name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.ShortComplex.moduleCat_exact_iff | Mathlib.Algebra.Homology.ShortComplex.ModuleCat | ∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)),
S.Exact ↔
∀ (x₂ : ↑S.X₂),
(CategoryTheory.ConcreteCategory.hom S.g) x₂ = 0 → ∃ x₁, (CategoryTheory.ConcreteCategory.hom S.f) x₁ = x₂ | null | true |
CategoryTheory.IsCofiltered.nonempty | Mathlib.CategoryTheory.Filtered.Basic | ∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} [self : CategoryTheory.IsCofiltered C], Nonempty C | a cofiltered category must be non-empty | true |
_private.Lean.Server.FileWorker.SemanticHighlighting.0.Lean.Server.FileWorker.splitStr | Lean.Server.FileWorker.SemanticHighlighting | Lean.FileMap → Lean.Syntax → Array Lean.Syntax | Split the token at newline boundaries to support LSP clients such as VS Code that can't deal with
newline-spanning tokens.
| true |
Lean.Widget.inst._@.Lean.Widget.Basic.2038268869._hygCtx._hyg.3 | Lean.Widget.Basic | TypeName Lean.Elab.InfoWithCtx | null | false |
_private.Lean.Meta.Tactic.Grind.Split.0.Lean.Meta.Grind.SplitCandidate.noConfusionType | Lean.Meta.Tactic.Grind.Split | Sort u → Lean.Meta.Grind.SplitCandidate✝ → Lean.Meta.Grind.SplitCandidate✝ → Sort u | null | false |
Computation.think.eq_1 | Mathlib.Data.Seq.Computation | ∀ {α : Type u} (c : Computation α), c.think = ⟨Stream'.cons none ↑c, ⋯⟩ | null | true |
CategoryTheory.Retract.op_i | Mathlib.CategoryTheory.Retract | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (h : CategoryTheory.Retract X Y), h.op.i = h.r.op | null | true |
HeytingHom.mk | Mathlib.Order.Heyting.Hom | {α : Type u_6} →
{β : Type u_7} →
[inst : HeytingAlgebra α] →
[inst_1 : HeytingAlgebra β] →
(toLatticeHom : LatticeHom α β) →
toLatticeHom.toFun ⊥ = ⊥ →
(∀ (a b : α), toLatticeHom.toFun (a ⇨ b) = toLatticeHom.toFun a ⇨ toLatticeHom.toFun b) → HeytingHom α β | null | true |
_private.Std.Data.DHashMap.Internal.WF.0.Std.DHashMap.Internal.Raw₀.isHashSelf_filterMapₘ._simp_1_2 | Std.Data.DHashMap.Internal.WF | ∀ {α : Type u_1} {b : α} {α_1 : Type u_2} {x : Option α_1} {f : α_1 → α},
(Option.map f x = some b) = ∃ a, x = some a ∧ f a = b | null | false |
Std.DHashMap.isEmpty_insertMany_list | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} [EquivBEq α] [LawfulHashable α]
{l : List ((a : α) × β a)}, (m.insertMany l).isEmpty = (m.isEmpty && l.isEmpty) | null | true |
Matrix.eq_zero_of_vecMul_eq_zero | Mathlib.LinearAlgebra.Matrix.Nondegenerate | ∀ {m : Type u_1} {A : Type u_4} [inst : Fintype m] [inst_1 : CommRing A] [IsDomain A] [inst_3 : DecidableEq m]
{M : Matrix m m A}, M.det ≠ 0 → ∀ {v : m → A}, Matrix.vecMul v M = 0 → v = 0 | null | true |
CategoryTheory.Functor.WellOrderInductionData.Extension.mk.injEq | Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | ∀ {J : Type u} [inst : LinearOrder J] [inst_1 : SuccOrder J] {F : CategoryTheory.Functor Jᵒᵖ (Type v)}
{d : F.WellOrderInductionData} [inst_2 : OrderBot J] {val₀ : F.obj (Opposite.op ⊥)} {j : J}
(val : F.obj (Opposite.op j))
(map_zero : (CategoryTheory.ConcreteCategory.hom (F.map (CategoryTheory.homOfLE ⋯).op)) v... | null | true |
Std.TreeMap.Raw.Equiv.insertManyIfNewUnit_list | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} [Std.TransCmp cmp] {t₁ t₂ : Std.TreeMap.Raw α Unit cmp},
t₁.WF → t₂.WF → t₁.Equiv t₂ → ∀ (l : List α), (t₁.insertManyIfNewUnit l).Equiv (t₂.insertManyIfNewUnit l) | null | true |
CategoryTheory.Limits.spanExt_inv_app_left | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z X' Y' Z' : C} (iX : X ≅ X') (iY : Y ≅ Y') (iZ : Z ≅ Z')
{f : X ⟶ Y} {g : X ⟶ Z} {f' : X' ⟶ Y'} {g' : X' ⟶ Z'}
(wf : CategoryTheory.CategoryStruct.comp iX.hom f' = CategoryTheory.CategoryStruct.comp f iY.hom)
(wg : CategoryTheory.CategoryStruct.comp i... | null | true |
_private.Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs.0.IsLinearSet.closure._simp_1_4 | Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + (b + c) = a + b + c | null | false |
CategoryTheory.IsAccessibleCategory.rec | Mathlib.CategoryTheory.Presentable.LocallyPresentable | {C : Type u} →
[hC : CategoryTheory.Category.{v, u} C] →
{motive : CategoryTheory.IsAccessibleCategory.{w, v, u} C → Sort u_1} →
((exists_cardinal : ∃ κ, ∃ (x : Fact κ.IsRegular), CategoryTheory.IsCardinalAccessibleCategory C κ) → motive ⋯) →
(t : CategoryTheory.IsAccessibleCategory.{w, v, u} C) → m... | null | false |
Std.Net.SocketAddressV6.mk | Std.Net.Addr | Std.Net.IPv6Addr → UInt16 → Std.Net.SocketAddressV6 | null | true |
IntermediateField.isIntegral_iff | Mathlib.FieldTheory.IntermediateField.Algebraic | ∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {S : IntermediateField K L}
{x : ↥S}, IsIntegral K x ↔ IsIntegral K ↑x | null | true |
_private.Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit.0.CochainComplex.mappingConeHomOfDegreewiseSplitIso._proof_4 | Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | ∀ (p : ℤ), p + 1 + 1 + -1 = p + 1 | null | false |
Nat.gcd_sub_mul_right_right | Init.Data.Nat.Gcd | ∀ {m n k : ℕ}, k * m ≤ n → m.gcd (n - k * m) = m.gcd n | null | true |
SimpleGraph.pathGraph3ComplEmbeddingOf._proof_1 | Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | ∀ {α : Type u_1} {G : SimpleGraph α} (h : ¬G.IsCompleteMultipartite), ∃ w₁ w₂, G.IsPathGraph3Compl ⋯.choose w₁ w₂ | null | false |
_private.Std.Data.Iterators.Lemmas.Producers.Monadic.List.0.Std.Iterators.Types.ListIterator.instIterator.match_3.splitter | Std.Data.Iterators.Lemmas.Producers.Monadic.List | {m : Type u_1 → Type u_2} →
{α : Type u_1} →
(motive : Std.IterM m α → Sort u_3) →
(it : Std.IterM m α) →
(Unit → motive { internalState := { list := [] } }) →
((x : α) → (xs : List α) → motive { internalState := { list := x :: xs } }) → motive it | null | true |
AlgebraicGeometry.Scheme.IdealSheafData.support_antitone | Mathlib.AlgebraicGeometry.IdealSheaf.Basic | ∀ {X : AlgebraicGeometry.Scheme}, Antitone AlgebraicGeometry.Scheme.IdealSheafData.support | null | true |
Int64.toInt_sub | Init.Data.SInt.Lemmas | ∀ (a b : Int64), (a - b).toInt = (a.toInt - b.toInt).bmod (2 ^ 64) | null | true |
CategoryTheory.is_coprod_iff_isPushout | Mathlib.CategoryTheory.Adhesive.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X E Y YE : C} (c : CategoryTheory.Limits.BinaryCofan X E)
(hc : CategoryTheory.Limits.IsColimit c) {f : X ⟶ Y} {iY : Y ⟶ YE} {fE : c.pt ⟶ YE},
CategoryTheory.CommSq f c.inl iY fE →
(Nonempty
(CategoryTheory.Limits.IsColimit
(CategoryThe... | null | true |
Algebra.TensorProduct.algHomOfLinearMapTensorProduct._proof_1 | Mathlib.RingTheory.TensorProduct.Maps | ∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Algebra R S]
[inst_3 : Semiring A] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [IsScalarTower R S A], SMulCommClass R S A | null | false |
_private.Mathlib.LinearAlgebra.Goursat.0.Submodule.goursat._simp_1_6 | Mathlib.LinearAlgebra.Goursat | ∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [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] {τ₁₂ : R →+* R₂}
[inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {x : M₂}, (x ∈ f.range) = ∃ y, f y = x | null | false |
CategoryTheory.CatCenter.localizationRingHom | Mathlib.CategoryTheory.Center.Localization | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(L : CategoryTheory.Functor C D) →
(W : CategoryTheory.MorphismProperty C) →
[L.IsLocalization W] →
[inst_3 : CategoryTheory.Preadditive C... | The morphism of rings `CatCenter C →+* CatCenter D` when `L : C ⥤ D`
is an additive localization functor between preadditive categories. | true |
_private.Std.Time.Date.ValidDate.0.Std.Time.ValidDate.ofOrdinal.go._unary._proof_3 | Std.Time.Date.ValidDate | ∀ {leap : Bool} (ordinal : Std.Time.Day.Ordinal.OfYear leap) (idx : Std.Time.Month.Ordinal) (acc : ℤ),
acc + ↑(Std.Time.Month.Ordinal.days leap idx) - acc = ↑(Std.Time.Month.Ordinal.days leap idx) | null | false |
Matroid.contract_inter_ground_eq | Mathlib.Combinatorics.Matroid.Minor.Contract | ∀ {α : Type u_1} (M : Matroid α) (C : Set α), M.contract (C ∩ M.E) = M.contract C | null | true |
Commute.isNilpotent_mul_left_iff | Mathlib.RingTheory.Nilpotent.Basic | ∀ {R : Type u_1} {x y : R} [inst : Semiring R],
Commute x y → x ∈ nonZeroDivisorsLeft R → (IsNilpotent (x * y) ↔ IsNilpotent y) | null | true |
Std.DHashMap.Internal.Raw₀.insertIfNew_equiv_congr | Std.Data.DHashMap.Internal.RawLemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] (m₁ m₂ : Std.DHashMap.Internal.Raw₀ α β)
[EquivBEq α] [LawfulHashable α],
(↑m₁).WF → (↑m₂).WF → (↑m₁).Equiv ↑m₂ → ∀ {k : α} {v : β k}, (↑(m₁.insertIfNew k v)).Equiv ↑(m₂.insertIfNew k v) | null | true |
_private.Mathlib.Data.Set.Image.0.Set.preimage_eq_empty_iff._simp_1_1 | Mathlib.Data.Set.Image | ∀ {α : Type u} {s : Set α}, (s = ∅) = ∀ (x : α), x ∉ s | null | false |
CategoryTheory.AddMon.Hom.recOn | Mathlib.CategoryTheory.Monoidal.Mon | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{M N : CategoryTheory.AddMon C} →
{motive : M.Hom N → Sort u} →
(t : M.Hom N) →
((hom : M.X ⟶ N.X) →
[isAddMonHom_hom : CategoryTheory.IsAddMonHom hom] →... | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Equiv.inter_left._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 |
AddCircle.gcd_mul_addOrderOf_div_eq | Mathlib.Topology.Instances.AddCircle.Defs | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] (p : 𝕜) [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [hp : Fact (0 < p)] {n : ℕ}
(m : ℕ), 0 < n → m.gcd n * addOrderOf ↑(↑m / ↑n * p) = n | null | true |
IsCompact.locallyCompactSpace_of_mem_nhds_of_group | Mathlib.Topology.Algebra.Group.Pointwise | ∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : Group G] [IsTopologicalGroup G] {K : Set G},
IsCompact K → ∀ {x : G}, K ∈ nhds x → LocallyCompactSpace G | If a point in a topological group has a compact neighborhood, then the group is
locally compact. | true |
ringChar.of_eq | Mathlib.Algebra.CharP.Defs | ∀ {R : Type u_1} [inst : NonAssocSemiring R] {p : ℕ}, ringChar R = p → CharP R p | null | true |
Std.Internal.Parsec.ParseResult.success.noConfusion | Std.Internal.Parsec.Basic | {α ι : Type} →
{P : Sort u} →
{pos : ι} →
{res : α} →
{pos' : ι} →
{res' : α} →
Std.Internal.Parsec.ParseResult.success pos res = Std.Internal.Parsec.ParseResult.success pos' res' →
(pos ≍ pos' → res ≍ res' → P) → P | null | false |
WittVector.poly_eq_of_wittPolynomial_bind_eq | Mathlib.RingTheory.WittVector.IsPoly | ∀ (p : ℕ) [Fact (Nat.Prime p)] (f g : ℕ → MvPolynomial ℕ ℤ),
(∀ (n : ℕ), (MvPolynomial.bind₁ f) (wittPolynomial p ℤ n) = (MvPolynomial.bind₁ g) (wittPolynomial p ℤ n)) → f = g | null | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter.0.SimplexCategory.σ_σ₀Iter'._proof_1_11 | Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | ∀ (i : ℕ) {n : ℕ}, i = 0 → n + 1 + i = n + 1 | null | false |
TwoSidedIdeal.orderIsoIsTwoSided_symm_apply | Mathlib.RingTheory.TwoSidedIdeal.Operations | ∀ {R : Type u_1} [inst : Ring R] (I : { I // I.IsTwoSided }),
(RelIso.symm TwoSidedIdeal.orderIsoIsTwoSided) I =
have this := ⋯;
(↑I).toTwoSided | null | true |
Finset.mem_addAntidiagonal._simp_1 | Mathlib.Data.Finset.MulAntidiagonal | ∀ {α : Type u_1} [inst : AddCommMonoid α] [inst_1 : PartialOrder α] [inst_2 : IsOrderedCancelAddMonoid α] {s t : Set α}
{hs : s.IsPWO} {ht : t.IsPWO} {a : α} {x : α × α},
(x ∈ Finset.addAntidiagonal hs ht a) = (x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 + x.2 = a) | null | false |
IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq | Mathlib.Topology.Algebra.Field | ∀ {α : Type u_2} {𝕜 : Type u_3} {f : α → 𝕜} {S : Set α} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace 𝕜]
[T1Space 𝕜] [inst_3 : Ring 𝕜] [NoZeroDivisors 𝕜],
IsPreconnected S → ContinuousOn f S → Set.EqOn (f ^ 2) 1 S → Set.EqOn f 1 S ∨ Set.EqOn f (-1) S | If `f` is a function `α → 𝕜` which is continuous on a preconnected set `S`, and
`f ^ 2 = 1` on `S`, then either `f = 1` on `S`, or `f = -1` on `S`. | true |
Std.DHashMap.Raw.toList_insert_perm | Std.Data.DHashMap.RawLemmas | ∀ {α : Type u} {β : α → Type v} {m : Std.DHashMap.Raw α β} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α]
[LawfulHashable α],
m.WF →
∀ {k : α} {v : β k},
(m.insert k v).toList.Perm (⟨k, v⟩ :: List.filter (fun x => decide ¬(k == x.fst) = true) m.toList) | null | true |
Fin.preimage_natAdd_uIoc_natAdd | Mathlib.Order.Interval.Set.Fin | ∀ {n : ℕ} (m : ℕ) (i j : Fin n), Fin.natAdd m ⁻¹' Set.uIoc (Fin.natAdd m i) (Fin.natAdd m j) = Set.uIoc i j | null | true |
CategoryTheory.Functor.CoconeTypes.IsColimit.equiv.congr_simp | Mathlib.CategoryTheory.Limits.Types.ColimitType | ∀ {J : Type u} [inst : CategoryTheory.Category.{v, u} J] {F : CategoryTheory.Functor J (Type w₀)} {c : F.CoconeTypes}
(hc : c.IsColimit), hc.equiv = hc.equiv | null | true |
Lean.Parser.TokenCacheEntry.startPos._default | Lean.Parser.Types | String.Pos.Raw | null | false |
_private.Mathlib.Util.AtomM.Recurse.0.Mathlib.Tactic.AtomM.Recurse.instBEqConfig.beq.match_1 | Mathlib.Util.AtomM.Recurse | (motive : Mathlib.Tactic.AtomM.Recurse.Config → Mathlib.Tactic.AtomM.Recurse.Config → Sort u_1) →
(x x_1 : Mathlib.Tactic.AtomM.Recurse.Config) →
((a : Lean.Meta.TransparencyMode) →
(a_1 a_2 : Bool) →
(b : Lean.Meta.TransparencyMode) →
(b_1 b_2 : Bool) →
motive { red :=... | null | false |
ContMDiffWithinAt.clm_postcomp | Mathlib.Geometry.Manifold.ContMDiff.NormedSpace | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F₁ : Type u_8} [inst_6 : NormedAddComm... | null | true |
FirstOrder.Language.Substructure.closure_induction | Mathlib.ModelTheory.Substructures | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {s : Set M} {p : M → Prop} {x : M},
x ∈ (FirstOrder.Language.Substructure.closure L).toFun s →
(∀ x ∈ s, p x) → (∀ {n : ℕ} (f : L.Functions n), FirstOrder.Language.ClosedUnder f (setOf p)) → p x | An induction principle for closure membership. If `p` holds for all elements of `s`, and
is preserved under function symbols, then `p` holds for all elements of the closure of `s`. | true |
Equiv.compl | Mathlib.Order.OrderDual | {α : Type u_1} → {β : Type u_2} → α ≃ β → [Compl β] → Compl α | Transfer `Compl` across an `Equiv`. | true |
Lean.Meta.Tactic.Cbv.CbvSimprocs.mk._flat_ctor | Lean.Meta.Tactic.Cbv.CbvSimproc | Lean.Meta.DiscrTree Lean.Meta.Tactic.Cbv.CbvSimprocEntry →
Lean.Meta.DiscrTree Lean.Meta.Tactic.Cbv.CbvSimprocEntry →
Lean.Meta.DiscrTree Lean.Meta.Tactic.Cbv.CbvSimprocEntry →
Lean.PHashSet Lean.Name → Lean.PHashSet Lean.Name → Lean.Meta.Tactic.Cbv.CbvSimprocs | null | false |
String.utf8ByteSize_sliceFrom | Init.Data.String.Basic | ∀ {s : String} {p : s.Pos}, (s.sliceFrom p).utf8ByteSize = s.utf8ByteSize - p.offset.byteIdx | null | true |
CategoryTheory.CartesianClosed.uncurry | Mathlib.CategoryTheory.Monoidal.Closed.Cartesian | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{A X Y : C} →
[inst_2 : CategoryTheory.Closed A] → (Y ⟶ A ⟹ X) → (CategoryTheory.MonoidalCategoryStruct.tensorObj A Y ⟶ X) | **Alias** of `CategoryTheory.MonoidalClosed.uncurry`. | true |
Mathlib.Tactic.Conv.Path.ctorElimType | Mathlib.Tactic.Widget.Conv | {motive : Mathlib.Tactic.Conv.Path → Sort u} → ℕ → Sort (max 1 u) | null | false |
OrderMonoidHom.fst._proof_1 | Mathlib.Algebra.Order.Monoid.Lex | ∀ (α : Type u_1) (β : Type u_2) [inst : Monoid α] [inst_1 : PartialOrder α] [inst_2 : Monoid β] [inst_3 : Preorder β],
Monotone ⇑(MonoidHom.fst α β) | null | false |
_private.Std.Data.TreeSet.Lemmas.0.Std.TreeSet.size_toArray._simp_1_1 | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp}, t.toArray = t.toList.toArray | null | false |
Set.preimage_const | Mathlib.Data.Set.Image | ∀ {α : Type u_1} {β : Type u_2} (b : β) (s : Set β) [inst : Decidable (b ∈ s)],
(fun x => b) ⁻¹' s = if b ∈ s then Set.univ else ∅ | null | true |
Multiset.union_le_iff | Mathlib.Data.Multiset.UnionInter | ∀ {α : Type u_1} [inst : DecidableEq α] {s t u : Multiset α}, s ∪ t ≤ u ↔ s ≤ u ∧ t ≤ u | null | true |
genericPoint_specializes | Mathlib.Topology.Sober | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : QuasiSober α] [inst_2 : IrreducibleSpace α] (x : α),
genericPoint α ⤳ x | null | true |
OrderMonoidWithZeroHom._sizeOf_inst | Mathlib.Algebra.Order.Hom.MonoidWithZero | (α : Type u_6) →
(β : Type u_7) →
{inst : Preorder α} →
{inst_1 : Preorder β} →
{inst_2 : MulZeroOneClass α} → {inst_3 : MulZeroOneClass β} → [SizeOf α] → [SizeOf β] → SizeOf (α →*₀o β) | null | false |
Mathlib.Meta.NormNum.Result.toSimpResult.match_1 | Mathlib.Tactic.NormNum.Result | {u : Lean.Level} →
{α : Q(Type u)} →
{e : Q(«$α»)} →
(motive : (e' : Q(«$α»)) × Q(«$e» = «$e'») → Sort u_1) →
(x : (e' : Q(«$α»)) × Q(«$e» = «$e'»)) →
((expr : Q(«$α»)) → (proof? : Q(«$e» = «$expr»)) → motive ⟨expr, proof?⟩) → motive x | null | false |
PositiveLinearMap.gnsStarAlgHom._proof_13 | Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | ∀ {A : Type u_1} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [inst_2 : StarOrderedRing A] (f : A →ₚ[ℂ] ℂ),
IsScalarTower ℂ ℂ (UniformSpace.Completion f.PreGNS) | null | false |
Batteries.BinomialHeap.Imp.Heap.headD._f | Batteries.Data.BinomialHeap.Basic | {α : Type u_1} →
(α → α → Bool) →
(x : Batteries.BinomialHeap.Imp.Heap α) → Batteries.BinomialHeap.Imp.Heap.below (motive := fun x => α → α) x → α → α | null | false |
SSet.N.mk_surjective | Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | ∀ {X : SSet} (x : X.N), ∃ n y, x = SSet.N.mk ↑y ⋯ | null | true |
_private.Std.Data.DTreeMap.Internal.Model.0.Std.DTreeMap.Internal.Cell.Const.get?.match_1.splitter | Std.Data.DTreeMap.Internal.Model | {α : Type u_2} →
{β : Type u_1} →
(motive : Option ((_ : α) × β) → Sort u_3) →
(x : Option ((_ : α) × β)) → (Unit → motive none) → ((p : (_ : α) × β) → motive (some p)) → motive x | null | true |
_private.Lean.Compiler.ExternAttr.0.Lean.expandExternPatternAux._unary.eq_def | Lean.Compiler.ExternAttr | ∀ (args : List String) (pattern : String) (_x : (_ : pattern.Pos) ×' String),
Lean.expandExternPatternAux._unary args pattern _x =
PSigma.casesOn _x fun it r =>
if h : it.IsAtEnd then r
else
have c := it.get h;
if c ≠ '#' then Lean.expandExternPatternAux._unary args pattern ⟨it.next h,... | null | false |
Algebra.Extension.algebraBaseChange._proof_3 | Mathlib.RingTheory.Extension.Basic | ∀ {R : Type u_1} [inst : CommRing R] (T : Type u_2) [inst_1 : CommRing T] [inst_2 : Algebra R T], SMulCommClass R R T | null | false |
MeasureTheory.ae_mem_iff_measure_eq | Mathlib.MeasureTheory.Measure.Typeclasses.Finite | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {s : Set α},
MeasureTheory.NullMeasurableSet s μ → ((∀ᵐ (a : α) ∂μ, a ∈ s) ↔ μ s = μ Set.univ) | null | true |
Filter.HasBasis.inf_neBot_iff | Mathlib.Order.Filter.Bases.Basic | ∀ {α : Type u_1} {ι : Sort u_4} {l l' : Filter α} {p : ι → Prop} {s : ι → Set α},
l.HasBasis p s → ((l ⊓ l').NeBot ↔ ∀ ⦃i : ι⦄, p i → ∀ ⦃s' : Set α⦄, s' ∈ l' → (s i ∩ s').Nonempty) | null | true |
BialgEquiv.ofAlgEquiv._proof_7 | Mathlib.RingTheory.Bialgebra.Equiv | ∀ {R : Type u_3} {A : Type u_1} {B : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
[inst_3 : Bialgebra R A] [inst_4 : Bialgebra R B] (f : A ≃ₐ[R] B), Function.LeftInverse f.invFun f.toFun | null | false |
_private.Mathlib.NumberTheory.LSeries.Nonvanishing.0.DirichletCharacter.zetaMul_prime_pow_nonneg._simp_1_2 | Mathlib.NumberTheory.LSeries.Nonvanishing | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {a : M₀} {n : ℕ} [IsReduced M₀] [Nontrivial M₀],
(a ^ n = 0) = (a = 0 ∧ n ≠ 0) | null | false |
Filter.mem_sup | Mathlib.Order.Filter.Basic | ∀ {α : Type u} {f g : Filter α} {s : Set α}, s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g | null | true |
CategoryTheory.Functor.PreservesRightHomologyOf.mk._flat_ctor | Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | ∀ {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.Limits.HasZeroMorphisms C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] {F : CategoryTheory.Functor C D}
[inst_4 : F.PreservesZeroMorphisms] {S : CategoryTh... | null | false |
_private.Aesop.Forward.State.0.Aesop.VariableMap.modifyM.match_3 | Aesop.Forward.State | (motive : Option Aesop.InstMap → Sort u_1) →
(x : Option Aesop.InstMap) → (Unit → motive none) → ((m : Aesop.InstMap) → motive (some m)) → motive x | null | false |
Matroid.mapSetEmbedding_indep_iff' | Mathlib.Combinatorics.Matroid.Map | ∀ {α : Type u_1} {β : Type u_2} {M : Matroid α} {f : ↑M.E ↪ β} {I : Set β},
(M.mapSetEmbedding f).Indep I ↔ ∃ I₀, M.Indep (Subtype.val '' I₀) ∧ I = ⇑f '' I₀ | null | true |
CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso._proof_1 | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C]
[inst_4 : CategoryTheory.Limits.HasKernels Cᵒᵖ] [inst_5 : CategoryTheory.Limits.HasCokernels Cᵒᵖ]
{X Y : ... | null | false |
AffineMap.instFunLike | Mathlib.LinearAlgebra.AffineSpace.AffineMap | (k : Type u_1) →
{V1 : Type u_2} →
(P1 : Type u_3) →
{V2 : Type u_4} →
(P2 : Type u_5) →
[inst : Ring k] →
[inst_1 : AddCommGroup V1] →
[inst_2 : Module k V1] →
[inst_3 : AddTorsor V1 P1] →
[inst_4 : AddCommGroup V2] →
... | null | true |
Topology.IsEmbedding.comapUniformSpace | Mathlib.Topology.UniformSpace.UniformEmbedding | {α : Type u_1} →
{β : Type u_2} →
[inst : TopologicalSpace α] → [u : UniformSpace β] → (f : α → β) → Topology.IsEmbedding f → UniformSpace α | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. | true |
_private.Mathlib.NumberTheory.DirichletCharacter.Orthogonality.0.DirichletCharacter.sum_char_inv_mul_char_eq._simp_1_1 | Mathlib.NumberTheory.DirichletCharacter.Orthogonality | ∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]
[MulHomClass F M N] (f : F) (x y : M), f x * f y = f (x * y) | null | false |
Nat.add_mod_add_ite | Mathlib.Data.Nat.ModEq | ∀ (a b c : ℕ), ((a + b) % c + if c ≤ a % c + b % c then c else 0) = a % c + b % c | null | true |
continuousAt_nsmul | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : AddMonoid M] [ContinuousAdd M] (x : M) (n : ℕ),
ContinuousAt (fun x => n • x) x | null | true |
List.idxOf_cons_ne | Mathlib.Data.List.Basic | ∀ {α : Type u} [inst : BEq α] [LawfulBEq α] {a b : α} (l : List α),
b ≠ a → List.idxOf a (b :: l) = (List.idxOf a l).succ | null | true |
_private.Mathlib.RingTheory.Nullstellensatz.0.MvPolynomial.eq_vanishingIdeal_singleton_of_isMaximal._simp_1_1 | Mathlib.RingTheory.Nullstellensatz | ∀ {α : Type u} [inst : Semiring α] {I J : Ideal α}, (I = J) = ∀ (x : α), x ∈ I ↔ x ∈ J | null | false |
Algebra.idealMap._proof_1 | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : Semiring S] [inst_2 : Algebra R S] (I : Ideal R),
∀ x ∈ I, (algebraMap R S) x ∈ Ideal.map (algebraMap R S) I | null | false |
WeierstrassCurve.variableChange_a₂ | Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange | ∀ {R : Type u} [inst : CommRing R] (W : WeierstrassCurve R) (C : WeierstrassCurve.VariableChange R),
(C • W).a₂ = ↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2) | null | true |
_private.Mathlib.RingTheory.WittVector.TeichmullerSeries.0.WittVector._aux_Mathlib_RingTheory_WittVector_TeichmullerSeries___unexpand_WittVector_1 | Mathlib.RingTheory.WittVector.TeichmullerSeries | Lean.PrettyPrinter.Unexpander | null | false |
Finset.toRight_union | Mathlib.Data.Finset.Sum | ∀ {α : Type u_1} {β : Type u_2} {u v : Finset (α ⊕ β)} [inst : DecidableEq α] [inst_1 : DecidableEq β],
(u ∪ v).toRight = u.toRight ∪ v.toRight | null | true |
CategoryTheory.Span.id.congr_simp | Mathlib.CategoryTheory.Bicategory.Span.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {Wₗ Wᵣ : CategoryTheory.MorphismProperty C}
[inst_1 : Wₗ.ContainsIdentities] [inst_2 : Wᵣ.ContainsIdentities] (c : C),
CategoryTheory.Span.id c = CategoryTheory.Span.id c | null | true |
Lean.JsonRpc.MessageMetaData.response.elim | Lean.Data.JsonRpc | {motive : Lean.JsonRpc.MessageMetaData → Sort u} →
(t : Lean.JsonRpc.MessageMetaData) →
t.ctorIdx = 2 → ((id : Lean.JsonRpc.RequestID) → motive (Lean.JsonRpc.MessageMetaData.response id)) → motive t | null | false |
CategoryTheory.Limits.imageSubobject_arrow_comp | Mathlib.CategoryTheory.Subobject.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : CategoryTheory.Limits.HasImage f],
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.factorThruImageSubobject f)
(CategoryTheory.Limits.imageSubobject f).arrow =
f | null | true |
HasSum.mul_of_nonarchimedean | Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | ∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : UniformSpace R] [IsUniformAddGroup R]
[NonarchimedeanRing R] {f : α → R} {g : β → R} {a b : R},
HasSum f a → HasSum g b → HasSum (fun i => f i.1 * g i.2) (a * b) | Let `R` be a nonarchimedean ring, let `f : α → R` be a function that sums to `a : R`,
and let `g : β → R` be a function that sums to `b : R`. Then `fun i : α × β ↦ f i.1 * g i.2`
sums to `a * b`. | true |
RootPairing.EmbeddedG2.longRoot | Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {ι : Type u_1} →
{R : Type u_2} →
{M : Type u_3} →
{N : Type u_4} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
[inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (P : RootPairing ι R M N) → [P.EmbeddedG2] → M | The long root `β`. | true |
Ordinal.exists_lsub_cof | Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | ∀ (o : Ordinal.{u}), ∃ ι f, Ordinal.lsub f = o ∧ Cardinal.mk ι = o.cof | null | true |
Asymptotics.isEquivalent_of_tendsto_one | Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | ∀ {α : Type u_1} {β : Type u_2} [inst : NormedField β] {u v : α → β} {l : Filter α},
Filter.Tendsto (u / v) l (nhds 1) → Asymptotics.IsEquivalent l u v | null | true |
Int.ediv_of_neg_of_pos | Mathlib.Data.Int.Init | ∀ {a b : ℤ}, a < 0 → 0 < b → a.ediv b = -((-a - 1) / b + 1) | null | true |
TopologicalSpace.Clopens.coe_inf | Mathlib.Topology.Sets.Closeds | ∀ {α : Type u_2} [inst : TopologicalSpace α] (s t : TopologicalSpace.Clopens α), ↑(s ⊓ t) = ↑s ∩ ↑t | null | true |
Aesop.Queue.mk._flat_ctor | Aesop.Search.Queue.Class | {Q : Type} → BaseIO Q → (Q → Array Aesop.GoalRef → BaseIO Q) → (Q → BaseIO (Option Aesop.GoalRef × Q)) → Aesop.Queue Q | null | false |
Complex.norm_natCast_cpow_of_pos | Mathlib.Analysis.SpecialFunctions.Pow.Real | ∀ {n : ℕ}, 0 < n → ∀ (s : ℂ), ‖↑n ^ s‖ = ↑n ^ s.re | null | true |
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