name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.0.BitVec.reduceAppend._regBuiltin.BitVec.reduceAppend.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.279275497._hygCtx._hyg.24 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec | IO Unit | null | false |
_private.Mathlib.GroupTheory.Commutator.Basic.0.Subgroup.commutator_prod_prod._simp_1_1 | Mathlib.GroupTheory.Commutator.Basic | ∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] {H : Subgroup G} {K : Subgroup N}
{J : Subgroup (G × N)},
(J ≤ H.prod K) = (Subgroup.map (MonoidHom.fst G N) J ≤ H ∧ Subgroup.map (MonoidHom.snd G N) J ≤ K) | null | false |
Std.Http.Status.expectationFailed.elim | Std.Http.Data.Status | {motive : Std.Http.Status → Sort u} →
(t : Std.Http.Status) → t.ctorIdx = 40 → motive Std.Http.Status.expectationFailed → motive t | null | false |
Real.antitone_rpow_of_base_le_one | Mathlib.Analysis.SpecialFunctions.Pow.Real | ∀ {b : ℝ}, 0 < b → b ≤ 1 → Antitone fun x => b ^ x | null | true |
Lean.Widget.InteractiveGoal | Lean.Widget.InteractiveGoal | Type | An interactive tactic-mode goal. | true |
_private.Mathlib.Probability.Process.Stopping.0.MeasureTheory.measurable_stoppedValue._simp_1_4 | Mathlib.Probability.Process.Stopping | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) | null | false |
Function.comp_def._simp_2 | Mathlib.Order.OmegaCompletePartialOrder | ∀ {α : Sort u_1} {β : Sort u_2} {δ : Sort u_3} (f : β → δ) (g : α → β), (fun x => f (g x)) = f ∘ g | null | false |
Membership.ctorIdx | Init.Prelude | {α : outParam (Type u)} → {γ : Type v} → Membership α γ → ℕ | null | false |
TangentBundle.mem_chart_target_iff._simp_1 | Mathlib.Geometry.Manifold.VectorBundle.Tangent | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_4} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [inst_6 : IsManifold I 1 M] (p : H × E)... | null | false |
CategoryTheory.MorphismProperty.under | Mathlib.CategoryTheory.MorphismProperty.Comma | {T : Type u_3} →
[inst : CategoryTheory.Category.{v_3, u_3} T] →
CategoryTheory.MorphismProperty T → {X : T} → CategoryTheory.MorphismProperty (CategoryTheory.Under X) | The morphism property on `Under X` induced by a morphism property on `C`. | true |
AddGrpCat.hasLimits | Mathlib.Algebra.Category.Grp.Limits | CategoryTheory.Limits.HasLimits AddGrpCat | null | true |
Interval.coe_le_coe | Mathlib.Order.Interval.Basic | ∀ {α : Type u_1} [inst : Preorder α] {s t : NonemptyInterval α}, ↑s ≤ ↑t ↔ s ≤ t | null | true |
HomologicalComplex.instIsStrictlySupportedTruncLE | Mathlib.Algebra.Homology.Embedding.TruncLE | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
[inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(K : HomologicalComplex C c') (e : c.Embedding c') [inst_2 : e.IsTruncLE] [inst_3 : ∀ (i' : ι'), K.HasHomology i']
[inst_4 :... | null | true |
Hyperreal.epsilon_ne_zero | Mathlib.Analysis.Real.Hyperreal | Hyperreal.epsilon ≠ 0 | null | true |
String.Slice.ByteIterator.s | Init.Data.String.Iterate | String.Slice.ByteIterator → String.Slice | null | true |
Circle.instIsTopologicalGroup | Mathlib.Analysis.Complex.Circle | IsTopologicalGroup Circle | null | true |
CategoryTheory.ObjectProperty.full_ι | Mathlib.CategoryTheory.ObjectProperty.FullSubcategory | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.ObjectProperty C), P.ι.Full | null | true |
imageToKernel_arrow_apply | Mathlib.Algebra.Homology.ImageToKernel | ∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {A B C : V}
(f : A ⟶ B) [inst_2 : CategoryTheory.Limits.HasImage f] (g : B ⟶ C) [inst_3 : CategoryTheory.Limits.HasKernel g]
(w : CategoryTheory.CategoryStruct.comp f g = 0) {F : V → V → Type uF} {carrier : ... | null | true |
CategoryTheory.Groupoid.isoEquivHom._proof_6 | Mathlib.CategoryTheory.Groupoid | ∀ {C : Type u_2} [inst : CategoryTheory.Groupoid C] (X Y : C),
Function.RightInverse (fun f => { hom := f, inv := CategoryTheory.Groupoid.inv f, hom_inv_id := ⋯, inv_hom_id := ⋯ })
CategoryTheory.Iso.hom | null | false |
QuadraticForm.isometryEquivWeightedSumSquaresWeightedSumSquares._proof_4 | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | ∀ {ι : Type u_1} {S : Type u_3} {R : Type u_2} [inst : CommSemiring R] [inst_1 : Monoid S]
[inst_2 : DistribMulAction S R] (u : ι → Sˣ) (x : ι → R), (fun x => u • x) ((fun x => u⁻¹ • x) x) = x | null | false |
MulEquiv.instMulEquivClass | Mathlib.Algebra.Group.Equiv.Defs | ∀ {M : Type u_4} {N : Type u_5} [inst : Mul M] [inst_1 : Mul N], MulEquivClass (M ≃* N) M N | null | true |
_private.Std.Time.Format.Basic.0.Std.Time.parseQuarterNumber | Std.Time.Format.Basic | Std.Internal.Parsec.String.Parser Std.Time.Month.Quarter | null | true |
TopologicalSpace.Opens.functor_map_eq_inf | Mathlib.Topology.Category.TopCat.Opens | ∀ {X : TopCat} (U V : TopologicalSpace.Opens ↑X),
⋯.functor.obj ((TopologicalSpace.Opens.map U.inclusion').obj V) = V ⊓ U | null | true |
Set.subset_accumulate | Mathlib.Data.Set.Accumulate | ∀ {α : Type u_1} {β : Type u_2} {s : α → Set β} [inst : Preorder α] {x : α}, s x ⊆ Set.accumulate s x | null | true |
Filter.Tendsto.limsup_comp_le_limsup._auto_3 | Mathlib.Order.LiminfLimsup | Lean.Syntax | null | false |
WithAbs.map_apply | Mathlib.Analysis.Normed.Ring.WithAbs | ∀ {R : Type u_1} {S : Type u_2} [inst : Semiring S] [inst_1 : PartialOrder S] [inst_2 : Semiring R]
(v : AbsoluteValue R S) {T : Type u_3} [inst_3 : Semiring T] (w : AbsoluteValue T S) (f : R →+* T) (x : WithAbs v),
(WithAbs.map v w f) x = WithAbs.toAbs w (f x.ofAbs) | null | true |
contDiff_prodMk_right | Mathlib.Analysis.Calculus.ContDiff.Operations | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : WithTop ℕ∞}
(e₀ : E), ContDiff 𝕜 n fun f => (e₀, f) | null | true |
_private.Init.Data.Vector.Lemmas.0.Array.eraseIdx!.eq_1 | Init.Data.Vector.Lemmas | ∀ {α : Type u} (xs : Array α) (i : ℕ),
xs.eraseIdx! i =
if h : i < xs.size then xs.eraseIdx i h
else panicWithPosWithDecl "Init.Data.Array.Basic" "Array.eraseIdx!" 1820 47 "invalid index" | null | true |
_private.Mathlib.Geometry.Convex.ConvexSpace.Defs.0.Convexity.StdSimplex.support_weights_restrict._simp_1_3 | Mathlib.Geometry.Convex.ConvexSpace.Defs | ∀ {α : Type u_1} {M : Type u_4} [inst : Zero M] {f g : α →₀ M}, (f ≠ g) = ∃ a, f a ≠ g a | null | false |
Finset.mem_filter_univ | Mathlib.Data.Fintype.Defs | ∀ {α : Type u_1} [inst : Fintype α] {p : α → Prop} [inst_1 : DecidablePred p] (x : α),
x ∈ Finset.filter p Finset.univ ↔ p x | null | true |
SSet.anodyneExtensions | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | CategoryTheory.MorphismProperty SSet | In the category of simplicial sets, an anodyne extension is a morphism
that has the left lifting property with respect to fibrations, where
a fibration is a morphism that has the right lifting property with respect
to horn inclusions. We do not introduce a typeclass for anodyne extensions
because when the Quillen model... | true |
_private.Mathlib.Data.Bool.Count.0.List.IsChain.count_not_le_count_add_one._proof_1_27 | Mathlib.Data.Bool.Count | ∀ (b head : Bool) (tail : List Bool) (h : List.count b (head :: tail) + 2 ≤ List.count (!b) (head :: tail)),
(List.findIdxs (fun x => decide (x = !b)) (head :: tail))[List.count b (head :: tail) + 1] + 1 ≤
List.findIdx (fun x => decide (x = !b)) (head :: tail) →
(List.findIdxs (fun x => decide (x = !b)) (he... | null | false |
_private.Mathlib.Tactic.Linter.TextBased.0.Mathlib.Linter.TextBased.StyleError.noConfusionType | Mathlib.Tactic.Linter.TextBased | Sort u → Mathlib.Linter.TextBased.StyleError✝ → Mathlib.Linter.TextBased.StyleError✝ → Sort u | null | false |
ContMDiffOn.iterate | Mathlib.Geometry.Manifold.ContMDiff.Basic | ∀ {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] {s : Set M} {n : WithTop ℕ∞} {f : M → M... | The iterates of `C^n` functions on domains are `C^n`. | true |
PEquiv.mk.inj | Mathlib.Data.PEquiv | ∀ {α : Type u} {β : Type v} {toFun : α → Option β} {invFun : β → Option α}
{inv : ∀ (a : α) (b : β), invFun b = some a ↔ toFun a = some b} {toFun_1 : α → Option β} {invFun_1 : β → Option α}
{inv_1 : ∀ (a : α) (b : β), invFun_1 b = some a ↔ toFun_1 a = some b},
{ toFun := toFun, invFun := invFun, inv := inv } = { ... | null | true |
CategoryTheory.Bicategory.OplaxTrans.ComonadBicat | Mathlib.CategoryTheory.Bicategory.Monad.Basic | (B : Type u) → [CategoryTheory.Bicategory B] → Type (max (max (max (max 0 u) v) 0) w) | The bicategory of comonads in `B`. | true |
CategoryTheory.MorphismProperty.overObj | Mathlib.CategoryTheory.MorphismProperty.Comma | {T : Type u_3} →
[inst : CategoryTheory.Category.{v_3, u_3} T] →
CategoryTheory.MorphismProperty T → {X : T} → CategoryTheory.ObjectProperty (CategoryTheory.Over X) | The object property on `Over X` induced by a morphism property. | true |
_private.Mathlib.Analysis.Matrix.Normed.0.Matrix.norm_unitOf | Mathlib.Analysis.Matrix.Normed | ∀ {α : Type u_5} [inst : NormedDivisionRing α] [inst_1 : NormedAlgebra ℝ α] (a : α), ‖Matrix.unitOf✝ a‖₊ = 1 | null | true |
PiNat.cylinder_eq_pi | Mathlib.Topology.MetricSpace.PiNat | ∀ {E : ℕ → Type u_1} (x : (n : ℕ) → E n) (n : ℕ), PiNat.cylinder x n = (↑(Finset.range n)).pi fun i => {x i} | null | true |
instTotallyDisconnectedSpaceSum | Mathlib.Topology.Connected.TotallyDisconnected | ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [TotallyDisconnectedSpace α]
[TotallyDisconnectedSpace β], TotallyDisconnectedSpace (α ⊕ β) | null | true |
Topology.CWComplex.instRelCWComplex._proof_5 | Mathlib.Topology.CWComplex.Classical.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] (C : Set X) [inst_1 : Topology.CWComplex C],
∀ A ⊆ C,
(∀ (n : ℕ) (j : Topology.CWComplex.cell C n),
IsClosed (A ∩ ↑(Topology.CWComplex.map n j) '' Metric.closedBall 0 1)) ∧
IsClosed (A ∩ ∅) →
IsClosed A | null | false |
Nat.fermatNumber | Mathlib.NumberTheory.Fermat | ℕ → ℕ | Fermat numbers: the `n`-th Fermat number is defined as `2^(2^n) + 1`. | true |
Std.Time.instHAddOffsetOffset | Std.Time.Date.Basic | HAdd Std.Time.Nanosecond.Offset Std.Time.Millisecond.Offset Std.Time.Nanosecond.Offset | null | true |
Pi.nnnorm_def | Mathlib.Analysis.Normed.Group.Constructions | ∀ {ι : Type u_1} {G : ι → Type u_4} [inst : Fintype ι] [inst_1 : (i : ι) → SeminormedAddGroup (G i)]
(f : (i : ι) → G i), ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ | null | true |
_private.Mathlib.Order.Filter.Map.0.Filter.comap_comap._simp_1_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α}, (sᶜ ∈ Filter.comap f l) = ((f '' s)ᶜ ∈ l) | null | false |
ωCPO.instLargeCategory._proof_3 | Mathlib.Order.Category.OmegaCompletePartialOrder | ∀ {W X Y Z : ωCPO} (f : W.carrier →𝒄 X.carrier) (g : X.carrier →𝒄 Y.carrier) (h : Y.carrier →𝒄 Z.carrier),
h.comp (g.comp f) = (h.comp g).comp f | null | false |
SimpleGraph.chromaticNumber_le_sum_right | Mathlib.Combinatorics.SimpleGraph.Sum | ∀ {V : Type u_3} {W : Type u_5} {G : SimpleGraph V} {H : SimpleGraph W}, H.chromaticNumber ≤ (G ⊕g H).chromaticNumber | null | true |
MulAction.fixedBy | Mathlib.GroupTheory.GroupAction.Defs | {M : Type u_1} → (α : Type u_3) → [inst : Monoid M] → [MulAction M α] → M → Set α | `fixedBy m` is the set of elements fixed by `m`. | true |
WithTop.top_ne_natCast._simp_1 | Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | ∀ {α : Type u} [inst : AddMonoidWithOne α] (n : ℕ), (⊤ = ↑n) = False | null | false |
DerivedCategory.triangleOfSESδ_naturality | Mathlib.Algebra.Homology.DerivedCategory.ShortExact | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] {S₁ S₂ : CategoryTheory.ShortComplex (CochainComplex C ℤ)} (hS₁ : S₁.ShortExact)
(hS₂ : S₂.ShortExact) (f : S₁ ⟶ S₂),
CategoryTheory.CategoryStruct.comp (DerivedCategory.triangleOfSESδ hS₁)... | null | true |
NonUnitalStarAlgebra.elemental.isClosed | Mathlib.Topology.Algebra.NonUnitalStarAlgebra | ∀ (R : Type u_1) {A : Type u_2} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
[inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : IsScalarTower R A A] [inst_6 : SMulCommClass R A A]
[inst_7 : StarModule R A] [inst_8 : TopologicalSpace A] [inst_9 : IsSemitopologicalSemiring A]
[ins... | null | true |
CategoryTheory.Linear.homCongr_symm_apply | Mathlib.CategoryTheory.Linear.Basic | ∀ (k : Type u_1) {C : Type u_2} [inst : CategoryTheory.Category.{v_1, u_2} C] [inst_1 : Semiring k]
[inst_2 : CategoryTheory.Preadditive C] [inst_3 : CategoryTheory.Linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z)
(f : Y ⟶ Z),
(CategoryTheory.Linear.homCongr k f₁ f₂).symm f =
CategoryTheory.CategoryStruct.... | null | true |
_private.Lean.Meta.Tactic.Grind.PP.0.Lean.Meta.Grind.Result.and.match_1 | Lean.Meta.Tactic.Grind.PP | (motive : Lean.Meta.Grind.Result✝ → Lean.Meta.Grind.Result✝ → Sort u_1) →
(x x_1 : Lean.Meta.Grind.Result✝) →
((x : Lean.Meta.Grind.Result✝) → motive Lean.Meta.Grind.Result.no✝ x) →
((x : Lean.Meta.Grind.Result✝) → motive x Lean.Meta.Grind.Result.no✝) →
((x : Lean.Meta.Grind.Result✝) → motive Lean.M... | null | false |
Std.TreeMap.compare_maxKey?_modify_eq | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
{f : β → β} {km kmm : α} (hkm : t.maxKey? = some km), (t.modify k f).maxKey?.get ⋯ = kmm → cmp kmm km = Ordering.eq | null | true |
ONote.NF.mk | Mathlib.SetTheory.Ordinal.Notation | ∀ {o : ONote}, Exists o.NFBelow → o.NF | null | true |
CategoryTheory.Functor.CoreMonoidal.associativity._autoParam | Mathlib.CategoryTheory.Monoidal.Functor | Lean.Syntax | null | false |
MeasureTheory.definition._@.Mathlib.MeasureTheory.Function.ConditionalLExpectation.118845607._hygCtx._hyg.8 | Mathlib.MeasureTheory.Function.ConditionalLExpectation | {Ω : Type u_1} → {mΩ₀ : MeasurableSpace Ω} → MeasurableSpace Ω → MeasureTheory.Measure Ω → (Ω → ENNReal) → Ω → ENNReal | null | false |
MultilinearMap.domCoprod_apply | Mathlib.LinearAlgebra.Multilinear.TensorProduct | ∀ {R : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} [inst : CommSemiring R] {N₁ : Type u_6} [inst_1 : AddCommMonoid N₁]
[inst_2 : Module R N₁] {N₂ : Type u_7} [inst_3 : AddCommMonoid N₂] [inst_4 : Module R N₂] {N : Type u_8}
[inst_5 : AddCommMonoid N] [inst_6 : Module R N] (a : MultilinearMap R (fun x => N) N₁)
(b :... | null | true |
AddAction.compHom._proof_1 | Mathlib.Algebra.Group.Action.Hom | ∀ {M : Type u_3} {N : Type u_2} (α : Type u_1) [inst : AddMonoid M] [inst_1 : AddAction M α] [inst_2 : AddMonoid N]
(g : N →+ M) (x x_1 : N) (x_2 : α), (x + x_1) +ᵥ x_2 = x +ᵥ x_1 +ᵥ x_2 | null | false |
_private.Init.Meta.Defs.0.Lean.Syntax.structEq._sparseCasesOn_2 | Init.Meta.Defs | {motive_1 : Lean.Syntax → Sort u} →
(t : Lean.Syntax) →
((info : Lean.SourceInfo) →
(kind : Lean.SyntaxNodeKind) → (args : Array Lean.Syntax) → motive_1 (Lean.Syntax.node info kind args)) →
(Nat.hasNotBit 2 t.ctorIdx → motive_1 t) → motive_1 t | null | false |
CategoryTheory.Triangulated.TStructure.truncGELTToLTGE_app_pentagon | 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 |
_private.Mathlib.Order.Category.BddOrd.0.BddOrd.Hom.mk._flat_ctor | Mathlib.Order.Category.BddOrd | {X Y : BddOrd} → BoundedOrderHom ↑X.toPartOrd ↑Y.toPartOrd → X.Hom Y | null | false |
_private.Mathlib.Topology.Connected.Clopen.0.IsClopen.isPreconnected_iff._proof_1_5 | Mathlib.Topology.Connected.Clopen | ∀ {α : Type u_1} {s : Set α} (a b : Set α), s ∩ (a ∩ b) = ∅ → Disjoint (s ∩ a) (s ∩ b) | null | false |
Nat.digits_of_two_le_of_pos | Mathlib.Data.Nat.Digits.Defs | ∀ {n b : ℕ}, 2 ≤ b → 0 < n → b.digits n = n % b :: b.digits (n / b) | null | true |
_private.Mathlib.CategoryTheory.Monoidal.Mod.0.CategoryTheory.AddMod.Hom.ext.match_1 | Mathlib.CategoryTheory.Monoidal.Mod | ∀ {C : Type u_3} {inst : CategoryTheory.Category.{u_1, u_3} C} {inst_1 : CategoryTheory.MonoidalCategory C}
{D : Type u_4} {inst_2 : CategoryTheory.Category.{u_2, u_4} D}
{inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {A : C} {inst_4 : CategoryTheory.AddMonObj A}
{M N : CategoryTheory.AddMod D ... | null | false |
DerivedCategory.isLE_Q_obj_iff | Mathlib.Algebra.Homology.DerivedCategory.TStructure | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] (K : CochainComplex C ℤ) (n : ℤ), (DerivedCategory.Q.obj K).IsLE n ↔ K.IsLE n | null | true |
Lean.instBEqExtraModUse.beq | Lean.ExtraModUses | Lean.ExtraModUse → Lean.ExtraModUse → Bool | null | true |
NonUnitalAlgHom.inr_apply | Mathlib.Algebra.Algebra.NonUnitalHom | ∀ {R : Type u} [inst : Monoid R] {A : Type v} {B : Type w} [inst_1 : NonUnitalNonAssocSemiring A]
[inst_2 : DistribMulAction R A] [inst_3 : NonUnitalNonAssocSemiring B] [inst_4 : DistribMulAction R B] (x : B),
(NonUnitalAlgHom.inr R A B) x = (0, x) | null | true |
Ideal.comap_le_comap_iff_of_surjective | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S] (f : F)
[inst_3 : RingHomClass F R S], Function.Surjective ⇑f → ∀ (I J : Ideal S), Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J | null | true |
Pi.instLattice._proof_2 | Mathlib.Order.Lattice | ∀ {ι : Type u_1} {α' : ι → Type u_2} [inst : (i : ι) → Lattice (α' i)] (a b : (i : ι) → α' i),
SemilatticeInf.inf a b ≤ b | null | false |
OpenPartialHomeomorph.lift_openEmbedding_symm | Mathlib.Topology.OpenPartialHomeomorph.Constructions | ∀ {X : Type u_7} {X' : Type u_8} {Z : Type u_9} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace X']
[inst_2 : TopologicalSpace Z] [inst_3 : Nonempty Z] {f : X → X'} (e : OpenPartialHomeomorph X Z)
(hf : Topology.IsOpenEmbedding f), ↑(e.lift_openEmbedding hf).symm = f ∘ ↑e.symm | null | true |
Std.TreeSet.toList_roo | Std.Data.TreeSet.Slice | ∀ {α : Type u} (cmp : autoParam (α → α → Ordering) Std.TreeSet.toList_roo._auto_1) [Std.TransCmp cmp]
{t : Std.TreeSet α cmp} {lowerBound upperBound : α},
Std.Slice.toList (Std.Roo.Sliceable.mkSlice t lowerBound<...upperBound) =
List.filter (fun e => decide ((cmp e lowerBound).isGT = true ∧ (cmp e upperBound).i... | null | true |
CategoryTheory.Functor.LeftLinear.μₗIso._proof_2 | Mathlib.CategoryTheory.Monoidal.Action.LinearFunctor | ∀ {D : Type u_6} {D' : Type u_2} [inst : CategoryTheory.Category.{u_5, u_6} D]
[inst_1 : CategoryTheory.Category.{u_1, u_2} D'] (F : CategoryTheory.Functor D D') {C : Type u_4}
[inst_2 : CategoryTheory.Category.{u_3, u_4} C] [inst_3 : CategoryTheory.MonoidalCategory C]
[inst_4 : CategoryTheory.MonoidalCategory.Mo... | null | false |
AEMeasurable.iInf | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α]
{mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α]
{ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α},
(∀ (i : ι), AEMeasura... | null | true |
_private.Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction.0.UpperHalfPlane.denom_cocycle._simp_1_2 | Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False | null | false |
Batteries.Tactic._aux_Batteries_Tactic_NoMatch___elabRules_Batteries_Tactic_matchWithDot_1 | Batteries.Tactic.NoMatch | Lean.Elab.Term.TermElab | The syntax `match ⋯ with.` has been deprecated in favor of `nomatch ⋯`.
Both now support multiple discriminants.
| false |
_private.Init.Data.Range.Polymorphic.PRange.0.Std.instDecidableEqRoc.decEq._proof_1 | Init.Data.Range.Polymorphic.PRange | ∀ {α : Type u_1} (a a_1 : α), (a<...=a_1) = a<...=a_1 | null | false |
ProbabilityTheory.«termEVar[_]» | Mathlib.Probability.Moments.Variance | Lean.ParserDescr | The `ℝ≥0∞`-valued variance of the real-valued random variable `X` according to the volume
measure.
This is defined as the Lebesgue integral of `(X - 𝔼[X])^2`. | true |
Array.all_filterMap | Init.Data.Array.Lemmas | ∀ {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Option β} {p : β → Bool},
(Array.filterMap f xs).all p =
xs.all fun a =>
match f a with
| some b => p b
| none => true | null | true |
List.findIdx_add_mem_findIdxs | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {xs : List α} {p : α → Bool} (s : ℕ),
List.findIdx p xs < xs.length → List.findIdx p xs + s ∈ List.findIdxs p xs s | null | true |
Mathlib.Tactic.Ring.Common.evalPow._sunfold | Mathlib.Tactic.Ring.Common | {u : Lean.Level} →
{α : Q(Type u)} →
{bt : Q(«$α») → Type} →
{sα : Q(CommSemiring «$α»)} →
Mathlib.Tactic.Ring.Common.RingCompute bt sα →
Mathlib.Tactic.Ring.Common.RingCompute Mathlib.Tactic.Ring.Common.btℕ Mathlib.Tactic.Ring.Common.sℕ →
{a : Q(«$α»)} →
{b : Q(ℕ... | null | false |
CategoryTheory.Pseudofunctor.StrongTrans.naturality_comp._autoParam | Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo | Lean.Syntax | null | false |
String.getUTF8Byte | Init.Data.String.PosRaw | (s : String) → (p : String.Pos.Raw) → p < s.rawEndPos → UInt8 | Accesses the indicated byte in the UTF-8 encoding of a string.
At runtime, this function is implemented by efficient, constant-time code.
| true |
instBooleanAlgebraSubtypeIsIdempotentElem._proof_15 | Mathlib.Algebra.Order.Ring.Idempotent | ∀ {R : Type u_1} [inst : CommRing R] (x x_1 : { a // IsIdempotentElem a }), x ⊓ ⟨1 - ↑x_1, ⋯⟩ = x ⊓ ⟨1 - ↑x_1, ⋯⟩ | null | false |
_private.Mathlib.Tactic.Translate.Core.0.Mathlib.Tactic.Translate.shouldTranslateUnsafe.visit.match_1 | Mathlib.Tactic.Translate.Core | (motive : Lean.Expr → Sort u_1) →
(f : Lean.Expr) →
((n : Lean.Name) → (us : List Lean.Level) → motive (Lean.Expr.const n us)) →
((fvarId : Lean.FVarId) → motive (Lean.Expr.fvar fvarId)) → ((x : Lean.Expr) → motive x) → motive f | null | false |
Geometry.SimplicialComplex.facets | Mathlib.Analysis.Convex.SimplicialComplex.Basic | {𝕜 : Type u_1} →
{E : Type u_2} →
[inst : Ring 𝕜] →
[inst_1 : PartialOrder 𝕜] →
[inst_2 : AddCommGroup E] → [inst_3 : Module 𝕜 E] → Geometry.SimplicialComplex 𝕜 E → Set (Finset E) | A facet of a simplicial complex is a maximal face. | true |
CategoryTheory.MonoidalCategory.tensorHom_comp_whiskerLeft | Mathlib.CategoryTheory.Monoidal.Category | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.MonoidalCategory C] {V W X Y Z : C}
(f : V ⟶ W) (g : X ⟶ Y) (h : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f g)
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft W h) =
Cate... | null | true |
CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac_assoc | Mathlib.Algebra.Homology.ExactSequenceFour | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Balanced C] {n : ℕ} {S : CategoryTheory.ComposableArrows C (n + 3)} (hS : S.Exact) (k : ℕ)
(hk : autoParam (k ≤ n) CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac._auto_1)
... | null | true |
_private.Mathlib.Combinatorics.Matroid.Basic.0.Matroid.singleton_subset_ground | Mathlib.Combinatorics.Matroid.Basic | ∀ {α : Type u_1} {M : Matroid α} {e : α}, e ∈ M.E → {e} ⊆ M.E | null | true |
CategoryTheory.Presieve.isSheaf_sup | Mathlib.CategoryTheory.Sites.Coverage | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{v_1, u_2} C] (K L : CategoryTheory.Coverage C)
(P : CategoryTheory.Functor Cᵒᵖ (Type u_1)),
CategoryTheory.Presieve.IsSheaf (K ⊔ L).toGrothendieck P ↔
CategoryTheory.Presieve.IsSheaf K.toGrothendieck P ∧ CategoryTheory.Presieve.IsSheaf L.toGrothendieck P | A presheaf is a sheaf for the Grothendieck topology generated by a union of coverages iff it is a
sheaf for the Grothendieck topology generated by each coverage separately.
| true |
Lean.Meta.NormCast.Label.ofNat_ctorIdx | Lean.Meta.Tactic.NormCast | ∀ (x : Lean.Meta.NormCast.Label), Lean.Meta.NormCast.Label.ofNat x.ctorIdx = x | null | true |
_private.Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv.0.StarAlgEquiv.eq_linearIsometryEquivConjStarAlgEquiv._simp_1_3 | Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | ∀ {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [inst : Semiring R₁] [inst_1 : Semiring R₂] [inst_2 : Semiring R₃]
{σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [inst_3 : TopologicalSpace M₁]
[inst_4 : AddCommMonoid M₁] {M₂ : Type u_6} [inst_5 : TopologicalSpace M₂] [inst_6 : AddCommMonoid... | null | false |
TopPair.incl._proof_2 | Mathlib.Topology.Category.TopPair | ∀ (X : TopCat),
TopPair.ofHom (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id TopPair.snd) ⋯ =
CategoryTheory.CategoryStruct.id (TopPair.ofTopCat X) | null | false |
ContinuousMultilinearMap.toContinuousLinearMap | Mathlib.Topology.Algebra.Module.Multilinear.Basic | {R : Type u} →
{ι : Type v} →
{M₁ : ι → Type w₁} →
{M₂ : Type w₂} →
[inst : Semiring R] →
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] →
[inst_2 : AddCommMonoid M₂] →
[inst_3 : (i : ι) → Module R (M₁ i)] →
[inst_4 : Module R M₂] →
[i... | If `f` is a continuous multilinear map, then `f.toContinuousLinearMap m i` is the continuous
linear map obtained by fixing all coordinates but `i` equal to those of `m`, and varying the
`i`-th coordinate. | true |
Array.uget.eq_1 | Init.Data.Array.Basic | ∀ {α : Type u} (xs : Array α) (i : USize) (h : i.toNat < xs.size), xs.uget i h = xs[i.toNat] | null | true |
HomologicalComplex₂.ιTotal_totalFlipIso_f_hom_assoc | Mathlib.Algebra.Homology.TotalComplexSymmetry | ∀ {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂}
(K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [inst_2 : TotalComplexShape c₁ c₂ c]
[inst_3 : TotalComplexShap... | null | true |
CategoryTheory.MonoidalCategory.MonoidalLeftAction.monoidalOppositeLeftAction._proof_6 | Mathlib.CategoryTheory.Monoidal.Action.Opposites | ∀ (C : Type u_1) (D : Type u_4) [inst : CategoryTheory.Category.{u_2, u_1} C]
[inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Category.{u_3, u_4} D]
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalRightAction C D] {c c' c'' : Cᴹᵒᵖ} {d d' d'' : D} (f₁ : c ⟶ c')
(f₂ : c' ⟶ c'') (g₁ : d ⟶ d')... | null | false |
Nat.coprime_two_right._simp_1 | Mathlib.Data.Nat.Prime.Basic | ∀ {n : ℕ}, n.Coprime 2 = Odd n | null | false |
List.eraseP_subset | Init.Data.List.Erase | ∀ {α : Type u_1} {p : α → Bool} {l : List α}, List.eraseP p l ⊆ l | null | true |
instAddCommGroupWithOneGradedTensorProduct._proof_30 | Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | ∀ (R : Type u_3) {ι : Type u_4} {A : Type u_1} {B : Type u_2} [inst : CommSemiring ι] [inst_1 : DecidableEq ι]
[inst_2 : CommRing R] [inst_3 : Ring A] [inst_4 : Ring B] [inst_5 : Algebra R A] [inst_6 : Algebra R B]
(𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [inst_7 : GradedAlgebra 𝒜] [inst_8 : GradedAlgebra ... | null | false |
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