name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CFC.nnrpow_eq_nnrpow_prod._auto_3 | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | Lean.Syntax | null | false |
CategoryTheory.MorphismProperty.inverseImage_inverseImage | Mathlib.CategoryTheory.MorphismProperty.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} D]
{E : Type u_2} [inst_2 : CategoryTheory.Category.{v_2, u_2} E] (P : CategoryTheory.MorphismProperty E)
(F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E),
(P.inverseImage G).inver... | null | true |
_private.Mathlib.Logic.Equiv.Basic.0.Equiv.swapCore_swapCore._proof_1_3 | Mathlib.Logic.Equiv.Basic | ∀ {α : Sort u_1} (r b : α), ¬b = b → b = r | null | false |
Lean.Grind.Linarith.instReprExpr.repr._sunfold | Init.Grind.Ordered.Linarith | Lean.Grind.Linarith.Expr → ℕ → Std.Format | null | false |
_private.Mathlib.Analysis.Convex.Side.0.Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_left_iff._simp_1_3 | Mathlib.Analysis.Convex.Side | ∀ {a b : Prop}, (a ∨ b) = (b ∨ a) | null | false |
_private.Mathlib.AlgebraicGeometry.AffineScheme.0.AlgebraicGeometry.iSup_basicOpen_of_span_eq_top._simp_1_1 | Mathlib.AlgebraicGeometry.AffineScheme | ∀ {α : Type u} {β : Type v} (f : α → β) (s : Set α) (y : β), (y ∈ f '' s) = ∃ x ∈ s, f x = y | null | false |
CategoryTheory.Discrete.equivalence_counitIso | Mathlib.CategoryTheory.Discrete.Basic | ∀ {I : Type u₁} {J : Type u₂} (e : I ≃ J),
(CategoryTheory.Discrete.equivalence e).counitIso = CategoryTheory.Discrete.natIso fun j => CategoryTheory.eqToIso ⋯ | null | true |
Polynomial.toFinsupp_nsmul | Mathlib.Algebra.Polynomial.Basic | ∀ {R : Type u} [inst : Semiring R] (a : ℕ) (b : Polynomial R), (a • b).toFinsupp = a • b.toFinsupp | null | true |
Finset.imageAddMonoidHom.eq_1 | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : DecidableEq α] [inst_1 : DecidableEq β] [inst_2 : AddZeroClass α]
[inst_3 : AddZeroClass β] [inst_4 : FunLike F α β] [inst_5 : AddMonoidHomClass F α β] (f : F),
Finset.imageAddMonoidHom f = { toFun := (Finset.imageAddHom f).toFun, map_zero' := ⋯, map_add' := ⋯ ... | null | true |
CategoryTheory.Limits.snd_of_isColimit | Mathlib.CategoryTheory.Preadditive.Biproducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C] {X Y : C}
{t : CategoryTheory.Limits.BinaryBicone X Y} (ht : CategoryTheory.Limits.IsColimit t.toCocone),
t.snd = CategoryTheory.Limits.BinaryCofan.IsColimit.desc ht 0 (CategoryTheory.CategoryStruct.id Y) | null | true |
WithTopology.casesOn | Mathlib.Topology.Defs.Basic | {X : Type u_1} →
{t : TopologicalSpace X} →
{motive : WithTopology X t → Sort u} →
(t_1 : WithTopology X t) → ((ofTopology : X) → motive (WithTopology.toTopology t ofTopology)) → motive t_1 | null | false |
ComplexShape.instIsTruncGEEmbeddingUp'Add | Mathlib.Algebra.Homology.Embedding.Basic | ∀ {A : Type u_3} [inst : AddCommSemigroup A] [inst_1 : IsRightCancelAdd A] (a b : A),
(ComplexShape.embeddingUp'Add a b).IsTruncGE | null | true |
Dyadic.toReal_mul | Mathlib.Data.Dyadic | ∀ (a b : Dyadic), (a * b).toReal = a.toReal * b.toReal | null | true |
PolynomialModule.coeffEquiv_symm_apply | Mathlib.Algebra.Polynomial.Module.Basic | ∀ (R : Type u_2) {M : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (coeff : ℕ →₀ M),
(PolynomialModule.coeffEquiv R).symm coeff = { coeff := coeff } | null | true |
Int.cast_mem_Ici_iff | Mathlib.Order.Interval.Finset.Floor | ∀ {α : Type u_1} [inst : Ring α] [inst_1 : LinearOrder α] [inst_2 : FloorRing α] {a : α} {n : ℤ},
↑n ∈ Set.Ici a ↔ n ∈ Set.Ici ⌈a⌉ | null | true |
TemperedDistribution.MemSobolev.neg | Mathlib.Analysis.Distribution.Sobolev | ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
[inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E]
[inst_6 : NormedSpace ℂ F] [inst_7 : CompleteSpace F] {s : ℝ} {p : ENNReal} [hp : Fact (1 ≤ p)]
{f : ... | null | true |
CompHausLike.HasProp.casesOn | Mathlib.Topology.Category.CompHausLike.Basic | {P : TopCat → Prop} →
{X : Type u} →
[inst : TopologicalSpace X] →
{motive : CompHausLike.HasProp P X → Sort u_1} →
(t : CompHausLike.HasProp P X) → ((hasProp : P ↧X) → motive ⋯) → motive t | null | false |
PiTensorProduct.map_comp | Mathlib.LinearAlgebra.PiTensorProduct.Basic | ∀ {ι : Type u_1} {R : Type u_4} [inst : CommSemiring R] {s : ι → Type u_7} [inst_1 : (i : ι) → AddCommMonoid (s i)]
[inst_2 : (i : ι) → Module R (s i)] {t : ι → Type u_11} {t' : ι → Type u_12} [inst_3 : (i : ι) → AddCommMonoid (t i)]
[inst_4 : (i : ι) → Module R (t i)] [inst_5 : (i : ι) → AddCommMonoid (t' i)] [ins... | null | true |
LieAlgebra.ExtendScalars.instLieRingModule | Mathlib.Algebra.Lie.BaseChange | (R : Type u_1) →
(A : Type u_2) →
(L : Type u_3) →
(M : Type u_4) →
[inst : CommRing R] →
[inst_1 : CommRing A] →
[inst_2 : Algebra R A] →
[inst_3 : LieRing L] →
[inst_4 : LieAlgebra R L] →
[inst_5 : AddCommGroup M] →
... | null | true |
OnePoint.IsZeroAt.eq_1 | Mathlib.NumberTheory.ModularForms.BoundedAtCusp | ∀ (c : OnePoint ℝ) (f : UpperHalfPlane → ℂ) (k : ℤ),
c.IsZeroAt f k = ∀ (g : GL (Fin 2) ℝ), g • OnePoint.infty = c → UpperHalfPlane.IsZeroAtImInfty (SlashAction.map k g f) | null | true |
Lean.Omega.Int.le_of_ge | Init.Omega.Int | ∀ {x y : ℤ}, x ≥ y → y ≤ x | null | true |
Lean.Expr.NumObjs.State.casesOn | Lean.Util.NumObjs | {motive : Lean.Expr.NumObjs.State → Sort u} →
(t : Lean.Expr.NumObjs.State) →
((visited : Lean.PtrSet Lean.Expr) → (counter : ℕ) → motive { visited := visited, counter := counter }) → motive t | null | false |
CategoryTheory.Bicategory.associator_hom_congr._proof_2 | Mathlib.CategoryTheory.Bicategory.EqToHom | ∀ {B : Type u_2} [inst : CategoryTheory.Bicategory B] {x y z t : B} {f f' : x ⟶ y} {g g' : y ⟶ z} {h h' : z ⟶ t},
f = f' →
g = g' →
h = h' →
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h =
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.com... | null | false |
WeierstrassCurve.Jacobian.dblXYZ_of_Z_ne_zero | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | ∀ {F : Type u} [inst : Field F] {W : WeierstrassCurve.Jacobian F} [inst_1 : DecidableEq F] {P Q : Fin 3 → F},
W.Equation P →
W.Equation Q →
P 2 ≠ 0 →
Q 2 ≠ 0 →
P 0 * Q 2 ^ 2 = Q 0 * P 2 ^ 2 →
P 1 * Q 2 ^ 3 ≠ W.negY Q * P 2 ^ 3 →
W.dblXYZ P =
W.dblZ... | null | true |
Fin.finsetImage_addNat_Ioc | Mathlib.Order.Interval.Finset.Fin | ∀ {n : ℕ} (m : ℕ) (i j : Fin n),
Finset.image (fun x => x.addNat m) (Finset.Ioc i j) = Finset.Ioc (i.addNat m) (j.addNat m) | null | true |
Lean.RArray.getImpl._sunfold | Init.Data.RArray | {α : Type u} → Lean.RArray α → ℕ → α | null | false |
norm_mul_le' | Mathlib.Analysis.Normed.Group.Basic | ∀ {E : Type u_4} [inst : SeminormedGroup E] (a b : E), ‖a * b‖ ≤ ‖a‖ + ‖b‖ | **Triangle inequality** for the norm. | true |
_private.Batteries.Data.Vector.Basic.0.Vector.scanlMFast.loop._proof_4 | Batteries.Data.Vector.Basic | ∀ (i : USize), (i + 1).toNat = i.toNat + 1 → i.toNat + 1 = (i + 1).toNat | null | false |
SheafOfModules.pushforwardCompForgetToSheafModuleCat._proof_1 | Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous | ∀ {C : Type u_6} [inst : CategoryTheory.Category.{u_5, u_6} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] {J : CategoryTheory.GrothendieckTopology C}
{K : CategoryTheory.GrothendieckTopology D} {F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat}
{R : CategoryTheory.Sheaf K Ri... | null | false |
HurwitzZeta.hasSum_int_sinZeta | Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | ∀ (a : ℝ) {s : ℂ},
1 < s.re →
HasSum (fun n => -Complex.I * ↑n.sign * Complex.exp (2 * ↑Real.pi * Complex.I * ↑a * ↑n) / ↑|n| ^ s / 2)
(HurwitzZeta.sinZeta (↑a) s) | Formula for `sinZeta` as a Dirichlet series in the convergence range, with sum over `ℤ`. | true |
Aesop.Options.noConfusionType | Aesop.Options.Public | Sort u → Aesop.Options → Aesop.Options → Sort u | null | false |
Rack.PreEnvelGroupRel'.congr_inv.inj | Mathlib.Algebra.Quandle | ∀ {R : Type u} {inst : Rack R} {a a' : Rack.PreEnvelGroup R} {ha ha_1 : Rack.PreEnvelGroupRel' R a a'},
ha.congr_inv = ha_1.congr_inv → ha = ha_1 | null | true |
Subfield.completableTopField | Mathlib.Topology.Algebra.UniformField | ∀ (L : Type u_2) [inst : Field L] [inst_1 : UniformSpace L] [CompletableTopField L] (K : Subfield L),
CompletableTopField ↥K | null | true |
Num.linearOrder._proof_4 | Mathlib.Data.Num.Lemmas | ∀ (a b : Num), (if a ≤ b then b else a) = if a ≤ b then b else a | null | false |
NonUnitalRingHom.snd._proof_1 | Mathlib.Algebra.Ring.Prod | ∀ (R : Type u_1) (S : Type u_2) [inst : NonUnitalNonAssocSemiring R] [inst_1 : NonUnitalNonAssocSemiring S]
(x y : R × S), (MulHom.snd R S).toFun (x * y) = (MulHom.snd R S).toFun x * (MulHom.snd R S).toFun y | null | false |
makeGCongrString | Mathlib.Tactic.Widget.GCongr | Array Lean.SubExpr.GoalsLocation →
Lean.Expr → SelectInsertParams → Lean.MetaM (String × String × Option (String.Pos.Raw × String.Pos.Raw)) | Return the link text and inserted text above and below of the gcongr widget. | true |
MulOpposite.instSeminormedAddGroup._proof_1 | Mathlib.Analysis.Normed.Group.Constructions | ∀ {E : Type u_1} [inst : SeminormedAddGroup E] (x x_1 : Eᵐᵒᵖ),
dist (MulOpposite.unop x) (MulOpposite.unop x_1) = ‖-MulOpposite.unop x + MulOpposite.unop x_1‖ | null | false |
MeasureTheory.NullMeasurableSet.inter._simp_1 | Mathlib.MeasureTheory.Measure.NullMeasurable | ∀ {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},
MeasureTheory.NullMeasurableSet s μ →
MeasureTheory.NullMeasurableSet t μ → MeasureTheory.NullMeasurableSet (s ∩ t) μ = True | null | false |
Booleanisation.comp_inf_lift | Mathlib.Order.Booleanisation | ∀ {α : Type u_1} [inst : GeneralizedBooleanAlgebra α] (a b : α),
Booleanisation.comp a ⊓ Booleanisation.lift b = Booleanisation.lift (b \ a) | null | true |
Algebra.TensorProduct.algEquivIncludeRange._proof_1 | Mathlib.LinearAlgebra.TensorProduct.Subalgebra | ∀ (R : Type u_3) (S : Type u_1) (T : Type u_2) [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S]
[inst_3 : Semiring T] [inst_4 : Algebra R T] (a₁ a₂ : S) (b₁ b₂ : T),
(Algebra.TensorProduct.linearEquivIncludeRange R S T) ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) =
(Algebra.TensorProduct.linearEquivIncludeR... | null | false |
_private.Init.Data.SInt.Lemmas.0.ISize.toInt_ofNat_of_lt._proof_1_1 | Init.Data.SInt.Lemmas | ∀ {n : ℕ}, n < 2 ^ 31 → ¬-2 ^ 31 ≤ ↑n → False | null | false |
CategoryTheory.Abelian.SpectralObject.HasSpectralSequence | Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {C : Type u_1} →
{ι : Type u_2} →
{κ : Type u_3} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Abelian C] →
[inst_2 : Preorder ι] →
{c : ℤ → ComplexShape κ} →
{r₀ : ℤ} →
CategoryTheory.Abelian.SpectralObject C ι →
... | Given `X : SpectralObject C ι` and `data : SpectralSequenceDataCore ι c r₀`, this is
the property which allows to construct a spectral sequence by using the recipe given
by `data`. The conditions given allow to show that the homology of a page identifies
to the next page. | true |
SSet.relativeCellComplexOfMono.sup_range_r_range_b | Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | ∀ {X Y : SSet} (i : X ⟶ Y) (d : ℕ),
SSet.Subcomplex.range (SSet.relativeCellComplexOfMono.r i d) ⊔
SSet.Subcomplex.range (SSet.relativeCellComplexOfMono.b i d) =
⊤ | null | true |
SchwartzMap.instNeg._proof_2 | Mathlib.Analysis.Distribution.SchwartzSpace.Basic | ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℝ F] (f : SchwartzMap E F) (k n : ℕ), ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ ≤ C | null | false |
Std.DTreeMap.Internal.Impl.getEntryGT! | Std.Data.DTreeMap.Internal.Queries | {α : Type u} →
{β : α → Type v} → [Ord α] → [Inhabited ((a : α) × β a)] → α → Std.DTreeMap.Internal.Impl α β → (a : α) × β a | Implementation detail of the tree map | true |
MeasureTheory.ae_dirac_iff | Mathlib.MeasureTheory.Measure.Dirac.Basic | ∀ {α : Type u_1} [inst : MeasurableSpace α] {a : α} {p : α → Prop},
MeasurableSet {x | p x} → ((∀ᵐ (x : α) ∂MeasureTheory.Measure.dirac a, p x) ↔ p a) | null | true |
Lean.FuzzyMatching.CharRole._sizeOf_1 | Lean.Data.FuzzyMatching | Lean.FuzzyMatching.CharRole → ℕ | null | false |
Bornology.ofBounded._proof_2 | Mathlib.Topology.Bornology.Basic | ∀ {α : Type u_1} (B : Set (Set α)) (empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B)
(union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B),
(∀ (x : α), {x} ∈ B) → Filter.comk (fun x => x ∈ B) empty_mem subset_mem union_mem ≤ Filter.cofinite | null | false |
ComplexShape.Embedding.AreComplementary.casesOn | Mathlib.Algebra.Homology.Embedding.AreComplementary | {ι : Type u_1} →
{ι₁ : Type u_2} →
{ι₂ : Type u_3} →
{c : ComplexShape ι} →
{c₁ : ComplexShape ι₁} →
{c₂ : ComplexShape ι₂} →
{e₁ : c₁.Embedding c} →
{e₂ : c₂.Embedding c} →
{motive : e₁.AreComplementary e₂ → Sort u} →
(t : e₁.Are... | null | false |
DirichletCharacter.LFunctionTrivChar | Mathlib.NumberTheory.LSeries.DirichletContinuation | (N : ℕ) → [NeZero N] → ℂ → ℂ | The `L`-function of the trivial character mod `N`. | true |
Std.Time.Second.instInhabitedOffset | Std.Time.Time.Unit.Second | Inhabited Std.Time.Second.Offset | null | true |
_private.Batteries.Data.BinaryHeap.WF.0.Batteries.BinaryHeap.WF.parent_ge_subtree._proof_1_25 | Batteries.Data.BinaryHeap.WF | ∀ {sz j k : ℕ} {hk : 2 * k + 2 < sz}, k + 1 ≤ sz → ↑⟨k, ⋯⟩ < sz | null | false |
Continuous.if_ge | Mathlib.Topology.Order.OrderClosed | ∀ {α : Type u} {β : Type v} {γ : Type w} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderClosedTopology α]
{f g : β → α} [inst_3 : TopologicalSpace β] [inst_4 : TopologicalSpace γ] [inst_5 : (x : β) → Decidable (g x ≤ f x)]
{f' g' : β → γ},
Continuous f' →
Continuous g' →
Continuous f →
... | null | true |
Lean.Elab.Tactic.Grind.State._sizeOf_1 | Lean.Elab.Tactic.Grind.Basic | Lean.Elab.Tactic.Grind.State → ℕ | null | false |
AffineEquiv.inv_def | Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | ∀ {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [inst : Ring k] [inst_1 : AddCommGroup V₁] [inst_2 : Module k V₁]
[inst_3 : AddTorsor V₁ P₁] (e : P₁ ≃ᵃ[k] P₁), e⁻¹ = e.symm | null | true |
ContinuousLinearMap.apply'.congr_simp | Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | ∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E' : Type u_6} (F' : Type u_10) [inst : NontriviallyNormedField 𝕜]
[inst_1 : NontriviallyNormedField 𝕜₂] (σ₁₂ : 𝕜 →+* 𝕜₂) [inst_2 : AddCommGroup E'] [inst_3 : Module 𝕜 E']
[inst_4 : TopologicalSpace E'] [inst_5 : AddCommGroup F'] [inst_6 : Module 𝕜₂ F'] [inst_7 : Topologica... | null | true |
SequentiallyComplete.seq_pair_mem | Mathlib.Topology.UniformSpace.Cauchy | ∀ {α : Type u} [uniformSpace : UniformSpace α] {f : Filter α} (hf : Cauchy f) {U : ℕ → SetRel α α}
(U_mem : ∀ (n : ℕ), U n ∈ uniformity α) ⦃N m n : ℕ⦄,
N ≤ m → N ≤ n → (SequentiallyComplete.seq hf U_mem m, SequentiallyComplete.seq hf U_mem n) ∈ U N | null | true |
_private.Lean.DocString.Markdown.0.Lean.Doc.trim | Lean.DocString.Markdown | {i : Type u_1} → Lean.Doc.Inline i → String × Lean.Doc.Inline i × String | Strips leading and trailing whitespace from `inline`, returning the leading whitespace, the stripped
middle (still an `Inline`), and the trailing whitespace, in order. The caller can then re-emit the
leading and trailing whitespace *outside* of emphasis.
| true |
Lean.Lsp.LeanDiagnosticTag.ctorIdx | Lean.Data.Lsp.Diagnostics | Lean.Lsp.LeanDiagnosticTag → ℕ | null | false |
LieSubmodule.normalizer.congr_simp | Mathlib.Algebra.Lie.Normalizer | ∀ {R : Type u_1} {L : Type u_2} {M : Type u_3} [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] [inst_6 : LieModule R L M]
(N N_1 : LieSubmodule R L M), N = N_1 → N.normalizer = N_1.normalizer | null | true |
IsOpenMap.adjunction | Mathlib.Topology.Category.TopCat.Opens | {X Y : TopCat} →
{f : X ⟶ Y} → (hf : IsOpenMap ⇑(CategoryTheory.ConcreteCategory.hom f)) → hf.functor ⊣ TopologicalSpace.Opens.map f | An open map `f : X ⟶ Y` induces an adjunction between `Opens X` and `Opens Y`.
| true |
Lean.ReducibilityHints.regular.injEq | Lean.Declaration | ∀ (a a_1 : UInt32), (Lean.ReducibilityHints.regular a = Lean.ReducibilityHints.regular a_1) = (a = a_1) | null | true |
CommRingCat.Colimits.Prequotient.add.noConfusion | Mathlib.Algebra.Category.Ring.Colimits | {J : Type v} →
{inst : CategoryTheory.SmallCategory J} →
{F : CategoryTheory.Functor J CommRingCat} →
{P : Sort u} →
{a a_1 a' a'_1 : CommRingCat.Colimits.Prequotient F} → a.add a_1 = a'.add a'_1 → (a ≍ a' → a_1 ≍ a'_1 → P) → P | null | false |
MvPolynomial.mkDerivation._proof_1 | Mathlib.Algebra.MvPolynomial.Derivation | ∀ {σ : Type u_2} (R : Type u_3) {A : Type u_1} [inst : CommSemiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A]
[inst_3 : Module (MvPolynomial σ R) A] (f : σ → A) (s : σ →₀ ℕ) (i : σ),
(MvPolynomial.mkDerivationₗ R f) ((MvPolynomial.monomial s) 1 * MvPolynomial.X i) =
(MvPolynomial.monomial s) 1 • (MvPo... | null | false |
Set.image_inter_nonempty_iff | Mathlib.Data.Set.Image | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {t : Set β}, (f '' s ∩ t).Nonempty ↔ (s ∩ f ⁻¹' t).Nonempty | null | true |
RelHom.coe_ofOnFun | Mathlib.Order.RelIso.Basic | ∀ {α : Type u_1} {β : Type u_2} (r : β → β → Prop) (f : α → β), ⇑(RelHom.ofOnFun r f) = f | null | true |
CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso | Mathlib.CategoryTheory.Sites.CompatiblePlus | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(J : CategoryTheory.GrothendieckTopology C) →
{D : Type u_1} →
[inst_1 : CategoryTheory.Category.{v_1, u_1} D] →
{E : Type u_2} →
[inst_2 : CategoryTheory.Category.{v_2, u_2} E] →
[inst_3 :
... | The isomorphism between `P⁺ ⋙ F` and `(P ⋙ F)⁺`, functorially in `F`. | true |
Finset.Nonempty.vadd_finset | Mathlib.Algebra.Group.Pointwise.Finset.Scalar | ∀ {α : Type u_1} {β : Type u_2} [inst : DecidableEq β] [inst_1 : VAdd α β] {s : Finset β} {a : α},
s.Nonempty → (a +ᵥ s).Nonempty | null | true |
AddSubgroup.instInhabited.eq_1 | Mathlib.Algebra.Group.Subgroup.Lattice | ∀ {G : Type u_1} [inst : AddGroup G], AddSubgroup.instInhabited = { default := ⊥ } | null | true |
AddSemigroup.isAddFG_iff_finite | Mathlib.GroupTheory.Finiteness | ∀ {M : Type u_1} [inst : Add M], IsAddFG M ↔ ∃ S, S.Finite ∧ AddSubsemigroup.closure S = ⊤ | null | true |
Algebra.IsStandardSmoothOfRelativeDimension.trans | Mathlib.RingTheory.Smooth.StandardSmooth | ∀ (n m : ℕ) (R : Type u) (S : Type v) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (T : Type u_1)
[inst_3 : CommRing T] [inst_4 : Algebra R T] [inst_5 : Algebra S T] [IsScalarTower R S T]
[Algebra.IsStandardSmoothOfRelativeDimension n R S] [Algebra.IsStandardSmoothOfRelativeDimension m S T],
A... | null | true |
FirstOrder.Language.StrongHomClass.casesOn | Mathlib.ModelTheory.Basic | {L : FirstOrder.Language} →
{F : Type u_3} →
{M : Type u_4} →
{N : Type u_5} →
[inst : FunLike F M N] →
[inst_1 : L.Structure M] →
[inst_2 : L.Structure N] →
{motive : L.StrongHomClass F M N → Sort u} →
(t : L.StrongHomClass F M N) →
... | null | false |
BoundedContinuousFunction.instSeminormedAddCommGroup._proof_1 | Mathlib.Topology.ContinuousMap.Bounded.Normed | ∀ {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst_1 : SeminormedAddCommGroup β]
(f g : BoundedContinuousFunction α β), dist f g = ‖-f + g‖ | null | false |
CategoryTheory.BasedNatTrans.id._proof_1 | Mathlib.CategoryTheory.FiberedCategory.BasedCategory | ∀ {𝒮 : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} 𝒮] {𝒳 : CategoryTheory.BasedCategory 𝒮}
{𝒴 : CategoryTheory.BasedCategory 𝒮} (F : CategoryTheory.BasedFunctor 𝒳 𝒴) (a : 𝒳.obj),
CategoryTheory.CategoryStruct.id (𝒳.p.obj a) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯)
... | null | false |
DividedPowers.Quotient.dpow | Mathlib.RingTheory.DividedPowers.SubDPIdeal | {A : Type u_1} → [inst : CommRing A] → {I : Ideal A} → DividedPowers I → (J : Ideal A) → ℕ → A ⧸ J → A ⧸ J | The definition of divided powers on `A ⧸ J`. Tagged as noncomputable because it makes use of
`Function.extend`, but under `IsSubDPIdeal hI (J ⊓ I)`, `dividedPowers_unique` proves that no
choices are involved. | true |
Int.ceil | Mathlib.Algebra.Order.Floor.Defs | {α : Type u_1} → [inst : Ring α] → [inst_1 : LinearOrder α] → [FloorRing α] → α → ℤ | `Int.ceil a` is the smallest integer `z` such that `a ≤ z`. It is denoted with `⌈a⌉`. | true |
Lean.Meta.Grind.Arith.Cutsat.UnsatProof._sizeOf_inst | Lean.Meta.Tactic.Grind.Arith.Cutsat.Types | SizeOf Lean.Meta.Grind.Arith.Cutsat.UnsatProof | null | false |
Homeomorph.mulRight₀_symm_apply | Mathlib.Topology.Algebra.GroupWithZero | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : GroupWithZero α] [inst_2 : SeparatelyContinuousMul α] (c : α)
(hc : c ≠ 0), ⇑(Homeomorph.mulRight₀ c hc).symm = fun x => x * c⁻¹ | null | true |
ite_div_ite | Mathlib.Algebra.Notation.Defs | ∀ {α : Type u_1} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c d : α),
((if P then a else b) / if P then c else d) = if P then a / c else b / d | null | true |
instCoeTCLatticeHomOfLatticeHomClass._proof_1 | Mathlib.Order.Hom.Lattice | ∀ {F : Type u_2} {α : Type u_3} {β : Type u_1} [inst : FunLike F α β] [inst_1 : Lattice α] [inst_2 : Lattice β]
[LatticeHomClass F α β] (f : F) (a b : α), f (a ⊔ b) = f a ⊔ f b | null | false |
Std.TreeMap.mem_insertManyIfNewUnit_list._simp_1 | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeMap α Unit cmp} [Std.TransCmp cmp] [inst : BEq α]
[Std.LawfulBEqCmp cmp] {l : List α} {k : α}, (k ∈ t.insertManyIfNewUnit l) = (k ∈ t ∨ l.contains k = true) | null | false |
CategoryTheory.ShortComplex.LeftHomologyData.unop_g' | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S : CategoryTheory.ShortComplex Cᵒᵖ} (h : S.LeftHomologyData), h.unop.g' = h.f'.unop | null | true |
Lean.Compiler.LCNF.Decl.inlineAttr | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} → Lean.Compiler.LCNF.Decl pu → Bool | null | true |
_private.Lean.Compiler.LCNF.ExpandResetReuse.0.Lean.Compiler.LCNF.isSelfSset | Lean.Compiler.LCNF.ExpandResetReuse | Lean.FVarId → ℕ → ℕ → Lean.FVarId → Lean.Compiler.LCNF.CompilerM Bool | null | true |
_private.Mathlib.NumberTheory.Multiplicity.0.odd_sq_dvd_geom_sum₂_sub._simp_1_3 | Mathlib.NumberTheory.Multiplicity | ∀ {α : Type u} [inst : CommSemiring α] {x y : α}, (y ∣ x) = (x ∈ Ideal.span {y}) | null | false |
AddSubgroup.isComplement'_top_bot | Mathlib.GroupTheory.Complement | ∀ {G : Type u_1} [inst : AddGroup G], ⊤.IsComplement' ⊥ | null | true |
Lean.Parser.ParserCacheEntry.mk.sizeOf_spec | Lean.Parser.Types | ∀ (stx : Lean.Syntax) (lhsPrec : ℕ) (newPos : String.Pos.Raw) (errorMsg : Option Lean.Parser.Error),
sizeOf { stx := stx, lhsPrec := lhsPrec, newPos := newPos, errorMsg := errorMsg } =
1 + sizeOf stx + sizeOf lhsPrec + sizeOf newPos + sizeOf errorMsg | null | true |
Lean.DelayedMetavarAssignment | Lean.MetavarContext | Type | A delayed assignment for a metavariable `?m`. It represents an assignment of the form `?m := (fun fvars => (mkMVar mvarIdPending))`.
`mvarIdPending` is a `syntheticOpaque` metavariable that has not been synthesized yet. The delayed assignment becomes a real one
as soon as `mvarIdPending` has been fully synthesized.
`fv... | true |
Prod.map_rightInverse._simp_1 | Mathlib.Data.Prod.Basic | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty α] [Nonempty γ] {f₁ : α → β} {g₁ : γ → δ}
{f₂ : β → α} {g₂ : δ → γ},
Function.RightInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂) = (Function.RightInverse f₁ f₂ ∧ Function.RightInverse g₁ g₂) | null | false |
_private.Mathlib.RingTheory.Localization.FractionRing.0.IsFractionRing.fieldEquivOfAlgEquiv._simp_1 | Mathlib.RingTheory.Localization.FractionRing | ∀ (R : Type u) (S : Type v) (A : Type w) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]
[inst_3 : Algebra R S] [inst_4 : Algebra S A] [inst_5 : Algebra R A] [IsScalarTower R S A] (x : R),
(algebraMap S A) ((algebraMap R S) x) = (algebraMap R A) x | null | false |
MeasureTheory.lintegral_const_lt_top | Mathlib.MeasureTheory.Integral.Lebesgue.Countable | ∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ]
{c : ENNReal}, c ≠ ⊤ → ∫⁻ (x : α), c ∂μ < ⊤ | null | true |
MeasureTheory.ae_dirac_eq | Mathlib.MeasureTheory.Measure.Dirac.Basic | ∀ {α : Type u_1} [inst : MeasurableSpace α] [MeasurableSingletonClass α] (a : α),
MeasureTheory.ae (MeasureTheory.Measure.dirac a) = pure a | null | true |
ReaderT.pure | Init.Prelude | {ρ : Type u} → {m : Type u → Type v} → [Monad m] → {α : Type u} → α → ReaderT ρ m α | Returns the provided value `a`, ignoring the reader monad's local value. Typically used via
`Pure.pure`.
| true |
WittVector.Tactic._aux_Mathlib_RingTheory_WittVector_IsPoly___macroRules_WittVector_Tactic_ghostSimp_1 | Mathlib.RingTheory.WittVector.IsPoly | Lean.Macro | null | false |
SemiNormedGrp.delabOf | Mathlib.Analysis.Normed.Group.SemiNormedGrp | Lean.PrettyPrinter.Delaborator.Delab | This prints `SemiNormedGrp.of X` as `↧X`. | true |
_private.Mathlib.Order.Notation.0.Max.ext.match_1 | Mathlib.Order.Notation | ∀ {α : Type u_1} (motive : Max α → Prop) (h : Max α), (∀ (max : α → α → α), motive { max := max }) → motive h | null | false |
Mathlib.Tactic.Push.not_iff._simp_2 | Mathlib.Tactic.Push | ∀ (p q : Prop), (p ∧ ¬q ∨ ¬p ∧ q) = ¬(p ↔ q) | null | false |
ComplexShape.Embedding.truncGEFunctor | Mathlib.Algebra.Homology.Embedding.TruncGE | {ι : Type u_1} →
{ι' : Type u_2} →
{c : ComplexShape ι} →
{c' : ComplexShape ι'} →
(e : c.Embedding c') →
[e.IsTruncGE] →
(C : Type u_4) →
[inst : CategoryTheory.Category.{v_2, u_4} C] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
... | Given an embedding `e : Embedding c c'` of complex shapes which satisfy `e.IsTruncGE`,
this is the (canonical) truncation functor
`HomologicalComplex C c' ⥤ HomologicalComplex C c'`. | true |
_private.Mathlib.Algebra.Lie.Cochain.0.LieModule.Cohomology.d₂₃._abel_7 | Mathlib.Algebra.Lie.Cochain | ∀ (R : Type u_3) [inst : CommRing R] (L : Type u_2) [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (M : Type u_1)
[inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M]
(a : ↥(LieModule.Cohomology.twoCochain R L M)) (x x_1 x_2 x_3 : L),
⁅x, (a x_2) x_3⁆ + ⁅x_1, (a x_2) x_3⁆ - (⁅x_2, (a x) x_... | null | false |
Std.DTreeMap.getKey!_filter | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [inst : Std.TransCmp cmp]
[inst_1 : Std.LawfulEqCmp cmp] [inst_2 : Inhabited α] {f : (a : α) → β a → Bool} {k : α},
(Std.DTreeMap.filter f t).getKey! k = ((t.getKey? k).pfilter fun x h' => f x (t.get x ⋯)).get! | null | true |
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