name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.Analysis.Calculus.Deriv.Slope.0.hasDerivAtFilter_iff_tendsto_slope._simp_1_2 | Mathlib.Analysis.Calculus.Deriv.Slope | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : β → γ} {g : α → β} {x : Filter α} {y : Filter γ},
Filter.Tendsto f (Filter.map g x) y = Filter.Tendsto (f ∘ g) x y | null | false |
AddGroupSeminorm.toFun_eq_coe | Mathlib.Analysis.Normed.Group.Seminorm | ∀ {E : Type u_3} [inst : AddGroup E] {p : AddGroupSeminorm E}, p.toFun = ⇑p | null | true |
DirectSum.instCommRingOfNat._proof_9 | Mathlib.Algebra.DirectSum.Ring | ∀ {ι : Type u_1} [inst : DecidableEq ι] (A : ι → Type u_2) [inst_1 : (i : ι) → AddCommGroup (A i)]
[inst_2 : AddCommMonoid ι] [inst_3 : DirectSum.GCommRing A] (x : ℤ) (x_1 : A 0),
(DirectSum.of A 0) (x • x_1) = x • (DirectSum.of A 0) x_1 | null | false |
CategoryTheory.MonoidalCategory._aux_Mathlib_CategoryTheory_Monoidal_Category___unexpand_CategoryTheory_MonoidalCategory_whiskerLeftIso_1 | Mathlib.CategoryTheory.Monoidal.Category | Lean.PrettyPrinter.Unexpander | null | false |
Std.Time.Duration.addSeconds | Std.Time.Duration | Std.Time.Duration → Std.Time.Second.Offset → Std.Time.Duration | Adds a `Second.Offset` to a `Duration`
| true |
Matrix.l2OpNormedAddCommGroupAux._proof_5 | Mathlib.Analysis.CStarAlgebra.Matrix | ∀ {𝕜 : Type u_1} [inst : RCLike 𝕜], RingHomCompTriple (RingHom.id 𝕜) (RingHom.id 𝕜) (RingHom.id 𝕜) | null | false |
Mathlib.Tactic.Rify.ratCast_lt._simp_1 | Mathlib.Tactic.Rify | ∀ (a b : ℚ), (a < b) = (↑a < ↑b) | null | false |
Computation.liftRelAux_inr_inl | Mathlib.Data.Seq.Computation | ∀ {α : Type u} {β : Type v} {R : α → β → Prop} {C : Computation α → Computation β → Prop} {b : β} {ca : Computation α},
Computation.LiftRelAux R C (Sum.inr ca) (Sum.inl b) = ∃ a ∈ ca, R a b | null | true |
DivInvOneMonoid.inv_one | Mathlib.Algebra.Group.Defs | ∀ {G : Type u_2} [self : DivInvOneMonoid G], 1⁻¹ = 1 | null | true |
CategoryTheory.Functor.preimageIso._proof_4 | Mathlib.CategoryTheory.Functor.FullyFaithful | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (F : CategoryTheory.Functor C D) {X Y : C} [inst_2 : F.Full]
[F.Faithful] (f : F.obj X ≅ F.obj Y),
CategoryTheory.CategoryStruct.comp (F.preimage f.inv) (F.preimage f.hom) = CategoryTheory... | null | false |
ClassGroup.mk0_eq_quotientMk | Mathlib.RingTheory.ClassGroup.Basic | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] [inst_2 : IsDedekindDomain R]
(I : ↥(nonZeroDivisors (Ideal R))), ClassGroup.mk0 I = ↑((FractionalIdeal.mk0 (FractionRing R)) I) | `ClassGroup.mk0` factors through the canonical quotient projection on
`(FractionalIdeal R⁰ (FractionRing R))ˣ`. | true |
_private.Mathlib.LinearAlgebra.QuadraticForm.Radical.0.QuadraticForm.radical_weightedSumSquares._simp_1_3 | Mathlib.LinearAlgebra.QuadraticForm.Radical | ∀ {M : Type u_4} [inst : AddMonoid M] [IsRightCancelAdd M] {a b : M}, (a + b = b) = (a = 0) | null | false |
BddAbove.isBounded_inter | Mathlib.Topology.Order.Bornology | ∀ {α : Type u_1} {s t : Set α} [inst : Bornology α] [inst_1 : Preorder α] [IsOrderBornology α],
BddAbove s → BddBelow t → Bornology.IsBounded (s ∩ t) | null | true |
CategoryTheory.Triangulated.TStructure.instIsLEObjTruncGE | Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]
[inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C]
(t : CategoryTheory.... | null | true |
AlgHom.cancel_left | Mathlib.Algebra.Algebra.Hom | ∀ {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : Semiring B] [inst_3 : Semiring C] [inst_4 : Algebra R A] [inst_5 : Algebra R B] [inst_6 : Algebra R C]
{g₁ g₂ : A →ₐ[R] B} {f : B →ₐ[R] C}, Function.Injective ⇑f → (f.comp g₁ = f.comp g₂ ↔ g₁ = g₂) | null | true |
Ideal.quotTorsionOfEquivSpanSingleton | Mathlib.Algebra.Module.Torsion.Basic | (R : Type u_1) →
(M : Type u_2) →
[inst : Ring R] →
[inst_1 : AddCommGroup M] → [inst_2 : Module R M] → (x : M) → (R ⧸ Ideal.torsionOf R M x) ≃ₗ[R] ↥(R ∙ x) | The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`. | true |
_private.Init.Data.Iterators.Lemmas.Consumers.Loop.0.Std.Iter.toArray_eq_fold._simp_1_1 | Init.Data.Iterators.Lemmas.Consumers.Loop | ∀ {α β : Type u_1} [inst : Std.Iterator α Id β] [Std.Iterators.Finite α Id] {it : Std.Iter β},
it.toArray = it.toList.toArray | null | false |
CommAlgCat.hasForgetToCommRingCat._proof_4 | Mathlib.Algebra.Category.CommAlgCat.Basic | ∀ {R : Type u_1} [inst : CommRing R],
{ obj := fun A => CommRingCat.of ↑A, map := fun {X Y} f => CommRingCat.ofHom (CommAlgCat.Hom.hom f).toRingHom,
map_id := ⋯, map_comp := ⋯ }.comp
(CategoryTheory.forget CommRingCat) =
CategoryTheory.forget (CommAlgCat R) | null | false |
Aesop.LIFOQueue | Aesop.Search.Queue | Type | null | true |
CategoryTheory.Comonad.Coalgebra.Hom.ext' | Mathlib.CategoryTheory.Monad.Algebra | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {G : CategoryTheory.Comonad C} (X Y : G.Coalgebra)
(f g : X ⟶ Y), f.f = g.f → f = g | null | true |
BoundedLatticeHom.mk.congr_simp | Mathlib.Order.Category.BddDistLat | ∀ {α : Type u_6} {β : Type u_7} [inst : Lattice α] [inst_1 : Lattice β] [inst_2 : BoundedOrder α]
[inst_3 : BoundedOrder β] (toLatticeHom toLatticeHom_1 : LatticeHom α β)
(e_toLatticeHom : toLatticeHom = toLatticeHom_1) (map_top' : toLatticeHom.toFun ⊤ = ⊤)
(map_bot' : toLatticeHom.toFun ⊥ = ⊥),
{ toLatticeHom ... | null | true |
_private.Lean.CoreM.0.Lean.mkAuxDeclName.match_1 | Lean.CoreM | (motive : Lean.Name × Lean.DeclNameGenerator → Sort u_1) →
(x : Lean.Name × Lean.DeclNameGenerator) →
((n : Lean.Name) → (ngen : Lean.DeclNameGenerator) → motive (n, ngen)) → motive x | null | false |
LocallyConstant.constₗ._proof_2 | Mathlib.Topology.LocallyConstant.Algebra | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] (R : Type u_3) [inst_1 : Semiring R]
[inst_2 : AddCommMonoid Y] [inst_3 : Module R Y] (x : R) (x_1 : Y),
LocallyConstant.const X (x • x_1) = LocallyConstant.const X (x • x_1) | null | false |
CategoryTheory.ObjectProperty.isLocal_of_isIso | Mathlib.CategoryTheory.Localization.Bousfield | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (P : CategoryTheory.ObjectProperty C) {X Y : C}
(f : X ⟶ Y) [CategoryTheory.IsIso f], P.isLocal f | null | true |
smulMonoidWithZeroHom._proof_3 | Mathlib.Algebra.GroupWithZero.Action.Basic | ∀ {M₀ : Type u_1} {N₀ : Type u_2} [inst : MonoidWithZero M₀] [inst_1 : MulZeroOneClass N₀]
[inst_2 : MulActionWithZero M₀ N₀] [inst_3 : IsScalarTower M₀ N₀ N₀] [inst_4 : SMulCommClass M₀ N₀ N₀]
(x y : M₀ × N₀), (↑smulMonoidHom).toFun (x * y) = (↑smulMonoidHom).toFun x * (↑smulMonoidHom).toFun y | null | false |
algEquivEquivAlgHom._proof_1 | Mathlib.RingTheory.Algebraic.Integral | ∀ (K : Type u_1) [inst : Field K], IsDomain K | null | false |
SimpleGraph.Iso.mapEdgeSet | Mathlib.Combinatorics.SimpleGraph.Maps | {V : Type u_1} → {W : Type u_2} → {G : SimpleGraph V} → {G' : SimpleGraph W} → G ≃g G' → ↑G.edgeSet ≃ ↑G'.edgeSet | An isomorphism of graphs induces an equivalence of edge sets. | true |
CategoryTheory.Cokleisli.Adjunction.fromCokleisli._proof_1 | Mathlib.CategoryTheory.Monad.Kleisli | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (U : CategoryTheory.Comonad C)
(x : CategoryTheory.Cokleisli U),
CategoryTheory.CategoryStruct.comp (U.δ.app x.of) (U.map (U.ε.app x.of)) =
CategoryTheory.CategoryStruct.id (U.obj x.of) | null | false |
Sym.erase_mk._proof_1 | Mathlib.Data.Sym.Basic | ∀ {α : Type u_1} {n : ℕ} [inst : DecidableEq α] (m : Multiset α), m.card = n + 1 → ∀ a ∈ m, (m.erase a).card = n | null | false |
BddDistLat.Hom.noConfusion | Mathlib.Order.Category.BddDistLat | {P : Sort u_1} →
{X Y : BddDistLat} →
{t : X.Hom Y} →
{X' Y' : BddDistLat} → {t' : X'.Hom Y'} → X = X' → Y = Y' → t ≍ t' → BddDistLat.Hom.noConfusionType P t t' | null | false |
MonoidHom.noncommCoprod_apply' | Mathlib.GroupTheory.NoncommCoprod | ∀ {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : MulOneClass M] [inst_1 : MulOneClass N] [inst_2 : Monoid P]
(f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) (mn : M × N),
(f.noncommCoprod g comm) mn = g mn.2 * f mn.1 | Variant of `MonoidHom.noncommCoprod_apply` with the product written in the other direction. | true |
CategoryTheory.ObjectProperty.SerreClassLocalization.inverseImage_monomorphisms | Mathlib.CategoryTheory.Abelian.SerreClass.Localization | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] {D : Type u'}
[inst_2 : CategoryTheory.Category.{v', u'} D] (L : CategoryTheory.Functor C D) (P : CategoryTheory.ObjectProperty C)
[inst_3 : P.IsSerreClass] [L.IsLocalization P.isoModSerre] [inst_5 : CategoryTheory.Preaddit... | null | true |
DoResultSBC.recOn | Init.Core | {α σ : Type u} →
{motive : DoResultSBC α σ → Sort u_1} →
(t : DoResultSBC α σ) →
((a : α) → (a_1 : σ) → motive (DoResultSBC.pureReturn a a_1)) →
((a : σ) → motive (DoResultSBC.break a)) → ((a : σ) → motive (DoResultSBC.continue a)) → motive t | null | false |
Fin.dfoldlM_succ | Batteries.Data.Fin.Fold | ∀ {m : Type u_1 → Type u_2} {n : ℕ} {α : Fin (n + 1 + 1) → Type u_1} [inst : Monad m]
(f : (i : Fin (n + 1)) → α i.castSucc → m (α i.succ)) (x : α 0),
Fin.dfoldlM (n + 1) α f x = do
let x ← f 0 x
Fin.dfoldlM n (α ∘ Fin.succ) (fun x1 x2 => f x1.succ x2) x | null | true |
Set.range_list_getI | Mathlib.Data.Set.List | ∀ {α : Type u_1} [inst : Inhabited α] (l : List α), Set.range l.getI = insert default {x | x ∈ l} | null | true |
CategoryTheory.Limits.prod.braiding | Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(P Q : C) →
[inst_1 : CategoryTheory.Limits.HasBinaryProduct P Q] →
[inst_2 : CategoryTheory.Limits.HasBinaryProduct Q P] → P ⨯ Q ≅ Q ⨯ P | The braiding isomorphism which swaps a binary product. | true |
_private.Init.Data.Vector.Lemmas.0.Vector.sum_reverse._simp_1_1 | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n : ℕ} [inst : Add α] [inst_1 : Zero α] {xs : Vector α n}, xs.sum = xs.toList.sum | null | false |
Lean.Syntax.Traverser._sizeOf_inst | Lean.Syntax | SizeOf Lean.Syntax.Traverser | null | false |
AddCommGroup.DirectLimit.map._proof_1 | Mathlib.Algebra.Colimit.Module | ∀ {ι : Type u_1} [inst : Preorder ι] {G : ι → Type u_3} [inst_1 : (i : ι) → AddCommMonoid (G i)]
{f : (i j : ι) → i ≤ j → G i →+ G j} [inst_2 : DecidableEq ι] {G' : ι → Type u_2}
[inst_3 : (i : ι) → AddCommMonoid (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (g : (i : ι) → G i →+ G' i),
(∀ (i j : ι) (h : i ≤ j)... | null | false |
HasSubset.Subset.diff_ssubset_of_nonempty | Mathlib.Order.BooleanAlgebra.Set | ∀ {α : Type u_1} {s t : Set α}, s ⊆ t → s.Nonempty → t \ s ⊂ t | **Alias** of `HasSubset.Subset.sdiff_ssubset_of_nonempty`. | true |
HahnSeries.SummableFamily.smul_apply | Mathlib.RingTheory.HahnSeries.Summable | ∀ {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} {V : Type u_4} {α : Type u_5} [inst : PartialOrder Γ]
[inst_1 : PartialOrder Γ'] [inst_2 : AddCommMonoid V] [inst_3 : AddCommMonoid R] [inst_4 : SMulWithZero R V]
[inst_5 : VAdd Γ Γ'] [inst_6 : IsOrderedCancelVAdd Γ Γ'] {x : HahnSeries Γ R} {s : HahnSeries.SummableFam... | null | true |
CategoryTheory.evaluationAdjunctionLeft._proof_9 | Mathlib.CategoryTheory.Adjunction.Evaluation | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] (D : Type u_4)
[inst_1 : CategoryTheory.Category.{u_2, u_4} D]
[inst_2 : ∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) {X : CategoryTheory.Functor C D}
{Y Y' : D} (f : ((CategoryTheory.evaluation C D).obj c).obj X ⟶ Y) (g : ... | null | false |
_private.Mathlib.Analysis.Analytic.Within.0.analyticOn_of_locally_analyticOn._simp_1_4 | Mathlib.Analysis.Analytic.Within | ∀ {α : Type u_1} {x a : α} {s : Set α}, (x ∈ insert a s) = (x = a ∨ x ∈ s) | null | false |
Lean.LeanOptions.mk.noConfusion | Lean.Util.LeanOptions | {P : Sort u} →
{values values' : Lean.NameMap Lean.LeanOptionValue} →
{ values := values } = { values := values' } → (values = values' → P) → P | null | false |
NormedCommGroup.ofMulDist' | Mathlib.Analysis.Normed.Group.Defs | {E : Type u_5} →
[inst : Norm E] →
[inst_1 : CommGroup E] →
[inst_2 : MetricSpace E] →
(∀ (x : E), ‖x‖ = dist 1 x) → (∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y) → NormedCommGroup E | Construct a normed group from a multiplication-invariant pseudodistance. | true |
_private.Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit.0.CategoryTheory.coherentTopology.preimage._proof_1 | Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [CategoryTheory.Preregular C]
[CategoryTheory.FinitaryExtensive C], CategoryTheory.Precoherent C | null | false |
_private.Lean.Elab.Tactic.Grind.Sym.0.Lean.Elab.Tactic.Grind.elabOptDSimproc | Lean.Elab.Tactic.Grind.Sym | Option Lean.Syntax → Lean.Elab.Tactic.Grind.GrindTacticM Lean.Meta.Sym.DSimp.DSimproc | null | true |
Subgroup.instNormalSubtypeMemFocalSubgroupOf | Mathlib.GroupTheory.Focal | ∀ {G : Type u_1} [inst : Group G] (H : Subgroup G), H.focalSubgroupOf.Normal | Lemma: H* is a normal subgroup of H. | true |
FintypeCat.toLightProfinite | Mathlib.Topology.Category.LightProfinite.Basic | CategoryTheory.Functor FintypeCat LightProfinite | The natural functor from `Fintype` to `LightProfinite`, endowing a finite type with the
discrete topology. | true |
CochainComplex.mappingConeCompTriangleh_comm₁_assoc | Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] {X₁ X₂ X₃ : CochainComplex C ℤ} (f : X₁ ⟶ X₂) (g : X₂ ⟶ X₃)
{Z : HomotopyCategory C (ComplexShape.up ℤ)}
(h :
(HomotopyCategory.quotient C (ComplexShape.u... | null | true |
FloatSpec.ctorIdx | Init.Data.Float | FloatSpec → ℕ | null | false |
LeanSearchClient.SearchServer.rec | LeanSearchClient.Syntax | {motive : LeanSearchClient.SearchServer → Sort u} →
((name url cmd : String) →
(query : String → ℕ → Lean.MetaM (Array LeanSearchClient.SearchResult)) →
(queryNum : Lean.CoreM ℕ) →
motive { name := name, url := url, cmd := cmd, query := query, queryNum := queryNum }) →
(t : LeanSearchClien... | null | false |
Algebra.IsEffective.of_section | Mathlib.RingTheory.TensorProduct.IncludeLeftSubRight | ∀ {R : Type u_1} [inst : CommSemiring R] {S : Type u_2} [inst_1 : Ring S] [inst_2 : Algebra R S] (g : S →ₐ[R] R),
Algebra.IsEffective R S | `IsEffective` is true for any `R`-algebra `S` having an `R`-algebra section of
`Algebra.ofId _ _ : R →ₐ[R] S`. | true |
_private.Init.Data.List.ToArray.0.Break.runK.match_1.splitter | Init.Data.List.ToArray | {α : Type u_1} →
(motive : Option α → Sort u_2) → (x : Option α) → ((a : α) → motive (some a)) → (Unit → motive none) → motive x | null | true |
Matrix.liftLinear_comp_singleLinearMap | Mathlib.Data.Matrix.Basis | ∀ {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [inst : DecidableEq m]
[inst_1 : DecidableEq n] [inst_2 : Fintype m] [inst_3 : Fintype n] [inst_4 : Semiring R] [inst_5 : Semiring S]
[inst_6 : AddCommMonoid α] [inst_7 : AddCommMonoid β] [inst_8 : Module R α] [inst_9 : Modu... | null | true |
ULift.recOn | Init.Prelude | {α : Type s} →
{motive : ULift.{r, s} α → Sort u} → (t : ULift.{r, s} α) → ((down : α) → motive { down := down }) → motive t | null | false |
ISize.toInt16_not | Init.Data.SInt.Bitwise | ∀ (a : ISize), (~~~a).toInt16 = ~~~a.toInt16 | null | true |
_private.Mathlib.Data.Seq.Parallel.0.Computation.map_parallel._proof_1_10 | Mathlib.Data.Seq.Parallel | ∀ {α : Type u_2} {β : Type u_1} (f : α → β) ⦃c1 c2 : Computation β⦄,
(∃ l S,
c1 = Computation.map f (Computation.corec Computation.parallel.aux1✝ (l, S)) ∧
c2 =
Computation.corec Computation.parallel.aux1✝
(List.map (Computation.map f) l, Stream'.WSeq.map (Computation.map f) S)) →
... | null | false |
LinOrd.ext | Mathlib.Order.Category.LinOrd | ∀ {X Y : LinOrd} {f g : X ⟶ Y},
(∀ (x : ↑X), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x) → f = g | null | true |
AffineEquiv.instCoeOutEquiv | Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | {k : Type u_1} →
{P₁ : Type u_2} →
{P₂ : Type u_3} →
{V₁ : Type u_6} →
{V₂ : Type u_7} →
[inst : Ring k] →
[inst_1 : AddCommGroup V₁] →
[inst_2 : AddCommGroup V₂] →
[inst_3 : Module k V₁] →
[inst_4 : Module k V₂] →
... | null | true |
CategoryTheory.Limits.DiagramOfCones.conePoints_map | Mathlib.CategoryTheory.Limits.Fubini | ∀ {J : Type u_1} {K : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} J]
[inst_1 : CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
{F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} (D : CategoryTheory.Limits.DiagramOfCones F) {X Y : J}
(f : X ⟶ Y... | null | true |
_private.Mathlib.CategoryTheory.SmallObject.Iteration.Basic.0.CategoryTheory.SmallObject.SuccStruct.Iteration.subsingleton._simp_5 | Mathlib.CategoryTheory.SmallObject.Iteration.Basic | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a ≤ b) = (b < a) | null | false |
CategoryTheory.DifferentialObject.instHasShift._proof_1 | Mathlib.CategoryTheory.DifferentialObject | ∀ {S : Type u_3} [inst : AddCommGroupWithOne S] (C : Type u_2) [inst_1 : CategoryTheory.Category.{u_1, u_2} C]
[inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_3 : CategoryTheory.HasShift C S] (m₁ m₂ m₃ : S)
(X : CategoryTheory.DifferentialObject S C),
CategoryTheory.CategoryStruct.comp ((CategoryTheory.... | null | false |
MonCat.Colimits.instInhabitedColimitType | Mathlib.Algebra.Category.MonCat.Colimits | {J : Type u_1} →
[inst : CategoryTheory.Category.{u_2, u_1} J] →
(F : CategoryTheory.Functor J MonCat) → Inhabited (MonCat.Colimits.ColimitType F) | null | true |
List.Cursor.current.eq_1 | Std.Do.Triple.SpecLemmas | ∀ {α : Type u_1} {l : List α} (c : l.Cursor) (h : 0 < c.suffix.length), c.current h = c.suffix[0] | null | true |
LinearMap.exists_mem_center_apply_eq_smul_of_forall_notLinearIndependent | Mathlib.LinearAlgebra.Center | ∀ {R : Type u_1} {V : Type u_2} [inst : Ring R] [IsDomain R] [StrongRankCondition R] [inst_3 : AddCommGroup V]
[inst_4 : Module R V] [Module.Free R V] {f : V →ₗ[R] V},
Module.finrank R V ≠ 1 → (∀ (v : V), ¬LinearIndependent R ![v, f v]) → ∃ a, f = a • 1 | Over a domain `R`, an endomorphism `f` of a free module `V`
of rank ≠ 1 such that `f v` and `v` are collinear, for all `v : V`,
consists of homotheties with central ratio.
When `R` does not satisfy `StrongRankCondition`, use
`LinearMap.exists_mem_center_apply_eq_smul_of_basis`.
When `finrank R V = 1`, up to a linear ... | true |
CategoryTheory.Limits.Cofork.IsColimit.desc'.congr_simp | Mathlib.CategoryTheory.Monad.Monadicity | ∀ {C : Type u} {X Y : C} [inst : CategoryTheory.Category.{v, u} C] {f g : X ⟶ Y} {s : CategoryTheory.Limits.Cofork f g}
(hs hs_1 : CategoryTheory.Limits.IsColimit s),
hs = hs_1 →
∀ {W : C} (k : Y ⟶ W) (h : CategoryTheory.CategoryStruct.comp f k = CategoryTheory.CategoryStruct.comp g k),
CategoryTheory.Lim... | null | true |
Polynomial.modByMonic_eq_sub_mul_div | Mathlib.Algebra.Polynomial.Div | ∀ {R : Type u} [inst : Ring R] (p q : Polynomial R), p %ₘ q = p - q * (p /ₘ q) | null | true |
Lean.Widget.MsgEmbed.brecOn_3.go | Lean.Widget.InteractiveDiagnostic | {motive_1 : Lean.Widget.MsgEmbed → Sort u} →
{motive_2 : Lean.Widget.TaggedText Lean.Widget.MsgEmbed → Sort u} →
{motive_3 :
Lean.Widget.StrictOrLazy (Array (Lean.Widget.TaggedText Lean.Widget.MsgEmbed))
(Lean.Server.WithRpcRef Lean.Widget.LazyTraceChildren) →
Sort u} →
{motive... | null | true |
Prod.map_comp_map | Init.Data.Prod | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} {ζ : Type u_6} (f : α → β) (f' : γ → δ)
(g : β → ε) (g' : δ → ζ), Prod.map g g' ∘ Prod.map f f' = Prod.map (g ∘ f) (g' ∘ f') | Composing a `Prod.map` with another `Prod.map` is equal to
a single `Prod.map` of composed functions.
| true |
SubAddAction.fixingAddSubgroupInsertEquiv._proof_6 | Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | ∀ {M : Type u_1} {α : Type u_2} [inst : AddGroup M] [inst_1 : AddAction M α] (a : α)
(s : Set ↥(SubAddAction.ofStabilizer M a)) (x : ↥(fixingAddSubgroup M (insert a (Subtype.val '' s)))),
(fun m => ⟨↑↑m, ⋯⟩) ((fun m => ⟨⟨↑m, ⋯⟩, ⋯⟩) x) = x | null | false |
Mathlib.instReprIneq | Mathlib.Data.Ineq | Repr Mathlib.Ineq | null | true |
CategoryTheory.InjectiveResolution.definition._proof_2._@.Mathlib.CategoryTheory.Abelian.Injective.Resolution.4211954440._hygCtx._hyg.8 | Mathlib.CategoryTheory.Abelian.Injective.Resolution | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : CategoryTheory.EnoughInjectives C] (Z : C),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Injective.ι Z)
((CategoryTheory.InjectiveResolution.ofCocomplex Z).d 0 1) =
0 | null | false |
Shrink.instAdd | Mathlib.Algebra.Group.Shrink | {α : Type u_2} → [inst : Small.{v, u_2} α] → [Add α] → Add (Shrink.{v, u_2} α) | null | true |
Pi.commMonoidWithZero._proof_3 | Mathlib.Algebra.GroupWithZero.Pi | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → CommMonoidWithZero (α i)] (a : (i : ι) → α i), a * 0 = 0 | null | false |
Lean.Elab.InlayHintLabel | Lean.Elab.InfoTree.InlayHints | Type | null | true |
_private.Mathlib.Order.Interval.Set.Basic.0.Set.Iio_True._simp_1_1 | Mathlib.Order.Interval.Set.Basic | ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, (a < b) = (a ≤ b ∧ ¬b ≤ a) | null | false |
ZeroHom.coe_copy | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} {x : Zero M} {x_1 : Zero N} (f : ZeroHom M N) (f' : M → N) (h : f' = ⇑f),
⇑(f.copy f' h) = f' | null | true |
Real.fourier_continuousMultilinearMap_apply | Mathlib.Analysis.Fourier.FourierTransform | ∀ {V : Type u_1} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : NormedAddCommGroup V]
[inst_3 : InnerProductSpace ℝ V] [inst_4 : MeasurableSpace V] [inst_5 : BorelSpace V] [inst_6 : FiniteDimensional ℝ V]
{ι : Type u_4} [inst_7 : Fintype ι] {M : ι → Type u_5} [inst_8 : (i : ι) → N... | null | true |
Lean.Meta.Grind.Arith.Linear.RingIneqCnstrProof.cancelDen.noConfusion | Lean.Meta.Tactic.Grind.Arith.Linear.Types | {P : Sort u} →
{c : Lean.Meta.Grind.Arith.Linear.RingIneqCnstr} →
{val : ℤ} →
{x n : Lean.Grind.Linarith.Var} →
{c' : Lean.Meta.Grind.Arith.Linear.RingIneqCnstr} →
{val' : ℤ} →
{x' n' : Lean.Grind.Linarith.Var} →
Lean.Meta.Grind.Arith.Linear.RingIneqCnstrProof.can... | null | false |
Set.iUnion_setOf | Mathlib.Data.Set.Lattice | ∀ {α : Type u_1} {ι : Sort u_5} (P : ι → α → Prop), ⋃ i, {x | P i x} = {x | ∃ i, P i x} | null | true |
String.Slice.Pattern.ForwardSliceSearcher.startsWith | Init.Data.String.Pattern.String | String.Slice → String.Slice → Bool | null | true |
_private.Mathlib.Data.Analysis.Filter.0.Filter.Realizer.bind.match_14 | Mathlib.Data.Analysis.Filter | ∀ {α : Type u_1} {β : Type u_3} {m : α → Filter β} (x : Set β) (σ : Type u_2) (F : CFilter (Set α) σ)
(motive :
(∃ t ∈ { sets := {a | ∃ b, F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ },
∀ x_1 ∈ t, x ∈ m x_1) →
Prop)
(x_1 :
∃ t ∈ { sets := {a | ∃ b, F.f b ⊆ a}, univ_sets ... | null | false |
_private.Mathlib.LinearAlgebra.Span.Basic.0.LinearMap.submoduleOf_span_singleton_of_mem._simp_1_1 | Mathlib.LinearAlgebra.Span.Basic | ∀ {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂}
{x : M} {f : M →ₛₗ[σ₁₂] M₂} {p : Submodule R₂ M₂}, (x ∈ Submodule.comap f p) = (f x ∈ p) | null | false |
Affine.Simplex.Equilateral.angle_eq_pi_div_three | Mathlib.Geometry.Euclidean.Simplex | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] {n : ℕ} {s : Affine.Simplex ℝ P n},
s.Equilateral →
∀ {i₁ i₂ i₃ : Fin (n + 1)},
i₁ ≠ i₂ → i₁ ≠ i₃ → i₂ ≠ i₃ → EuclideanGeometry.angle (s.points i₁) (s.poin... | null | true |
summable_of_absolute_convergence_real | Mathlib.Analysis.Normed.Ring.InfiniteSum | ∀ {f : ℕ → ℝ}, (∃ r, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, |f i|) Filter.atTop (nhds r)) → Summable f | null | true |
_private.Lean.Elab.ConfigEval.DeriveEvalConfigItem.0.Lean.Elab.ConfigEval.HandlerTrie.exact? | Lean.Elab.ConfigEval.DeriveEvalConfigItem | Lean.Elab.ConfigEval.HandlerTrie✝ → Option Lean.Elab.ConfigEval.EvalConfigItemHandler | The `EvalConfigItemHandlerKind.exact` handler for this trie position's key. | true |
Nonneg.linearOrderedCommGroupWithZero._proof_2 | Mathlib.Algebra.Order.Nonneg.Field | ∀ {α : Type u_1} [inst : Field α] [inst_1 : LinearOrder α] [inst_2 : IsStrictOrderedRing α],
autoParam (∀ (a : { x // 0 ≤ x }), ⟨↑a ^ 0, ⋯⟩ = 1) DivInvMonoid.zpow_zero'._autoParam | null | false |
Mathlib.Tactic.ClickSuggestions.Context | Mathlib.Tactic.ClickSuggestions.Util | Type | The information required for pasting a suggestion into the editor. | true |
LinearMap.BilinForm.tmul.eq_1 | Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | ∀ {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} [inst : CommSemiring R] [inst_1 : CommSemiring A]
[inst_2 : AddCommMonoid M₁] [inst_3 : AddCommMonoid M₂] [inst_4 : Algebra R A] [inst_5 : Module R M₁]
[inst_6 : Module A M₁] [inst_7 : SMulCommClass R A M₁] [inst_8 : IsScalarTower R A M₁] [inst_9 : Modul... | null | true |
_private.Lean.Meta.Basic.0.Lean.Meta.DefEqCacheKey.mk.noConfusion | Lean.Meta.Basic | {P : Sort u} →
{lhs rhs : Lean.Expr} →
{configKey : UInt64} →
{lhs' rhs' : Lean.Expr} →
{configKey' : UInt64} →
{ lhs := lhs, rhs := rhs, configKey := configKey } = { lhs := lhs', rhs := rhs', configKey := configKey' } →
(lhs = lhs' → rhs = rhs' → configKey = configKey' → P) → ... | null | false |
List.Cursor.tail.congr_simp | Std.Do.Triple.SpecLemmas | ∀ {α : Type u_1} {l : List α} (s s_1 : l.Cursor) (e_s : s = s_1) (h : 0 < s.suffix.length), s.tail h = s_1.tail ⋯ | null | true |
CategoryTheory.Functor.isoCopyObj | Mathlib.CategoryTheory.NatIso | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(F : CategoryTheory.Functor C D) → (obj : C → D) → (e : (X : C) → F.obj X ≅ obj X) → F ≅ F.copyObj obj e | The functor constructed with `copyObj` is isomorphic to the given functor. | true |
_private.Lean.LibrarySuggestions.Basic.0.Lean.LibrarySuggestions.elabSetLibrarySuggestions._regBuiltin.Lean.LibrarySuggestions.elabSetLibrarySuggestions_1 | Lean.LibrarySuggestions.Basic | IO Unit | null | false |
Finset.mulETransformLeft_inv | Mathlib.Combinatorics.Additive.ETransform | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : CommGroup α] (e : α) (x : Finset α × Finset α),
Finset.mulETransformLeft e⁻¹ x = (Finset.mulETransformRight e x.swap).swap | null | true |
Fin.predAbove_le_predAbove | Mathlib.Order.Fin.Basic | ∀ {n : ℕ} {p q : Fin n}, p ≤ q → ∀ {i j : Fin (n + 1)}, i ≤ j → p.predAbove i ≤ q.predAbove j | null | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.toList_toArray._simp_1_2 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {a b : α}, (compare a b ≠ Ordering.eq) = ((a == b) = false) | null | false |
Int.divisorsAntidiag.eq_2 | Mathlib.NumberTheory.Divisors | ∀ (n : ℕ),
(Int.negSucc n).divisorsAntidiag =
(Finset.map (Nat.castEmbedding.prodMap (Nat.castEmbedding.trans (Equiv.toEmbedding (Equiv.neg ℤ))))
(n + 1).divisorsAntidiagonal).disjUnion
(Finset.map ((Nat.castEmbedding.trans (Equiv.toEmbedding (Equiv.neg ℤ))).prodMap Nat.castEmbedding)
(n +... | null | true |
CategoryTheory.LocalizerMorphism.liftingLocalizedFunctor._aux_1 | Mathlib.CategoryTheory.Localization.LocalizerMorphism | {C₁ : Type u_1} →
{C₂ : Type u_8} →
{D₁ : Type u_6} →
{D₂ : Type u_3} →
[inst : CategoryTheory.Category.{u_4, u_1} C₁] →
[inst_1 : CategoryTheory.Category.{u_7, u_8} C₂] →
[inst_2 : CategoryTheory.Category.{u_5, u_6} D₁] →
[inst_3 : CategoryTheory.Category.{u_2, u... | null | false |
OrthonormalBasis.fromOrthogonalSpanSingleton._proof_1 | Mathlib.Analysis.InnerProductSpace.PiL2 | ∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(n : ℕ) [Fact (Module.finrank 𝕜 E = n + 1)], FiniteDimensional 𝕜 E | null | false |
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