name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
_private.Mathlib.Tactic.Linter.Style.0.Mathlib.Linter.Style.longLine.longLineLinter | Mathlib.Tactic.Linter.Style | Lean.Linter |
SemimoduleCat.Hom._sizeOf_1 | Mathlib.Algebra.Category.ModuleCat.Semi | {R : Type u} → {inst : Semiring R} → {M N : SemimoduleCat R} → [SizeOf R] → M.Hom N → ℕ |
UInt16.fromExpr | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | Lean.Expr → Lean.Meta.SimpM (Option UInt16) |
Action.instConcreteCategoryHomSubtypeV | Mathlib.CategoryTheory.Action.Basic | (V : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} V] →
(G : Type u_2) →
[inst_1 : Monoid G] →
{FV : V → V → Type u_3} →
{CV : V → Type u_4} →
[inst_2 : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] →
[inst_3 : CategoryTheory.ConcreteCategory V FV] →
CategoryTheory.ConcreteCategory (Action V G) (Action.HomSubtype V G) |
SemidirectProduct.inr_splitting | Mathlib.GroupTheory.GroupExtension.Defs | {N : Type u_1} →
{G : Type u_3} →
[inst : Group G] → [inst_1 : Group N] → (φ : G →* MulAut N) → (SemidirectProduct.toGroupExtension φ).Splitting |
TensorAlgebra.GradedAlgebra.ι_apply._proof_1 | Mathlib.LinearAlgebra.TensorAlgebra.Grading | ∀ (R : Type u_1) (M : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (m : M),
(TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1 |
CartanMatrix.E₈ | Mathlib.Data.Matrix.Cartan | Matrix (Fin 8) (Fin 8) ℤ |
Mathlib.Tactic.Translate.Config.doc._default | Mathlib.Tactic.Translate.Core | Option String |
_private.Mathlib.Combinatorics.SetFamily.AhlswedeZhang.0.Finset.infs_aux | Mathlib.Combinatorics.SetFamily.AhlswedeZhang | ∀ {α : Type u_1} [inst : DistribLattice α] [inst_1 : DecidableEq α] {s t : Finset α} {a : α},
a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t |
PowerSeries.instInhabited | Mathlib.RingTheory.PowerSeries.Basic | {R : Type u_1} → [Inhabited R] → Inhabited (PowerSeries R) |
NonAssocRing.toAddCommGroupWithOne | Mathlib.Algebra.Ring.Defs | {α : Type u_1} → [self : NonAssocRing α] → AddCommGroupWithOne α |
ContDiffWithinAt.contDiffBump | Mathlib.Analysis.Calculus.BumpFunction.Basic | ∀ {E : Type u_1} {X : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup X]
[inst_3 : NormedSpace ℝ X] [inst_4 : HasContDiffBump E] {n : ℕ∞} {c g : X → E} {s : Set X}
{f : (x : X) → ContDiffBump (c x)} {x : X},
ContDiffWithinAt ℝ (↑n) c s x →
ContDiffWithinAt ℝ (↑n) (fun x => (f x).rIn) s x →
ContDiffWithinAt ℝ (↑n) (fun x => (f x).rOut) s x →
ContDiffWithinAt ℝ (↑n) g s x → ContDiffWithinAt ℝ (↑n) (fun x => ↑(f x) (g x)) s x |
WithCStarModule.norm_apply_le_norm | Mathlib.Analysis.CStarAlgebra.Module.Constructions | ∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] {ι : Type u_2} {E : ι → Type u_3}
[inst_2 : Fintype ι] [inst_3 : (i : ι) → NormedAddCommGroup (E i)] [inst_4 : (i : ι) → Module ℂ (E i)]
[inst_5 : (i : ι) → SMul A (E i)] [inst_6 : (i : ι) → CStarModule A (E i)] [StarOrderedRing A]
(x : WithCStarModule A ((i : ι) → E i)) (i : ι), ‖x i‖ ≤ ‖x‖ |
Nat.xor_right_injective | Batteries.Data.Nat.Bitwise | ∀ {x : ℕ}, Function.Injective fun x_1 => x ^^^ x_1 |
TopologicalSpace.le_def | Mathlib.Topology.Order | ∀ {α : Type u_1} {t s : TopologicalSpace α}, t ≤ s ↔ IsOpen ≤ IsOpen |
String.Slice.Pattern.Model.SlicesFrom.extend | Init.Data.String.Lemmas.Pattern.Split | {s : String.Slice} →
(p₁ : s.Pos) →
{p₂ : s.Pos} → p₁ ≤ p₂ → String.Slice.Pattern.Model.SlicesFrom p₂ → String.Slice.Pattern.Model.SlicesFrom p₁ |
List.destutter'_of_chain | Mathlib.Data.List.Destutter | ∀ {α : Type u_1} (l : List α) (R : α → α → Prop) [inst : DecidableRel R] {a : α},
List.IsChain R (a :: l) → List.destutter' R a l = a :: l |
ValuationSubring.one_mem | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] (A : ValuationSubring K), 1 ∈ A |
TrivSqZeroExt.instAlgebra._proof_2 | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ (R' : Type u_1) (M : Type u_2) [inst : CommSemiring R'] [inst_1 : AddCommMonoid M] [inst_2 : Module R' M],
IsScalarTower R' R' M |
Lean.Elab.Command.InductiveElabStep3.finalize | Lean.Elab.MutualInductive | Lean.Elab.Command.InductiveElabStep3 → Lean.Elab.TermElabM Unit |
CategoryTheory.PullbackShift.adjunction | Mathlib.CategoryTheory.Shift.Pullback | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{A : Type u_2} →
{B : Type u_3} →
[inst_1 : AddMonoid A] →
[inst_2 : AddMonoid B] →
(φ : A →+ B) →
[inst_3 : CategoryTheory.HasShift C B] →
{D : Type u_4} →
[inst_4 : CategoryTheory.Category.{v_2, u_4} D] →
[inst_5 : CategoryTheory.HasShift D B] →
{F : CategoryTheory.Functor C D} →
{G : CategoryTheory.Functor D C} →
(F ⊣ G) →
(CategoryTheory.PullbackShift.functor φ F ⊣ CategoryTheory.PullbackShift.functor φ G) |
MeasureTheory.SimpleFunc.ofIsEmpty._proof_1 | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {α : Type u_1} [IsEmpty α], Finite α |
Turing.TM0.Machine.map_step | Mathlib.Computability.PostTuringMachine | ∀ {Γ : Type u_1} [inst : Inhabited Γ] {Γ' : Type u_2} [inst_1 : Inhabited Γ'] {Λ : Type u_3} [inst_2 : Inhabited Λ]
{Λ' : Type u_4} [inst_3 : Inhabited Λ'] (M : Turing.TM0.Machine Γ Λ) (f₁ : Turing.PointedMap Γ Γ')
(f₂ : Turing.PointedMap Γ' Γ) (g₁ : Λ → Λ') (g₂ : Λ' → Λ) {S : Set Λ},
Function.RightInverse f₁.f f₂.f →
(∀ q ∈ S, g₂ (g₁ q) = q) →
∀ (c : Turing.TM0.Cfg Γ Λ),
c.q ∈ S →
Option.map (Turing.TM0.Cfg.map f₁ g₁) (Turing.TM0.step M c) =
Turing.TM0.step (M.map f₁ f₂ g₁ g₂) (Turing.TM0.Cfg.map f₁ g₁ c) |
CategoryTheory.NatTrans.CommShift.verticalComposition | Mathlib.CategoryTheory.Shift.CommShift | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} {D₃ : Type u_6}
[inst : CategoryTheory.Category.{v_1, u_1} C₁] [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂]
[inst_2 : CategoryTheory.Category.{v_3, u_3} C₃] [inst_3 : CategoryTheory.Category.{v_4, u_4} D₁]
[inst_4 : CategoryTheory.Category.{v_5, u_5} D₂] [inst_5 : CategoryTheory.Category.{v_6, u_6} D₃]
{F₁₂ : CategoryTheory.Functor C₁ C₂} {F₂₃ : CategoryTheory.Functor C₂ C₃} {F₁₃ : CategoryTheory.Functor C₁ C₃}
(α : F₁₃ ⟶ F₁₂.comp F₂₃) {G₁₂ : CategoryTheory.Functor D₁ D₂} {G₂₃ : CategoryTheory.Functor D₂ D₃}
{G₁₃ : CategoryTheory.Functor D₁ D₃} (β : G₁₂.comp G₂₃ ⟶ G₁₃) {L₁ : CategoryTheory.Functor C₁ D₁}
{L₂ : CategoryTheory.Functor C₂ D₂} {L₃ : CategoryTheory.Functor C₃ D₃} (e₁₂ : F₁₂.comp L₂ ⟶ L₁.comp G₁₂)
(e₂₃ : F₂₃.comp L₃ ⟶ L₂.comp G₂₃) (e₁₃ : F₁₃.comp L₃ ⟶ L₁.comp G₁₃) (A : Type u_7) [inst_6 : AddMonoid A]
[inst_7 : CategoryTheory.HasShift C₁ A] [inst_8 : CategoryTheory.HasShift C₂ A]
[inst_9 : CategoryTheory.HasShift C₃ A] [inst_10 : CategoryTheory.HasShift D₁ A]
[inst_11 : CategoryTheory.HasShift D₂ A] [inst_12 : CategoryTheory.HasShift D₃ A] [inst_13 : F₁₂.CommShift A]
[inst_14 : F₂₃.CommShift A] [inst_15 : F₁₃.CommShift A] [CategoryTheory.NatTrans.CommShift α A]
[inst_17 : G₁₂.CommShift A] [inst_18 : G₂₃.CommShift A] [inst_19 : G₁₃.CommShift A]
[CategoryTheory.NatTrans.CommShift β A] [inst_21 : L₁.CommShift A] [inst_22 : L₂.CommShift A]
[inst_23 : L₃.CommShift A] [CategoryTheory.NatTrans.CommShift e₁₂ A] [CategoryTheory.NatTrans.CommShift e₂₃ A],
e₁₃ =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight α L₃)
(CategoryTheory.CategoryStruct.comp ⋯ ⋯) →
CategoryTheory.NatTrans.CommShift e₁₃ A |
CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.casesOn | Mathlib.CategoryTheory.Monoidal.DayConvolution | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{V : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} V] →
[inst_2 : CategoryTheory.MonoidalCategory C] →
[inst_3 : CategoryTheory.MonoidalCategory V] →
{D : Type u₃} →
[inst_4 : CategoryTheory.Category.{v₃, u₃} D] →
[inst_5 : CategoryTheory.MonoidalCategoryStruct D] →
{motive : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D → Sort u} →
(t : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D) →
((ι : CategoryTheory.Functor D (CategoryTheory.Functor C V)) →
(convolutionExtensionUnit :
(d d' : D) →
CategoryTheory.MonoidalCategory.externalProduct (ι.obj d) (ι.obj d') ⟶
(CategoryTheory.MonoidalCategory.tensor C).comp
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))) →
(isPointwiseLeftKanExtensionConvolutionExtensionUnit :
(d d' : D) →
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))
(convolutionExtensionUnit d d')).IsPointwiseLeftKanExtension) →
(unitUnit :
CategoryTheory.MonoidalCategoryStruct.tensorUnit V ⟶
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D)).obj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) →
(isPointwiseLeftKanExtensionUnitUnit :
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D))
{ app := fun x => unitUnit, naturality := ⋯ }).IsPointwiseLeftKanExtension) →
(faithful_ι : ι.Faithful) →
(convolutionExtensionUnit_comp_ι_map_tensorHom_app :
∀ {d₁ d₂ d₁' d₂' : D} (f₁ : d₁ ⟶ d₁') (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.tensorHom f₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.tensorHom ((ι.map f₁).app x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁' d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :
∀ (d₁ : D) {d₂ d₂' : D} (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.whiskerLeft d₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft ((ι.obj d₁).obj x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁ d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerRight_app :
∀ {d₁ d₁' : D} (f₁ : d₁ ⟶ d₁') (d₂ : D) (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f₁ d₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight ((ι.map f₁).app x)
((ι.obj d₂).obj y))
((convolutionExtensionUnit d₁' d₂).app (x, y))) →
(associator_hom_unit_unit :
∀ (d d' d'' : D) (x y z : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight
((convolutionExtensionUnit d d').app (x, y)) ((ι.obj d'').obj z))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorObj d d')
d'').app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y, z))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.associator d d'
d'').hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y) z))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.associator ((ι.obj d).obj x)
((ι.obj d', ι.obj d'').1.obj (y, z).1)
((ι.obj d', ι.obj d'').2.obj (y, z).2)).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj x)
((convolutionExtensionUnit d' d'').app (y, z)))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d'')).app
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z))
((ι.obj
(CategoryTheory.MonoidalCategoryStruct.tensorObj d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d''))).map
(CategoryTheory.MonoidalCategoryStruct.associator
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z).1
y z).inv)))) →
(leftUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight unitUnit
((ι.obj d).obj y))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorUnit D)
d).app
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C) y))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor y).inv)) →
(rightUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj y) unitUnit)
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorUnit
D)).app
(y, CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj y
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor y).inv)) →
motive
{ ι := ι, convolutionExtensionUnit := convolutionExtensionUnit,
isPointwiseLeftKanExtensionConvolutionExtensionUnit :=
isPointwiseLeftKanExtensionConvolutionExtensionUnit,
unitUnit := unitUnit,
isPointwiseLeftKanExtensionUnitUnit :=
isPointwiseLeftKanExtensionUnitUnit,
faithful_ι := faithful_ι,
convolutionExtensionUnit_comp_ι_map_tensorHom_app :=
convolutionExtensionUnit_comp_ι_map_tensorHom_app,
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :=
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app,
convolutionExtensionUnit_comp_ι_map_whiskerRight_app :=
convolutionExtensionUnit_comp_ι_map_whiskerRight_app,
associator_hom_unit_unit := associator_hom_unit_unit,
leftUnitor_hom_unit_app := leftUnitor_hom_unit_app,
rightUnitor_hom_unit_app := rightUnitor_hom_unit_app }) →
motive t |
Lean.Json.instCoeArrayStructured | Lean.Data.Json.Basic | Coe (Array Lean.Json) Lean.Json.Structured |
groupCohomology.map_one_fst_of_isCocycle₂ | Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | ∀ {G : Type u_1} {A : Type u_2} [inst : Monoid G] [inst_1 : AddCommGroup A] [inst_2 : MulAction G A] {f : G × G → A},
groupCohomology.IsCocycle₂ f → ∀ (g : G), f (1, g) = f (1, 1) |
_private.Mathlib.MeasureTheory.VectorMeasure.AddContent.0.MeasureTheory.VectorMeasure.exists_extension_of_isSetRing_of_le_measure_of_dense._simp_1_6 | Mathlib.MeasureTheory.VectorMeasure.AddContent | ∀ {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α},
MeasurableSet s₁ → MeasurableSet s₂ → MeasurableSet (s₁ ∪ s₂) = True |
Ordinal.iterate_veblen_lt_gamma_zero | Mathlib.SetTheory.Ordinal.Veblen | ∀ (n : ℕ), (fun a => Ordinal.veblen a 0)^[n] 0 < Ordinal.gamma 0 |
GaloisCoinsertion.isAtom_of_image | Mathlib.Order.Atoms | ∀ {α : Type u_2} {β : Type u_3} [inst : PartialOrder α] [inst_1 : PartialOrder β] [inst_2 : OrderBot α]
[inst_3 : OrderBot β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) {a : α}, IsAtom (l a) → IsAtom a |
Mathlib.Tactic.Widget.StringDiagram.Kind.monoidal.sizeOf_spec | Mathlib.Tactic.Widget.StringDiagram | sizeOf Mathlib.Tactic.Widget.StringDiagram.Kind.monoidal = 1 |
_private.Lean.Meta.Tactic.Contradiction.0.Lean.Meta.isGenDiseq | Lean.Meta.Tactic.Contradiction | Lean.Expr → Bool |
DifferentiableOn.sinh | Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {s : Set E},
DifferentiableOn ℝ f s → DifferentiableOn ℝ (fun x => Real.sinh (f x)) s |
Orientation.inner_smul_rotation_pi_div_two_smul_right | Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | ∀ {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 : V) (r₁ r₂ : ℝ), inner ℝ (r₂ • x) (r₁ • (o.rotation ↑(Real.pi / 2)) x) = 0 |
TopCat.Sheaf.interUnionPullbackCone._proof_3 | Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | ∀ {X : TopCat} (U V : TopologicalSpace.Opens ↑X), U ⊓ V ≤ V |
Commute.zpow_right | Mathlib.Algebra.Group.Commute.Basic | ∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → ∀ (m : ℤ), Commute a (b ^ m) |
Filter.IsCobounded.mk | Mathlib.Order.Filter.IsBounded | ∀ {α : Type u_1} {r : α → α → Prop} {f : Filter α} [IsTrans α r] (a : α),
(∀ s ∈ f, ∃ x ∈ s, r a x) → Filter.IsCobounded r f |
SSet.stdSimplex.spineId | Mathlib.AlgebraicTopology.SimplicialSet.Path | (n : ℕ) → (SSet.stdSimplex.obj (SimplexCategory.mk n)).Path n |
instSemilatticeSupENNReal | Mathlib.Data.ENNReal.Basic | SemilatticeSup ENNReal |
Polynomial.Nontrivial.of_polynomial_ne | Mathlib.Algebra.Polynomial.Basic | ∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R}, p ≠ q → Nontrivial R |
Subfield.instIsScalarTowerSubtypeMem | Mathlib.Algebra.Field.Subfield.Basic | ∀ {K : Type u} [inst : DivisionRing K] {X : Type u_1} {Y : Type u_2} [inst_1 : SMul X Y] [inst_2 : SMul K X]
[inst_3 : SMul K Y] [IsScalarTower K X Y] (F : Subfield K), IsScalarTower (↥F) X Y |
AlgebraicGeometry.Scheme.Hom.mem_smoothLocus | Mathlib.AlgebraicGeometry.Morphisms.Smooth | ∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [inst : AlgebraicGeometry.LocallyOfFinitePresentation f] {x : ↥X},
x ∈ AlgebraicGeometry.Scheme.Hom.smoothLocus f ↔
(CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x)).FormallySmooth |
Complex.arg_exp_mul_I | Mathlib.Analysis.SpecialFunctions.Complex.Arg | ∀ (θ : ℝ), (Complex.exp (↑θ * Complex.I)).arg = toIocMod Real.two_pi_pos (-Real.pi) θ |
ContinuousMultilinearMap.smulRight | Mathlib.Topology.Algebra.Module.Multilinear.Basic | {R : Type u} →
{ι : Type v} →
{M₁ : ι → Type w₁} →
{M₂ : Type w₂} →
[inst : CommSemiring R] →
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] →
[inst_2 : AddCommMonoid M₂] →
[inst_3 : (i : ι) → Module R (M₁ i)] →
[inst_4 : Module R M₂] →
[inst_5 : TopologicalSpace R] →
[inst_6 : (i : ι) → TopologicalSpace (M₁ i)] →
[inst_7 : TopologicalSpace M₂] →
[ContinuousSMul R M₂] → ContinuousMultilinearMap R M₁ R → M₂ → ContinuousMultilinearMap R M₁ M₂ |
AdjoinRoot.liftHom_mk | Mathlib.RingTheory.AdjoinRoot | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] (f : Polynomial R) [inst_1 : CommRing S] {a : S}
[inst_2 : Algebra R S] (hfx : (Polynomial.aeval a) f = 0) {g : Polynomial R},
(AdjoinRoot.liftAlgHom f (Algebra.ofId R S) a hfx) ((AdjoinRoot.mk f) g) = (Polynomial.aeval a) g |
_private.Mathlib.Data.EReal.Operations.0.EReal.add_ne_top_iff_ne_top₂._simp_1_2 | Mathlib.Data.EReal.Operations | ∀ (x : ℝ), (↑x = ⊤) = False |
List.prod_mul_prod_eq_prod_zipWith_mul_prod_drop | Mathlib.Algebra.BigOperators.Group.List.Basic | ∀ {M : Type u_4} [inst : CommMonoid M] (l l' : List M),
l.prod * l'.prod =
(List.zipWith (fun x1 x2 => x1 * x2) l l').prod * (List.drop l'.length l).prod * (List.drop l.length l').prod |
Lean.ScopedEnvExtension.State.rec | Lean.ScopedEnvExtension | {σ : Type} →
{motive : Lean.ScopedEnvExtension.State σ → Sort u} →
((state : σ) →
(activeScopes : Lean.NameSet) →
(delimitsLocal : Bool) →
motive { state := state, activeScopes := activeScopes, delimitsLocal := delimitsLocal }) →
(t : Lean.ScopedEnvExtension.State σ) → motive t |
Std.LawfulOrderMin.mk | Init.Data.Order.Classes | ∀ {α : Type u} [inst : Min α] [inst_1 : LE α] [toMinEqOr : Std.MinEqOr α] [toLawfulOrderInf : Std.LawfulOrderInf α],
Std.LawfulOrderMin α |
Algebra.tensorH1CotangentOfIsLocalization._proof_2 | Mathlib.RingTheory.Etale.Kaehler | ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],
MonoidHomClass ((Algebra.Generators.self R S).toExtension.Ring →+* S) (Algebra.Generators.self R S).toExtension.Ring S |
Int.le_floor_add | Mathlib.Algebra.Order.Floor.Ring | ∀ {R : Type u_2} [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : FloorRing R] [IsStrictOrderedRing R] (a b : R),
⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋ |
Std.Internal.List.containsKey_maxKey? | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
∀ {km : α}, Std.Internal.List.maxKey? l = some km → Std.Internal.List.containsKey km l = true |
Lean.Language.SnapshotBundle.mk | Lean.Language.Basic | {α : Type} →
Option (Lean.Language.SyntaxGuarded (Lean.Language.SnapshotTask α)) → IO.Promise α → Lean.Language.SnapshotBundle α |
Std.IterM.TerminationMeasures.Productive.mk.injEq | Init.Data.Iterators.Basic | ∀ {α : Type w} {m : Type w → Type w'} {β : Type w} [inst : Std.Iterator α m β] (it it_1 : Std.IterM m β),
({ it := it } = { it := it_1 }) = (it = it_1) |
CategoryTheory.PreOneHypercover.cylinderX._proof_1 | Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {S : C} {E : CategoryTheory.PreOneHypercover S}
{F : CategoryTheory.PreOneHypercover S} (f g : E.Hom F) {i : E.I₀},
CategoryTheory.CategoryStruct.comp (f.h₀ i) (F.f (f.s₀ i)) =
CategoryTheory.CategoryStruct.comp (g.h₀ i) (F.f (g.s₀ i)) |
ContinuousMultilinearMap.compContinuousLinearMap._proof_1 | Mathlib.Topology.Algebra.Module.Multilinear.Basic | ∀ {R : Type u_5} {ι : Type u_1} {M₁ : ι → Type u_2} {M₁' : ι → Type u_4} {M₄ : Type u_3} [inst : Semiring R]
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : (i : ι) → AddCommMonoid (M₁' i)] [inst_3 : AddCommMonoid M₄]
[inst_4 : (i : ι) → Module R (M₁ i)] [inst_5 : (i : ι) → Module R (M₁' i)] [inst_6 : Module R M₄]
[inst_7 : (i : ι) → TopologicalSpace (M₁ i)] [inst_8 : (i : ι) → TopologicalSpace (M₁' i)]
[inst_9 : TopologicalSpace M₄] (g : ContinuousMultilinearMap R M₁' M₄) (f : (i : ι) → M₁ i →L[R] M₁' i),
Continuous (g.toFun ∘ fun x i => ↑(f i) (x i)) |
CategoryTheory.Bicategory.prod._proof_22 | Mathlib.CategoryTheory.Bicategory.Product | ∀ (B : Type u_1) [inst : CategoryTheory.Bicategory B] (C : Type u_2) [inst_1 : CategoryTheory.Bicategory C]
{a b c : B × C} (f : a ⟶ b) (g : b ⟶ c),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Bicategory.associator f.1 (CategoryTheory.CategoryStruct.id b).1 g.1).prod
(CategoryTheory.Bicategory.associator f.2 (CategoryTheory.CategoryStruct.id b).2 g.2)).hom
(CategoryTheory.Prod.mkHom
(CategoryTheory.Bicategory.whiskerLeft f.1
((CategoryTheory.Bicategory.leftUnitor g.1).prod (CategoryTheory.Bicategory.leftUnitor g.2)).hom.1)
(CategoryTheory.Bicategory.whiskerLeft f.2
((CategoryTheory.Bicategory.leftUnitor g.1).prod (CategoryTheory.Bicategory.leftUnitor g.2)).hom.2)) =
CategoryTheory.Prod.mkHom
(CategoryTheory.Bicategory.whiskerRight
((CategoryTheory.Bicategory.rightUnitor f.1).prod (CategoryTheory.Bicategory.rightUnitor f.2)).hom.1 g.1)
(CategoryTheory.Bicategory.whiskerRight
((CategoryTheory.Bicategory.rightUnitor f.1).prod (CategoryTheory.Bicategory.rightUnitor f.2)).hom.2 g.2) |
AddCon.list_sum | Mathlib.GroupTheory.Congruence.BigOperators | ∀ {ι : Type u_1} {M : Type u_2} [inst : AddZeroClass M] (c : AddCon M) {l : List ι} {f g : ι → M},
(∀ x ∈ l, c (f x) (g x)) → c (List.map f l).sum (List.map g l).sum |
List.nil_eq_flatten_iff | Init.Data.List.Lemmas | ∀ {α : Type u_1} {L : List (List α)}, [] = L.flatten ↔ ∀ l ∈ L, l = [] |
_private.Lean.Meta.Tactic.Grind.Types.0.Lean.Meta.Grind.PendingSolverPropagationsData.rec | Lean.Meta.Tactic.Grind.Types | {motive : Lean.Meta.Grind.PendingSolverPropagationsData✝ → Sort u} →
motive Lean.Meta.Grind.PendingSolverPropagationsData.nil✝ →
((solverId : ℕ) →
(lhs rhs : Lean.Expr) →
(rest : Lean.Meta.Grind.PendingSolverPropagationsData✝¹) →
motive rest → motive (Lean.Meta.Grind.PendingSolverPropagationsData.eq✝ solverId lhs rhs rest)) →
((solverId : ℕ) →
(ps : Lean.Meta.Grind.ParentSet) →
(rest : Lean.Meta.Grind.PendingSolverPropagationsData✝²) →
motive rest → motive (Lean.Meta.Grind.PendingSolverPropagationsData.diseqs✝ solverId ps rest)) →
(t : Lean.Meta.Grind.PendingSolverPropagationsData✝³) → motive t |
_private.Lean.Meta.DiscrTree.Main.0.Lean.Meta.DiscrTree.reduceUntilBadKey.step._unsafe_rec | Lean.Meta.DiscrTree.Main | Lean.Expr → Lean.MetaM Lean.Expr |
Cardinal.mk_set_nat | Mathlib.SetTheory.Cardinal.Continuum | Cardinal.mk (Set ℕ) = Cardinal.continuum |
Submodule.comap_equiv_self_of_inj_of_le.match_1 | Mathlib.Algebra.Module.Submodule.Equiv | ∀ {R : Type u_2} {M : Type u_1} {N : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid N] [inst_4 : Module R N] {f : M →ₗ[R] N} {p : Submodule R N}
(motive : ↥(Submodule.comap f p) → Prop) (x : ↥(Submodule.comap f p)),
(∀ (val : M) (hx : val ∈ Submodule.comap f p), motive ⟨val, hx⟩) → motive x |
Std.DHashMap.Const.mem_ofList | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} [EquivBEq α] [LawfulHashable α] {l : List (α × β)} {k : α},
k ∈ Std.DHashMap.Const.ofList l ↔ (List.map Prod.fst l).contains k = true |
CategoryTheory.Localization.Preadditive.add.congr_simp | Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | ∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Preadditive C]
{L : CategoryTheory.Functor C D} (W W_1 : CategoryTheory.MorphismProperty C) (e_W : W = W_1)
[inst_3 : L.IsLocalization W] [inst_4 : W.HasLeftCalculusOfFractions] {X Y : C} {X' Y' : D} (eX eX_1 : L.obj X ≅ X'),
eX = eX_1 →
∀ (eY eY_1 : L.obj Y ≅ Y'),
eY = eY_1 →
∀ (f₁ f₁_1 : X' ⟶ Y'),
f₁ = f₁_1 →
∀ (f₂ f₂_1 : X' ⟶ Y'),
f₂ = f₂_1 →
CategoryTheory.Localization.Preadditive.add W eX eY f₁ f₂ =
CategoryTheory.Localization.Preadditive.add W_1 eX_1 eY_1 f₁_1 f₂_1 |
_private.Mathlib.Order.SupIndep.0.iSupIndep.of_coe_Iic_comp._simp_1_1 | Mathlib.Order.SupIndep | ∀ {ι : Sort u_1} {α : Type u_2} [inst : CompleteLattice α] {a : α} (f : ι → ↑(Set.Iic a)), ⨆ i, ↑(f i) = ↑(⨆ i, f i) |
LinearIndepOn.image_of_comp | Mathlib.LinearAlgebra.LinearIndependent.Basic | ∀ {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {s : Set ι} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M] (f : ι → ι') (g : ι' → M), LinearIndepOn R (g ∘ f) s → LinearIndepOn R g (f '' s) |
CategoryTheory.Presieve.IsSheafFor.functorInclusion_comp_extend | Mathlib.CategoryTheory.Sites.IsSheafFor | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X : C} {S : CategoryTheory.Sieve X}
{P : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (h : CategoryTheory.Presieve.IsSheafFor P S.arrows) (f : S.functor ⟶ P),
CategoryTheory.CategoryStruct.comp S.functorInclusion (h.extend f) = f |
CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.hasOfPrecompProperty_epimorphisms | Mathlib.CategoryTheory.MorphismProperty.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C}
[P.IsStableUnderCobaseChange], P.HasOfPrecompProperty (CategoryTheory.MorphismProperty.epimorphisms C) |
LeanSearchClient.SearchResult.mk.noConfusion | LeanSearchClient.Syntax | {P : Sort u} →
{name : String} →
{type? docString? doc_url? kind? : Option String} →
{name' : String} →
{type?' docString?' doc_url?' kind?' : Option String} →
{ name := name, type? := type?, docString? := docString?, doc_url? := doc_url?, kind? := kind? } =
{ name := name', type? := type?', docString? := docString?', doc_url? := doc_url?', kind? := kind?' } →
(name = name' → type? = type?' → docString? = docString?' → doc_url? = doc_url?' → kind? = kind?' → P) → P |
Lean.Server.FileWorker.WorkerContext.modifyGetPartialHandler | Lean.Server.FileWorker | {α : Type} →
Lean.Server.FileWorker.WorkerContext →
String → (Lean.Server.FileWorker.PartialHandlerInfo → α × Lean.Server.FileWorker.PartialHandlerInfo) → BaseIO α |
ContinuousMap.HomotopyRel.symm_bijective | Mathlib.Topology.Homotopy.Basic | ∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f₀ f₁ : C(X, Y)} {S : Set X},
Function.Bijective ContinuousMap.HomotopyRel.symm |
isDedekindRing_iff | Mathlib.RingTheory.DedekindDomain.Basic | ∀ (A : Type u_2) [inst : CommRing A] (K : Type u_4) [inst_1 : CommRing K] [inst_2 : Algebra A K] [IsFractionRing A K],
IsDedekindRing A ↔
IsNoetherianRing A ∧ Ring.DimensionLEOne A ∧ ∀ {x : K}, IsIntegral A x → ∃ y, (algebraMap A K) y = x |
_private.Std.Data.DTreeMap.Internal.Model.0.Std.DTreeMap.Internal.Impl.some_getEntryLE_eq_getEntryLE?._simp_1_9 | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u_1} {a : α} {o : Option α}, some (o.getD a) = o.or (some a) |
Complex.equivRealProd | Mathlib.Data.Complex.Basic | ℂ ≃ ℝ × ℝ |
IsLocalization.AtPrime.mk'_mem_maximal_iff | Mathlib.RingTheory.Localization.AtPrime.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [inst_3 : IsLocalization.AtPrime S I] (x : R) (y : ↥I.primeCompl) (h : optParam (IsLocalRing S) ⋯),
IsLocalization.mk' S x y ∈ IsLocalRing.maximalIdeal S ↔ x ∈ I |
CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.recOn | Mathlib.CategoryTheory.Monoidal.DayConvolution | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{V : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} V] →
[inst_2 : CategoryTheory.MonoidalCategory C] →
[inst_3 : CategoryTheory.MonoidalCategory V] →
{D : Type u₃} →
[inst_4 : CategoryTheory.Category.{v₃, u₃} D] →
[inst_5 : CategoryTheory.MonoidalCategoryStruct D] →
{motive : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D → Sort u} →
(t : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D) →
((ι : CategoryTheory.Functor D (CategoryTheory.Functor C V)) →
(convolutionExtensionUnit :
(d d' : D) →
CategoryTheory.MonoidalCategory.externalProduct (ι.obj d) (ι.obj d') ⟶
(CategoryTheory.MonoidalCategory.tensor C).comp
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))) →
(isPointwiseLeftKanExtensionConvolutionExtensionUnit :
(d d' : D) →
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))
(convolutionExtensionUnit d d')).IsPointwiseLeftKanExtension) →
(unitUnit :
CategoryTheory.MonoidalCategoryStruct.tensorUnit V ⟶
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D)).obj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) →
(isPointwiseLeftKanExtensionUnitUnit :
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D))
{ app := fun x => unitUnit, naturality := ⋯ }).IsPointwiseLeftKanExtension) →
(faithful_ι : ι.Faithful) →
(convolutionExtensionUnit_comp_ι_map_tensorHom_app :
∀ {d₁ d₂ d₁' d₂' : D} (f₁ : d₁ ⟶ d₁') (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.tensorHom f₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.tensorHom ((ι.map f₁).app x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁' d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :
∀ (d₁ : D) {d₂ d₂' : D} (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.whiskerLeft d₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft ((ι.obj d₁).obj x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁ d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerRight_app :
∀ {d₁ d₁' : D} (f₁ : d₁ ⟶ d₁') (d₂ : D) (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f₁ d₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight ((ι.map f₁).app x)
((ι.obj d₂).obj y))
((convolutionExtensionUnit d₁' d₂).app (x, y))) →
(associator_hom_unit_unit :
∀ (d d' d'' : D) (x y z : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight
((convolutionExtensionUnit d d').app (x, y)) ((ι.obj d'').obj z))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorObj d d')
d'').app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y, z))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.associator d d'
d'').hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y) z))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.associator ((ι.obj d).obj x)
((ι.obj d', ι.obj d'').1.obj (y, z).1)
((ι.obj d', ι.obj d'').2.obj (y, z).2)).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj x)
((convolutionExtensionUnit d' d'').app (y, z)))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d'')).app
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z))
((ι.obj
(CategoryTheory.MonoidalCategoryStruct.tensorObj d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d''))).map
(CategoryTheory.MonoidalCategoryStruct.associator
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z).1
y z).inv)))) →
(leftUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight unitUnit
((ι.obj d).obj y))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorUnit D)
d).app
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C) y))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor y).inv)) →
(rightUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj y) unitUnit)
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorUnit
D)).app
(y, CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj y
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor y).inv)) →
motive
{ ι := ι, convolutionExtensionUnit := convolutionExtensionUnit,
isPointwiseLeftKanExtensionConvolutionExtensionUnit :=
isPointwiseLeftKanExtensionConvolutionExtensionUnit,
unitUnit := unitUnit,
isPointwiseLeftKanExtensionUnitUnit :=
isPointwiseLeftKanExtensionUnitUnit,
faithful_ι := faithful_ι,
convolutionExtensionUnit_comp_ι_map_tensorHom_app :=
convolutionExtensionUnit_comp_ι_map_tensorHom_app,
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :=
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app,
convolutionExtensionUnit_comp_ι_map_whiskerRight_app :=
convolutionExtensionUnit_comp_ι_map_whiskerRight_app,
associator_hom_unit_unit := associator_hom_unit_unit,
leftUnitor_hom_unit_app := leftUnitor_hom_unit_app,
rightUnitor_hom_unit_app := rightUnitor_hom_unit_app }) →
motive t |
_private.Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone.0.CochainComplex.mappingCocone.δ_descCochain._proof_1_6 | Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone | ∀ (p : ℤ), p + -1 + 0 = p + -1 |
Lean.PrettyPrinter.parenthesizeTerm | Lean.PrettyPrinter.Parenthesizer | Lean.Syntax → Lean.CoreM Lean.Syntax |
Lean.Parser.suppressInsideQuot | Lean.Parser.Basic | Lean.Parser.Parser → Lean.Parser.Parser |
ENNReal.ofReal_rpow_of_pos | Mathlib.Analysis.SpecialFunctions.Pow.NNReal | ∀ {x p : ℝ}, 0 < x → ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) |
Path.Homotopic.equivalence | Mathlib.Topology.Homotopy.Path | ∀ {X : Type u} [inst : TopologicalSpace X] {x₀ x₁ : X}, Equivalence Path.Homotopic |
completeLatticeOfInf._proof_12 | Mathlib.Order.CompleteLattice.Defs | ∀ (α : Type u_1) [H1 : PartialOrder α] [H2 : InfSet α],
(∀ (s : Set α), IsGLB s (sInf s)) → ∀ (s : Set α) (x : α), (∀ b ∈ s, x ≤ b) → x ≤ sInf s |
Lean.Elab.Term.LetIdDeclView.recOn | Lean.Elab.Binders | {motive : Lean.Elab.Term.LetIdDeclView → Sort u} →
(t : Lean.Elab.Term.LetIdDeclView) →
((id : Lean.Syntax) →
(binders : Array Lean.Syntax) →
(type value : Lean.Syntax) → motive { id := id, binders := binders, type := type, value := value }) →
motive t |
ContFract.instCoeGenContFract | Mathlib.Algebra.ContinuedFractions.Basic | {α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Coe (ContFract α) (GenContFract α) |
CategoryTheory.MonoidalCategory.DayConvolutionUnit.noConfusion | Mathlib.CategoryTheory.Monoidal.DayConvolution | {P : Sort u} →
{C : Type u₁} →
{inst : CategoryTheory.Category.{v₁, u₁} C} →
{V : Type u₂} →
{inst_1 : CategoryTheory.Category.{v₂, u₂} V} →
{inst_2 : CategoryTheory.MonoidalCategory C} →
{inst_3 : CategoryTheory.MonoidalCategory V} →
{F : CategoryTheory.Functor C V} →
{t : CategoryTheory.MonoidalCategory.DayConvolutionUnit F} →
{C' : Type u₁} →
{inst' : CategoryTheory.Category.{v₁, u₁} C'} →
{V' : Type u₂} →
{inst'_1 : CategoryTheory.Category.{v₂, u₂} V'} →
{inst'_2 : CategoryTheory.MonoidalCategory C'} →
{inst'_3 : CategoryTheory.MonoidalCategory V'} →
{F' : CategoryTheory.Functor C' V'} →
{t' : CategoryTheory.MonoidalCategory.DayConvolutionUnit F'} →
C = C' →
inst ≍ inst' →
V = V' →
inst_1 ≍ inst'_1 →
inst_2 ≍ inst'_2 →
inst_3 ≍ inst'_3 →
F ≍ F' →
t ≍ t' →
CategoryTheory.MonoidalCategory.DayConvolutionUnit.noConfusionType P t
t' |
Nat.range_nth_of_infinite | Mathlib.Data.Nat.Nth | ∀ {p : ℕ → Prop}, (setOf p).Infinite → Set.range (Nat.nth p) = setOf p |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.containsKey_filter_iff._simp_1_1 | Std.Data.Internal.List.Associative | ∀ {α : Type u_1} (p : α → Bool) (x : Option α), (Option.any p x = true) = ∃ y, x = some y ∧ p y = true |
LinearMap.toMatrix._proof_1 | Mathlib.LinearAlgebra.Matrix.ToLin | ∀ {R : Type u_1} [inst : CommSemiring R] {M₂ : Type u_2} [inst_1 : AddCommMonoid M₂] [inst_2 : Module R M₂],
SMulCommClass R R M₂ |
continuous_iff_ultrafilter | Mathlib.Topology.Ultrafilter | ∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f ↔ ∀ (x : X) (g : Ultrafilter X), ↑g ≤ nhds x → Filter.Tendsto f (↑g) (nhds (f x)) |
_private.Lean.Elab.Tactic.BVDecide.Frontend.Attr.0.Lean.Elab.Tactic.BVDecide.Frontend.elabBVDecideConfig.match_1 | Lean.Elab.Tactic.BVDecide.Frontend.Attr | (motive :
DoResultPR Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig
PUnit.{1} →
Sort u_1) →
(r :
DoResultPR Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig
PUnit.{1}) →
((a : Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig) → (u : PUnit.{1}) → motive (DoResultPR.pure a u)) →
((b : Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig) → (u : PUnit.{1}) → motive (DoResultPR.return b u)) →
motive r |
NumberField.Units.regOfFamily_div_regulator | Mathlib.NumberTheory.NumberField.Units.Regulator | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
(u : Fin (NumberField.Units.rank K) → (NumberField.RingOfIntegers K)ˣ),
NumberField.Units.regOfFamily u / NumberField.Units.regulator K =
↑(Subgroup.closure (Set.range u) ⊔ NumberField.Units.torsion K).index |
_private.Mathlib.CategoryTheory.WithTerminal.Basic.0.CategoryTheory.WithTerminal.comp.match_1.eq_4 | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ {C : Type u_1}
(motive : CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → Sort u_2)
(x : CategoryTheory.WithTerminal C) (_Y : C)
(h_1 :
(_X _Y _Z : C) →
motive (CategoryTheory.WithTerminal.of _X) (CategoryTheory.WithTerminal.of _Y)
(CategoryTheory.WithTerminal.of _Z))
(h_2 :
(_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive (CategoryTheory.WithTerminal.of _X) x CategoryTheory.WithTerminal.star)
(h_3 :
(_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _X) x)
(h_4 :
(x : CategoryTheory.WithTerminal C) →
(_Y : C) → motive x CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _Y))
(h_5 :
Unit → motive CategoryTheory.WithTerminal.star CategoryTheory.WithTerminal.star CategoryTheory.WithTerminal.star),
(match x, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _Y with
| CategoryTheory.WithTerminal.of _X, CategoryTheory.WithTerminal.of _Y, CategoryTheory.WithTerminal.of _Z =>
h_1 _X _Y _Z
| CategoryTheory.WithTerminal.of _X, x, CategoryTheory.WithTerminal.star => h_2 _X x
| CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _X, x => h_3 _X x
| x, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _Y => h_4 x _Y
| CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star => h_5 ()) =
h_4 x _Y |
NNReal.exists_pow_lt_of_lt_one | Mathlib.Data.NNReal.Defs | ∀ {a b : NNReal}, 0 < a → b < 1 → ∃ n, b ^ n < a |
IsFractionRing.isAlgebraic_iff | Mathlib.RingTheory.Localization.Integral | ∀ (A : Type u_3) (K : Type u_4) (C : Type u_5) [inst : CommRing A] [IsDomain A] [inst_2 : Field K]
[inst_3 : Algebra A K] [IsFractionRing A K] [inst_5 : CommRing C] [inst_6 : Algebra A C] [inst_7 : Algebra K C]
[IsScalarTower A K C] {x : C}, IsAlgebraic A x ↔ IsAlgebraic K x |
Sym2.IsDiag._proof_1 | Mathlib.Data.Sym.Sym2 | ∀ {α : Type u_1} (x x_1 : α), (x = x_1) = (x_1 = x) |
Quiver.Path.length_eq_zero_iff._simp_1 | Mathlib.Combinatorics.Quiver.Path.Vertices | ∀ {V : Type u_1} [inst : Quiver V] {a : V} (p : Quiver.Path a a), (p.length = 0) = (p = Quiver.Path.nil) |
idRestrGroupoid._proof_3 | Mathlib.Geometry.Manifold.StructureGroupoid | ∀ {H : Type u_1} [inst : TopologicalSpace H],
∃ s, ∃ (h : IsOpen s), OpenPartialHomeomorph.refl H ≈ OpenPartialHomeomorph.ofSet s h |
BitVec.ushiftRight_eq_zero | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {x : BitVec w} {n : ℕ}, w ≤ n → x >>> n = 0#w |
_private.Mathlib.Tactic.Abel.0.Mathlib.Tactic.Abel.eval._sparseCasesOn_5 | Mathlib.Tactic.Abel | {motive : Lean.Literal → Sort u} →
(t : Lean.Literal) →
((val : ℕ) → motive (Lean.Literal.natVal val)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.