔸' : Type u_3} [inst_1 : NormedCommRing 𝔸']
[inst_2 : NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'},
(∀ i ∈ u, DifferentiableAt 𝕜 (f i) x) → DifferentiableAt 𝕜 (∏ i ∈ u, f i) x | null | true |
Ultrafilter.mem_or_compl_mem | Mathlib.Order.Filter.Ultrafilter.Defs | ∀ {α : Type u} (f : Ultrafilter α) (s : Set α), s ∈ f ∨ sᶜ ∈ f | null | true |
_private.Mathlib.Algebra.Star.Subalgebra.0.StarSubalgebra.mem_centralizer_iff._simp_1_2 | Mathlib.Algebra.Star.Subalgebra | ∀ {α : Type u_1} {s : Set α} [inst : InvolutiveStar α], star s = star '' s | null | false |
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