name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ImportGraph.Shake.ConstLocation.ctorIdx | ImportGraph.Shake.DeclNeeds | ImportGraph.Shake.ConstLocation → ℕ | null | false |
Metric.tendstoUniformlyOnFilter_iff | Mathlib.Topology.MetricSpace.Pseudo.Basic | ∀ {α : Type u} {β : Type v} {ι : Type u_1} [inst : PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι}
{p' : Filter β}, TendstoUniformlyOnFilter F f p p' ↔ ∀ ε > 0, ∀ᶠ (n : ι × β) in p ×ˢ p', dist (f n.2) (F n.1 n.2) < ε | Expressing uniform convergence using `dist` | true |
tsub_pos_iff_lt | Mathlib.Algebra.Order.Sub.Basic | ∀ {α : Type u_1} [inst : AddCommMonoid α] [inst_1 : LinearOrder α] [CanonicallyOrderedAdd α] [inst_3 : Sub α]
[OrderedSub α] {a b : α}, 0 < a - b ↔ b < a | null | true |
TopologicalSpace.PositiveCompacts.interior_nonempty | Mathlib.Topology.Sets.Compacts | ∀ {α : Type u_1} [inst : TopologicalSpace α] (s : TopologicalSpace.PositiveCompacts α), (interior ↑s).Nonempty | null | true |
_private.Init.Data.String.Pattern.String.0.String.Slice.Pattern.ForwardSliceSearcher.buildTable._proof_10 | Init.Data.String.Pattern.String | ∀ (pat : String.Slice) (table : Array ℕ) (guess : ℕ) (hg : guess < table.size),
table[guess - 1] < guess → ¬guess - 1 < table.size → False | null | false |
_private.Mathlib.Tactic.ComputeDegree.0.Mathlib.Tactic.ComputeDegree.getCongrLemma._sparseCasesOn_3 | Mathlib.Tactic.ComputeDegree | {motive : Lean.Name → Sort u} →
(t : Lean.Name) →
((pre : Lean.Name) → (str : String) → motive (pre.str str)) →
motive Lean.Name.anonymous → (Nat.hasNotBit 3 t.ctorIdx → motive t) → motive t | null | false |
Lean.instInhabitedImport | Lean.Setup | Inhabited Lean.Import | null | true |
CategoryTheory.Pretriangulated.Triangle.add_hom₁ | Mathlib.CategoryTheory.Triangulated.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.HasShift C ℤ]
{T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C} [inst_2 : CategoryTheory.Preadditive C]
[inst_3 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (f g : T₁ ⟶ T₂), (f + g).hom₁ = f.hom₁ + g.hom₁ | null | true |
exists_nhds_one_split | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : MulOneClass M] [ContinuousMul M] {s : Set M},
s ∈ nhds 1 → ∃ V ∈ nhds 1, ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | null | true |
_private.Mathlib.Geometry.Convex.ConvexSpace.Defs.0.Convexity.StdSimplex.restrict_singleton._simp_1_6 | Mathlib.Geometry.Convex.ConvexSpace.Defs | ∀ {α : Type u_1} {M : Type u_4} [inst : Zero M] {f : α →₀ M} {a : α}, (a ∈ f.support) = (f a ≠ 0) | null | false |
List.pairwise_of_forall_sublist | Init.Data.List.Pairwise | ∀ {α : Type u_1} {l : List α} {R : α → α → Prop}, (∀ {a b : α}, [a, b].Sublist l → R a b) → List.Pairwise R l | null | true |
catalan_one | Mathlib.Combinatorics.Enumerative.Catalan.Basic | catalan 1 = 1 | null | true |
cfc_predicate_algebraMap | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | ∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
[inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
[inst_7 : StarRing A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R ... | null | true |
_private.Mathlib.Analysis.Meromorphic.TrailingCoefficient.0.MeromorphicAt.meromorphicTrailingCoeffAt_add_eq_add._proof_1_1 | Mathlib.Analysis.Meromorphic.TrailingCoefficient | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {x : 𝕜} {f₁ : 𝕜 → E}, ¬meromorphicOrderAt f₁ x = ⊤ → ¬meromorphicOrderAt f₁ x = ⊤ | null | false |
AddSubmonoid.topologicalClosure._proof_2 | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_1} [inst : TopologicalSpace M] [inst_1 : AddMonoid M] (s : AddSubmonoid M), 0 ∈ closure ↑s | null | false |
Graph.IsSubgraph.casesOn | Mathlib.Combinatorics.Graph.Subgraph | {α : Type u_1} →
{β : Type u_2} →
{H G : Graph α β} →
{motive : H.IsSubgraph G → Sort u} →
(t : H.IsSubgraph G) →
((vertexSet_mono : H.vertexSet ⊆ G.vertexSet) →
(isLink_mono : ∀ ⦃e : β⦄ ⦃x y : α⦄, H.IsLink e x y → G.IsLink e x y) → motive ⋯) →
motive t | null | false |
Mathlib.Linter.linter.style.show | Mathlib.Tactic.Linter.Style | Lean.Option Bool | The "show" linter emits a warning if the `show` tactic changed the goal. `show` should only be used
to indicate intermediate goal states for proof readability. When the goal is actually changed,
`change` should be preferred.
| true |
SetLike.GradeZero.instSemiring._proof_3 | Mathlib.Algebra.DirectSum.Internal | ∀ {ι : Type u_3} {σ : Type u_2} {R : Type u_1} [inst : Semiring R] [inst_1 : AddMonoid ι] [inst_2 : SetLike σ R]
[inst_3 : AddSubmonoidClass σ R] (A : ι → σ) [inst_4 : SetLike.GradedMonoid A] (a b c : ↥(A 0)),
a * (b + c) = a * b + a * c | null | false |
SetLike.GradeZero.submonoid._proof_1 | Mathlib.Algebra.GradedMonoid | ∀ {ι : Type u_3} {R : Type u_1} {S : Type u_2} [inst : SetLike S R] [inst_1 : Monoid R] [inst_2 : AddMonoid ι]
(A : ι → S) [SetLike.GradedMonoid A] {a b : R}, a ∈ ↑(A 0) → b ∈ ↑(A 0) → a * b ∈ ↑(A 0) | null | false |
Rep.resFunctor | Mathlib.RepresentationTheory.Rep.Res | {k : Type u} →
[inst : Semiring k] →
{G : Type v1} →
{H : Type v2} →
[inst_1 : Monoid G] →
[inst_2 : Monoid H] → (H →* G) → CategoryTheory.Functor (Rep.{t, u, v1} k G) (Rep.{t, u, v2} k H) | The restriction functor `Rep R G ⥤ Rep R H` for a subgroup `H` of `G`. | true |
_private.Mathlib.CategoryTheory.Presentable.IsCardinalFiltered.0.CategoryTheory.IsCardinalFiltered.of_final._simp_1_1 | Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | ∀ {α : Type u_1} {β : Type u_2} {p : α × β → Prop}, (∀ (x : α × β), p x) = ∀ (a : α) (b : β), p (a, b) | null | false |
AffineIsometryEquiv.comp_continuousOn_iff._simp_1 | Mathlib.Analysis.Normed.Affine.Isometry | ∀ {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_7} {P₂ : Type u_8} [inst : NormedField 𝕜]
[inst_1 : SeminormedAddCommGroup V] [inst_2 : NormedSpace 𝕜 V] [inst_3 : PseudoMetricSpace P]
[inst_4 : NormedAddTorsor V P] [inst_5 : SeminormedAddCommGroup V₂] [inst_6 : NormedSpace 𝕜 V₂]
[inst_7 : PseudoMe... | null | false |
CompleteSublattice.comap._proof_2 | Mathlib.Order.CompleteSublattice | ∀ {α : Type u_1} {β : Type u_2} [inst : CompleteLattice α] [inst_1 : CompleteLattice β] (f : CompleteLatticeHom α β)
(L : CompleteSublattice β), InfClosed (⇑f ⁻¹' ↑L.toSublattice) | null | false |
Finset.Iic_mul_Iic_subset' | Mathlib.Algebra.Group.Pointwise.Finset.Interval | ∀ {α : Type u_1} [inst : Mul α] [inst_1 : Preorder α] [inst_2 : DecidableEq α] [MulLeftMono α] [MulRightMono α]
[inst_5 : LocallyFiniteOrderBot α] (a b : α), Finset.Iic a * Finset.Iic b ⊆ Finset.Iic (a * b) | null | true |
FractionalIdeal.zero_le | Mathlib.RingTheory.FractionalIdeal.Basic | ∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
(I : FractionalIdeal S P), 0 ≤ I | null | true |
_private.Batteries.Data.List.Lemmas.0.List.exists_of_replaceF.match_1_1 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} (x : α) (motive : none = some x → Prop) (x : none = some x), motive x | null | false |
_private.Lean.Parser.Term.0.Lean.Parser.Term.borrowed._regBuiltin.Lean.Parser.Term.borrowed.formatter_9 | Lean.Parser.Term | IO Unit | null | false |
Matrix.rank.eq_1 | Mathlib.LinearAlgebra.Matrix.Rank | ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Fintype n] [inst_1 : CommSemiring R] (A : Matrix m n R),
A.rank = Module.finrank R ↥A.mulVecLin.range | null | true |
instNoZeroDivisorsENat._proof_1 | Mathlib.Data.ENat.Monoid | NoZeroDivisors ℕ∞ | null | false |
MinTropical.instZeroOfTop.eq_1 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u} [inst : Top R], MinTropical.instZeroOfTop = { zero := MinTropical.trop ⊤ } | null | true |
String.startsWith_string_iff | Init.Data.String.Lemmas.Pattern.TakeDrop.String | ∀ {pat s : String}, s.startsWith pat = true ↔ pat.toList <+: s.toList | null | true |
CategoryTheory.Pi.δ_def | Mathlib.CategoryTheory.Pi.Monoidal | ∀ {I : Type w₁} {C : I → Type u₁} [inst : (i : I) → CategoryTheory.Category.{v₁, u₁} (C i)]
[inst_1 : (i : I) → CategoryTheory.MonoidalCategory (C i)] (i : I) (X Y : (i : I) → C i),
CategoryTheory.Functor.OplaxMonoidal.δ (CategoryTheory.Pi.eval C i) X Y =
CategoryTheory.CategoryStruct.id (CategoryTheory.Monoida... | null | true |
MeasurableDiv.measurable_const_div._autoParam | Mathlib.MeasureTheory.Group.Arithmetic | Lean.Syntax | null | false |
sbtw_sub_const_iff | Mathlib.Analysis.Convex.Between | ∀ {R : Type u_1} {V : Type u_2} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : AddCommGroup V]
[inst_3 : Module R V] {x y z : V} (v : V), Sbtw R (x - v) (y - v) (z - v) ↔ Sbtw R x y z | null | true |
groupHomology.H0 | Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {k G : Type u} → [inst : CommRing k] → [inst_1 : Group G] → Rep.{u, u, u} k G → ModuleCat k | Shorthand for the 0th group homology of a `k`-linear `G`-representation `A`, `H₀(G, A)`,
defined as the 0th homology of the complex of inhomogeneous chains of `A`. | true |
SheafOfModules.IsLocallyFree.recOn | Mathlib.Algebra.Category.ModuleCat.Sheaf.LocallyFree | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{J : CategoryTheory.GrothendieckTopology C} →
{R : CategoryTheory.Sheaf J RingCat} →
[inst_1 : ∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrpCat] →
[inst_2 : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddComm... | null | false |
tendsto_birkhoffAverage_apply_sub_birkhoffAverage | Mathlib.Dynamics.BirkhoffSum.NormedSpace | ∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] (𝕜 : Type u_3) [inst_1 : RCLike 𝕜]
[inst_2 : NormedSpace 𝕜 E] {f : α → α} {g : α → E} {x : α},
Bornology.IsBounded (Set.range fun x_1 => g (f^[x_1] x)) →
Filter.Tendsto (fun n => birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) Filter.... | If a function `g` is bounded along the positive orbit of `x` under `f`,
then the difference between Birkhoff averages of `g`
along the orbit of `f x` and along the orbit of `x`
tends to zero.
See also `tendsto_birkhoffAverage_apply_sub_birkhoffAverage'`. | true |
Lat.recOn | Mathlib.Order.Category.Lat | {motive : Lat → Sort u} →
(t : Lat) → ((carrier : Type u_1) → [str : Lattice carrier] → motive { carrier := carrier, str := str }) → motive t | null | false |
Sym2.sortEquiv._proof_1 | Mathlib.Order.Sym2 | ∀ {α : Type u_1} [inst : LinearOrder α] (a b : α),
(↑((fun s => ⟨(s.inf, s.sup), ⋯⟩) s(a, b))).1 = a ∧ (↑((fun s => ⟨(s.inf, s.sup), ⋯⟩) s(a, b))).2 = b ∨
(↑((fun s => ⟨(s.inf, s.sup), ⋯⟩) s(a, b))).1 = b ∧ (↑((fun s => ⟨(s.inf, s.sup), ⋯⟩) s(a, b))).2 = a | null | false |
_private.Mathlib.Analysis.CStarAlgebra.Extreme.0.isStarProjection_iff_mem_extremePoints_setOfPred_nonneg_inter_unitClosedBall._proof_1_11 | Mathlib.Analysis.CStarAlgebra.Extreme | ∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] (a : A),
0 ≤ a → ‖a‖ ≤ 1 → ∀ (b : A), 0 ≤ b → ‖b‖ ≤ 1 → ↑a ∈ Set.Icc 0 1 ∧ ↑b ∈ Set.Icc 0 1 | null | false |
WeierstrassCurve.isCharThreeJNeZeroNF_iff | Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | ∀ {R : Type u_1} [inst : CommRing R] (W : WeierstrassCurve R), W.IsCharThreeJNeZeroNF ↔ W.a₁ = 0 ∧ W.a₃ = 0 ∧ W.a₄ = 0 | null | true |
EuclideanGeometry.Sphere.nonempty_iff | Mathlib.Geometry.Euclidean.Sphere.Basic | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [NormedSpace ℝ V] [inst_2 : MetricSpace P]
[NormedAddTorsor V P] [Nontrivial V] {s : EuclideanGeometry.Sphere P},
(Metric.sphere s.center s.radius).Nonempty ↔ 0 ≤ s.radius | null | true |
AddEquiv.coprodAssoc | Mathlib.GroupTheory.Coprod.Basic | (M : Type u_1) →
(N : Type u_2) →
(P : Type u_3) →
[inst : AddMonoid M] →
[inst_1 : AddMonoid N] →
[inst_2 : AddMonoid P] →
AddMonoid.Coprod (AddMonoid.Coprod M N) P ≃+ AddMonoid.Coprod M (AddMonoid.Coprod N P) | An additive equivalence between `AddMonoid.Coprod (AddMonoid.Coprod M N) P` and
`AddMonoid.Coprod M (AddMonoid.Coprod N P)`. | true |
CategoryTheory.Pairwise.diagramMap._proof_1 | Mathlib.CategoryTheory.Category.Pairwise | ∀ {ι : Type u_2} {α : Type u_1} (U : ι → α) [inst : SemilatticeInf α] (i j : ι), U i ⊓ U j ≤ U i | null | false |
Std.DTreeMap.Internal.Impl.Const.get_insertMany!_list_of_contains_eq_false | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {β : Type v} {t : Std.DTreeMap.Internal.Impl α fun x => β} [inst : Std.TransOrd α]
[inst_1 : BEq α] [inst_2 : Std.LawfulBEqOrd α] (h : t.WF) {l : List (α × β)} {k : α}
(h₁ : (List.map Prod.fst l).contains k = false)
{h' : Std.DTreeMap.Internal.Impl.contains k ↑(Std.DTreeMap.Intern... | null | true |
Computation.runFor | Mathlib.Data.Seq.Computation | {α : Type u} → Computation α → ℕ → Option α | `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`
if it did not terminate after `n` steps. | true |
FirstOrder.Language.Term.ctorIdx | Mathlib.ModelTheory.Syntax | {L : FirstOrder.Language} → {α : Type u'} → L.Term α → ℕ | null | false |
Algebra.intNorm_eq_zero._simp_1 | Mathlib.RingTheory.IntegralClosure.IntegralRestrict | ∀ {A : Type u_1} {B : Type u_6} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B]
[inst_3 : IsIntegrallyClosed A] [inst_4 : IsDomain A] [inst_5 : IsDomain B] [inst_6 : IsIntegrallyClosed B]
[inst_7 : Algebra.IsIntegral A B] [inst_8 : Module.IsTorsionFree A B]
[FiniteDimensional (FractionRing A) (Fr... | null | false |
Subgroup.conj_smul_subgroupOf | Mathlib.Algebra.Group.Subgroup.Pointwise | ∀ {G : Type u_2} [inst : Group G] {P H : Subgroup G},
P ≤ H → ∀ (h : ↥H), MulAut.conj h • P.subgroupOf H = (MulAut.conj ↑h • P).subgroupOf H | null | true |
Lean.Meta.Grind.Arith.CommRing.EqCnstrProof.mul.sizeOf_spec | Lean.Meta.Tactic.Grind.Arith.CommRing.Types | ∀ (k : ℤ) (e : Lean.Meta.Grind.Arith.CommRing.EqCnstr),
sizeOf (Lean.Meta.Grind.Arith.CommRing.EqCnstrProof.mul k e) = 1 + sizeOf k + sizeOf e | null | true |
Filter.ofCountableUnion.congr_simp | Mathlib.Topology.Compactness.Lindelof | ∀ {α : Type u_2} (l l_1 : Set (Set α)) (e_l : l = l_1)
(hUnion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l) (hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l),
Filter.ofCountableUnion l hUnion hmono = Filter.ofCountableUnion l_1 ⋯ ⋯ | null | true |
Submodule.toAddSubgroup_eq_top | Mathlib.Algebra.Module.Submodule.Lattice | ∀ {R : Type u_4} {M : Type u_5} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {p : Submodule R M},
p.toAddSubgroup = ⊤ ↔ p = ⊤ | null | true |
Matrix.Nonsingular.linearIndependent_col | Mathlib.LinearAlgebra.Matrix.Nonsingular | ∀ {R : Type u_1} {n : Type u_3} [inst : CommSemiring R] [inst_1 : Fintype n] [inst_2 : DecidableEq n] {A : Matrix n n R}
[IsCancelAdd R], A.Nonsingular → LinearIndependent R A.col | If a matrix over a commutative semiring with cancellative addition is nonsingular, then its
columns are linearly independent. Generalizes `Matrix.linearIndependent_cols_of_det_ne_zero`. | true |
AddAction.isPretransitive_compHom | Mathlib.Algebra.Group.Action.Hom | ∀ {E : Type u_4} {F : Type u_5} {G : Type u_6} [inst : AddMonoid E] [inst_1 : AddMonoid F] [inst_2 : AddAction F G]
[AddAction.IsPretransitive F G] {f : E →+ F}, Function.Surjective ⇑f → AddAction.IsPretransitive E G | null | true |
IsAddCyclic.card_addOrderOf_eq_totient | Mathlib.GroupTheory.SpecificGroups.Cyclic | ∀ {α : Type u_1} [inst : AddGroup α] [IsAddCyclic α] [inst_2 : Fintype α] {d : ℕ},
d ∣ Fintype.card α → {a | addOrderOf a = d}.card = d.totient | null | true |
Std.Time.Modifier.k.injEq | Std.Time.Format.Modifier | ∀ (presentation presentation_1 : Std.Time.Number),
(Std.Time.Modifier.k presentation = Std.Time.Modifier.k presentation_1) = (presentation = presentation_1) | null | true |
CategoryTheory.StrongEpiCategory.recOn | Mathlib.CategoryTheory.Limits.Shapes.StrongEpi | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{motive : CategoryTheory.StrongEpiCategory C → Sort u_1} →
(t : CategoryTheory.StrongEpiCategory C) →
((strongEpi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f], CategoryTheory.StrongEpi f) → motive ⋯) →
motive t | null | false |
OrderDual.instAddCommMonoid | Mathlib.Algebra.Order.Group.Synonym | {α : Type u_1} → [AddCommMonoid α] → AddCommMonoid αᵒᵈ | null | true |
ContinuousMap.UniqueHom.rec | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {R : Type u_1} →
{A : Type u_2} →
[inst : CommSemiring R] →
[inst_1 : StarRing R] →
[inst_2 : MetricSpace R] →
[inst_3 : IsTopologicalSemiring R] →
[inst_4 : ContinuousStar R] →
[inst_5 : Ring A] →
[inst_6 : StarRing A] →
[inst_7 ... | null | false |
Function.bijective_of_subsingleton._simp_1 | Mathlib.Logic.Function.Basic | ∀ {α : Sort u_1} [Subsingleton α] (f : α → α), Function.Bijective f = True | null | false |
_private.Lean.Widget.UserWidget.0.Lean.Widget.initFn._@.Lean.Widget.UserWidget.1579414551._hygCtx._hyg.2 | Lean.Widget.UserWidget | IO Lean.AttributeImpl | null | false |
Lean.instInhabitedFVarIdSet | Lean.Expr | Inhabited Lean.FVarIdSet | null | true |
_private.Mathlib.Topology.Compactness.Lindelof.0.IsLindelof.adherence_nhdset.match_1_1 | Mathlib.Topology.Compactness.Lindelof | ∀ {X : Type u_1} [inst : TopologicalSpace X] {s t : Set X} {f : Filter X}
(motive : (∃ x ∈ s, ClusterPt x (f ⊓ Filter.principal tᶜ)) → Prop)
(x : ∃ x ∈ s, ClusterPt x (f ⊓ Filter.principal tᶜ)),
(∀ (x : X) (hx : x ∈ s) (hfx : ClusterPt x (f ⊓ Filter.principal tᶜ)), motive ⋯) → motive x | null | false |
Vector.Perm.extract | Init.Data.Vector.Perm | ∀ {α : Type u_1} {n : ℕ} {xs ys : Vector α n},
xs.Perm ys →
∀ {lo hi : ℕ},
(∀ i < lo, xs[i]? = ys[i]?) → (∀ (i : ℕ), hi ≤ i → xs[i]? = ys[i]?) → (xs.extract lo hi).Perm (ys.extract lo hi) | null | true |
AlgebraicGeometry.Scheme.Modules.instAbelian._proof_8 | Mathlib.AlgebraicGeometry.Modules.Sheaf | ∀ {X : AlgebraicGeometry.Scheme}, CategoryTheory.Limits.HasKernels X.Modules | null | false |
_private.Batteries.Data.Fin.Coding.0.Fin.encodeSubtype._proof_1 | Batteries.Data.Fin.Coding | ∀ (n : ℕ) (P : Fin (n + 1) → Prop) (inst : DecidablePred P),
∀ k < Fin.countP fun x => decide (P x.succ), k.succ < (Fin.countP fun i => decide (P i.succ)).succ | null | false |
Std.ExtDHashMap.get?_diff | Std.Data.ExtDHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : α → Type v} {m₁ m₂ : Std.ExtDHashMap α β} [inst : LawfulBEq α]
{k : α}, (m₁ \ m₂).get? k = if k ∈ m₂ then none else m₁.get? k | null | true |
irrational_div_intCast_iff._simp_1 | Mathlib.NumberTheory.Real.Irrational | ∀ {m : ℤ} {x : ℝ}, Irrational (x / ↑m) = (m ≠ 0 ∧ Irrational x) | null | false |
MeasureTheory.VectorMeasure.setIntegral_eq_zero_of_ae_eq_zero | Mathlib.MeasureTheory.VectorMeasure.SetIntegral | ∀ {X : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E]
[inst_1 : NormedAddCommGroup F] [inst_2 : NormedAddCommGroup G] {μ : MeasureTheory.VectorMeasure X F} {f : X → E}
{t : Set X} [inst_3 : NormedSpace ℝ E] [inst_4 : NormedSpace ℝ F] [inst_5 : NormedSpa... | null | true |
_private.Mathlib.Topology.VectorBundle.Riemannian.0.eventually_norm_symmL_trivializationAt_self_comp_lt._simp_1_1 | Mathlib.Topology.VectorBundle.Riemannian | ∀ {𝕜₁ : Type u_1} [inst : NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [inst_1 : NontriviallyNormedField 𝕜₂]
(σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedSpace 𝕜₁ F₁]
{E₁ : B → Type u_5} [inst_4 : (x : B) → AddCommGroup (E₁ x)] [inst_5 : (x : B) → Module 𝕜₁ ... | null | false |
CategoryTheory.GlueData.ofGlueData'._proof_6 | Mathlib.CategoryTheory.GlueData | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (D : CategoryTheory.GlueData' C) (i j k : D.J),
CategoryTheory.CategoryStruct.comp (D.t'' i j k) (CategoryTheory.Limits.pullback.snd (D.f' j k) (D.f' j i)) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst (D.f' i j) (D.f' i k))... | null | false |
Filter.not_isBoundedUnder_of_tendsto_atBot | Mathlib.Order.Filter.IsBounded | ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α} [l.NeBot],
Filter.Tendsto f l Filter.atBot → ¬Filter.IsBoundedUnder (fun x1 x2 => x2 ≤ x1) l f | null | true |
CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.fac._proof_8 | Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | ∀ (n₁ n₂ : ℤ),
autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.fac._auto_3 → n₁ + 1 = n₂ | null | false |
Algebra.Presentation.instCommRingCore._proof_71 | Mathlib.RingTheory.Extension.Presentation.Core | ∀ {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : Algebra R S] {P : Algebra.Presentation R S ι σ},
autoParam (∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)) AddGroupWithOne.intCast_negSucc._autoParam | null | false |
_private.Lean.Elab.Tactic.Simp.0.Lean.Elab.Tactic.instEvalExprConfigCtx | Lean.Elab.Tactic.Simp | Lean.Elab.ConfigEval.EvalExpr Lean.Meta.Simp.ConfigCtx | null | true |
TopologicalSpace.MetrizableSpace.recOn | Mathlib.Topology.Metrizable.Basic | {X : Type u_5} →
[t : TopologicalSpace X] →
{motive : TopologicalSpace.MetrizableSpace X → Sort u} →
(t_1 : TopologicalSpace.MetrizableSpace X) →
([toPseudoMetrizableSpace : TopologicalSpace.PseudoMetrizableSpace X] → [toT0Space : T0Space X] → motive ⋯) →
motive t_1 | null | false |
Lean.Meta.Grind.AC.EqCnstrProof.superpose_tail_idempotent | Lean.Meta.Tactic.Grind.AC.Types | Lean.Grind.AC.Var → Lean.Meta.Grind.AC.EqCnstr → Lean.Meta.Grind.AC.EqCnstrProof | null | true |
_private.Mathlib.Analysis.SpecificLimits.Normed.0.TFAE_exists_lt_isLittleO_pow.match_1_14 | Mathlib.Analysis.SpecificLimits.Normed | ∀ (f : ℕ → ℝ) (R : ℝ) (motive : (∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n) → Prop)
(x : ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n),
(∀ (a : ℝ) (ha : a ∈ Set.Ioo 0 R) (H : ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n), motive ⋯) → motive x | null | false |
HahnSeries.SummableFamily.powers._proof_2 | Mathlib.RingTheory.HahnSeries.Summable | ∀ {Γ : Type u_2} {R : Type u_1} [inst : AddCommMonoid Γ] [inst_1 : LinearOrder Γ] [inst_2 : IsOrderedCancelAddMonoid Γ]
[inst_3 : CommRing R] (x : HahnSeries Γ R) (g : Γ), {a | ((if 0 < x.orderTop then x else 0) ^ a).coeff g ≠ 0}.Finite | null | false |
SSet.prodStdSimplex.pairingCore.IsType₂.simplex_snd_apply | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | ∀ {m : ℕ} {k : Fin (m + 1)} {n : ℕ} {x : ((SSet.horn (m + 1) k.castSucc).unionProd (SSet.boundary n)).N}
(hx : SSet.prodStdSimplex.pairingCore.IsType₂ x) {d : ℕ} (hd : x.dim = d) (i : Fin (d + 2)),
(hx.simplex hd).2 i = (SSet.prodStdSimplex.pairingCore.IsType₂.φ x hd i).2 | null | true |
OrderDual.instBooleanAlgebra._proof_2 | Mathlib.Order.BooleanAlgebra.Basic | ∀ {α : Type u_1} [inst : BooleanAlgebra α] (a : αᵒᵈ), OrderDual.ofDual a ⊓ (OrderDual.ofDual a)ᶜ ≤ ⊥ | null | false |
Algebra.Extension.tensorCotangent.congr_simp | Mathlib.RingTheory.Etale.Kaehler | ∀ {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T]
[inst_3 : Algebra R S] [inst_4 : Algebra R T] [inst_5 : Algebra S T] {P : Algebra.Extension R S}
{Q : Algebra.Extension R T} (f f_1 : P.Hom Q) (e_f : f = f_1) [alg : Algebra P.Ring Q.Ring]
(halg : algebr... | null | true |
_private.Mathlib.CategoryTheory.Functor.KanExtension.Dense.0.CategoryTheory.Functor.IsDense.of_fullyFaithful_restrictedULiftYoneda.match_1_1.splitter | Mathlib.CategoryTheory.Functor.KanExtension.Dense | {C : Type u_4} →
{D : Type u_5} →
[inst : CategoryTheory.Category.{u_1, u_4} C] →
[inst_1 : CategoryTheory.Category.{u_2, u_5} D] →
{F : CategoryTheory.Functor C D} →
(Y : D) →
(X : C) →
(motive : ((CategoryTheory.Presheaf.restrictedULiftYoneda F).obj Y).obj (Oppo... | null | true |
ContDiffWithinAt.neg | 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] {x : E}
{n : WithTop ℕ∞} {s : Set E} {f : E → F}, ContDiffWithinAt 𝕜 n f s x → ContDiffWithinAt 𝕜 n (fun x => -f... | The negative of a `C^n` function within a domain at a point is `C^n` within this domain at
this point. | true |
_private.Mathlib.MeasureTheory.Group.Arithmetic.0.IsUnit.aemeasurable_const_smul_iff.match_1_1 | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_1} [inst : Monoid M] {c : M} (motive : IsUnit c → Prop) (hc : IsUnit c),
(∀ (u : Mˣ) (hu : ↑u = c), motive ⋯) → motive hc | null | false |
MeromorphicOn.logDeriv_fun_zpow_eventuallyEq | Mathlib.Analysis.Meromorphic.LogDeriv | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [inst_1 : NontriviallyNormedField 𝕜']
[inst_2 : NormedAlgebra 𝕜 𝕜'] {f : 𝕜 → 𝕜'} {U : Set 𝕜},
MeromorphicOn f U → ∀ (n : ℤ), (logDeriv fun i => f i ^ n) =ᶠ[Filter.codiscreteWithin U] fun i => n • logDeriv f i | Eta-expanded form of `MeromorphicOn.logDeriv_zpow_eventuallyEq`
---
Away from a codiscrete subset of `U`, the logarithmic derivative of the `n`-th power of a
meromorphic function is `n` times the logarithmic derivative.
| true |
_private.Lake.Util.Version.0.Lake.ToolchainVer.blt._sparseCasesOn_1 | Lake.Util.Version | {motive : Lake.ToolchainVer → Sort u} →
(t : Lake.ToolchainVer) →
((ver : Lake.LeanVer) → motive (Lake.ToolchainVer.release ver)) →
((date : Lake.Date) → (rev : Option ℕ) → motive (Lake.ToolchainVer.nightly date rev)) →
(Nat.hasNotBit 3 t.ctorIdx → motive t) → motive t | null | false |
_private.Mathlib.Order.Partition.Finpartition.0.Finpartition.pairwiseDisjoint_apply._simp_1_1 | Mathlib.Order.Partition.Finpartition | ∀ {α : Type u_1} [inst : SemilatticeInf α] [inst_1 : OrderBot α] {a b : α}, Disjoint a b = (a ⊓ b = ⊥) | null | false |
_private.Mathlib.Tactic.LinearCombination.Lemmas.0.Mathlib.Ineq.mulRelConstData.match_1 | Mathlib.Tactic.LinearCombination.Lemmas | (motive : Mathlib.Ineq.WithStrictness → Sort u_1) →
(x : Mathlib.Ineq.WithStrictness) →
(Unit → motive Mathlib.Ineq.WithStrictness.eq) →
(Unit → motive Mathlib.Ineq.WithStrictness.le) →
(Unit → motive (Mathlib.Ineq.WithStrictness.lt true)) →
(Unit → motive (Mathlib.Ineq.WithStrictness.lt f... | null | false |
CategoryTheory.Limits.IsTerminal.isTerminalObj | Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(G : CategoryTheory.Functor C D) →
(X : C) →
[CategoryTheory.Limits.PreservesLimit (CategoryTheory.Functor.empty C) G] →
CategoryTheory.Li... | The property of preserving terminal objects expressed in terms of `IsTerminal`. | true |
NNReal.sqrtHom.eq_1 | Mathlib.Analysis.Real.Sqrt | NNReal.sqrtHom =
{ toFun := ⇑NNReal.sqrt, map_zero' := NNReal.sqrt_zero, map_one' := NNReal.sqrt_one, map_mul' := NNReal.sqrt_mul } | null | true |
MeasureTheory.memLp_indicator_iff_restrict | Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} {ε : Type u_6}
[inst : TopologicalSpace ε] [inst_1 : ESeminormedAddMonoid ε] {s : Set α} {f : α → ε},
MeasureTheory.NullMeasurableSet s μ →
(MeasureTheory.MemLp (s.indicator f) p μ ↔ MeasureTheory.MemLp f p (μ.restrict s)) | null | true |
Std.Time.TimeZone.TZif.TZifV2.noConfusion | Std.Time.Zoned.Database.TzIf | {P : Sort u} → {t t' : Std.Time.TimeZone.TZif.TZifV2} → t = t' → Std.Time.TimeZone.TZif.TZifV2.noConfusionType P t t' | null | false |
IsStarNormal.pow | Mathlib.Algebra.Star.SelfAdjoint | ∀ {R : Type u_1} [inst : Monoid R] [inst_1 : StarMul R] {x : R} [IsStarNormal x] (n : ℕ), IsStarNormal (x ^ n) | null | true |
UInt16.toUInt32_ofNatClamp_of_le._proof_1 | Init.Data.UInt.Lemmas | UInt16.size - 1 < UInt32.size | null | false |
ChevalleyThm.MvPolynomialC.numBound_casesOn_succ | Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | ∀ (k₀ k : ℕ) (D : ℕ → ℕ) (n : ℕ),
ChevalleyThm.MvPolynomialC.numBound k₀ (fun x => Nat.casesOn x k D) (n + 1) =
ChevalleyThm.MvPolynomialC.numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n | null | true |
Std.Do.ExceptConds.imp.eq_3 | Std.Do.PostCond | ∀ (ε : Type u) (ps_2 : Std.Do.PostShape) (x_2 y_2 : Std.Do.ExceptConds (Std.Do.PostShape.except ε ps_2)),
(x_2 →ₑ y_2) = (fun e => spred(x_2.1 e → y_2.1 e), x_2.2 →ₑ y_2.2) | null | true |
LinearMap.BilinForm.Isometry.ext_iff | Mathlib.LinearAlgebra.BilinearForm.Isometry | ∀ {R : Type u_1} {M₁ : Type u_3} {M₂ : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M₁]
[inst_2 : AddCommMonoid M₂] [inst_3 : Module R M₁] [inst_4 : Module R M₂] {B₁ : LinearMap.BilinForm R M₁}
{B₂ : LinearMap.BilinForm R M₂} {f g : B₁ →bᵢ B₂}, f = g ↔ ∀ (x : M₁), f x = g x | null | true |
_private.Mathlib.Topology.Instances.ENNReal.Lemmas.0.ENNReal.biInf_le_nhds._simp_1_3 | Mathlib.Topology.Instances.ENNReal.Lemmas | 1 = ↑1 | null | false |
CategoryTheory.Limits.coequalizer.desc.eq_1 | Mathlib.CategoryTheory.Limits.Shapes.Equalizers | ∀ {C : Type u} {X Y : C} [inst : CategoryTheory.Category.{v, u} C] {f g : Y ⟶ X}
[inst_1 : CategoryTheory.Limits.HasCoequalizer f g] {W : C} (k : X ⟶ W)
(h : CategoryTheory.CategoryStruct.comp f k = CategoryTheory.CategoryStruct.comp g k),
CategoryTheory.Limits.coequalizer.desc k h =
CategoryTheory.Limits.col... | null | true |
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