name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Set.Ioo_sub_one_left_eq_Ioc | Mathlib.Algebra.Order.Interval.Set.SuccPred | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : One α] [inst_2 : Sub α] [PredSubOrder α] [NoMinOrder α] (a b : α),
Set.Ioo (a - 1) b = Set.Ico a b | **Alias** of `Set.Ioo_sub_one_left_eq_Ico`. | true |
LieModule.nontrivial_lowerCentralSeriesLast | Mathlib.Algebra.Lie.Nilpotent | ∀ (R : Type u) (L : Type v) (M : Type w) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
[inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [LieModule R L M] [Nontrivial M]
[LieModule.IsNilpotent L M], Nontrivial ↥(LieModule.lowerCentralSeriesLast R L M) | null | true |
FractionalIdeal.map_coeIdeal | Mathlib.RingTheory.FractionalIdeal.Operations | ∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
{P' : Type u_3} [inst_3 : CommRing P'] [inst_4 : Algebra R P'] (g : P →ₐ[R] P') (I : Ideal R),
FractionalIdeal.map g ↑I = ↑I | null | true |
Std.DHashMap.Internal.Raw₀.get!ₘ_eq_getValueCast! | Std.Data.DHashMap.Internal.WF | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] [inst_2 : LawfulBEq α]
{m : Std.DHashMap.Internal.Raw₀ α β},
Std.DHashMap.Internal.Raw.WFImp ↑m →
∀ {a : α} [inst_3 : Inhabited (β a)],
m.get!ₘ a = Std.Internal.List.getValueCast! a (Std.DHashMap.Internal.toListModel (↑m).buckets) | null | true |
Std.ExtTreeSet.get?_max? | Std.Data.ExtTreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.ExtTreeSet α cmp} [inst : Std.TransCmp cmp] {km : α},
t.max? = some km → t.get? km = some km | null | true |
_private.Init.Data.Array.Find.0.Array.of_findIdx?_eq_some.match_1.splitter | Init.Data.Array.Find | {α : Type u_1} →
(motive : Option α → Sort u_2) → (x : Option α) → ((a : α) → motive (some a)) → (Unit → motive none) → motive x | null | true |
CategoryTheory.Functor.Monoidal.ε_η | Mathlib.CategoryTheory.Monoidal.Functor | ∀ {C : Type u₁} {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {D : Type u₂}
{inst_2 : CategoryTheory.Category.{v₂, u₂} D} {inst_3 : CategoryTheory.MonoidalCategory D}
(F : CategoryTheory.Functor C D) [self : F.Monoidal],
CategoryTheory.CategoryStruct.comp (CategoryTheory... | null | true |
FirstOrder.Field.FieldAxiom.toSentence.eq_9 | Mathlib.ModelTheory.Algebra.Field.Basic | FirstOrder.Field.FieldAxiom.existsPairNE.toSentence =
(((FirstOrder.Language.var ∘ Sum.inr) 0).bdEqual ((FirstOrder.Language.var ∘ Sum.inr) 1)).not.ex.ex | null | true |
SeparationQuotient.t2Space_iff | Mathlib.Topology.Separation.Hausdorff | ∀ {X : Type u_1} [inst : TopologicalSpace X], T2Space (SeparationQuotient X) ↔ R1Space X | null | true |
BitVec.getLsbD_ofBool | Init.Data.BitVec.Lemmas | ∀ (b : Bool) (i : ℕ), (BitVec.ofBool b).getLsbD i = (decide (i = 0) && b) | null | true |
Submodule.span_singleton_mul | Mathlib.Algebra.Algebra.Operations | ∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A}
{p : Submodule R A}, (R ∙ x) * p = x • p | null | true |
_private.Mathlib.RingTheory.Nilpotent.Exp.0.IsNilpotent.exp_add_of_commute._proof_1_12 | Mathlib.RingTheory.Nilpotent.Exp | ∀ (x₁ x₂ y₂ : ℕ), y₂ ≤ x₁ → y₂ ≤ x₂ → x₁ - y₂ = x₂ - y₂ → x₁ = x₂ | null | false |
div_self_eq_one₀._simp_1 | Mathlib.Algebra.GroupWithZero.Units.Basic | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, (a / a = 1) = (a ≠ 0) | null | false |
HNNExtension.NormalWord.TransversalPair.noConfusion | Mathlib.GroupTheory.HNNExtension | {P : Sort u} →
{G : Type u_1} →
{inst : Group G} →
{A B : Subgroup G} →
{t : HNNExtension.NormalWord.TransversalPair G A B} →
{G' : Type u_1} →
{inst' : Group G'} →
{A' B' : Subgroup G'} →
{t' : HNNExtension.NormalWord.TransversalPair G' A' B'} →
... | null | false |
_private.Mathlib.Probability.Independence.Basic.0.ProbabilityTheory.iIndepFun_iff._simp_1_2 | Mathlib.Probability.Independence.Basic | ∀ {Ω : Type u_1} {ι : Type u_2} (m : ι → MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω),
ProbabilityTheory.iIndep m μ =
∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, MeasurableSet (f i)) → μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) | null | false |
Std.Roo.toList_eq_match_rco | Init.Data.Range.Polymorphic.Lemmas | ∀ {α : Type u} {r : Std.Roo α} [inst : Std.PRange.UpwardEnumerable α] [inst_1 : LT α] [inst_2 : DecidableLT α]
[inst_3 : Std.PRange.LawfulUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLT α]
[inst_5 : Std.Rxo.IsAlwaysFinite α],
r.toList =
match Std.PRange.succ? r.lower with
| none => []
| some ... | null | true |
Simps.ProjectionData.mk._flat_ctor | Mathlib.Tactic.Simps.Basic | Lean.Name → Lean.Expr → List ℕ → Bool → Bool → Simps.ProjectionData | null | false |
_private.Std.Data.String.ToNat.0.String.Slice.isNat_iff'._simp_1_1 | Std.Data.String.ToNat | ∀ {a b : Bool}, ((!a) = b) = (a = !b) | null | false |
SchwartzMap.smulRightCLM._proof_1 | Mathlib.Analysis.Distribution.SchwartzSpace.Basic | ∀ {E : Type u_1} (F : Type u_2) {G : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
[inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G]
(L : E →L[ℝ] G →L[ℝ] ℝ) (f g : SchwartzMap E F) (x : E),
(L x).smulRight ((f + g) x) = (L x)... | null | false |
BitVec.getElem_eq_true_of_lt_of_le | Init.Data.BitVec.Lemmas | ∀ {w k : ℕ} {x : BitVec w} (hk' : k < w), x.toNat < 2 ^ (k + 1) → 2 ^ k ≤ x.toNat → x[k] = true | If a bitvector interpreted as a natural number is strictly smaller than `2 ^ (k + 1)` and greater than or
equal to 2 ^ k, then the bit at position `k` must be `true` | true |
Complex.instCoeReal | Mathlib.Basic.Complex.Basic | Coe ℝ ℂ | null | true |
Std.Iterators.Types.FilterMap.mk.inj | Init.Data.Iterators.Combinators.Monadic.FilterMap | ∀ {α β γ : Type w} {m : Type w → Type w'} {n : Type w → Type w''} {lift : ⦃α : Type w⦄ → m α → n α}
{f : β → Std.Iterators.PostconditionT n (Option γ)} {inner inner_1 : Std.IterM m β},
{ inner := inner } = { inner := inner_1 } → inner = inner_1 | null | true |
Subalgebra.instSubringClass | Mathlib.Algebra.Algebra.Subalgebra.Basic | ∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A],
SubringClass (Subalgebra R A) A | null | true |
iSupIndep.linearEquiv_symm_apply | Mathlib.LinearAlgebra.DFinsupp | ∀ {ι : Type u_1} {R : Type u_3} {N : Type u_6} [inst : DecidableEq ι] [inst_1 : Ring R] [inst_2 : AddCommGroup N]
[inst_3 : Module R N] {p : ι → Submodule R N} (ind : iSupIndep p) (iSup_top : ⨆ i, p i = ⊤) {i : ι} {x : N}
(h : x ∈ p i), (ind.linearEquiv iSup_top).symm x = fun₀ | i => ⟨x, h⟩ | null | true |
Metric.ediam_image_le_iff | Mathlib.Topology.EMetricSpace.Diam | ∀ {α : Type u_1} {X : Type u_2} [inst : TopologicalSpace X] [inst_1 : WeakPseudoEMetricSpace X] {d : ENNReal}
{f : α → X} {s : Set α}, Metric.ediam (f '' s) ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) ≤ d | null | true |
_private.Mathlib.Topology.UnitInterval.0.unitInterval.image_coe_preimage_symm._simp_1_1 | Mathlib.Topology.UnitInterval | Function.Involutive unitInterval.symm = True | null | false |
Dense.exists_seq_strictAnti_tendsto_of_lt | Mathlib.Topology.Order.IsLUB | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] [DenselyOrdered α]
[FirstCountableTopology α] {s : Set α},
Dense s →
∀ {x y : α}, x < y → ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Set.Ioo x y ∩ s) ∧ Filter.Tendsto u Filter.atTop (nhds x) | null | true |
HahnSeries.SummableFamily.instAddCommGroup | Mathlib.RingTheory.HahnSeries.Summable | {Γ : Type u_1} →
{R : Type u_3} →
{α : Type u_5} →
[inst : PartialOrder Γ] → [inst_1 : AddCommGroup R] → AddCommGroup (HahnSeries.SummableFamily Γ R α) | null | true |
_private.Mathlib.Data.Quot.0.Quot.surjective_lift.match_1_1 | Mathlib.Data.Quot | ∀ {α : Sort u_1} {γ : Sort u_2} {f : α → γ} (y : γ) (motive : (∃ a, f a = y) → Prop) (x : ∃ a, f a = y),
(∀ (x : α) (hx : f x = y), motive ⋯) → motive x | null | false |
_private.Mathlib.NumberTheory.RamificationInertia.Basic.0.Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial._simp_1_6 | Mathlib.NumberTheory.RamificationInertia.Basic | ∀ {F : Type u_8} {M : Type u_9} {X : Type u_10} {Y : Type u_11} [inst : SMul M X] [inst_1 : SMul M Y]
[inst_2 : FunLike F X Y] [MulActionHomClass F M X Y] (f : F) (c : M) (x : X), c • f x = f (c • x) | null | false |
Real.one_le_exp | Mathlib.Analysis.Complex.Exponential | ∀ {x : ℝ}, 0 ≤ x → 1 ≤ Real.exp x | null | true |
withSeminorms_iInf | Mathlib.Analysis.LocallyConvex.WithSeminorms | ∀ {𝕜 : Type u_2} {E : Type u_6} {ι : Type u_9} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
{κ : ι → Type u_11} {p : (i : ι) → SeminormFamily 𝕜 E (κ i)} {t : ι → TopologicalSpace E},
(∀ (i : ι), WithSeminorms (p i)) → WithSeminorms (SeminormFamily.sigma p) | null | true |
with_gaugeSeminormFamily | Mathlib.Analysis.LocallyConvex.AbsConvexOpen | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E]
[inst_3 : Module 𝕜 E] [inst_4 : Module ℝ E] [inst_5 : IsScalarTower ℝ 𝕜 E] [inst_6 : ContinuousSMul ℝ E]
[IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [LocallyConvexSpace 𝕜 E], WithSeminorms (gaugeSemino... | The topology of a locally convex space is induced by the gauge seminorm family. | true |
ergodic_mul_left_of_denseRange_zpow | Mathlib.Dynamics.Ergodic.Action.OfMinimal | ∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] [IsTopologicalGroup G] [inst_3 : MeasurableSpace G]
[SecondCountableTopology G] [BorelSpace G] {g : G},
(DenseRange fun x => g ^ x) →
∀ (μ : MeasureTheory.Measure G) [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant],
E... | null | true |
Language.map | Mathlib.Computability.Language | {α : Type u_1} → {β : Type u_2} → (α → β) → Language α →+* Language β | Maps the alphabet of a language. | true |
_private.Lean.Parser.Attr.0.Lean.Parser.Attr.specialize._regBuiltin.Lean.Parser.Attr.specialize.parenthesizer_11 | Lean.Parser.Attr | IO Unit | null | false |
Polynomial.iterate_derivative_mul | Mathlib.Algebra.Polynomial.Derivative | ∀ {R : Type u} [inst : Semiring R] {n : ℕ} (p q : Polynomial R),
(⇑Polynomial.derivative)^[n] (p * q) =
∑ k ∈ Finset.range n.succ, n.choose k • ((⇑Polynomial.derivative)^[n - k] p * (⇑Polynomial.derivative)^[k] q) | null | true |
CategoryTheory.Functor.mapArrowEquivalence._proof_2 | Mathlib.CategoryTheory.Comma.Arrow | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (e : C ≌ D) (X : CategoryTheory.Arrow C),
CategoryTheory.CategoryStruct.comp
(e.functor.mapArrow.map (((CategoryTheory.Functor.mapArrowFunctor C C).mapIso e.unitIso).hom.app X))
... | null | false |
List.cyclicPermutations_of_ne_nil | Mathlib.Data.List.Rotate | ∀ {α : Type u} (l : List α),
l ≠ [] → l.cyclicPermutations = (List.zipWith (fun x1 x2 => x1 ++ x2) l.tails l.inits).dropLast | null | true |
_private.Mathlib.Combinatorics.Extremal.RuzsaSzemeredi.0.ruzsaSzemerediNumberNat_asymptotic_lower_bound._simp_1_2 | Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | ∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],
(∀ᶠ (x : α) in Filter.atTop, p x) = ∃ a, ∀ (b : α), a ≤ b → p b | null | false |
IsCoprime.isRelPrime | Mathlib.RingTheory.Coprime.Basic | ∀ {R : Type u} [inst : CommSemiring R] {a b : R}, IsCoprime a b → IsRelPrime a b | null | true |
Submonoid.unitsEquivUnitsType._proof_6 | Mathlib.Algebra.Group.Submonoid.Units | ∀ {M : Type u_1} [inst : Monoid M] (S : Submonoid M) (x x_1 : ↥S.units),
(match x * x_1 with
| ⟨val, h⟩ =>
{ val := ⟨(Units.coeHom M) val, ⋯⟩, inv := ⟨(Units.coeHom M) val⁻¹, ⋯⟩, val_inv := ⋯, inv_val := ⋯ }) =
match x * x_1 with
| ⟨val, h⟩ => { val := ⟨(Units.coeHom M) val, ⋯⟩, inv := ⟨(Units.coeHo... | null | false |
PresheafOfModules.instAddCommGroupModuleColimit._proof_20 | Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat}
{cR : CategoryTheory.Limits.Cocone R} (hcR : CategoryTheory.Limits.IsColimit cR) {M : PresheafOfModules R}
{cM : CategoryTheory.Limits.Cocone M.presheaf} (hcM : CategoryTheory.Limits.IsColimit cM),
autoParam (∀... | null | false |
_private.Mathlib.Analysis.SpecialFunctions.Complex.CircleMap.0.eq_of_circleMap_eq._simp_1_3 | Mathlib.Analysis.SpecialFunctions.Complex.CircleMap | (Complex.I = 0) = False | null | false |
List.mem_reverse | Init.Data.List.Lemmas | ∀ {α : Type u_1} {x : α} {as : List α}, x ∈ as.reverse ↔ x ∈ as | null | true |
_private.Mathlib.RingTheory.Valuation.Basic.0.Valuation.IsEquiv.valueGroup₀Fun_spec._simp_1_2 | Mathlib.RingTheory.Valuation.Basic | ∀ {α : Type u} {a b : α}, (↑a = ↑b) = (a = b) | null | false |
Std.DTreeMap.Raw.WF.emptyc | Std.Data.DTreeMap.Raw.WF | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering}, ∅.WF | null | true |
finsum_eq_indicator_apply | Mathlib.Algebra.BigOperators.Finprod | ∀ {α : Type u_1} {M : Type u_5} [inst : AddCommMonoid M] (s : Set α) (f : α → M) (a : α),
∑ᶠ (_ : a ∈ s), f a = s.indicator f a | null | true |
CommBialgCat._sizeOf_1 | Mathlib.Algebra.Category.CommBialgCat | {R : Type u} → {inst : CommRing R} → [SizeOf R] → CommBialgCat R → ℕ | null | false |
_private.Mathlib.NumberTheory.Padics.PadicIntegers.0.PadicInt.coe_sum._simp_1_1 | Mathlib.NumberTheory.Padics.PadicIntegers | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] (self : ↥(PadicInt.subring p)), ↑self = PadicInt.Coe.ringHom self | null | false |
Lean.Elab.Term.LetIdDeclView.binders | Lean.Elab.Binders | Lean.Elab.Term.LetIdDeclView → Array Lean.Syntax | null | true |
UpperSemicontinuousOn.add | Mathlib.Topology.Semicontinuity.Basic | ∀ {α : Type u_1} [inst : TopologicalSpace α] {s : Set α} {γ : Type u_5} [inst_1 : AddCommMonoid γ]
[inst_2 : LinearOrder γ] [IsOrderedAddMonoid γ] [inst_4 : TopologicalSpace γ] [OrderTopology γ] [ContinuousAdd γ]
{f g : α → γ}, UpperSemicontinuousOn f s → UpperSemicontinuousOn g s → UpperSemicontinuousOn (fun z => ... | The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. | true |
_private.Init.Data.List.Pairwise.0.List.Pairwise.imp.match_1_1 | Init.Data.List.Pairwise | ∀ {α : Type u_1} {R : α → α → Prop} (motive : (x : List α) → List.Pairwise R x → Prop) (x : List α)
(x_1 : List.Pairwise R x),
(∀ (_ : Unit), motive [] ⋯) →
(∀ (a : α) (l : List α) (h₁ : ∀ a' ∈ l, R a a') (h₂ : List.Pairwise R l), motive (a :: l) ⋯) → motive x x_1 | null | false |
Lean.Name.MatchUpToIndexSuffix.suffixMatch.elim | Batteries.Lean.Meta.UnusedNames | {motive : Lean.Name.MatchUpToIndexSuffix → Sort u} →
(t : Lean.Name.MatchUpToIndexSuffix) →
t.ctorIdx = 2 → ((i : ℕ) → motive (Lean.Name.MatchUpToIndexSuffix.suffixMatch i)) → motive t | null | false |
Int64.ofIntLE_eq_ofIntTruncate | Init.Data.SInt.Lemmas | ∀ {x : ℤ} {h₁ : Int64.minValue.toInt ≤ x} {h₂ : x ≤ Int64.maxValue.toInt}, Int64.ofIntLE x h₁ h₂ = Int64.ofIntClamp x | null | true |
_private.Mathlib.Lean.Meta.RefinedDiscrTree.Encode.0.Lean.Meta.RefinedDiscrTree.encodingStepAux.match_5 | Mathlib.Lean.Meta.RefinedDiscrTree.Encode | (motive : Lean.Expr → Sort u_1) →
(x : Lean.Expr) →
((n : Lean.Name) → (us : List Lean.Level) → motive (Lean.Expr.const n us)) →
((n : Lean.Name) → (i : ℕ) → (struct : Lean.Expr) → motive (Lean.Expr.proj n i struct)) →
((fvarId : Lean.FVarId) → motive (Lean.Expr.fvar fvarId)) →
((mvarId : ... | null | false |
Stream'.Seq.update_cons_succ | Mathlib.Data.Seq.Basic | ∀ {α : Type u} (hd : α) (tl : Stream'.Seq α) (f : α → α) (n : ℕ),
(Stream'.Seq.cons hd tl).update (n + 1) f = Stream'.Seq.cons hd (tl.update n f) | null | true |
Mathlib.Explode.Entry.type | Mathlib.Tactic.Explode.Datatypes | Mathlib.Explode.Entry → Lean.MessageData | A type of this expression as a `MessageData`. Make sure to use `addMessageContext`. | true |
_private.Mathlib.Analysis.Normed.Affine.Simplex.0.Affine.Simplex.scalene_reindex_iff._proof_1_8 | Mathlib.Analysis.Normed.Affine.Simplex | ∀ {m n : ℕ} (e : Fin (m + 1) ≃ Fin (n + 1)) (fst snd : Fin (m + 1)) (property : fst < snd),
e fst < e snd →
(e.symm (↑(if h : e fst < e snd then ⟨(e fst, e snd), ⋯⟩ else ⟨(e snd, e fst), ⋯⟩)).1,
e.symm (↑(if h : e fst < e snd then ⟨(e fst, e snd), ⋯⟩ else ⟨(e snd, e fst), ⋯⟩)).2).1 <
(e.symm (↑(if... | null | false |
_private.Mathlib.Algebra.SkewMonoidAlgebra.Basic.0.SkewMonoidAlgebra.instAddGroup._simp_6 | Mathlib.Algebra.SkewMonoidAlgebra.Basic | ∀ {k : Type u_1} {G : Type u_2} [inst : AddMonoid k] {a b : G →₀ k},
{ coeff := a } + { coeff := b } = { coeff := a + b } | null | false |
_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs.0.EisensteinSeries.denom_aux | Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs | ∀ (A B : Matrix.SpecialLinearGroup (Fin 2) ℤ) (z : UpperHalfPlane),
↑(↑(A * B) 1 0) *
UpperHalfPlane.denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) B)) ↑z =
↑(↑A 1 0) * ↑(↑B).det +
↑(↑B 1 0) *
UpperHalfPlane.denom
(Matrix.SpecialLinearGroup... | null | true |
Lean.Elab.Tactic.Omega.Fact.mk.sizeOf_spec | Lean.Elab.Tactic.Omega.Core | ∀ (coeffs : Lean.Omega.Coeffs) (constraint : Lean.Omega.Constraint)
(justification : Lean.Elab.Tactic.Omega.Justification constraint coeffs),
sizeOf { coeffs := coeffs, constraint := constraint, justification := justification } =
1 + sizeOf coeffs + sizeOf constraint + sizeOf justification | null | true |
_private.Mathlib.Topology.Instances.EReal.Lemmas.0.EReal.nhdsWithin_bot.match_1_1 | Mathlib.Topology.Instances.EReal.Lemmas | ∀ (x : ℝ) (x_1 : EReal) (motive : x_1 ∈ Set.Ioc ⊥ ↑x → Prop) (x_2 : x_1 ∈ Set.Ioc ⊥ ↑x),
(∀ (h1 : ⊥ < x_1) (h2 : x_1 ≤ ↑x), motive ⋯) → motive x_2 | null | false |
Set.image_sub_const_uIcc | Mathlib.Algebra.Order.Group.Pointwise.Interval | ∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : LinearOrder α] [IsOrderedAddMonoid α] (a b c : α),
(fun x => x - a) '' Set.uIcc b c = Set.uIcc (b - a) (c - a) | null | true |
WeierstrassCurve.natDegree_Φ_pos | Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | ∀ {R : Type u} [inst : CommRing R] (W : WeierstrassCurve R) [Nontrivial R] {n : ℤ}, n ≠ 0 → 0 < (W.Φ n).natDegree | null | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.0.Int64.reduceNe._regBuiltin.Int64.reduceNe.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.922157756._hygCtx._hyg.9 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt | IO Unit | null | false |
_private.Batteries.Recycling.RBTree.WF.0.RBTree.RBNode.Ordered.balLeft.match_1_3 | Batteries.Recycling.RBTree.WF | ∀ {α : Type u_1} {cmp : α → α → Ordering} {v : α} (a : RBTree.RBNode α) (y : α) (b : RBTree.RBNode α)
(motive :
RBTree.RBNode.All (fun x => RBTree.RBNode.cmpLT cmp v x) (RBTree.RBNode.node RBTree.RBColor.black a y b) → Prop)
(x : RBTree.RBNode.All (fun x => RBTree.RBNode.cmpLT cmp v x) (RBTree.RBNode.node RBTre... | null | false |
SSet.Truncated.HomotopyCategory.homMk_comp_homMk_assoc | Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | ∀ {V : SSet.Truncated 2}
{x₀ x₁ x₂ : V.obj (Opposite.op { obj := { len := 0 }, property := SSet.OneTruncation₂._proof_1 })}
{e₀₁ : SSet.Truncated.Edge x₀ x₁} {e₁₂ : SSet.Truncated.Edge x₁ x₂} {e₀₂ : SSet.Truncated.Edge x₀ x₂}
(h : e₀₁.CompStruct e₁₂ e₀₂) {Z : V.HomotopyCategory} (h : SSet.Truncated.HomotopyCatego... | null | true |
_private.Lean.Meta.AppBuilder.0.Lean.Meta.mkNoConfusion.match_1 | Lean.Meta.AppBuilder | (motive : Option (Lean.ConstructorVal × Array Lean.Expr) → Sort u_1) →
(__do_lift : Option (Lean.ConstructorVal × Array Lean.Expr)) →
((ctorB : Lean.ConstructorVal) → (ys2 : Array Lean.Expr) → motive (some (ctorB, ys2))) →
((x : Option (Lean.ConstructorVal × Array Lean.Expr)) → motive x) → motive __do_lift | null | false |
LibraryNote.continuity_lemma_statement | Mathlib.Topology.Continuous | Batteries.Util.LibraryNote | The library contains many lemmas stating that functions/operations are continuous. There are many
ways to formulate the continuity of operations. Some are more convenient than others.
Note: for the most part this note also applies to other properties
(`Measurable`, `Differentiable`, `ContinuousOn`, ...).
### The tradi... | true |
Algebra.Extension.cotangentComplex | Mathlib.RingTheory.Extension.Cotangent.Basic | {R : Type u} →
{S : Type v} →
[inst : CommRing R] →
[inst_1 : CommRing S] → [inst_2 : Algebra R S] → (P : Algebra.Extension R S) → P.Cotangent →ₗ[S] P.CotangentSpace | The cotangent complex given by a presentation `R[X] → S` (i.e. a closed embedding `S ↪ Aⁿ`). | true |
FaithfulSMul.of_injective | Mathlib.GroupTheory.GroupAction.Hom | ∀ {M' : Type u_1} {X : Type u_5} [inst : SMul M' X] {Y : Type u_6} [inst_1 : SMul M' Y] {F : Type u_8}
[inst_2 : FunLike F X Y] [FaithfulSMul M' X] [MulActionHomClass F M' X Y] (f : F),
Function.Injective ⇑f → FaithfulSMul M' Y | null | true |
szemeredi_regularity | Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : Fintype α] (G : SimpleGraph α) [inst_2 : DecidableRel G.Adj] {ε : ℝ}
{l : ℕ},
0 < ε →
l ≤ Fintype.card α →
∃ P, P.IsEquipartition ∧ l ≤ P.parts.card ∧ P.parts.card ≤ SzemerediRegularity.bound ε l ∧ P.IsUniform G ε | Effective **Szemerédi Regularity Lemma**: For any sufficiently large graph, there is an
`ε`-uniform equipartition of bounded size (where the bound does not depend on the graph). | true |
MonoidWithZeroHom.fst_apply_coe | Mathlib.Algebra.GroupWithZero.ProdHom | ∀ {G₀ : Type u_1} {H₀ : Type u_2} [inst : GroupWithZero G₀] [inst_1 : GroupWithZero H₀] (x : G₀ˣ × H₀ˣ),
(MonoidWithZeroHom.fst G₀ H₀) ↑x = ↑x.1 | null | true |
DoubleCentralizer.natCast_toProd | Mathlib.Analysis.CStarAlgebra.Multiplier | ∀ {𝕜 : Type u_1} {A : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NonUnitalNormedRing A]
[inst_2 : NormedSpace 𝕜 A] [inst_3 : SMulCommClass 𝕜 A A] [inst_4 : IsScalarTower 𝕜 A A] (n : ℕ), (↑n).toProd = ↑n | null | true |
Vector.finIdxOf?_mk | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n : ℕ} [inst : BEq α] {xs : Array α} (h : xs.size = n) (x : α),
(Vector.mk xs h).finIdxOf? x = Option.map (Fin.cast h) (xs.finIdxOf? x) | null | true |
Finset.toRight | Mathlib.Data.Finset.Sum | {α : Type u_1} → {β : Type u_2} → Finset (α ⊕ β) → Finset β | Given a finset of elements `α ⊕ β`, extract all the elements of the form `β`. This
forms a quasi-inverse to `disjSum`, in that it recovers its right input.
See also `List.partitionMap`.
| true |
AddSubmonoid.isAddCommutative_topologicalClosure | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : AddMonoid M] [inst_2 : SeparatelyContinuousAdd M] [T2Space M]
(s : AddSubmonoid M) [IsAddCommutative ↥s], IsAddCommutative ↥s.topologicalClosure | If a submonoid of an additive topological monoid is commutative, then so is its
topological closure. | true |
CategoryTheory.Arrow.catCommSq | Mathlib.CategoryTheory.Comma.CatCommSq | {C₁ : Type u_1} →
{C₂ : Type u_2} →
{D₁ : Type u_3} →
{D₂ : Type u_4} →
[inst : CategoryTheory.Category.{u_5, u_1} C₁] →
[inst_1 : CategoryTheory.Category.{u_6, u_2} C₂] →
[inst_2 : CategoryTheory.Category.{u_7, u_3} D₁] →
[inst_3 : CategoryTheory.Category.{u_8, u... | null | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.toList_insert_perm._simp_1_2 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {a b : α}, (compare a b ≠ Ordering.eq) = ((a == b) = false) | null | false |
CategoryTheory.ComposableArrows.Mk₁.map.eq_3 | Mathlib.CategoryTheory.ComposableArrows.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X₀ X₁ : C} (f : X₀ ⟶ X₁) (isLt isLt_1 : 1 < 2)
(x_3 : ⟨1, isLt⟩ ≤ ⟨1, isLt_1⟩),
CategoryTheory.ComposableArrows.Mk₁.map f ⟨1, isLt⟩ ⟨1, isLt_1⟩ x_3 =
CategoryTheory.CategoryStruct.id (CategoryTheory.ComposableArrows.Mk₁.obj X₀ X₁ ⟨1, isLt⟩) | null | true |
AbsoluteValue.trivial._proof_3 | Mathlib.Algebra.Order.AbsoluteValue.Basic | ∀ {R : Type u_2} [inst : Semiring R] [inst_1 : DecidablePred fun x => x = 0] {S : Type u_1} [inst_2 : Semiring S]
[Nontrivial S] (x : R), (if x = 0 then 0 else 1) = 0 ↔ x = 0 | null | false |
HomologicalComplex.homology.congr_simp | Mathlib.Algebra.Homology.HomotopyCategory.Acyclic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{ι : Type u_2} {c : ComplexShape ι} (K K_1 : HomologicalComplex C c) (e_K : K = K_1) (i i_1 : ι) (e_i : i = i_1)
[inst_2 : K.HasHomology i], K.homology i = K_1.homology i_1 | null | true |
DirectedOn.le_of_minimal | Mathlib.Order.Bounds.Basic | ∀ {α : Type u_1} [inst : Preorder α] {s : Set α} {a b : α},
DirectedOn (fun x y => y ≤ x) s → Minimal (fun x => x ∈ s) a → b ∈ s → a ≤ b | null | true |
WittVector.instIsocrystalStandardOneDimIsocrystal._proof_2 | Mathlib.RingTheory.WittVector.Isocrystal | ∀ (p : ℕ) [inst : Fact (Nat.Prime p)] (k : Type u_1) [inst_1 : CommRing k] [inst_2 : CharP k p]
[inst_3 : PerfectRing k p],
RingHomCompTriple (↑(WittVector.FractionRing.frobenius p k)) (RingHom.id (FractionRing (WittVector p k)))
↑(WittVector.FractionRing.frobenius p k) | null | false |
Sum.instLocallyFiniteOrder._proof_1 | Mathlib.Data.Sum.Interval | ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : LocallyFiniteOrder α]
[inst_3 : LocallyFiniteOrder β] (a b x : α ⊕ β), x ∈ Finset.sumLift₂ Finset.Icc Finset.Icc a b ↔ a ≤ x ∧ x ≤ b | null | false |
CategoryTheory.HasInjectiveResolutions.mk._flat_ctor | Mathlib.CategoryTheory.Preadditive.Injective.Resolution | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C]
[inst_2 : CategoryTheory.Limits.HasZeroMorphisms C],
(∀ (Z : C), CategoryTheory.HasInjectiveResolution Z) → CategoryTheory.HasInjectiveResolutions C | null | false |
CanonicallyOrderedAddCommMonoid.toAddCancelCommMonoid | Mathlib.Algebra.Order.Sub.Basic | (α : Type u_1) →
[inst : AddCommMonoid α] →
[inst_1 : PartialOrder α] → [inst_2 : Sub α] → [OrderedSub α] → [AddLeftReflectLE α] → AddCancelCommMonoid α | A `CanonicallyOrderedAddCommMonoid` with ordered subtraction and order-reflecting addition is
cancellative. This is not an instance as it would form a typeclass loop.
See note [reducible non-instances]. | true |
continuous_pow._f | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : Monoid M] [ContinuousMul M] (x : ℕ) (f : Nat.below x),
Continuous fun a => a ^ x | null | false |
NumberField.ComplexEmbedding.LiesOver.mk | Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | ∀ {K : Type u_3} {L : Type u_4} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {φ : L →+* ℂ} {ψ : K →+* ℂ},
φ.comp (algebraMap K L) = ψ → NumberField.ComplexEmbedding.LiesOver φ ψ | null | true |
Submodule.rank_sup_add_rank_inf_eq | Mathlib.LinearAlgebra.Dimension.RankNullity | ∀ {R : Type u_1} {M : Type u} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[HasRankNullity.{u, u_1} R] (s t : Submodule R M),
Module.rank R ↥(s ⊔ t) + Module.rank R ↥(s ⊓ t) = Module.rank R ↥s + Module.rank R ↥t | null | true |
comp_partialSups | Mathlib.Order.PartialSups | ∀ {α : Type u_1} {β : Type u_2} {ι : Type u_3} [inst : SemilatticeSup α] [inst_1 : SemilatticeSup β]
[inst_2 : Preorder ι] [inst_3 : LocallyFiniteOrderBot ι] {F : Type u_4} [inst_4 : FunLike F α β] [SupHomClass F α β]
(f : ι → α) (g : F), ⇑(partialSups (⇑g ∘ f)) = ⇑g ∘ ⇑(partialSups f) | null | true |
Nat.isSome_getElem?_toArray_ric_eq | Init.Data.Range.Polymorphic.NatLemmas | ∀ {n i : ℕ}, ((*...=n).toArray[i]?.isSome = true) = (i ≤ n) | null | true |
Module.End.UnifEigenvalues.val | Mathlib.LinearAlgebra.Eigenspace.Basic | {R : Type v} →
{M : Type w} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] → [inst_2 : Module R M] → (f : Module.End R M) → (k : ℕ∞) → f.UnifEigenvalues k → R | The underlying value of a bundled eigenvalue. | true |
CategoryTheory.GradedObject.mapBifunctorLeftUnitor_naturality | Mathlib.CategoryTheory.GradedObject.Unitor | ∀ {C : Type u_1} {D : Type u_2} {I : Type u_3} {J : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : Zero I] [inst_3 : DecidableEq I]
[inst_4 : CategoryTheory.Limits.HasInitial C] (F : CategoryTheory.Functor C (CategoryTheory.Functor D D)) (X : C)
(... | null | true |
_private.Mathlib.Topology.MetricSpace.Thickening.0.Metric.thickening_eq_empty_iff._simp_1_1 | Mathlib.Topology.MetricSpace.Thickening | ∀ {α : Type u} [inst : PseudoEMetricSpace α] {ε : ℝ} {s : Set α}, 0 < ε → (Metric.thickening ε s = ∅) = (s = ∅) | null | false |
Finset.addConst_neg_left | Mathlib.Combinatorics.Additive.DoublingConst | ∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : DecidableEq G] (A B : Finset G), (-A).addConst B = A.subConst B | null | true |
AddMonoidAlgebra.coeff_divOf | Mathlib.Algebra.MonoidAlgebra.Division | ∀ {k : Type u_1} {G : Type u_2} [inst : Semiring k] [inst_1 : AddCommMonoid G] [inst_2 : IsCancelAdd G] (g : G)
(x : AddMonoidAlgebra k G) (g' : G), (x.divOf g).coeff g' = x.coeff (g + g') | null | true |
_private.Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer.0.CategoryTheory.Limits.WalkingMulticospan.instFinCategoryOfLOfDecidableEqR._simp_4 | Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | ∀ {α : Type u_1} {a : α} {s : Multiset α}, (a ::ₘ s).Nodup = (a ∉ s ∧ s.Nodup) | null | false |
CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback._proof_7 | Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | ∀ {T : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} T] {P : CategoryTheory.MorphismProperty T} {Y Z : T}
(g : Y ⟶ Z) [CategoryTheory.Limits.HasPullbacks T], P.HasPullbacksAlong g | null | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.