name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
MonoidHom.ext_mnat_iff | Mathlib.Algebra.Group.Nat.Hom | ∀ {M : Type u_1} [inst : Monoid M] {f g : Multiplicative ℕ →* M},
f = g ↔ f (Multiplicative.ofAdd 1) = g (Multiplicative.ofAdd 1) | null | true |
Fin.succFunEquiv_symm_apply | Mathlib.Logic.Equiv.Fin.Basic | ∀ (α : Type u_1) (n : ℕ) (a : (Fin n → α) × α) (a_1 : Fin (n + 1)),
(Fin.succFunEquiv α n).symm a a_1 = Fin.append a.1 (uniqueElim a.2) a_1 | null | true |
disjointed_le | Mathlib.Order.Disjointed | ∀ {α : Type u_1} {ι : Type u_2} [inst : GeneralizedBooleanAlgebra α] [inst_1 : Preorder ι]
[inst_2 : LocallyFiniteOrderBot ι] (f : ι → α), disjointed f ≤ f | null | true |
_private.Mathlib.Tactic.Widget.GCongr.0._aux_Mathlib_Tactic_Widget_GCongr___elabRules_tacticGcongr?_1._sparseCasesOn_1 | Mathlib.Tactic.Widget.GCongr | {α : Type u} →
{motive : Option α → Sort u_1} →
(t : Option α) → ((val : α) → motive (some val)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
CategoryTheory.Monoidal.instMonoidalTransportedInverseEquivalenceTransported._proof_3 | Mathlib.CategoryTheory.Monoidal.Transport | ∀ {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] (e : C ≌ D)
{X₁ X₂ : CategoryTheory.Monoidal.Transported e} (f : X₁ ⟶ X₂) (Y : CategoryTheory.Monoidal.Transported e),
e.inverse.map (CategoryT... | null | false |
Monoid.PushoutI.NormalWord.baseAction | Mathlib.GroupTheory.PushoutI | {ι : Type u_1} →
{G : ι → Type u_2} →
{H : Type u_3} →
[inst : (i : ι) → Group (G i)] →
[inst_1 : Group H] →
{φ : (i : ι) → H →* G i} →
{d : Monoid.PushoutI.NormalWord.Transversal φ} → MulAction H (Monoid.PushoutI.NormalWord d) | null | true |
Aesop.Percent.mk._flat_ctor | Aesop.Percent | Float → Aesop.Percent | null | false |
_private.Mathlib.Algebra.Group.Pointwise.Set.ListOfFn.0.Set.mem_list_prod._simp_1_5 | Mathlib.Algebra.Group.Pointwise.Set.ListOfFn | ∀ {a b c : Prop}, (a ∧ b ∧ c) = (b ∧ a ∧ c) | null | false |
ChainComplex.nullHomotopicMap_f_succ | Mathlib.Algebra.Homology.Homotopy | ∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Preadditive V]
{K L : ChainComplex V ℕ} (h : (i j : ℕ) → K.X i ⟶ L.X j) (n : ℕ),
(Homotopy.nullHomotopicMap h).f (n + 1) =
CategoryTheory.CategoryStruct.comp (K.d (n + 1) n) (h n (n + 1)) +
CategoryTheory.CategoryStruct.comp... | null | true |
Set.Icc.coe_pow | Mathlib.Algebra.Order.Interval.Set.Instances | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R] (x : ↑(Set.Icc 0 1)) (n : ℕ),
↑(x ^ n) = ↑x ^ n | null | true |
CategoryTheory.ShortComplex.shortExact_of_iso | Mathlib.Algebra.Homology.ShortComplex.ShortExact | ∀ {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), S₁.ShortExact → S₂.ShortExact | null | true |
CategoryTheory.associativity_app_assoc | Mathlib.CategoryTheory.Monoidal.End | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {M : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} M]
[inst_2 : CategoryTheory.MonoidalCategory M] (F : CategoryTheory.Functor M (CategoryTheory.Functor C C))
(m₁ m₂ m₃ : M) (X : C) [inst_3 : F.LaxMonoidal] {Z : C}
(h :
(F.obj
(Cate... | null | true |
permsOfList._unsafe_rec | Mathlib.Data.Fintype.Perm | {α : Type u_1} → [DecidableEq α] → List α → List (Equiv.Perm α) | null | false |
WithAbs.algebraLeft | Mathlib.Analysis.Normed.Ring.WithAbs | {S : Type u_2} →
[inst : Semiring S] →
[inst_1 : PartialOrder S] →
{R : Type u_3} →
(T : Type u_4) →
[inst_2 : CommSemiring R] →
[inst_3 : Semiring T] → [Algebra R T] → (v : AbsoluteValue R S) → Algebra (WithAbs v) T | null | true |
_private.Mathlib.Combinatorics.Quiver.Path.Vertices.0.Quiver.Path.verticesSet_nil._simp_1_2 | Mathlib.Combinatorics.Quiver.Path.Vertices | ∀ {α : Type u} {a b : Set α}, (a = b) = ∀ (x : α), x ∈ a ↔ x ∈ b | null | false |
instNonUnitalCStarAlgebraSubtypeMemNonUnitalStarSubalgebraComplexElemental._proof_7 | Mathlib.Analysis.CStarAlgebra.Classes | ∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] (x : A), IsClosed ↑(NonUnitalStarAlgebra.elemental ℂ x) | null | false |
topToLocale_map | Mathlib.Topology.Category.Locale | ∀ {X Y : TopCat} (f : X ⟶ Y), topToLocale.map f = (Frm.ofHom (TopologicalSpace.Opens.comap (TopCat.Hom.hom f))).op | null | true |
_private.Batteries.Data.BinaryHeap.WF.0.Batteries.BinaryHeap.WF.ParentGeChildren | Batteries.Data.BinaryHeap.WF | {α : Type u_1} → {sz : ℕ} → [Ord α] → Vector α sz → Fin sz → Prop | The parent of node `i` is >= node `i`'s children
Part of the invariant required/maintained by heapifyUp
| true |
Fin.succFunEquiv | Mathlib.Logic.Equiv.Fin.Basic | (α : Type u_1) → (n : ℕ) → (Fin (n + 1) → α) ≃ (Fin n → α) × α | `Fin (n + 1) → α` and `(Fin n → α) × α` are equivalent. | true |
NonUnitalSubring.ext | Mathlib.RingTheory.NonUnitalSubring.Defs | ∀ {R : Type u} [inst : NonUnitalNonAssocRing R] {S T : NonUnitalSubring R}, (∀ (x : R), x ∈ S ↔ x ∈ T) → S = T | Two non-unital subrings are equal if they have the same elements. | true |
Finset.mulAntidiagonal.eq_1 | Mathlib.Data.Finset.MulAntidiagonal | ∀ {α : Type u_1} [inst : CommMonoid α] [inst_1 : PartialOrder α] [inst_2 : IsOrderedCancelMonoid α] {s t : Set α}
(hs : s.IsPWO) (ht : t.IsPWO) (a : α), Finset.mulAntidiagonal hs ht a = ⋯.toFinset | null | true |
_private.Mathlib.Data.Fin.Tuple.NatAntidiagonal.0.List.Nat.antidiagonalTuple_one._simp_1_7 | Mathlib.Data.Fin.Tuple.NatAntidiagonal | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : List α}, (List.map f l = []) = (l = []) | null | false |
Equiv.coe_fn_mk | Mathlib.Logic.Equiv.Defs | ∀ {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) (l : Function.LeftInverse g f) (r : Function.RightInverse g f),
⇑{ toFun := f, invFun := g, left_inv := l, right_inv := r } = f | null | true |
CoxeterSystem.IsReflection.isLeftInversion_mul_right_iff | Mathlib.GroupTheory.Coxeter.Inversion | ∀ {B : Type u_1} {W : Type u_2} [inst : Group W] {M : CoxeterMatrix B} {cs : CoxeterSystem M W} {t : W},
cs.IsReflection t → ∀ {w : W}, cs.IsLeftInversion (t * w) t ↔ ¬cs.IsLeftInversion w t | null | true |
CochainComplex.HomComplex.Cochain.rightShiftAddEquiv._proof_4 | Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
(K L : CochainComplex C ℤ) (n a n' : ℤ) (hn' : n' + a = n)
(γ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'),
(fun γ => γ.rightShift a n' hn') ((fun γ => γ... | null | false |
MeasureTheory.SimpleFunc.map_mul | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : Mul β] [inst_2 : Mul γ] {g : β → γ},
(∀ (x y : β), g (x * y) = g x * g y) →
∀ (f₁ f₂ : MeasureTheory.SimpleFunc α β),
MeasureTheory.SimpleFunc.map g (f₁ * f₂) = MeasureTheory.SimpleFunc.map g f₁ * MeasureTheory.SimpleFunc.ma... | null | true |
CategoryTheory.Cat.freeMapIdIso_inv_app | Mathlib.CategoryTheory.Category.Quiv | ∀ (V : Type u_1) [inst : Quiver V] (X : CategoryTheory.Paths V),
(CategoryTheory.Cat.freeMapIdIso V).inv.app X = CategoryTheory.CategoryStruct.id X | null | true |
Submodule.FG.lTensor.directLimit.eq_1 | Mathlib.RingTheory.TensorProduct.DirectLimitFG | ∀ (R : Type u) (M : Type u_1) (N : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid N] [inst_4 : Module R N] [inst_5 : DecidableEq { Q // Q.FG }],
Submodule.FG.lTensor.directLimit R M N =
(TensorProduct.directLimitRight (fun x x_1 => Submodule.inclusion)... | null | true |
Int8.toInt32_toInt64 | Init.Data.SInt.Lemmas | ∀ (n : Int8), n.toInt64.toInt32 = n.toInt32 | null | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.map_id_equiv._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {x : Ord α} {x_1 : BEq α} [Std.LawfulBEqOrd α] {a b : α}, (compare a b = Ordering.eq) = ((a == b) = true) | null | false |
_private.Mathlib.Data.QPF.Multivariate.Constructions.Cofix.0.Mathlib.Tactic.MvBisim._aux_Mathlib_Data_QPF_Multivariate_Constructions_Cofix___elabRules_Mathlib_Tactic_MvBisim_tacticMv_bisim___With____1.match_1 | Mathlib.Data.QPF.Multivariate.Constructions.Cofix | (motive : Option (Lean.TSyntax `Lean.binderIdent) → Sort u_1) →
(x : Option (Lean.TSyntax `Lean.binderIdent)) →
((s : Lean.TSyntax `Lean.binderIdent) → motive (some s)) → (Unit → motive none) → motive x | null | false |
Array.findSome?_map | Init.Data.Array.Find | ∀ {β : Type u_1} {γ : Type u_2} {α : Type u_3} {p : γ → Option α} {f : β → γ} {xs : Array β},
Array.findSome? p (Array.map f xs) = Array.findSome? (p ∘ f) xs | null | true |
AddMonoidHom.compHom._proof_1 | Mathlib.Algebra.Group.Hom.Instances | ∀ {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : AddZeroClass M] [inst_1 : AddCommMonoid N]
[inst_2 : AddCommMonoid P], { toFun := AddMonoidHom.comp 0, map_zero' := ⋯, map_add' := ⋯ } = 0 | null | false |
_private.Init.Data.Array.Erase.0.Array.set_eraseIdx_le._proof_1 | Init.Data.Array.Erase | ∀ {α : Type u_1} {xs : Array α} {i : ℕ} {w : i < xs.size} {j : ℕ}, j < xs.size - 1 → ¬j + 1 < xs.size → False | null | false |
CategoryTheory.regularOfIsPullbackSndOfRegular._proof_4 | Mathlib.CategoryTheory.Limits.Shapes.RegularMono | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S}
{k : R ⟶ S} (hr : CategoryTheory.RegularMono h),
CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k →
∀ {X Y : CategoryTheory.Limits.WalkingParallelPair} (f : X ⟶ Y),
... | null | false |
Polynomial.IsSplittingField.rec | Mathlib.FieldTheory.SplittingField.IsSplittingField | {K : Type v} →
{L : Type w} →
[inst : Field K] →
[inst_1 : Field L] →
[inst_2 : Algebra K L] →
{f : Polynomial K} →
{motive : Polynomial.IsSplittingField K L f → Sort u} →
((splits' : (Polynomial.map (algebraMap K L) f).Splits) →
(adjoin_rootSet'... | null | false |
MeasureTheory.L1.integralCLM'.congr_simp | Mathlib.MeasureTheory.Integral.Bochner.L1 | ∀ {α : Type u_1} {E : Type u_2} (𝕜 : Type u_4) [inst : NormedAddCommGroup E] {m : MeasurableSpace α}
{μ : MeasureTheory.Measure α} [inst_1 : NormedSpace ℝ E] [inst_2 : NormedRing 𝕜] [inst_3 : Module 𝕜 E]
[inst_4 : IsBoundedSMul 𝕜 E] [inst_5 : SMulCommClass ℝ 𝕜 E] [inst_6 : CompleteSpace E],
MeasureTheory.L1.... | null | true |
CategoryTheory.Over.mapFunctor_obj | Mathlib.CategoryTheory.Comma.Over.Basic | ∀ (T : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} T] (X : T),
(CategoryTheory.Over.mapFunctor T).obj X = ↧(CategoryTheory.Over X) | null | true |
_private.Mathlib.Data.List.Basic.0.List.mem_dropLast_of_mem_of_ne_getLast._proof_1_5 | Mathlib.Data.List.Basic | ∀ {α : Type u_1} {l : List α} {a : α} (ha : a ∈ l), ¬l.dropLast ++ [l.getLast ⋯] = [] | null | false |
_private.Mathlib.Data.Set.Inclusion.0.Set.eq_of_inclusion_surjective._proof_1_2 | Mathlib.Data.Set.Inclusion | ∀ {α : Type u_1} {s t : Set α} {h : s ⊆ t}, Function.Surjective (Set.inclusion h) → ∀ x ∈ t, x ∈ s | null | false |
_private.Mathlib.Tactic.Polynomial.Basic.0.Mathlib.Tactic.Polynomial._aux_Mathlib_Tactic_Polynomial_Basic___elabRules_Mathlib_Tactic_Polynomial_polynomial_1.match_4 | Mathlib.Tactic.Polynomial.Basic | (motive : Option (Lean.Expr × Lean.Expr × Lean.Expr) → Sort u_1) →
(x : Option (Lean.Expr × Lean.Expr × Lean.Expr)) →
((α fst snd : Lean.Expr) → motive (some (α, fst, snd))) →
((x : Option (Lean.Expr × Lean.Expr × Lean.Expr)) → motive x) → motive x | null | false |
ISize.pow_succ | Init.Data.SInt.Lemmas | ∀ (x : ISize) (n : ℕ), x ^ (n + 1) = x ^ n * x | null | true |
CategoryTheory.shrinkCoyonedaEquiv_comp | Mathlib.CategoryTheory.ShrinkYoneda | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C] {X : Cᵒᵖ}
{P Q : CategoryTheory.Functor C (Type w)} (α : CategoryTheory.shrinkCoyoneda.{w, v, u}.obj X ⟶ P) (β : P ⟶ Q),
CategoryTheory.shrinkCoyonedaEquiv (CategoryTheory.CategoryStruct.comp α β) =
(Cate... | null | true |
Matrix.SpecialLinearGroup.mapGL_inj | Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs | ∀ {n : Type u} [inst : DecidableEq n] [inst_1 : Fintype n] {R : Type v} [inst_2 : CommRing R] {S : Type u_1}
[inst_3 : CommRing S] [inst_4 : Algebra R S] [FaithfulSMul R S] (g g' : Matrix.SpecialLinearGroup n R),
(Matrix.SpecialLinearGroup.mapGL S) g = (Matrix.SpecialLinearGroup.mapGL S) g' ↔ g = g' | null | true |
LocallyConstant.instAddMonoidWithOne._proof_4 | Mathlib.Topology.LocallyConstant.Algebra | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : AddMonoidWithOne Y] (x : ℕ)
(x_1 : LocallyConstant X Y), ⇑(x • x_1) = ⇑(x • x_1) | null | false |
Nat.stirlingSecond.eq_4 | Mathlib.Combinatorics.Enumerative.Stirling | ∀ (n k : ℕ), n.succ.stirlingSecond k.succ = (k + 1) * n.stirlingSecond (k + 1) + n.stirlingSecond k | null | true |
FiberBundle.extend_apply_self | Mathlib.Topology.FiberBundle.Basic | ∀ {B : Type u_2} (F : Type u_3) {E : B → Type u_5} [inst : TopologicalSpace B] [inst_1 : TopologicalSpace F]
[inst_2 : (x : B) → TopologicalSpace (E x)] [inst_3 : (x : B) → Zero (E x)]
[inst_4 : TopologicalSpace (Bundle.TotalSpace F E)] [inst_5 : FiberBundle F E] {x : B} (v : E x),
FiberBundle.extend F v x = v | null | true |
MvPolynomial.coeffAddMonoidHom_apply | Mathlib.Algebra.MvPolynomial.Basic | ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R] (m : σ →₀ ℕ) (x : MvPolynomial σ R),
(MvPolynomial.coeffAddMonoidHom m) x = x.coeff m | null | true |
Concept.instInfSet._proof_1 | Mathlib.Order.Concept | ∀ {α : Type u_1} {β : Type u_2} {r : α → β → Prop} (S : Set (Concept α β r)), Order.IsExtent r (⋂ i ∈ S, i.extent) | null | false |
Algebra.Presentation.ofAlgEquiv_toGenerators | Mathlib.RingTheory.Extension.Presentation.Basic | ∀ {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
(P : Algebra.Presentation R S ι σ) {T : Type u_1} [inst_3 : CommRing T] [inst_4 : Algebra R T] (e : S ≃ₐ[R] T),
(P.ofAlgEquiv e).toGenerators = P.ofAlgEquiv e | null | true |
PrimeSpectrum.closedsEmbedding | Mathlib.RingTheory.Spectrum.Prime.Topology | (R : Type u_1) → [inst : CommSemiring R] → (TopologicalSpace.Closeds (PrimeSpectrum R))ᵒᵈ ↪o Ideal R | The antitone order embedding of closed subsets of `Spec R` into ideals of `R`. | true |
Int16.ofInt_bitVecToInt | Init.Data.SInt.Lemmas | ∀ (n : BitVec 16), Int16.ofInt n.toInt = Int16.ofBitVec n | null | true |
String.Slice.Pattern.Model.IsValidRevSearchFrom.matched_of_eq | Init.Data.String.Lemmas.Pattern.Basic | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.Model.PatternModel pat] {s : String.Slice}
{startPos endPos endPos' : s.Pos} {l : List (String.Slice.Pattern.SearchStep s)},
String.Slice.Pattern.Model.IsValidRevSearchFrom pat startPos l →
String.Slice.Pattern.Model.IsLongestRevMatchAt pat startPos endPos' →
... | null | true |
Std.DHashMap.Const.beq_iff_equiv | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m₁ m₂ : Std.DHashMap α fun x => β} [inst : BEq β]
[LawfulBEq α] [LawfulBEq β], Std.DHashMap.Const.beq m₁ m₂ = true ↔ m₁.Equiv m₂ | null | true |
AlgebraicGeometry.Scheme.AffineOpenCover.ofIsOpenCover._proof_4 | Mathlib.AlgebraicGeometry.AffineScheme | ∀ {X : AlgebraicGeometry.Scheme} {ι : Type u_2} (U : ι → X.Opens) (hU : TopologicalSpace.IsOpenCover U),
(∀ (i : ι), AlgebraicGeometry.IsAffineOpen (U i)) → ∀ (x : ↥X), AlgebraicGeometry.IsAffineOpen (U ⋯.choose) | null | false |
Finsupp.degree_apply | Mathlib.Data.Finsupp.Weight | ∀ {σ : Type u_1} {R : Type u_4} [inst : AddCommMonoid R] (d : σ →₀ R), Finsupp.degree d = ∑ i ∈ d.support, d i | null | true |
continuous_norm' | Mathlib.Analysis.Normed.Group.Continuity | ∀ {E : Type u_4} [inst : SeminormedGroup E], Continuous fun a => ‖a‖ | null | true |
MonoidHom.map_mul | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} [inst : MulOne M] [inst_1 : MulOne N] (f : M →* N) (a b : M), f (a * b) = f a * f b | If `f` is a monoid homomorphism then `f (a * b) = f a * f b`. | true |
Set.powersetCard.faithfulSMul | Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] {n : ℕ} [inst_2 : DecidableEq α],
1 ≤ n → ↑n < ENat.card α → ∀ [FaithfulSMul G α], FaithfulSMul G ↑(Set.powersetCard α n) | If a group `G` acts faithfully on `α`, then
it acts faithfully on `powersetCard α n`, provided `1 ≤ n < ENat.card α`. | true |
SchwartzMap.smulLeftCLM_smul | Mathlib.Analysis.Distribution.SchwartzSpace.Basic | ∀ {𝕜 : Type u_2} {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] [inst_4 : NontriviallyNormedField 𝕜]
[inst_5 : NormedAlgebra ℝ 𝕜] [inst_6 : NormedSpace 𝕜 F] {g : E → 𝕜},
Function.HasTemperateGrowth g → ∀ (c : �... | null | true |
_private.Mathlib.Topology.Algebra.Group.Matrix.0.Matrix.SpecialLinearGroup.continuous_toGL._simp_1_1 | Mathlib.Topology.Algebra.Group.Matrix | ∀ {M : Type u_1} {X : Type u_3} [inst : TopologicalSpace M] [inst_1 : Monoid M] [inst_2 : TopologicalSpace X]
{f : X → Mˣ}, Continuous f = (Continuous (Units.val ∘ f) ∧ Continuous fun x => ↑(f x)⁻¹) | null | false |
Lean.Meta.getProdFields | Lean.Meta.ProdN | Lean.Expr → Lean.Expr → Lean.MetaM (Lean.Expr × Lean.Expr × Lean.Expr × Lean.Expr) | Given a product `(e₁, e₂)` of type `t₁ × t₂`, return `(e₁, t₁, e₂, t₂)`. | true |
Lean.Meta.Grind.Arith.Cutsat.CooperSplit.mk.noConfusion | Lean.Meta.Tactic.Grind.Arith.Cutsat.Types | {P : Sort u} →
{pred : Lean.Meta.Grind.Arith.Cutsat.CooperSplitPred} →
{k : ℕ} →
{h : Lean.Meta.Grind.Arith.Cutsat.CooperSplitProof} →
{pred' : Lean.Meta.Grind.Arith.Cutsat.CooperSplitPred} →
{k' : ℕ} →
{h' : Lean.Meta.Grind.Arith.Cutsat.CooperSplitProof} →
{ pred... | null | false |
MvPolynomial.optionEquivLeft_apply | Mathlib.Algebra.MvPolynomial.Equiv | ∀ (R : Type u) (S₁ : Type v) [inst : CommSemiring R] (a : MvPolynomial (Option S₁) R),
(MvPolynomial.optionEquivLeft R S₁) a =
(MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polynomial.C (MvPolynomial.X s)) a | null | true |
List.Ico.eq_1 | Mathlib.Data.List.Intervals | ∀ (n m : ℕ), List.Ico n m = List.range' n (m - n) | null | true |
CategoryTheory.Cat.comp_eq_comp | Mathlib.CategoryTheory.Category.Cat | ∀ {X Y Z : CategoryTheory.Cat} (F : X ⟶ Y) (G : Y ⟶ Z),
(CategoryTheory.CategoryStruct.comp F G).toFunctor = F.toFunctor.comp G.toFunctor | Composition in the category of categories equals functor composition. | true |
IsNowhereDense.isMeagre | Mathlib.Topology.GDelta.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] {s : Set X}, IsNowhereDense s → IsMeagre s | A nowhere dense set is meagre. | true |
Turing.PartrecToTM2.K'.ofNat_ctorIdx | Mathlib.Computability.TuringMachine.ToPartrec | ∀ (x : Turing.PartrecToTM2.K'), Turing.PartrecToTM2.K'.ofNat x.ctorIdx = x | null | true |
eVariation_mul_le | Mathlib.Analysis.BoundedVariation | ∀ {α : Type u_2} [inst : LinearOrder α] {f g : α → ℝ} {C D : ENNReal} {s : Set α},
(∀ x ∈ s, ‖f x‖ₑ ≤ C) → (∀ x ∈ s, ‖g x‖ₑ ≤ D) → eVariationOn (f * g) s ≤ C * eVariationOn g s + D * eVariationOn f s | null | true |
Matrix.blockDiag_add | Mathlib.Data.Matrix.Block | ∀ {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_11} [inst : Add α] (M N : Matrix (m × o) (n × o) α),
(M + N).blockDiag = M.blockDiag + N.blockDiag | null | true |
Lean.Meta.Grind.CheckResult.progress | Lean.Meta.Tactic.Grind.CheckResult | Lean.Meta.Grind.CheckResult | Updated basis, simplified equations. | true |
Set.FiniteExhaustion.finite' | Mathlib.Data.Set.FiniteExhaustion | ∀ {α : Type u_1} {s : Set α} (self : s.FiniteExhaustion) (n : ℕ), Finite ↑(self.toFun n) | Every set in a `FiniteExhaustion` is finite. | true |
MeasureTheory.Filtration.coeFn_inf | Mathlib.Probability.Process.Filtration | ∀ {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} [inst : Preorder ι] {f g : MeasureTheory.Filtration ι m},
↑(f ⊓ g) = ↑f ⊓ ↑g | null | true |
_private.Mathlib.Analysis.InnerProductSpace.Projection.Basic.0.Submodule.nnnorm_starProjection_eq_infNndist._simp_1_1 | Mathlib.Analysis.InnerProductSpace.Projection.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(U : Submodule 𝕜 E) [inst_3 : U.HasOrthogonalProjection] (y : E),
Metric.infNndist y ↑U = nndist y (U.starProjection y) | null | false |
Orientation.oangle_sign_sub_left_eq_neg | Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | ∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : Fact (Module.finrank ℝ V = 2)]
(o : Orientation ℝ V (Fin 2)) (x y : V), (o.oangle (y - x) y).sign = -(o.oangle x y).sign | Subtracting the first vector passed to `oangle` from the second vector negates the sign of
the angle. | true |
PNat.dvd_antisymm | Mathlib.Data.PNat.Basic | ∀ {m n : ℕ+}, m ∣ n → n ∣ m → m = n | null | true |
IsRetrocompact.preimage_of_isClosedEmbedding | Mathlib.Topology.Constructible | ∀ {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} {s : Set Y},
Topology.IsClosedEmbedding f → IsCompact (Set.range f)ᶜ → IsRetrocompact s → IsRetrocompact (f ⁻¹' s) | [Stacks Tag 09YE](https://stacks.math.columbia.edu/tag/09YE) (Extracted from the proof) | true |
Mathlib.Tactic.Linarith.PreprocessorBase.mk.inj | Mathlib.Tactic.Linarith.Datatypes | ∀ {name : autoParam Lean.Name Mathlib.Tactic.Linarith.PreprocessorBase.name._autoParam} {description : String}
{name_1 : autoParam Lean.Name Mathlib.Tactic.Linarith.PreprocessorBase.name._autoParam} {description_1 : String},
{ name := name, description := description } = { name := name_1, description := description... | null | true |
Disjoint.closure_right | Mathlib.Topology.Closure | ∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, Disjoint s t → IsOpen s → Disjoint s (closure t) | null | true |
NonUnitalSubring.map_id | Mathlib.RingTheory.NonUnitalSubring.Basic | ∀ {R : Type u} [inst : NonUnitalNonAssocRing R] (s : NonUnitalSubring R),
NonUnitalSubring.map (NonUnitalRingHom.id R) s = s | null | true |
HasDerivAt | Mathlib.Analysis.Calculus.Deriv.Basic | {𝕜 : Type u} →
[inst : NontriviallyNormedField 𝕜] →
{F : Type v} →
[inst_1 : AddCommGroup F] →
[inst_2 : Module 𝕜 F] → [inst_3 : TopologicalSpace F] → [ContinuousSMul 𝕜 F] → (𝕜 → F) → F → 𝕜 → Prop | `f` has the derivative `f'` at the point `x`.
That is, `f x' = f x + (x' - x) • f' + o(x' - x)` where `x'` converges to `x`.
| true |
MeasureTheory.FiniteMeasure.apply_iUnion_le | Mathlib.MeasureTheory.Measure.FiniteMeasure | ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : MeasureTheory.FiniteMeasure Ω} {f : ℕ → Set Ω},
(Summable fun n => μ (f n)) → μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n) | null | true |
Std.DTreeMap.Raw.WF.insertMany | Std.Data.DTreeMap.Raw.WF | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} [Std.TransCmp cmp] {ρ : Type u_1}
[inst : ForIn Id ρ ((a : α) × β a)] {l : ρ} {t : Std.DTreeMap.Raw α β cmp}, t.WF → (t.insertMany l).WF | null | true |
Tactic.ComputeAsymptotics.Monomial.ctorIdx | Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | Tactic.ComputeAsymptotics.Monomial → ℕ | null | false |
Finset.le_min' | Mathlib.Data.Finset.Max | ∀ {α : Type u_1} [inst : LinearOrder α] (s : Finset α) (H : s.Nonempty) (x : α), (∀ y ∈ s, x ≤ y) → x ≤ s.min' H | null | true |
List.splitOnPPrepend_cons_neg | Init.Data.List.SplitOn.Lemmas | ∀ {α : Type u_1} {p : α → Bool} {a : α} {l acc : List α},
p a = false → List.splitOnPPrepend p (a :: l) acc = List.splitOnPPrepend p l (a :: acc) | null | true |
CategoryTheory.Hom.ring._proof_11 | Mathlib.CategoryTheory.Monoidal.Ring | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] {R : C} [inst_3 : CategoryTheory.RingObj R] {X : C} (n : ℕ) (a : X ⟶ R),
Int.negSucc n • a = -(↑n.succ • a) | null | false |
Real.smoothTransition | Mathlib.Analysis.SpecialFunctions.SmoothTransition | ℝ → ℝ | An infinitely smooth function `f : ℝ → ℝ` such that `f x = 0` for `x ≤ 0`,
`f x = 1` for `1 ≤ x`, and `0 < f x < 1` for `0 < x < 1`. | true |
DomMulAct.symm_mk_one | Mathlib.GroupTheory.GroupAction.DomAct.Basic | ∀ {M : Type u_1} [inst : One M], DomMulAct.mk.symm 1 = 1 | null | true |
SSet.stdSimplex.faceRepresentableBy._proof_3 | Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | ∀ {n : ℕ} (S : Finset (Fin (n + 1))) (m : ℕ) (e : Fin (m + 1) ≃o ↥S) {j : SimplexCategory}
(x : (SSet.stdSimplex.face S).toSSet.obj (Opposite.op j)),
∀
x_1 ∈
Finset.image
⇑(SimplexCategory.Hom.toOrderHom
(SSet.stdSimplex.objEquiv
(SSet.stdSimplex.objMk
((O... | null | false |
Lean.Elab.Tactic.UnusedSimpArgsInfo._sizeOf_inst | Lean.Elab.Tactic.Simp | SizeOf Lean.Elab.Tactic.UnusedSimpArgsInfo | null | false |
Lean.Grind.CommRing.SPolResult.m₂ | Lean.Meta.Sym.Arith.Poly | Lean.Grind.CommRing.SPolResult → Lean.Grind.CommRing.Mon | Monomial factor applied to polynomial `p₂`. | true |
CategoryTheory.CostructuredArrow.prodInverse._proof_1 | Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | ∀ {C : Type u_7} [inst : CategoryTheory.Category.{u_5, u_7} C] {D : Type u_3}
[inst_1 : CategoryTheory.Category.{u_1, u_3} D] {C' : Type u_8} [inst_2 : CategoryTheory.Category.{u_6, u_8} C']
{D' : Type u_4} [inst_3 : CategoryTheory.Category.{u_2, u_4} D'] (S : CategoryTheory.Functor C D)
(S' : CategoryTheory.Func... | null | false |
_private.Lean.Compiler.LCNF.ExplicitBoxing.0.Lean.Compiler.LCNF.Ctx.mk._flat_ctor | Lean.Compiler.LCNF.ExplicitBoxing | Lean.Name → Lean.Expr → Lean.Compiler.LCNF.Ctx✝ | null | false |
_private.Mathlib.NumberTheory.Primorial.0.Nat.Prime.dvd_primorial_iff._proof_1_1 | Mathlib.NumberTheory.Primorial | ∀ {p n : ℕ}, Nat.Prime p → p ≤ n → p ∈ {p ∈ Finset.range (n + 1) | Nat.Prime p} | null | false |
Option.get_dite | Init.Data.Option.Lemmas | ∀ {β : Type u_1} {p : Prop} {x : Decidable p} (b : p → β) (w : (if h : p then some (b h) else none).isSome = true),
(if h : p then some (b h) else none).get w = b ⋯ | null | true |
CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_obj | Mathlib.CategoryTheory.Localization.Resolution | ∀ {C₁ : Type u_1} {C₂ : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁}
{W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂)
(L : (Φ.LeftResolution X₂)ᵒᵖ),
(CategoryThe... | null | true |
Finset.insert_inj_on | Mathlib.Data.Finset.Insert | ∀ {α : Type u_1} [inst : DecidableEq α] (s : Finset α), Set.InjOn (fun a => insert a s) (↑s)ᶜ | null | true |
_private.Init.Data.List.Zip.0.List.map_snd_zip.match_1_1 | Init.Data.List.Zip | ∀ {α : Type u_1} {β : Type u_2} (motive : (x : List α) → (x_1 : List β) → x_1.length ≤ x.length → Prop) (x : List α)
(x_1 : List β) (x_2 : x_1.length ≤ x.length),
(∀ (x : List α) (x_3 : [].length ≤ x.length), motive x [] x_3) →
(∀ (b : β) (bs : List β) (h : (b :: bs).length ≤ [].length), motive [] (b :: bs) h) ... | null | false |
Polynomial.nonempty_support_iff | Mathlib.Algebra.Polynomial.Degree.Support | ∀ {R : Type u} [inst : Semiring R] {p : Polynomial R}, p.support.Nonempty ↔ p ≠ 0 | null | true |
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