name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.Limits.MonoFactorisation.e | Mathlib.CategoryTheory.Limits.Shapes.Images | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{X Y : C} → {f : X ⟶ Y} → (self : CategoryTheory.Limits.MonoFactorisation f) → X ⟶ self.I | A factorisation of a morphism `f = e ≫ m`, with `m` monic. | true |
Mathlib.Meta.FunProp.DecompositionResult.failed.elim | Mathlib.Tactic.FunProp.FunctionData | {motive : Mathlib.Meta.FunProp.DecompositionResult → Sort u} →
(t : Mathlib.Meta.FunProp.DecompositionResult) →
t.ctorIdx = 2 → motive Mathlib.Meta.FunProp.DecompositionResult.failed → motive t | null | false |
List.min_singleton | Init.Data.List.MinMax | ∀ {α : Type u_1} [inst : Min α] {x : α}, [x].min ⋯ = x | null | true |
Multiset.zero_product | Mathlib.Data.Multiset.Bind | ∀ {α : Type u_1} {β : Type v} (t : Multiset β), 0 ×ˢ t = 0 | null | true |
PointedCone.mem_closure | Mathlib.Analysis.Convex.Cone.Closure | ∀ {𝕜 : Type u_1} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : IsOrderedRing 𝕜] {E : Type u_2}
[inst_3 : AddCommMonoid E] [inst_4 : TopologicalSpace E] [inst_5 : ContinuousAdd E] [inst_6 : Module 𝕜 E]
[inst_7 : ContinuousConstSMul 𝕜 E] {K : PointedCone 𝕜 E} {a : E}, a ∈ K.closure ↔ a ∈ closure ↑K | null | true |
ChevalleyThm.MvPolynomialC.degBound_casesOn_succ._mutual | Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | ∀ (k₀ k : ℕ) (D : ℕ → ℕ) (x : ℕ ⊕' ℕ),
PSum.casesOn x
(fun _x =>
ChevalleyThm.MvPolynomialC.degBound k₀ (fun t => Nat.casesOn t k D) (_x + 1) =
(k₀ * k) ^ (k₀ * k) * ChevalleyThm.MvPolynomialC.degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) _x)
fun _x =>
ChevalleyThm.MvPolynomialC.numBound k₀ (f... | null | false |
Continuous.fourier_inversion | Mathlib.Analysis.Fourier.Inversion | ∀ {V : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V]
[inst_2 : MeasurableSpace V] [inst_3 : BorelSpace V] [inst_4 : FiniteDimensional ℝ V] [inst_5 : NormedAddCommGroup E]
[inst_6 : NormedSpace ℂ E] {f : V → E} [CompleteSpace E],
Continuous f →
MeasureTheory.Integrable... | **Alias** of `Continuous.fourierInv_fourier_eq`.
---
**Fourier inversion formula**: If a function `f` on a finite-dimensional real inner product
space is continuous, integrable, and its Fourier transform `𝓕 f` is also integrable,
then `𝓕⁻ (𝓕 f) = f`. | true |
Directed.le_sequence | Mathlib.Logic.Encodable.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : Encodable α] [inst_1 : Inhabited α] [inst_2 : Preorder β] {f : α → β}
(hf : Directed (fun x1 x2 => x1 ≤ x2) f) (a : α), f a ≤ f (Directed.sequence f hf (Encodable.encode a + 1)) | null | true |
List.tailsTR.go.eq_def | Batteries.Data.List.Basic | ∀ {α : Type u_1} (l : List α) (acc : Array (List α)),
List.tailsTR.go l acc =
match l with
| [] => acc.toListAppend [[]]
| head :: xs => List.tailsTR.go xs (acc.push l) | null | true |
_private.Mathlib.Data.Finsupp.Indicator.0.Finsupp.indicator_indicator._proof_1_2 | Mathlib.Data.Finsupp.Indicator | ∀ {ι : Type u_1} {α : Type u_2} [inst : Zero α] (s : Finset ι) {t : Finset ι} (f : (i : ι) → i ∈ s → α)
[inst_1 : DecidableEq ι],
(Finsupp.indicator t fun i x => (Finsupp.indicator s f) i) = Finsupp.indicator (s ∩ t) fun i hi => f i ⋯ | null | false |
BialgCat.mk | Mathlib.Algebra.Category.BialgCat.Basic | {R : Type u} →
[inst : CommRing R] →
(carrier : Type v) → [instRing : Ring carrier] → [instBialgebra : Bialgebra R carrier] → BialgCat R | null | true |
Std.TreeSet.Raw.le_maxD_of_contains | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp],
t.WF → ∀ {k : α}, t.contains k = true → ∀ {fallback : α}, (cmp k (t.maxD fallback)).isLE = true | null | true |
Prod.instBornology._proof_1 | Mathlib.Topology.Bornology.Constructions | ∀ {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst_1 : Bornology β],
(Bornology.cobounded α).coprod (Bornology.cobounded β) ≤ Filter.cofinite | null | false |
AffineSubspace.SOppSide.trans_wSameSide | Mathlib.Analysis.Convex.Side | ∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Field R] [inst_1 : LinearOrder R]
[inst_2 : IsStrictOrderedRing R] [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : AddTorsor V P]
{s : AffineSubspace R P} {x y z : P}, s.SOppSide x y → s.WSameSide y z → s.WOppSide x z | null | true |
_private.Mathlib.Combinatorics.SimpleGraph.Triangle.Removal.0.Mathlib.Meta.Positivity.evalTriangleRemovalBound.match_4 | Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | (α : Q(Type)) →
(_zα : Q(Zero «$α»)) →
(_pα : Q(PartialOrder «$α»)) →
(ε : Q(ℝ)) →
(motive : Mathlib.Meta.Positivity.Strictness q(inferInstance) q(inferInstance) ε → Sort u_1) →
(__discr : Mathlib.Meta.Positivity.Strictness q(inferInstance) q(inferInstance) ε) →
((hε : Q(0 < «$... | null | false |
SupBotHom.dual_comp | Mathlib.Order.Hom.BoundedLattice | ∀ {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : Max α] [inst_1 : Bot α] [inst_2 : Max β] [inst_3 : Bot β]
[inst_4 : Max γ] [inst_5 : Bot γ] (g : SupBotHom β γ) (f : SupBotHom α β),
SupBotHom.dual (g.comp f) = (SupBotHom.dual g).comp (SupBotHom.dual f) | null | true |
Nat.mem_divisors_self | Mathlib.NumberTheory.Divisors | ∀ (n : ℕ), n ≠ 0 → n ∈ n.divisors | null | true |
UInt16.le_refl._simp_1 | Init.Data.UInt.Lemmas | ∀ (a : UInt16), (a ≤ a) = True | null | false |
AlexDisc.recOn | Mathlib.Topology.Order.Category.AlexDisc | {motive : AlexDisc → Sort u} →
(t : AlexDisc) →
((toTopCat : TopCat) →
[is_alexandrovDiscrete : AlexandrovDiscrete ↑toTopCat] →
motive { toTopCat := toTopCat, is_alexandrovDiscrete := is_alexandrovDiscrete }) →
motive t | null | false |
CochainComplex.mappingCone.δ_descCochain._proof_2 | Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | ∀ {n : ℤ} (n' : ℤ), n + 1 = n' → 1 + n = n' | null | false |
_private.Mathlib.Data.Fintype.Prod.0.Finset.product_eq_univ._simp_1_1 | Mathlib.Data.Fintype.Prod | ∀ {α : Type u_1} [inst : Fintype α] {s : Finset α}, (s = Finset.univ) = ∀ (x : α), x ∈ s | null | false |
PowerSeries.exp_pow_eq_rescale_exp | Mathlib.RingTheory.PowerSeries.Exp | ∀ {A : Type u_4} [inst : CommRing A] [inst_1 : Algebra ℚ A] (k : ℕ),
PowerSeries.exp A ^ k = (PowerSeries.rescale ↑k) (PowerSeries.exp A) | Shows that $(e^{X})^k = e^{kX}$. | true |
ContinuousMulEquiv.eq_symm_comp | Mathlib.Topology.Algebra.ContinuousMonoidHom | ∀ {M : Type u_1} {N : Type u_2} [inst : TopologicalSpace M] [inst_1 : TopologicalSpace N] [inst_2 : Mul M]
[inst_3 : Mul N] {α : Type u_3} (e : M ≃ₜ* N) (f : α → M) (g : α → N), f = ⇑e.symm ∘ g ↔ ⇑e ∘ f = g | null | true |
AlgebraicGeometry.Scheme.Cover.Over | Mathlib.AlgebraicGeometry.Cover.Over | (S : AlgebraicGeometry.Scheme) →
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} →
[P.IsStableUnderBaseChange] →
[AlgebraicGeometry.Scheme.IsJointlySurjectivePreserving P] →
{X : AlgebraicGeometry.Scheme} →
[X.Over S] → AlgebraicGeometry.Scheme.Cover (AlgebraicGeometry.Schem... | A `P`-cover of a scheme `X` over `S` is a cover, where the components are over `S` and the
component maps commute with the structure morphisms. | true |
QuotientGroup.preimage_image_mk | Mathlib.GroupTheory.Coset.Defs | ∀ {α : Type u_1} [inst : Group α] (N : Subgroup α) (s : Set α),
QuotientGroup.mk ⁻¹' QuotientGroup.mk '' s = ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' s | null | true |
ValuativeRel.ValueGroupWithZero.exact | Mathlib.RingTheory.Valuation.ValuativeRel.Basic | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : ValuativeRel R] {x y : R} {t s : ↥(ValuativeRel.posSubmonoid R)},
ValuativeRel.ValueGroupWithZero.mk x t = ValuativeRel.ValueGroupWithZero.mk y s → x * ↑s ≤ᵥ y * ↑t ∧ y * ↑t ≤ᵥ x * ↑s | null | true |
Ordering.swap.eq_3 | Std.Data.DTreeMap.Internal.Model | Ordering.gt.swap = Ordering.lt | null | true |
ZFSet.singleton_inj._simp_1 | Mathlib.SetTheory.ZFC.Basic | ∀ {x y : ZFSet.{u_1}}, ({x} = {y}) = (x = y) | null | false |
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