name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ComplexShape.noConfusionType | Mathlib.Algebra.Homology.ComplexShape | Sort u → {ι : Type u_1} → ComplexShape ι → {ι' : Type u_1} → ComplexShape ι' → Sort u | null | false |
Matrix.permanent.congr_simp | Mathlib.LinearAlgebra.Matrix.Permanent | ∀ {n : Type u_1} {inst : DecidableEq n} [inst_1 : DecidableEq n] [inst_2 : Fintype n] {R : Type u_2}
[inst_3 : CommSemiring R] (M M_1 : Matrix n n R), M = M_1 → M.permanent = M_1.permanent | null | true |
GeneralizedHeytingAlgebra | Mathlib.Order.Heyting.Basic | Type u_4 → Type u_4 | A generalized Heyting algebra is a lattice with an additional binary operation `⇨` called
Heyting implication such that `(a ⇨ ·)` is right adjoint to `(a ⊓ ·)`.
This generalizes `HeytingAlgebra` by not requiring a bottom element. | true |
Submodule.pow_induction_on_left | Mathlib.Algebra.Algebra.Operations | ∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] (M : Submodule R A)
{C : A → Prop},
(∀ (r : R), C ((algebraMap R A) r)) →
(∀ (x y : A), C x → C y → C (x + y)) → (∀ m ∈ M, ∀ (x : A), C x → C (m * x)) → ∀ {x : A} {n : ℕ}, x ∈ M ^ n → C x | To show a property on elements of `M ^ n` holds, it suffices to show that it holds for scalars,
is closed under addition, and holds for `m * x` where `m ∈ M` and it holds for `x` | true |
Set.Finite.of_inv | Mathlib.Algebra.Group.Pointwise.Set.Finite | ∀ {α : Type u_1} [inst : InvolutiveInv α] {s : Set α}, s⁻¹.Finite → s.Finite | **Alias** of the forward direction of `Set.finite_inv`. | true |
ContinuousAffineEquiv.preimage_image | Mathlib.Topology.Algebra.ContinuousAffineEquiv | ∀ {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 : Module k V₁] [inst_3 : AddTorsor V₁ P₁] [inst_4 : TopologicalSpace P₁]
[inst_5 : AddCommGroup V₂] [inst_6 : Module k V₂] [inst_7 : AddTorsor V₂ P₂] [inst_8 : TopologicalSpace P₂]
(... | null | true |
SimpleGraph.Path.loop_eq | Mathlib.Combinatorics.SimpleGraph.Paths | ∀ {V : Type u} {G : SimpleGraph V} {v : V} (p : G.Path v v), p = SimpleGraph.Path.nil | null | true |
_private.Mathlib.Algebra.Order.Module.HahnEmbedding.0.HahnEmbedding.Partial.eval_ne._simp_1_1 | Mathlib.Algebra.Order.Module.HahnEmbedding | ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - (b + c) = a - b - c | null | false |
instModuleQuotientAddSubgroupTorsion | Mathlib.GroupTheory.Torsion | {R : Type u_3} →
{M : Type u_4} → [inst : Ring R] → [inst_1 : AddCommGroup M] → [Module R M] → Module R (M ⧸ AddCommGroup.torsion M) | null | true |
List.findRev?._unsafe_rec | Init.Data.List.Basic | {α : Type u} → (α → Bool) → List α → Option α | null | false |
LinearIsometryEquiv.withLpProdAssoc._proof_4 | Mathlib.Analysis.Normed.Lp.ProdLp | ∀ (p : ENNReal) (α : Type u_1) (β : Type u_2) (γ : Type u_3) [hp : Fact (1 ≤ p)] [inst : SeminormedAddCommGroup α]
[inst_1 : SeminormedAddCommGroup β] [inst_2 : SeminormedAddCommGroup γ],
Function.RightInverse (IsometryEquiv.withLpProdAssoc p α β γ).invFun (IsometryEquiv.withLpProdAssoc p α β γ).toFun | null | false |
BoxIntegral.Box.continuousOn_face_Icc | Mathlib.Analysis.BoxIntegral.Box.Basic | ∀ {X : Type u_2} [inst : TopologicalSpace X] {n : ℕ} {f : (Fin (n + 1) → ℝ) → X} {I : BoxIntegral.Box (Fin (n + 1))},
ContinuousOn f (BoxIntegral.Box.Icc I) →
∀ {i : Fin (n + 1)} {x : ℝ},
x ∈ Set.Icc (I.lower i) (I.upper i) → ContinuousOn (f ∘ i.insertNth x) (BoxIntegral.Box.Icc (I.face i)) | null | true |
ContinuousMap.instInvolutiveStarOfContinuousStar | Mathlib.Topology.ContinuousMap.Star | {α : Type u_2} →
{β : Type u_3} →
[inst : TopologicalSpace α] →
[inst_1 : TopologicalSpace β] → [inst_2 : InvolutiveStar β] → [ContinuousStar β] → InvolutiveStar C(α, β) | null | true |
BarrelledSpace.mk | Mathlib.Analysis.LocallyConvex.Barrelled | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : SeminormedRing 𝕜] [inst_1 : AddGroup E] [inst_2 : SMul 𝕜 E]
[inst_3 : TopologicalSpace E], (∀ (p : Seminorm 𝕜 E), LowerSemicontinuous ⇑p → Continuous ⇑p) → BarrelledSpace 𝕜 E | null | true |
CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction._proof_11 | Mathlib.CategoryTheory.Monoidal.Action.Opposites | ∀ (C : Type u_1) (D : Type u_2) [inst : CategoryTheory.Category.{u_4, u_1} C]
[inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Category.{u_3, u_2} D]
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalRightAction C D] {c c' c'' : Cᵒᵖ} {d d' d'' : Dᵒᵖ} (f₁ : d ⟶ d')
(f₂ : d' ⟶ d'') (g₁ : c ⟶ c'... | null | false |
SimpleGraph.starGraph_adj_center_iff | Mathlib.Combinatorics.SimpleGraph.Star | ∀ {V : Type u_1} {r v : V}, (SimpleGraph.starGraph r).Adj r v ↔ r ≠ v | On (starGraph r), r is adjacent to v iff v ≠ r. | true |
_private.Mathlib.Combinatorics.SimpleGraph.Hamiltonian.0.SimpleGraph.isHamiltonian_iff_cycleGraph_isContained._proof_1_2 | Mathlib.Combinatorics.SimpleGraph.Hamiltonian | ∀ {α : Type u_1} [inst : Fintype α], 3 ≤ Fintype.card α → ¬Fintype.card α = 1 | null | false |
AlgHom.IsArithFrobAt.restrict._proof_1 | Mathlib.RingTheory.Frobenius | ∀ {R : Type u_2} {S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {φ : S →ₐ[R] S}
{Q : Ideal S} (H : φ.IsArithFrobAt Q) (x : R ⧸ Ideal.under R Q),
(↑↑(Ideal.quotientMap Q ↑φ ⋯)).toFun ((algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)) x) =
(algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)) x | null | false |
_private.Std.Data.DTreeMap.Internal.WF.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.alter!.match_1.eq_1 | Std.Data.DTreeMap.Internal.WF.Lemmas | ∀ {α : Type u_1} {β : Type u_2} (motive : (Std.DTreeMap.Internal.Impl α fun x => β) → Sort u_3)
(h_1 : Unit → motive Std.DTreeMap.Internal.Impl.leaf)
(h_2 :
(sz : ℕ) →
(k' : α) →
(v' : β) →
(l' r' : Std.DTreeMap.Internal.Impl α fun x => β) → motive (Std.DTreeMap.Internal.Impl.inner sz k'... | null | true |
Aesop.traverseUp | Aesop.Tree.Traversal | {m : Type → Type} →
[Monad m] →
[MonadLiftT (ST IO.RealWorld) m] →
(Aesop.GoalRef → m Bool) →
(Aesop.GoalRef → m Unit) →
(Aesop.RappRef → m Bool) →
(Aesop.RappRef → m Unit) →
(Aesop.MVarClusterRef → m Bool) → (Aesop.MVarClusterRef → m Unit) → Aesop.TreeRef → m Uni... | null | true |
AddZeroClass.toAddZero | Mathlib.Algebra.Group.Monoid | {M : Type u_2} → [self : AddZeroClass M] → AddZero M | null | true |
Subring.map_inf | Mathlib.Algebra.Ring.Subring.Basic | ∀ {R : Type u} {S : Type v} [inst : NonAssocRing R] [inst_1 : NonAssocRing S] (s t : Subring R) (f : R →+* S),
Function.Injective ⇑f → Subring.map f (s ⊓ t) = Subring.map f s ⊓ Subring.map f t | null | true |
Mathlib.Tactic.Translate.Reorder.mk.injEq | Mathlib.Tactic.Translate.Reorder | ∀ (univReorder : Mathlib.Tactic.Translate.Permutation) (reorder : Mathlib.Tactic.Translate.ArgReorder)
(univReorder_1 : Mathlib.Tactic.Translate.Permutation) (reorder_1 : Mathlib.Tactic.Translate.ArgReorder),
({ univReorder := univReorder, reorder := reorder } = { univReorder := univReorder_1, reorder := reorder_1 ... | null | true |
Complex.instModule | Mathlib.LinearAlgebra.Complex.Module | {R : Type u_1} → [inst : Semiring R] → [Module R ℝ] → Module R ℂ | null | true |
forall₃_congr | Init.PropLemmas | ∀ {α : Sort u_1} {β : α → Sort u_2} {γ : (a : α) → β a → Sort u_3} {p q : (a : α) → (b : β a) → γ a b → Prop},
(∀ (a : α) (b : β a) (c : γ a b), p a b c ↔ q a b c) →
((∀ (a : α) (b : β a) (c : γ a b), p a b c) ↔ ∀ (a : α) (b : β a) (c : γ a b), q a b c) | null | true |
ContinuousAt.finInsertNth | Mathlib.Topology.Constructions | ∀ {X : Type u} [inst : TopologicalSpace X] {n : ℕ} {A : Fin (n + 1) → Type u_6}
[inst_1 : (i : Fin (n + 1)) → TopologicalSpace (A i)] (i : Fin (n + 1)) {f : X → A i}
{g : X → (j : Fin n) → A (i.succAbove j)} {x : X},
ContinuousAt f x → ContinuousAt g x → ContinuousAt (fun a => i.insertNth (f a) (g a)) x | null | true |
_private.Mathlib.Combinatorics.SetFamily.AhlswedeZhang.0.AhlswedeZhang.infSum_compls_add_supSum._simp_1_3 | Mathlib.Combinatorics.SetFamily.AhlswedeZhang | ∀ {K : Type u_1} [inst : DivisionSemiring K] (a b c : K), a / c + b / c = (a + b) / c | null | false |
IsLocalization'.recOn | Mathlib.RingTheory.Localization.Defs | {R : Type u_1} →
[inst : CommSemiring R] →
{M : Submonoid R} →
{S : Type u_2} →
[inst_1 : CommSemiring S] →
[inst_2 : Algebra R S] →
{motive : IsLocalization' M S → Sort u} →
(t : IsLocalization' M S) →
((toIsLocalizationMap : M.IsLocalizationMap ⇑... | null | false |
CommRingCat.Colimits.Relation.add_zero | Mathlib.Algebra.Category.Ring.Colimits | ∀ {J : Type v} [inst : CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CommRingCat}
(x : CommRingCat.Colimits.Prequotient F),
CommRingCat.Colimits.Relation F (x.add CommRingCat.Colimits.Prequotient.zero) x | null | true |
TopologicalSpace.Opens.ext_iff | Mathlib.Topology.Sets.Opens | ∀ {α : Type u_2} [inst : TopologicalSpace α] {U V : TopologicalSpace.Opens α}, U = V ↔ ↑U = ↑V | null | true |
Qq.Impl.MVarSynth.noConfusion | Qq.Macro | {P : Sort u} → {t t' : Qq.Impl.MVarSynth} → t = t' → Qq.Impl.MVarSynth.noConfusionType P t t' | null | false |
PNat.sub_add_of_lt | Mathlib.Data.PNat.Basic | ∀ {a b : ℕ+}, b < a → a - b + b = a | null | true |
TensorProduct.instModuleFstSnd._proof_1 | Mathlib.LinearAlgebra.TensorProduct.Graded.External | ∀ {R : Type u_1} {ι : Type u_3} (𝒜 : ι → Type u_2) [inst : CommRing R] [inst_1 : (i : ι) → AddCommGroup (𝒜 i)]
[inst_2 : (i : ι) → Module R (𝒜 i)] (i : ι × ι), SMulCommClass R R (𝒜 i.1) | null | false |
CategoryTheory.MorphismProperty.Arrow.changeProp | Mathlib.CategoryTheory.MorphismProperty.Comma | {T : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} T] →
{P Q W : CategoryTheory.MorphismProperty T} →
[inst_1 : Q.IsMultiplicative] →
[inst_2 : W.IsMultiplicative] →
{P' Q' W' : CategoryTheory.MorphismProperty T} →
[inst_3 : Q'.IsMultiplicative] →
[inst_4... | The natural inclusion induced by implications of morphism properties. | true |
AlgebraicGeometry.QuasiCompact.mk | Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | ∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},
(∀ (U : Set ↥Y), IsOpen U → IsCompact U → IsCompact (⇑f ⁻¹' U)) → AlgebraicGeometry.QuasiCompact f | null | true |
CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom | Mathlib.CategoryTheory.Limits.Opposites | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} J]
(F : CategoryTheory.Functor Jᵒᵖ C) [inst_2 : CategoryTheory.Limits.HasLimit F] (j : J),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι F.rightOp j)
(CategoryTheory.Limit... | null | true |
ciSup_prod | Mathlib.Order.ConditionallyCompleteLattice.Indexed | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : ConditionallyCompleteLattice α] {f : β × γ → α},
BddAbove (Set.range f) → ⨆ p, f p = ⨆ b, ⨆ c, f (b, c) | null | true |
Module.Finite.trans | Mathlib.RingTheory.Finiteness.Basic | ∀ {R : Type u_5} (A : Type u_6) (M : Type u_7) [inst : Semiring R] [inst_1 : Semiring A] [inst_2 : Module R A]
[inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M] [IsScalarTower R A M] [Module.Finite R A]
[Module.Finite A M], Module.Finite R M | null | true |
CategoryTheory.Functor.preimageIso | Mathlib.CategoryTheory.Functor.FullyFaithful | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(F : CategoryTheory.Functor C D) → {X Y : C} → [F.Full] → [F.Faithful] → (F.obj X ≅ F.obj Y) → (X ≅ Y) | If `F : C ⥤ D` is fully faithful, every isomorphism `F.obj X ≅ F.obj Y` has a preimage. | true |
_private.Init.Data.Array.Sort.Lemmas.0.List.merge.match_1.eq_1 | Init.Data.Array.Sort.Lemmas | ∀ {α : Type u_1} (motive : List α → List α → Sort u_2) (ys : List α) (h_1 : (ys : List α) → motive [] ys)
(h_2 : (xs : List α) → motive xs [])
(h_3 : (x : α) → (xs : List α) → (y : α) → (ys : List α) → motive (x :: xs) (y :: ys)),
(match [], ys with
| [], ys => h_1 ys
| xs, [] => h_2 xs
| x :: xs, y :... | null | true |
_private.Mathlib.Analysis.BoxIntegral.UnitPartition.0.BoxIntegral.unitPartition._aux_Mathlib_Analysis_BoxIntegral_UnitPartition___macroRules__private_Mathlib_Analysis_BoxIntegral_UnitPartition_0_BoxIntegral_unitPartition_termL_1 | Mathlib.Analysis.BoxIntegral.UnitPartition | Lean.Macro | null | false |
Lean.Lsp.instFromJsonSymbolInformation | Lean.Data.Lsp.LanguageFeatures | Lean.FromJson Lean.Lsp.SymbolInformation | null | true |
_private.Mathlib.Tactic.Algebra.Basic.0.Mathlib.Tactic.Algebra.RingCompute.neg._proof_1 | Mathlib.Tactic.Algebra.Basic | ∀ {u : Lean.Level} {R : Q(Type u)} {sR : Q(CommSemiring «$R»)} (rR : Q(CommRing «$R»)), «$sR» =Q CommRing.toCommSemiring | null | false |
Real.holderConjugate_comm | Mathlib.Basic.Real.ConjExponents | ∀ {p q : ℝ}, p.HolderConjugate q ↔ q.HolderConjugate p | null | true |
_private.Mathlib.Algebra.BigOperators.Group.Finset.Basic.0.Finset.eq_of_card_le_one_of_sum_eq._proof_1_1 | Mathlib.Algebra.BigOperators.Group.Finset.Basic | ∀ {ι : Type u_2} {M : Type u_1} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → M} {b : M},
∑ x ∈ s, f x = b → ∀ x ∈ s, (∃ a, s = {a}) → f x = b | null | false |
Lean.Compiler.LCNF.CacheExtension.ctorIdx | Lean.Compiler.LCNF.CompilerM | {α β : Type} → {inst : BEq α} → {inst_1 : Hashable α} → Lean.Compiler.LCNF.CacheExtension α β → ℕ | null | false |
Filter.blimsup_congr | Mathlib.Order.LiminfLimsup | ∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompleteLattice α] {f : Filter β} {u v : β → α} {p : β → Prop},
(∀ᶠ (a : β) in f, p a → u a = v a) → Filter.blimsup u f p = Filter.blimsup v f p | null | true |
Multiset.le_ndinsert_self._simp_1 | Mathlib.Data.Multiset.FinsetOps | ∀ {α : Type u_1} [inst : DecidableEq α] (a : α) (s : Multiset α), (s ≤ Multiset.ndinsert a s) = True | null | false |
_private.Lean.Parser.Term.0.Lean.Parser.Term.inaccessible._regBuiltin.Lean.Parser.Term.inaccessible.formatter_9 | Lean.Parser.Term | IO Unit | null | false |
MultilinearMap.freeDFinsuppEquiv._proof_1 | Mathlib.LinearAlgebra.Multilinear.DFinsupp | ∀ {R : Type u_1} {ι' : Type u_2} [inst : CommSemiring R], SMulCommClass R R (Π₀ (i : ι'), R) | null | false |
_private.Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real.0.RealRMK.range_cut_partition._simp_1_8 | Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | ∀ {α : Type u} {s t : Set α}, (s ∩ t = t) = (t ⊆ s) | null | false |
Batteries.AssocList.find?.eq_def | Batteries.Data.AssocList.Lemmas | ∀ {α : Type u_1} {β : Type u_2} [inst : BEq α] (a : α) (x : Batteries.AssocList α β),
Batteries.AssocList.find? a x =
match x with
| Batteries.AssocList.nil => none
| Batteries.AssocList.cons k v es =>
match k == a with
| true => some v
| false => Batteries.AssocList.find? a es | null | true |
Plausible.NamedBinder.eq_1 | Plausible.Testable | ∀ (_n : String) (p : Prop), Plausible.NamedBinder _n p = p | null | true |
_private.Lean.Meta.WrapInstance.0.Lean.Meta.initFn._@.Lean.Meta.WrapInstance.2650342594._hygCtx._hyg.4 | Lean.Meta.WrapInstance | IO (Lean.Option Bool) | null | false |
CStarAlgebra.nnnorm_le_one_iff_of_nonneg | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | ∀ {A : Type u_1} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] (a : A),
autoParam (0 ≤ a) CStarAlgebra.nnnorm_le_one_iff_of_nonneg._auto_1 → (‖a‖₊ ≤ 1 ↔ a ≤ 1) | null | true |
_private.Mathlib.Topology.Algebra.Nonarchimedean.Basic.0.NonarchimedeanGroup.prod_self_subset.match_1_1 | Mathlib.Topology.Algebra.Nonarchimedean.Basic | ∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] {U : Set (G × G)}
(motive : (∃ V W, ↑V ×ˢ ↑W ⊆ U) → Prop) (x : ∃ V W, ↑V ×ˢ ↑W ⊆ U),
(∀ (V W : OpenSubgroup G) (h : ↑V ×ˢ ↑W ⊆ U), motive ⋯) → motive x | null | false |
_private.Mathlib.LinearAlgebra.FixedSubmodule.0.LinearMap.fixedSubmodule_eq_top_iff._simp_1_1 | Mathlib.LinearAlgebra.FixedSubmodule | ∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}
{f g : M →ₛₗ[σ] M₃}, (f = g) = ∀ (x : M), f x = g x | null | false |
IsLocalRing.ResidueField.mapAlgEquiv._proof_5 | Mathlib.RingTheory.LocalRing.ResidueField.Basic | ∀ {R : Type u_3} {S : Type u_1} {T : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsLocalRing S]
[inst_3 : CommRing T] [inst_4 : IsLocalRing T] [inst_5 : Algebra R S] [inst_6 : Algebra R T] (e : S ≃ₐ[R] T)
(x y : IsLocalRing.ResidueField S),
(↑↑(IsLocalRing.ResidueField.mapAlgHom ↑e).toRingHom).t... | null | false |
measurable_set_iff | Mathlib.MeasureTheory.MeasurableSpace.Constructions | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace β] {g : β → Set α},
Measurable g ↔ ∀ (a : α), Measurable fun x => a ∈ g x | null | true |
Lean.Meta.Grind.Arith.Linear.getCommRingInst | Lean.Meta.Tactic.Grind.Arith.Linear.Util | Lean.Meta.Grind.Arith.Linear.LinearM Lean.Expr | null | true |
groupCohomology.δ_naturality | Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence | ∀ {k G : Type u} [inst : CommRing k] [inst_1 : Group G] {X1 X2 : CategoryTheory.ShortComplex (Rep.{u, u, u} k G)}
(hX1 : X1.ShortExact) (hX2 : X2.ShortExact) (F : X1 ⟶ X2) (i j : ℕ) (hij : i + 1 = j),
CategoryTheory.CategoryStruct.comp (groupCohomology.δ hX1 i j hij) (groupCohomology.map (MonoidHom.id G) F.τ₁ j) =
... | The connecting homomorphism `δ` is actually a natural transformation between
`groupCohomology.funtor`s. | true |
TopCat.Presheaf.SubmonoidPresheaf.mk._flat_ctor | Mathlib.Topology.Sheaves.CommRingCat | {X : TopCat} →
{F : TopCat.Presheaf CommRingCat X} →
(obj : (U : (TopologicalSpace.Opens ↑X)ᵒᵖ) → Submonoid ↑(F.obj U)) →
(∀ {U V : (TopologicalSpace.Opens ↑X)ᵒᵖ} (i : U ⟶ V),
obj U ≤ Submonoid.comap (CommRingCat.Hom.hom (F.map i)) (obj V)) →
F.SubmonoidPresheaf | null | false |
CommBialgCat.noConfusionType | Mathlib.Algebra.Category.CommBialgCat | Sort u_1 →
{R : Type u} →
[inst : CommRing R] → CommBialgCat R → {R' : Type u} → [inst' : CommRing R'] → CommBialgCat R' → Sort u_1 | null | false |
_private.Batteries.Data.BinaryHeap.WF.0.Batteries.BinaryHeap.WF.ParentGeChildren.swap_parent._proof_1_23 | Batteries.Data.BinaryHeap.WF | ∀ {sz : ℕ} {i : Fin sz}, 2 * ((↑i - 1) / 2) + 1 < sz → 0 < (↑i - 1) / 2 → ¬(↑i - 1) / 2 < sz → False | null | false |
CategoryTheory.Preadditive.toCommGrp_map | Mathlib.CategoryTheory.Preadditive.CommGrp_ | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.CartesianMonoidalCategory C] [inst_3 : CategoryTheory.BraidedCategory C] {X Y : C}
(f : X ⟶ Y),
(CategoryTheory.Preadditive.toCommGrp C).map f =
CategoryTheory.InducedCategory.homMk (Categ... | null | true |
_private.Mathlib.Algebra.BigOperators.Group.Multiset.Basic.0.Multiset.prod_nsmul.match_1_1 | Mathlib.Algebra.BigOperators.Group.Multiset.Basic | ∀ (motive : ℕ → Prop) (x : ℕ), (∀ (_ : Unit), motive 0) → (∀ (n : ℕ), motive n.succ) → motive x | null | false |
Mathlib.Meta.FunProp.GeneralTheorem.mk.sizeOf_spec | Mathlib.Tactic.FunProp.Types | ∀ (funPropName thmName : Lean.Name)
(keys : List (Lean.Meta.RefinedDiscrTree.Key × Lean.Meta.RefinedDiscrTree.LazyEntry)) (priority : ℕ),
sizeOf { funPropName := funPropName, thmName := thmName, keys := keys, priority := priority } =
1 + sizeOf funPropName + sizeOf thmName + sizeOf keys + sizeOf priority | null | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.getKey?_insertMany_list_of_contains_eq_false._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {x : Ord α} {x_1 : BEq α} [Std.LawfulBEqOrd α] {a b : α}, (compare a b = Ordering.eq) = ((a == b) = true) | null | false |
_private.Batteries.Data.BinaryHeap.WF.0.Batteries.BinaryHeap.WF.ParentGeChildren.swap_parent._proof_1_15 | Batteries.Data.BinaryHeap.WF | ∀ {sz : ℕ} {i : Fin sz} (hside : 2 * ↑⟨(↑i - 1) / 2, ⋯⟩ + 1 + 1 < sz), 0 < ↑⟨2 * ((↑i - 1) / 2) + 2, hside⟩ | null | false |
PSet.brecOn.go | Mathlib.SetTheory.ZFC.PSet | {motive : PSet.{u} → Sort u_1} → (t : PSet.{u}) → ((t : PSet.{u}) → PSet.below t → motive t) → motive t ×' PSet.below t | null | true |
_private.Mathlib.Computability.Partrec.0.Nat.rfindX._proof_1 | Mathlib.Computability.Partrec | ∀ (p : ℕ →. Bool), (∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom) → ∀ (m : ℕ), (∀ n < m, false ∈ p n) → (p m).Dom | null | false |
measurableSet_iSup_of_mem_piiUnionInter | Mathlib.MeasureTheory.PiSystem | ∀ {α : Type u_3} {ι : Type u_4} (m : ι → MeasurableSpace α) (S : Set ι),
∀ t ∈ piiUnionInter (fun n => {s | MeasurableSet s}) S, MeasurableSet t | null | true |
analyticAt_pi_iff | Mathlib.Analysis.Analytic.Constructions | ∀ {𝕜 : Type u_2} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {ι : Type u_9} [inst_3 : Fintype ι] {e : E} {Fm : ι → Type u_10}
[inst_4 : (i : ι) → NormedAddCommGroup (Fm i)] [inst_5 : (i : ι) → NormedSpace 𝕜 (Fm i)] {f : (i : ι) → E → Fm i},
Anal... | null | true |
Topology.WithLower.instLinearOrder | Mathlib.Topology.Order.LowerUpperTopology | {α : Type u_1} → [LinearOrder α] → LinearOrder (Topology.WithLower α) | null | true |
Matroid.isBasis_union_iff_indep_closure | Mathlib.Combinatorics.Matroid.Closure | ∀ {α : Type u_2} {M : Matroid α} {X I : Set α}, M.IsBasis I (I ∪ X) ↔ M.Indep I ∧ X ⊆ M.closure I | null | true |
_private.Init.Data.Nat.Lemmas.0.Nat.one_mod_eq_one._proof_1_3 | Init.Data.Nat.Lemmas | ∀ (n : ℕ), n + 2 = 1 → False | null | false |
Nat.unpair_add_le | Mathlib.Data.Nat.Pairing | ∀ (n : ℕ), (Nat.unpair n).1 + (Nat.unpair n).2 ≤ n | null | true |
CategoryTheory.Functor.isoSum_hom_app_inl | Mathlib.CategoryTheory.Sums.Basic | ∀ {A : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} B]
{C : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor (A ⊕ B) C) (a : A),
F.isoSum.hom.app (Sum.inl a) = CategoryTheory.CategoryStruct.id (F.obj (Sum.inl a)) | null | true |
CategoryTheory.StructuredArrow.toCostructuredArrow._proof_4 | Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_2, u_4} D] (F : CategoryTheory.Functor C D) (d : D)
{X Y Z : (CategoryTheory.StructuredArrow d F)ᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z),
CategoryTheory.CostructuredArrow.homMk
(CategoryTheory.StructuredAr... | null | false |
Finset.interior_iInter | Mathlib.Topology.Closure | ∀ {X : Type u} [inst : TopologicalSpace X] {ι : Type u_1} (s : Finset ι) (f : ι → Set X),
interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) | null | true |
Equiv.Perm.not_solvable | Mathlib.GroupTheory.Solvable | ∀ (X : Type u_3), 5 ≤ Cardinal.mk X → ¬Group.IsSolvable (Equiv.Perm X) | **Alias** of `Equiv.Perm.not_isSolvable`. | true |
Lean.Lsp.ServerCapabilities.definitionProvider._default | Lean.Data.Lsp.Capabilities | Bool | null | false |
Inseparable.specializes | Mathlib.Topology.Inseparable | ∀ {X : Type u_1} [inst : TopologicalSpace X] {x y : X}, Inseparable x y → x ⤳ y | null | true |
Lean.Syntax.Range.contains | Lean.Syntax | Lean.Syntax.Range → String.Pos.Raw → optParam Bool false → Bool | null | true |
Submonoid.mem_sup_right | Mathlib.Algebra.Group.Submonoid.Membership | ∀ {M : Type u_1} [inst : MulOneClass M] {S T : Submonoid M} {x : M}, x ∈ T → x ∈ S ⊔ T | null | true |
_private.Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension.0.ExistsContDiffBumpBase.u_exists._simp_1_2 | Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | ∀ {a : Prop}, (¬¬a) = a | null | false |
AddMonCat.FilteredColimits.colimitZero | Mathlib.Algebra.Category.MonCat.FilteredColimits | {J : Type v} →
[inst : CategoryTheory.SmallCategory J] →
(F : CategoryTheory.Functor J AddMonCat) → [CategoryTheory.IsFiltered J] → Zero (AddMonCat.FilteredColimits.M F) | As `J` is nonempty, we can pick an arbitrary object `j₀ : J`. We use this object to
define the "zero" in the colimit as the equivalence class of `⟨j₀, 0 : F.obj j₀⟩`. | true |
DirectSum.IsInternal.addSubmonoid_iSup_eq_top | Mathlib.Algebra.DirectSum.Basic | ∀ {ι : Type v} {M : Type u_1} [inst : DecidableEq ι] [inst_1 : AddCommMonoid M] (A : ι → AddSubmonoid M),
DirectSum.IsInternal A → iSup A = ⊤ | null | true |
instCStarAlgebraProd._proof_7 | Mathlib.Analysis.CStarAlgebra.Classes | ∀ {A : Type u_1} {B : Type u_2} [inst : CStarAlgebra A] [inst_1 : CStarAlgebra B] (a : A × B), a * 0 = 0 | null | false |
isTopologicalGroup_induced | Mathlib.Topology.Algebra.Group.Basic | ∀ {G : Type u_1} {H : Type u_2} [inst : TopologicalSpace G] [inst_1 : Group G] [IsTopologicalGroup G] {F : Type u_5}
[inst_3 : Group H] [inst_4 : FunLike F H G] [MonoidHomClass F H G] (f : F), IsTopologicalGroup H | null | true |
lpBCFₗᵢ._proof_3 | Mathlib.Analysis.Normed.Lp.LpEquiv | ∀ (E : Type u_2) (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E], IsBoundedSMul 𝕜 E | null | false |
CategoryTheory.instIsDiscreteOpposite | Mathlib.CategoryTheory.Discrete.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.IsDiscrete C],
CategoryTheory.IsDiscrete Cᵒᵖ | null | true |
_private.Mathlib.Data.Finsupp.MonomialOrder.DegLex.0.Finsupp.DegLex.single_strictAnti._simp_1_2 | Mathlib.Data.Finsupp.MonomialOrder.DegLex | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, ((toLex fun₀ | b => 1) < toLex fun₀ | a => 1) = (a < b) | null | false |
Lake.StdVer.instLT | Lake.Util.Version | LT Lake.StdVer | null | true |
_private.Init.Data.List.Erase.0.List.eraseP_sublist.match_1_1 | Init.Data.List.Erase | ∀ {α : Type u_1} {p : α → Bool} {l : List α}
(motive :
(List.eraseP p l = l ∨
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b = true) ∧ p a = true ∧ l = l₁ ++ a :: l₂ ∧ List.eraseP p l = l₁ ++ l₂) →
Prop)
(x :
List.eraseP p l = l ∨
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b = true) ∧ p a = true ∧ l = l₁ ++ a :: l₂ ∧ List.er... | null | false |
AlgebraicGeometry.UniversallyOpen.rec | Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {X Y : AlgebraicGeometry.Scheme} →
{f : X ⟶ Y} →
{motive : AlgebraicGeometry.UniversallyOpen f → Sort u_1} →
((universally_isOpenMap : (AlgebraicGeometry.topologically @IsOpenMap).universally f) → motive ⋯) →
(t : AlgebraicGeometry.UniversallyOpen f) → motive t | null | false |
AlgebraicGeometry.Scheme.IdealSheafData.glueDataT'Aux._proof_1 | Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | ∀ {X : AlgebraicGeometry.Scheme} (I : X.IdealSheafData) (U V W : ↑X.affineOpens),
CategoryTheory.Limits.HasPullback (CategoryTheory.Limits.pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯))
(CategoryTheory.Limits.pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) | null | false |
CategoryTheory.Over.tensorHom_left_fst_assoc | Mathlib.CategoryTheory.Monoidal.Cartesian.Over | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasPullbacks C]
{X S U : C} {R T : CategoryTheory.Over X} (fS : S ⟶ X) (fU : U ⟶ X) (f : R ⟶ CategoryTheory.Over.mk fS)
(g : T ⟶ CategoryTheory.Over.mk fU) {Z : C} (h : S ⟶ Z),
CategoryTheory.CategoryStruct.comp
(... | null | true |
IsCyclotomicExtension.Rat.mem_subgroupGalEquivSubgroupChar_symm_iff._simp_1 | Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | ∀ (n : ℕ) [inst : NeZero n] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField K]
[hK : IsCyclotomicExtension {n} ℚ K] (R : Type u_2) [inst_3 : CommRing R]
[inst_4 : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)] (σ : Gal(K/ℚ)) (Y : Subgroup (DirichletCharacter R n)),
(σ ∈ (IsCyclotomicExtension.Rat.su... | null | false |
Lean.Elab.Tactic.SimpKind.ctorIdx | Lean.Elab.Tactic.Simp | Lean.Elab.Tactic.SimpKind → ℕ | null | false |
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