name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt.0.SummationFilter.symmetricIcc_eq_symmetricIoo_int._proof_1_13 | Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | ∀ (b : ℕ) (a_1 : ℤ), -↑b ≤ a_1 ∧ a_1 ≤ ↑b ↔ -1 < a_1 + ↑b ∧ a_1 < ↑b + 1 | null | false |
CategoryTheory.MorphismProperty.instIsStableUnderTransfiniteCompositionOfShapeFunctorFunctorCategoryOfHasIterationOfShape | Mathlib.CategoryTheory.MorphismProperty.FunctorCategory | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W : CategoryTheory.MorphismProperty C} (K : Type u')
[inst_1 : LinearOrder K] [inst_2 : SuccOrder K] [inst_3 : OrderBot K] [inst_4 : WellFoundedLT K]
[W.IsStableUnderTransfiniteCompositionOfShape K] (J : Type u'') [inst_6 : CategoryTheory.Category.{v'', u''}... | null | true |
_private.Mathlib.RingTheory.DedekindDomain.AdicValuation.0.IsDedekindDomain.HeightOneSpectrum.exists_intValuation_mul_sub_lt._simp_1_1 | Mathlib.RingTheory.DedekindDomain.AdicValuation | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x : M}
{p p' : Submodule R M}, (x ∈ p ⊔ p') = ∃ y ∈ p, ∃ z ∈ p', y + z = x | null | false |
_private.Mathlib.RingTheory.MvPowerSeries.LexOrder.0.MvPowerSeries.exists_finsupp_eq_lexOrder_of_ne_zero._simp_1_2 | Mathlib.RingTheory.MvPowerSeries.LexOrder | ∀ {α : Type u_1} {x : WithTop α}, (x ≠ ⊤) = ∃ a, ↑a = x | null | false |
RingHom.mk'._proof_4 | Mathlib.Algebra.Ring.Hom.Defs | ∀ {α : Type u_2} {β : Type u_1} [inst : NonAssocSemiring α] [inst_1 : NonAssocRing β] (f : α →* β)
(map_add : ∀ (a b : α), f (a + b) = f a + f b) (x y : α),
(↑(AddMonoidHom.mk' (⇑f) map_add)).toFun (x + y) =
(↑(AddMonoidHom.mk' (⇑f) map_add)).toFun x + (↑(AddMonoidHom.mk' (⇑f) map_add)).toFun y | null | false |
_private.Mathlib.Data.List.Basic.0.List.mem_getLast?_eq_getLast._proof_1_14 | Mathlib.Data.List.Basic | ∀ {α : Type u_1} (a b : α) (l : List α), ¬(a :: b :: l).length - 1 = 0 → (a :: b :: l).length - 2 < (b :: l).length | null | false |
Ring.KrullDimLE.subsingleton_primeSpectrum | Mathlib.RingTheory.KrullDimension.Zero | ∀ (R : Type u_1) [inst : CommSemiring R] [Ring.KrullDimLE 0 R] [IsLocalRing R], Subsingleton (PrimeSpectrum R) | null | true |
id_tensor_π_preserves_coequalizer_inv_colimMap_desc | Mathlib.CategoryTheory.Monoidal.Bimod | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.Limits.HasCoequalizers C]
[inst_3 :
∀ (X : C),
CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
(CategoryTheory.MonoidalCategory.tensorLeft X)]
{X... | null | true |
_private.Lean.Compiler.LCNF.Internalize.0.Lean.Compiler.LCNF.Internalize.internalizeCodeDecl.match_4 | Lean.Compiler.LCNF.Internalize | {pu : Lean.Compiler.LCNF.Purity} →
(motive : Lean.Compiler.LCNF.CodeDecl pu → Sort u_1) →
(decl : Lean.Compiler.LCNF.CodeDecl pu) →
((decl : Lean.Compiler.LCNF.LetDecl pu) → motive (Lean.Compiler.LCNF.CodeDecl.let decl)) →
((decl : Lean.Compiler.LCNF.FunDecl pu) →
(h : pu = Lean.Compiler... | null | false |
EReal.top_ne_coe._simp_1 | Mathlib.Data.EReal.Basic | ∀ (x : ℝ), (⊤ = ↑x) = False | null | false |
Subsemiring.coe_prod._simp_1 | Mathlib.Algebra.Ring.Subsemiring.Basic | ∀ {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst_1 : NonAssocSemiring S] (s : Subsemiring R)
(t : Subsemiring S), ↑s ×ˢ ↑t = ↑(s.prod t) | null | false |
QuadraticMap.zeroHomClass | Mathlib.LinearAlgebra.QuadraticForm.Basic | ∀ {R : Type u_3} {M : Type u_4} {N : Type u_5} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid N] [inst_4 : Module R N], ZeroHomClass (QuadraticMap R M N) M N | null | true |
_private.Std.Data.String.ToNat.0.not_noRepetition_append_singleton_of_suffix | Std.Data.String.ToNat | ∀ {α : Type u} {a : α} {l : List α}, [a] <:+ l → ¬NoRepetition✝ a (l ++ [a]) | null | true |
Std.DTreeMap.Internal.Impl.balanceL!_pair_congr | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} {k : α} {v : β k} {k' : α} {v' : β k'},
⟨k, v⟩ = ⟨k', v'⟩ →
∀ {l l' r r' : Std.DTreeMap.Internal.Impl α β},
l = l' → r = r' → Std.DTreeMap.Internal.Impl.balanceL! k v l r = Std.DTreeMap.Internal.Impl.balanceL! k' v' l' r' | null | true |
Module.mapEvalEquiv | Mathlib.LinearAlgebra.Dual.Defs | (R : Type u_1) →
(M : Type u_2) →
[inst : CommSemiring R] →
[inst_1 : AddCommMonoid M] →
[inst_2 : Module R M] →
[Module.IsReflexive R M] → Submodule R M ≃o Submodule R (Module.Dual R (Module.Dual R M)) | The isomorphism `Module.evalEquiv` induces an order isomorphism on subspaces. | true |
SimplexCategory.δ_comp_σ_of_le | Mathlib.AlgebraicTopology.SimplexCategory.Basic | ∀ {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 1)},
i ≤ j.castSucc →
CategoryTheory.CategoryStruct.comp (SimplexCategory.δ i.castSucc) (SimplexCategory.σ j.succ) =
CategoryTheory.CategoryStruct.comp (SimplexCategory.σ j) (SimplexCategory.δ i) | The second simplicial identity | true |
_private.Mathlib.Algebra.Polynomial.UnitTrinomial.0.Polynomial.isUnitTrinomial_iff'._simp_1_1 | Mathlib.Algebra.Polynomial.UnitTrinomial | ∀ {α : Type u} [inst : Monoid α] (a : αˣ) (n : ℕ), ↑a ^ n = ↑(a ^ n) | null | false |
Polynomial.SplittingField.instCharZero | Mathlib.FieldTheory.SplittingField.Construction | ∀ {K : Type v} [inst : Field K] (f : Polynomial K) [CharZero K], CharZero f.SplittingField | null | true |
_private.Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean.0.NNReal.bddAbove_range_agmSequences_fst._simp_1_3 | Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | ∀ {α : Sort u_2} {β : Sort u_1} {f : α → β} {p : β → Prop}, (∀ (b : β) (a : α), f a = b → p b) = ∀ (a : α), p (f a) | null | false |
FreeGroup.isReduced_iff_reduce_eq | Mathlib.GroupTheory.FreeGroup.Reduce | ∀ {α : Type u_1} {L : List (α × Bool)} [inst : DecidableEq α], FreeGroup.IsReduced L ↔ FreeGroup.reduce L = L | null | true |
instLTBitVec | Init.Prelude | {w : ℕ} → LT (BitVec w) | null | true |
PMF.uniformOfFintype_apply | Mathlib.Probability.Distributions.Uniform | ∀ {α : Type u_2} [inst : MeasurableSpace α] [MeasurableSingletonClass α] {s : Finset α} (a : α),
(MeasureTheory.Measure.uniformOfFinset s) {a} = if a ∈ s then (↑s.card)⁻¹ else 0 | **Alias** of `MeasureTheory.Measure.uniformOfFinset_apply_singleton`. | true |
ciSup_eq_ite | Mathlib.Order.ConditionallyCompletePartialOrder.Indexed | ∀ {α : Type u_1} [inst : ConditionallyCompletePartialOrderSup α] {p : Prop} [inst_1 : Decidable p] {f : p → α},
⨆ (h : p), f h = if h : p then f h else sSup ∅ | null | true |
_private.Mathlib.GroupTheory.Perm.Finite.0.Equiv.Perm.disjoint_support_closure_of_disjoint_support._simp_1_2 | Mathlib.GroupTheory.Perm.Finite | ∀ {α : Type u_1} {t : Set α} {ι : Sort u_5} {s : ι → Set α}, Disjoint t (⋃ i, s i) = ∀ (i : ι), Disjoint t (s i) | null | false |
CategoryTheory.MorphismProperty.IsStableUnderComonoid.recOn | Mathlib.CategoryTheory.CopyDiscardCategory.Widesubcategory | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{P : CategoryTheory.MorphismProperty C} →
{c : C} →
[inst_2 : CategoryTheory.ComonObj c] →
{motive : P.IsStableUnderComonoid c → Sort u} →
(t : P.IsStab... | null | false |
String.Slice.toList_split_prop | Init.Data.String.Lemmas.Pattern.Split.Pred | ∀ {s : String.Slice} {p : Char → Prop} [inst : DecidablePred p],
List.map String.Slice.copy (s.split p).toList =
List.map String.ofList (List.splitOnP (fun b => decide (p b)) s.copy.toList) | null | true |
ISize.toInt_maxValue | Init.Data.SInt.Lemmas | ISize.maxValue.toInt = 2 ^ (System.Platform.numBits - 1) - 1 | null | true |
_private.Mathlib.Tactic.Find.0.Mathlib.Tactic.Find.matchHyps | Mathlib.Tactic.Find | List Lean.Expr → List Lean.Expr → List Lean.Expr → Lean.MetaM Bool | null | true |
LaurentSeries.val_le_one_iff_eq_coe | Mathlib.RingTheory.LaurentSeries | ∀ (K : Type u_2) [inst : Field K] (f : LaurentSeries K), Valued.v f ≤ 1 ↔ ∃ F, (HahnSeries.ofPowerSeries ℤ K) F = f | Every Laurent series of valuation less than `(1 : ℤᵐ⁰)` comes from a power series. | true |
Lean.Lsp.DiagnosticCode | Lean.Data.Lsp.Diagnostics | Type | Some languages have specific codes for each error type. | true |
Ordnode.eraseMin._unsafe_rec | Mathlib.Data.Ordmap.Ordnode | {α : Type u_1} → Ordnode α → Ordnode α | null | false |
UpperSet.coe_nonempty | Mathlib.Order.UpperLower.CompleteLattice | ∀ {α : Type u_1} [inst : LE α] {s : UpperSet α}, (↑s).Nonempty ↔ s ≠ ⊤ | null | true |
_private.Mathlib.RingTheory.Lasker.0.Submodule.IsMinimalPrimaryDecomposition.comap_localized₀_eq_ite._simp_1_9 | Mathlib.RingTheory.Lasker | ∀ {α : Sort u} {p : α → Prop} {q : { a // p a } → Prop}, (∀ (x : { a // p a }), q x) = ∀ (a : α) (b : p a), q ⟨a, b⟩ | null | false |
ContinuousMap.instRegularSpace | Mathlib.Topology.CompactOpen | ∀ {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] [RegularSpace Y],
RegularSpace C(X, Y) | null | true |
Mathlib.Tactic.ClickSuggestions.SectionKind.hyp.elim | Mathlib.Tactic.ClickSuggestions.SectionState | {motive : Mathlib.Tactic.ClickSuggestions.SectionKind → Sort u} →
(t : Mathlib.Tactic.ClickSuggestions.SectionKind) →
t.ctorIdx = 0 → motive Mathlib.Tactic.ClickSuggestions.SectionKind.hyp → motive t | null | false |
Std.IterM.Partial.it | Init.Data.Iterators.Consumers.Monadic.Partial | {α : Type w} → {m : Type w → Type w'} → {β : Type w} → Std.IterM.Partial m β → Std.IterM m β | The wrapped iterator, which was wrapped by `IterM.allowNontermination`.
| true |
AlgebraicTopology.DoldKan.Γ₀.Obj.map._proof_1 | Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | ∀ {Δ' Δ : SimplexCategoryᵒᵖ} (θ : Δ ⟶ Δ') (A : CategoryTheory.SimplicialObject.Splitting.IndexSet Δ),
CategoryTheory.Limits.HasImage (CategoryTheory.CategoryStruct.comp θ.unop A.e) | null | false |
Algebra.TensorProduct.tensorQuotientEquiv_symm_apply_tmul | Mathlib.RingTheory.TensorProduct.Quotient | ∀ {R : Type u_1} (S : Type u_2) (T : Type u_3) (A : Type u_4) [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : Algebra R S] [inst_3 : CommRing T] [inst_4 : Algebra R T] [inst_5 : CommRing A] [inst_6 : Algebra R A]
[inst_7 : Algebra S A] [inst_8 : IsScalarTower R S A] (I : Ideal T) (a : A) (t : T),
(Algebra.Ten... | null | true |
Std.ExtTreeMap.union_insert_right_eq_insert_union | Std.Data.ExtTreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp]
{p : (_ : α) × β}, t₁ ∪ t₂.insert p.fst p.snd = (t₁ ∪ t₂).insert p.fst p.snd | null | true |
String.Slice.Pattern.Model.IsValidSearchFrom.mismatched_of_eq | Init.Data.String.Lemmas.Pattern.Basic | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.Model.PatternModel pat] {s : String.Slice}
{startPos startPos' endPos : s.Pos} {l : List (String.Slice.Pattern.SearchStep s)},
String.Slice.Pattern.Model.IsValidSearchFrom pat endPos l →
startPos' < endPos →
(∀ (pos : s.Pos), startPos' ≤ pos → pos < endP... | null | true |
Std.ExtDHashMap.mem_modify | Std.Data.ExtDHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : α → Type v} {m : Std.ExtDHashMap α β} [inst : LawfulBEq α] {k k' : α}
{f : β k → β k}, k' ∈ m.modify k f ↔ k' ∈ m | null | true |
instDecidableEqFloat.decEq._proof_3 | Init.Data.Float.Float | ∀ (a b : Float.Model), ¬a = b → ¬{ toModel := a } = { toModel := b } | null | false |
CategoryTheory.MonoidalCategory.whiskerLeftIso_trans | Mathlib.CategoryTheory.Monoidal.Category | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.MonoidalCategory C] (W : C)
{X Y Z : C} (f : X ≅ Y) (g : Y ≅ Z),
CategoryTheory.MonoidalCategory.whiskerLeftIso W (f ≪≫ g) =
CategoryTheory.MonoidalCategory.whiskerLeftIso W f ≪≫ CategoryTheory.MonoidalCategory.whiskerLeftIso W g | null | true |
instCommGroupAbelianization | Mathlib.GroupTheory.Abelianization.Defs | (G : Type u_1) → [inst : Group G] → CommGroup (Abelianization G) | null | true |
List.isEmpty_reverse | Init.Data.List.Lemmas | ∀ {α : Type u_1} {xs : List α}, xs.reverse.isEmpty = xs.isEmpty | null | true |
Std.TreeSet.Raw.insertMany_list_equiv_foldl | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t₁ : Std.TreeSet.Raw α cmp} {l : List α},
(t₁.insertMany l).Equiv (List.foldl (fun acc a => acc.insert a) t₁ l) | null | true |
CategoryTheory.ComposableArrows.homMkSucc | Mathlib.CategoryTheory.ComposableArrows.Basic | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{n : ℕ} →
{F G : CategoryTheory.ComposableArrows C (n + 1)} →
(α : F.obj' 0 ⋯ ⟶ G.obj' 0 ⋯) →
(β : F.δ₀ ⟶ G.δ₀) →
CategoryTheory.CategoryStruct.comp (F.map' 0 1 CategoryTheory.ComposableArrows.homMk₁._proof_4 ⋯)
... | Inductive construction of morphisms in `ComposableArrows C (n + 1)`: in order to construct
a morphism `F ⟶ G`, it suffices to provide `α : F.obj' 0 ⟶ G.obj' 0` and `β : F.δ₀ ⟶ G.δ₀`
such that `F.map' 0 1 ≫ app' β 0 = α ≫ G.map' 0 1`. | true |
_private.Mathlib.LinearAlgebra.Projection.0.LinearMap.IsIdempotentElem.commute_iff_of_isUnit._simp_1_4 | Mathlib.LinearAlgebra.Projection | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {f : M ≃ₗ[R] M}
{p : Submodule R M}, (p ≤ Submodule.map (↑f) p) = (p ∈ Module.End.invtSubmodule ↑f.symm) | null | false |
Ideal.quotientEquivDirectSum | Mathlib.LinearAlgebra.FreeModule.IdealQuotient | {ι : Type u_1} →
{R : Type u_2} →
{S : Type u_3} →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
[inst_3 : IsDomain R] →
[inst_4 : IsPrincipalIdealRing R] →
[inst_5 : IsDomain S] →
[inst_6 : Finite ι] →
... | Decompose `S⧸I` as a direct sum of cyclic `R`-modules
(quotients by the ideals generated by Smith coefficients of `I`). | true |
CategoryTheory.Functor.DenseAt.ofIso | Mathlib.CategoryTheory.Functor.KanExtension.DenseAt | {C : Type u₁} →
{D : Type u₂} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
{F : CategoryTheory.Functor C D} → {Y : D} → F.DenseAt Y → {Y' : D} → (Y ≅ Y') → F.DenseAt Y' | If `F : C ⥤ D` is dense at `Y : D`, then it is also at `Y'`
if `Y` and `Y'` are isomorphic. | true |
Option.merge.eq_4 | Init.Omega.Constraint | ∀ {α : Type u_1} (fn : α → α → α) (x_2 y : α), Option.merge fn (some x_2) (some y) = some (fn x_2 y) | null | true |
SemiRingCat.FilteredColimits.colimitCoconeIsColimit._proof_2 | Mathlib.Algebra.Category.Ring.FilteredColimits | ∀ {J : Type u_1} [inst : CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J SemiRingCat)
[inst_1 : CategoryTheory.IsFiltered J] (t : CategoryTheory.Limits.Cocone F),
(SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom t) 1 = 1 | null | false |
Lean.Server.RequestCancellation._sizeOf_1 | Lean.Server.RequestCancellation | Lean.Server.RequestCancellation → ℕ | null | false |
ContinuousLinearEquiv.equivLike._proof_1 | Mathlib.Topology.Algebra.Module.Equiv.Basic | ∀ {R₁ : Type u_3} {R₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁}
[inst_2 : RingHomInvPair σ₁₂ σ₂₁] [inst_3 : RingHomInvPair σ₂₁ σ₁₂] {M₁ : Type u_1} [inst_4 : TopologicalSpace M₁]
[inst_5 : AddCommMonoid M₁] {M₂ : Type u_2} [inst_6 : TopologicalSpace M₂] [inst_7 : Ad... | null | false |
Lean.FindLevelMVar.main._unsafe_rec | Lean.Util.FindLevelMVar | (Lean.LMVarId → Bool) → Lean.Expr → Lean.FindLevelMVar.Visitor | null | false |
CategoryTheory.Enriched.FunctorCategory.homEquiv_apply_π_assoc | Mathlib.CategoryTheory.Enriched.FunctorCategory | ∀ (V : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} V] [inst_1 : CategoryTheory.MonoidalCategory V] {C : Type u₂}
[inst_2 : CategoryTheory.Category.{v₂, u₂} C] {J : Type u₃} [inst_3 : CategoryTheory.Category.{v₃, u₃} J]
[inst_4 : CategoryTheory.EnrichedOrdinaryCategory V C] {F₁ F₂ : CategoryTheory.Functor J C}... | null | true |
FirstCountableTopology.frechetUrysohnSpace | Mathlib.Topology.Sequences | ∀ {X : Type u_1} [inst : TopologicalSpace X] [FirstCountableTopology X], FrechetUrysohnSpace X | Every first-countable space is a Fréchet-Urysohn space. | true |
Int.cast_le_neg_one_of_neg | Mathlib.Algebra.Order.Ring.Cast | ∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {a : ℤ}, a < 0 → ↑a ≤ -1 | null | true |
WithZero.instAddMonoidWithOne | Mathlib.Algebra.GroupWithZero.WithZero | {α : Type u_1} → [AddMonoidWithOne α] → AddMonoidWithOne (WithZero α) | null | true |
lt_of_mul_self_lt_mul_self₀ | Mathlib.Algebra.Order.GroupWithZero.Basic | ∀ {M₀ : Type u_2} [inst : MonoidWithZero M₀] [inst_1 : LinearOrder M₀] [PosMulStrictMono M₀] {a b : M₀} [MulPosMono M₀],
0 ≤ b → a * a < b * b → a < b | null | true |
Module.piEquiv | Mathlib.LinearAlgebra.StdBasis | (ι : Type u_1) →
(R : Type u_2) →
(M : Type u_3) →
[Finite ι] →
[inst : CommSemiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → (ι → M) ≃ₗ[R] (ι → R) →ₗ[R] M | The natural linear equivalence: `Mⁱ ≃ Hom(Rⁱ, M)` for an `R`-module `M`. | true |
Mathlib.Tactic.Determinant.Ctx.mk.injEq | Mathlib.Tactic.Determinant.Bird.Meta | ∀ {u : Lean.Level} {α : Q(Type u)} {rα : Q(CommRing «$α»)}
(cα : Mathlib.Tactic.Ring.Common.Cache (Mathlib.Tactic.Determinant.commSemiringOfCommRing rα))
(rc :
Mathlib.Tactic.Ring.Common.RingCompute Mathlib.Tactic.Ring.RatCoeff
(Mathlib.Tactic.Determinant.commSemiringOfCommRing rα))
(dimension : ℕ) (dim... | null | true |
Orientation.rotationAux._proof_1 | 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)) (θ : Real.Angle) (x y : V),
inner ℝ ((θ.cos • LinearMap.id + θ.sin • ↑o.rightAngleRotation.toLinearEquiv) x)
((θ.cos • LinearMap.id + θ.sin • ↑o.rightAngleRota... | null | false |
Valuation.RankLeOne.mk._flat_ctor | Mathlib.RingTheory.Valuation.RankOne | {R : Type u_1} →
{Γ₀ : Type u_2} →
[inst : Ring R] →
[inst_1 : LinearOrderedCommGroupWithZero Γ₀] →
{v : Valuation R Γ₀} → (hom' : (↑v).ValueGroup₀ →*₀ NNReal) → StrictMono ⇑hom' → v.RankLeOne | null | false |
ContinuousLinearMap.bilinear_hasTemperateGrowth | Mathlib.Analysis.Distribution.TemperateGrowth | ∀ {𝕜 : Type u_2} {D : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} [inst : NormedAddCommGroup E]
[inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F]
[inst_4 : NontriviallyNormedField 𝕜] [NormedAlgebra ℝ 𝕜] [inst_6 : NormedAddCommGroup D] [inst_7 : NormedSpace ℝ D]
[i... | The product of two functions of temperate growth is again of temperate growth.
Version for bilinear maps. | true |
_private.Mathlib.Data.List.NodupEquivFin.0.List.sublist_iff_exists_orderEmbedding_getElem?_eq._simp_1_1 | Mathlib.Data.List.NodupEquivFin | ∀ {a b : ℕ}, (a.succ ≤ b.succ) = (a ≤ b) | null | false |
Lean.Meta.Try.Collector.OrdSet.set | Lean.Meta.Tactic.Try.Collect | {α : Type} → [inst : Hashable α] → [inst_1 : BEq α] → Lean.Meta.Try.Collector.OrdSet α → Std.HashSet α | null | true |
_private.Mathlib.Topology.Algebra.InfiniteSum.Defs.0.hasProd_fintype_support._simp_1_1 | Mathlib.Topology.Algebra.InfiniteSum.Defs | ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋂ i, s i) = ∀ (i : ι), x ∈ s i | null | false |
hasStrictDerivAt_abs | Mathlib.Analysis.Calculus.Deriv.Abs | ∀ {x : ℝ}, x ≠ 0 → HasStrictDerivAt (fun x => |x|) (↑(SignType.sign x)) x | null | true |
DistribSMul.toAddMonoidHom_eq_zsmulAddGroupHom | Mathlib.Algebra.Module.NatInt | ∀ (M : Type u_3) [inst : AddCommGroup M], DistribSMul.toAddMonoidHom M = zsmulAddGroupHom | null | true |
Representation.IntertwiningMap.rTensor_zero | Mathlib.RepresentationTheory.Intertwining | ∀ {A : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} {U : Type u_5} [inst : CommSemiring A] [inst_1 : Monoid G]
[inst_2 : AddCommMonoid V] [inst_3 : AddCommMonoid W] [inst_4 : AddCommMonoid U] [inst_5 : Module A V]
[inst_6 : Module A W] [inst_7 : Module A U] {ρ : Representation A G V} {σ : Representation A... | null | true |
Fin.dfoldrM.loop._sunfold | Batteries.Data.Fin.Basic | {m : Type u_1 → Type u_2} →
[Monad m] →
(n : ℕ) →
(α : Fin (n + 1) → Type u_1) →
((i : Fin n) → α i.succ → m (α i.castSucc)) → (i : ℕ) → (h : i < n + 1) → α ⟨i, h⟩ → m (α 0) | null | false |
WellFounded.partialExtrinsicFix₂_eq_partialExtrinsicFix | Init.WFExtrinsicFix | ∀ {α : Sort u_2} {β : α → Sort u_3} {C₂ : (a : α) → β a → Sort u_1} [inst : ∀ (a : α) (b : β a), Nonempty (C₂ a b)]
{R : (a : α) ×' β a → (a : α) ×' β a → Prop}
{F : (a : α) → (b : β a) → ((a' : α) → (b' : β a') → R ⟨a', b'⟩ ⟨a, b⟩ → C₂ a' b') → C₂ a b} {a : α} {b : β a},
Acc R ⟨a, b⟩ →
WellFounded.partialExt... | null | true |
CategoryTheory.Functor.sheafInducedTopologyEquivOfIsCoverDense._proof_2 | Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_6, u_3} C] {D : Type u_1}
[inst_1 : CategoryTheory.Category.{u_4, u_1} D] (G : CategoryTheory.Functor C D)
(K : CategoryTheory.GrothendieckTopology D) (A : Type u_5) [inst_2 : CategoryTheory.Category.{u_2, u_5} A]
[inst_3 : G.LocallyCoverDense K] [inst_4 : G.IsL... | null | false |
matPolyEquiv_symm_map_eval | Mathlib.RingTheory.MatrixPolynomialAlgebra | ∀ {R : Type u_1} [inst : CommSemiring R] {n : Type w} [inst_1 : DecidableEq n] [inst_2 : Fintype n]
(M : Polynomial (Matrix n n R)) (r : R),
(matPolyEquiv.symm M).map (Polynomial.eval r) = Polynomial.eval ((Matrix.scalar n) r) M | null | true |
_private.Mathlib.Algebra.Module.LocalizedModule.Submodule.0.LinearMap.localizedMap_surjective_iff_subsingleton_localized_coker._simp_1_1 | Mathlib.Algebra.Module.LocalizedModule.Submodule | ∀ {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}, (f.range = ⊤) = Function.Surjective ⇑... | null | false |
MonoidWithZero.toOppositeMulActionWithZero | Mathlib.Algebra.GroupWithZero.Action.Defs | (M₀ : Type u_2) → [inst : MonoidWithZero M₀] → MulActionWithZero M₀ᵐᵒᵖ M₀ | Like `MonoidWithZero.toMulActionWithZero`, but multiplies on the right. See also
`Semiring.toOppositeModule` | true |
Module.Basis.dualBasis_coord_toDualEquiv_apply | Mathlib.LinearAlgebra.Dual.Basis | ∀ {R : Type uR} {M : Type uM} {ι : Type uι} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : DecidableEq ι] (b : Module.Basis ι R M) [inst_4 : Finite ι] (i : ι) (f : M),
(b.dualBasis.coord i) (b.toDualEquiv f) = (b.coord i) f | null | true |
Lean.Parser.Module.module.formatter | Lean.Parser.Module.Syntax | Lean.PrettyPrinter.Formatter | null | true |
Module.Basis.noConfusionType | Mathlib.LinearAlgebra.Basis.Defs | Sort u →
{ι : Type u_1} →
{R : Type u_3} →
{M : Type u_4} →
[inst : Semiring R] →
[inst_1 : AddCommMonoid M] →
[inst_2 : Module R M] →
Module.Basis ι R M →
{ι' : Type u_1} →
{R' : Type u_3} →
{M' : Type u_4} →
... | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Equiv.minKey!_eq._simp_1_3 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {k : α},
(k ∈ t) = (Std.DTreeMap.Internal.Impl.contains k t = true) | null | false |
Std.Internal.List.getValueCast_alterKey._proof_1 | Std.Data.Internal.List.Associative | ∀ {α : Type u_2} {β : α → Type u_1} [inst : BEq α] [inst_1 : LawfulBEq α] (k k' : α) (f : Option (β k) → Option (β k))
(l : List ((a : α) × β a)),
Std.Internal.List.DistinctKeys l →
Std.Internal.List.containsKey k' (Std.Internal.List.alterKey k f l) = true →
(k == k') = true → (f (Std.Internal.List.getVal... | null | false |
_private.Mathlib.LinearAlgebra.Matrix.Transvection.0.Matrix.Pivot.reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal._simp_1_2 | Mathlib.LinearAlgebra.Matrix.Transvection | ∀ {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 |
Colex.instSMul'.eq_1 | Mathlib.Algebra.Order.Group.Synonym | ∀ {β : Type u_2} {α : Type u_1} [inst : SMul β α], Colex.instSMul' = Colex.instSMul'._aux_1 | null | true |
_private.Mathlib.NumberTheory.NumberField.House.0.NumberField.house.asiegel.eq_1 | Mathlib.NumberTheory.NumberField.House | ∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] {α : Type u_2} {β : Type u_3}
(a : Matrix α β (NumberField.RingOfIntegers K)) (k : α × (K →+* ℂ)) (l : β × (K →+* ℂ)),
NumberField.house.asiegel✝ K a k l = NumberField.house.a'✝ K a k.1 l.1 l.2 k.2 | null | true |
Con.congr.eq_1 | Mathlib.GroupTheory.Congruence.Basic | ∀ {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst_1 : Mul N] {c : Con M} {d : Con N} (e : M ≃* N)
(h : c = Con.comap ⇑e ⋯ d), Con.congr e h = { toEquiv := Quotient.congr ↑e ⋯, map_mul' := ⋯ } | null | true |
Nat.le_sqrt | Mathlib.Data.Nat.Sqrt | ∀ {m n : ℕ}, m ≤ n.sqrt ↔ m * m ≤ n | null | true |
_private.Init.Data.Int.LemmasAux.0.Int.sub_max_sub_left._proof_1_1 | Init.Data.Int.LemmasAux | ∀ (a b c : ℤ), ¬max (a - b) (a - c) = a - min b c → False | null | false |
Int.lt_of_lt_of_le | Init.Data.Int.Order | ∀ {a b c : ℤ}, a < b → b ≤ c → a < c | null | true |
Qq.SortLocalDecls.State.ctorIdx | Qq.SortLocalDecls | Qq.SortLocalDecls.State → ℕ | null | false |
«_aux_Mathlib_Algebra_Star_StarAlgHom___macroRules_term_→⋆ₙₐ__1» | Mathlib.Algebra.Star.StarAlgHom | Lean.Macro | null | false |
CategoryTheory.Limits.fiberwiseColimitLimitIso._proof_12 | Mathlib.CategoryTheory.Limits.Preserves.Grothendieck | ∀ {C : Type u_5} [inst : CategoryTheory.Category.{u_6, u_5} C] {H : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} H] {J : Type u_2} [inst_2 : CategoryTheory.Category.{u_1, u_2} J]
{F : CategoryTheory.Functor C CategoryTheory.Cat}
(K : CategoryTheory.Functor J (CategoryTheory.Functor (CategoryTheory.Groth... | null | false |
AlgHom.card_le | Mathlib.FieldTheory.Fixed | ∀ {F : Type u_2} {K : Type u_3} [inst : Field F] [inst_1 : Field K] [inst_2 : Algebra F K]
[inst_3 : FiniteDimensional F K], Fintype.card (K →ₐ[F] K) ≤ Module.finrank F K | null | true |
CategoryTheory.finrank_hom_simple_simple_le_one | Mathlib.CategoryTheory.Preadditive.Schur | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] (𝕜 : Type u_2)
[inst_2 : Field 𝕜] [IsAlgClosed 𝕜] [inst_4 : CategoryTheory.Linear 𝕜 C] [CategoryTheory.Limits.HasKernels C] (X Y : C)
[FiniteDimensional 𝕜 (X ⟶ X)] [CategoryTheory.Simple X] [CategoryTheory.Si... | **Schur's lemma** for `𝕜`-linear categories:
if hom spaces are finite dimensional, then the hom space between simples is at most 1-dimensional.
See `finrank_hom_simple_simple_eq_one_iff` and `finrank_hom_simple_simple_eq_zero_iff` below
for the refinements when we know whether or not the simples are isomorphic.
| true |
CategoryTheory.Functor.isLeftAdjoint_comp_iff_right | Mathlib.CategoryTheory.Adjunction.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D)
(G : CategoryTheory.Functor D E) [F.IsEquivalence], (F.comp G).IsLeftAdjoint ↔ G.IsLeftAdjoint | null | true |
Lean.Meta.Grind.instHasAnchorSplitCandidateWithAnchor | Lean.Meta.Tactic.Grind.Split | Lean.Meta.Grind.HasAnchor Lean.Meta.Grind.SplitCandidateWithAnchor | null | true |
DyckWord.firstReturn.eq_1 | Mathlib.Combinatorics.Enumerative.DyckWord | ∀ (p : DyckWord),
p.firstReturn =
List.findIdx
(fun i => decide (List.count DyckStep.U (List.take (i + 1) ↑p) = List.count DyckStep.D (List.take (i + 1) ↑p)))
(List.range (↑p).length) | null | true |
Asymptotics.isBigOTVS_fun_neg_right._simp_1 | Mathlib.Analysis.Asymptotics.TVS | ∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
[inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E] [inst_3 : Module 𝕜 E] [inst_4 : AddCommGroup F]
[inst_5 : TopologicalSpace F] [inst_6 : Module 𝕜 F] {l : Filter α} {f : α → E} {g : α → F} [ContinuousNeg F],
... | null | false |
AddOpposite.forall | Mathlib.Algebra.Opposites | ∀ {α : Type u_1} {p : αᵃᵒᵖ → Prop}, (∀ (a : αᵃᵒᵖ), p a) ↔ ∀ (a : α), p (AddOpposite.op a) | null | true |
ProbabilityTheory.iIndepSets.piiUnionInter_of_notMem | Mathlib.Probability.Independence.Basic | ∀ {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {π : ι → Set (Set Ω)} {a : ι}
{S : Finset ι}, ProbabilityTheory.iIndepSets π μ → a ∉ S → ProbabilityTheory.IndepSets (piiUnionInter π ↑S) (π a) μ | null | true |
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