name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
LinearGrowth.linearGrowthInf_iInf | Mathlib.Analysis.Asymptotics.LinearGrowth | ∀ {ι : Type u_1} [Finite ι] (u : ι → ℕ → EReal),
LinearGrowth.linearGrowthInf (⨅ i, u i) = ⨅ i, LinearGrowth.linearGrowthInf (u i) | null | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal.0.AugmentedSimplexCategory.tensorObj.match_1.eq_2 | Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal | ∀ (motive : AugmentedSimplexCategory → AugmentedSimplexCategory → Sort u_1) (x : AugmentedSimplexCategory)
(h_1 : (m n : SimplexCategory) → motive (CategoryTheory.WithInitial.of m) (CategoryTheory.WithInitial.of n))
(h_2 : (x : AugmentedSimplexCategory) → motive CategoryTheory.WithInitial.star x)
(h_3 : (x : Augm... | null | true |
Array.instOrientedOrd | Init.Data.Order.Ord | ∀ {α : Type u_1} [inst : Ord α] [Std.OrientedOrd α], Std.OrientedOrd (Array α) | null | true |
_private.Lean.Meta.CongrTheorems.0.Lean.Meta.mkHCongrWithArity.mkProof.match_1 | Lean.Meta.CongrTheorems | (motive : Option (Lean.Expr × Lean.Expr × Lean.Expr × Lean.Expr) → Sort u_1) →
(x : Option (Lean.Expr × Lean.Expr × Lean.Expr × Lean.Expr)) →
((fst lhs fst_1 snd : Lean.Expr) → motive (some (fst, lhs, fst_1, snd))) →
((x : Option (Lean.Expr × Lean.Expr × Lean.Expr × Lean.Expr)) → motive x) → motive x | null | false |
Even.pow_nonneg | Mathlib.Algebra.Order.Ring.Basic | ∀ {R : Type u_3} [inst : Semiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {n : ℕ},
Even n → ∀ (a : R), 0 ≤ a ^ n | null | true |
Algebra.IsSeparablyGenerated | Mathlib.FieldTheory.TranscendentalSeparable | (k : Type u_1) → (K : Type u_2) → [inst : Field k] → [inst_1 : Field K] → [Algebra k K] → Prop | A field extension is separably generated if there exists a transcendence basis such that
the extension above it is separable. | true |
selfAdjoint.instNNRatCast._proof_1 | Mathlib.Algebra.Star.SelfAdjoint | ∀ {R : Type u_1} [inst : Field R] [inst_1 : StarRing R] (q : ℚ≥0), IsSelfAdjoint ↑q | null | false |
Array.count_eq_zero_of_not_mem | Init.Data.Array.Count | ∀ {α : Type u_1} [inst : BEq α] [LawfulBEq α] {a : α} {xs : Array α}, a ∉ xs → Array.count a xs = 0 | null | true |
Set.countable_iUnion_iff._simp_1 | Mathlib.Data.Set.Countable | ∀ {α : Type u} {ι : Sort x} [Countable ι] {t : ι → Set α}, (⋃ i, t i).Countable = ∀ (i : ι), (t i).Countable | null | false |
MeasureTheory.Lp._proof_4 | Mathlib.MeasureTheory.Function.LpSpace.Basic | ∀ (E : Type u_1) [inst : NormedAddCommGroup E], IsTopologicalAddGroup E | null | false |
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 |
Std.WP.Assertion.NondetFun.total | Std.WP.Assertion | ∀ {Pred : Type u} {Fun : Type v} {α : outParam (Type w)} {inst : Std.WP.Assertion Pred}
[self : Std.WP.Assertion.NondetFun Pred Fun α] (f : Fun),
(Lean.Order.iSup fun a => Std.WP.Assertion.NondetFun.EvalsTo f a) = Lean.Order.top | Every function hits some value. | 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 |
_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 |
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 |
_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 |
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 |
_private.Lean.Compiler.LCNF.ToLCNF.0.Lean.Compiler.LCNF.ToLCNF.isTypeFormerType.match_1 | Lean.Compiler.LCNF.ToLCNF | (motive : Option Bool → Sort u_1) →
(x : Option Bool) → ((result : Bool) → motive (some result)) → ((x : Option Bool) → motive x) → motive x | null | false |
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 |
_private.Mathlib.Topology.UnitInterval.0.unitInterval.image_coe_preimage_symm._simp_1_1 | Mathlib.Topology.UnitInterval | Function.Involutive unitInterval.symm = True | null | false |
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 |
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 |
_private.Lean.Parser.Attr.0.Lean.Parser.Attr.specialize._regBuiltin.Lean.Parser.Attr.specialize.parenthesizer_11 | Lean.Parser.Attr | IO Unit | null | false |
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 |
_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 |
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 |
CommBialgCat._sizeOf_1 | Mathlib.Algebra.Category.CommBialgCat | {R : Type u} → {inst : CommRing R} → [SizeOf R] → CommBialgCat R → ℕ | 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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
Nat.isSome_getElem?_toArray_ric_eq | Init.Data.Range.Polymorphic.NatLemmas | ∀ {n i : ℕ}, ((*...=n).toArray[i]?.isSome = true) = (i ≤ n) | null | true |
Lean.IR.EmitLLVM.getEnv | Lean.Compiler.IR.EmitLLVM | {llvmctx : Lean.LLVM.Context} → Lean.IR.EmitLLVM.M llvmctx Lean.Environment | 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 |
Function.Injective.leftCancelMonoid | Mathlib.Algebra.Group.InjSurj | {M₁ : Type u_1} →
{M₂ : Type u_2} →
[inst : Mul M₁] →
[inst_1 : One M₁] →
[inst_2 : Pow M₁ ℕ] →
[inst_3 : LeftCancelMonoid M₂] →
(f : M₁ → M₂) →
Function.Injective f →
f 1 = 1 →
(∀ (x y : M₁), f (x * y) = f x * f y) →
... | A type endowed with `1` and `*` is a left cancel monoid, if it admits an injective map that
preserves `1` and `*` to a left cancel monoid. See note [reducible non-instances]. | true |
Std.DTreeMap.Internal.Impl.toList_filter! | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {f : (a : α) → β a → Bool},
t.WF → (Std.DTreeMap.Internal.Impl.filter! f t).toList = List.filter (fun p => f p.fst p.snd) t.toList | null | true |
CategoryTheory.AddMonObj.zero_associator | Mathlib.CategoryTheory.Monoidal.Mon | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C] {M N P : C}
[inst_2 : CategoryTheory.AddMonObj M] [inst_3 : CategoryTheory.AddMonObj N] [inst_4 : CategoryTheory.AddMonObj P],
CategoryTheory.CategoryStruct.comp
(CategoryTheory.CategoryStruct.comp
... | null | true |
Equiv.sumArrowEquivProdArrow._proof_1 | Mathlib.Logic.Equiv.Prod | ∀ (α : Type u_3) (β : Type u_1) (γ : Type u_2) (f : α ⊕ β → γ),
(fun p => Sum.elim p.1 p.2) ((fun f => (f ∘ Sum.inl, f ∘ Sum.inr)) f) = f | null | false |
Module.Baer.ExtensionOfMaxAdjoin.ideal._proof_1 | Mathlib.Algebra.Module.Injective | ∀ {R : Type u_1} [inst : Ring R] {N : Type u_2} [inst_1 : AddCommGroup N] [inst_2 : Module R N], IsScalarTower R R N | null | false |
Aesop.RuleApplication.successProbability? | Aesop.RuleTac.Basic | Aesop.RuleApplication → Option Aesop.Percent | null | true |
Nat.range_succ_eq_Iic | Mathlib.Order.Interval.Finset.Nat | ∀ (n : ℕ), Finset.range (n + 1) = Finset.Iic n | null | true |
_private.Mathlib.Algebra.Group.Submonoid.Operations.0.AddSubmonoid.map_id.match_1_1 | Mathlib.Algebra.Group.Submonoid.Operations | ∀ {M : Type u_1} [inst : AddZeroClass M] (S : AddSubmonoid M) (x : M)
(motive : x ∈ AddSubmonoid.map (AddMonoidHom.id M) S → Prop) (x_1 : x ∈ AddSubmonoid.map (AddMonoidHom.id M) S),
(∀ (h : x ∈ ↑S), motive ⋯) → motive x_1 | null | false |
Lean.Elab.Structural.IndGroupInst.toMessageData | Lean.Elab.PreDefinition.Structural.IndGroupInfo | Lean.Elab.Structural.IndGroupInst → Lean.MessageData | null | true |
List.all_one_of_le_one_le_of_prod_eq_one | Mathlib.Algebra.Order.BigOperators.Group.List | ∀ {M : Type u_3} [inst : CommMonoid M] [inst_1 : PartialOrder M] [IsOrderedMonoid M] {l : List M},
(∀ x ∈ l, 1 ≤ x) → l.prod = 1 → ∀ {x : M}, x ∈ l → x = 1 | null | true |
ClopenUpperSet.noConfusion | Mathlib.Topology.Sets.Order | {P : Sort u} →
{α : Type u_2} →
{inst : TopologicalSpace α} →
{inst_1 : LE α} →
{t : ClopenUpperSet α} →
{α' : Type u_2} →
{inst' : TopologicalSpace α'} →
{inst'_1 : LE α'} →
{t' : ClopenUpperSet α'} →
α = α' → inst ≍ inst' → inst... | null | false |
_private.Mathlib.Algebra.FiniteSupport.Basic.0.Function.HasFiniteMulSupport.pi._simp_1_1 | Mathlib.Algebra.FiniteSupport.Basic | ∀ {ι : Type u_1} {M : Type u_3} [inst : One M] {f : ι → M} {x : ι}, (x ∈ Function.mulSupport f) = (f x ≠ 1) | null | false |
MeasureTheory.VectorMeasure.enorm_integral_le_lintegral_enorm | Mathlib.MeasureTheory.VectorMeasure.Integral | ∀ {X : Type u_2} {E : Type u_4} {F : Type u_5} {G : Type u_6} {mX : MeasurableSpace X} [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] {f : X → E} {μ : MeasureTheory.VectorMeasure X F} {B :... | null | true |
_private.Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable.0.MDifferentiableWithinAt.sum_section._simp_1_1 | Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | ∀ {α : Type u_1} [inst : DecidableEq α] {s : Finset α} {a b : α}, (a ∈ insert b s) = (a = b ∨ a ∈ s) | null | false |
Urysohns.CU.approx.match_1 | Mathlib.Topology.UrysohnsLemma | {X : Type u_1} →
[inst : TopologicalSpace X] →
{P : Set X → Set X → Prop} →
(motive : ℕ → Urysohns.CU P → X → Sort u_2) →
(x : ℕ) →
(x_1 : Urysohns.CU P) →
(x_2 : X) →
((c : Urysohns.CU P) → (x : X) → motive 0 c x) →
((n : ℕ) → (c : Urysohns.CU P) ... | null | false |
CategoryTheory.PreOneHypercover.isoMk_inv_s₀ | Mathlib.CategoryTheory.Sites.Hypercover.One | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {S : C} {E F : CategoryTheory.PreOneHypercover S}
(s₀ : E.I₀ ≃ F.I₀) (h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)) (s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j))
(h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k))
(w₀ :
autoParam (∀ (i : E.I₀), Cate... | null | true |
CategoryTheory.Limits.SingleObj.Types.sections.equivFixedPoints._proof_4 | Mathlib.CategoryTheory.Limits.Shapes.SingleObj | ∀ {M : Type u_1} [inst : Monoid M] (J : CategoryTheory.Functor (CategoryTheory.SingleObj M) (Type u_2))
(p : ↑(MulAction.fixedPoints M (J.obj (CategoryTheory.SingleObj.star M)))) {j j' : CategoryTheory.SingleObj M},
↑p ∈ MulAction.fixedPoints M (J.obj (CategoryTheory.SingleObj.star M)) | null | false |
_private.Mathlib.Probability.Kernel.Composition.MeasureCompProd.0.MeasureTheory.Measure.absolutelyContinuous_of_compProd._simp_1_1 | Mathlib.Probability.Kernel.Composition.MeasureCompProd | ∀ {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}, (μ Set.univ = 0) = (μ = 0) | null | false |
_private.Mathlib.RingTheory.LaurentSeries.0.LaurentSeries.coe_range_dense._simp_1_2 | Mathlib.RingTheory.LaurentSeries | ∀ {α : Type u} (x : α), (x ∈ Set.univ) = True | null | false |
CategoryTheory.Category.mk._proof_6 | Mathlib.CategoryTheory.Category.Basic | ∀ {obj : Type u_2} [toCategoryStruct : CategoryTheory.CategoryStruct.{u_1, u_2} obj],
autoParam
(∀ {W X Y Z : obj} (f : X ⟶ W) (g : Y ⟶ X) (h : Z ⟶ Y),
CategoryTheory.CategoryStruct.comp h (CategoryTheory.CategoryStruct.comp g f) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStr... | null | false |
Lean.Compiler.LCNF.instantiateRangeArgs | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} → Lean.Expr → ℕ → ℕ → Array (Lean.Compiler.LCNF.Arg pu) → Lean.Expr | null | true |
SpectrumRestricts.starAlgHom_injective | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | ∀ {R : Type u_1} {S : Type u_2} {A : Type u_3} [inst : Semifield R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
[inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : Semifield S] [inst_6 : StarRing S]
[inst_7 : MetricSpace S] [inst_8 : IsTopologicalSemiring S] [inst_9 : ContinuousStar S] [inst... | null | true |
AddCircle.homeomorphAddCircle_symm_apply_mk | Mathlib.Topology.Instances.AddCircle.Defs | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] (p q : 𝕜) [inst_1 : LinearOrder 𝕜] [inst_2 : IsStrictOrderedRing 𝕜]
[inst_3 : TopologicalSpace 𝕜] [inst_4 : OrderTopology 𝕜] (hp : p ≠ 0) (hq : q ≠ 0) (x : 𝕜),
(AddCircle.homeomorphAddCircle p q hp hq).symm ↑x = ↑(x * (q⁻¹ * p)) | null | true |
ZeroAtInftyContinuousMap.coe_zero | Mathlib.Topology.ContinuousMap.ZeroAtInfty | ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : Zero β], ⇑0 = 0 | null | true |
_private.Init.Grind.Module.Envelope.0.Lean.Grind.IntModule.OfNatModule.rel | Init.Grind.Module.Envelope | {α : Type u} →
[inst : Lean.Grind.NatModule α] →
Equivalence (Lean.Grind.IntModule.OfNatModule.r α) →
Lean.Grind.IntModule.OfNatModule.Q α → Lean.Grind.IntModule.OfNatModule.Q α → Prop | null | true |
_private.Mathlib.RingTheory.Flat.EquationalCriterion.0.Module.Flat.projective_of_finitePresentation.match_1_1 | Mathlib.RingTheory.Flat.EquationalCriterion | ∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
(motive : (∃ k h₂ h₃, LinearMap.id = h₃ ∘ₗ h₂) → Prop) (x : ∃ k h₂ h₃, LinearMap.id = h₃ ∘ₗ h₂),
(∀ (w : ℕ) (f : M →ₗ[R] Fin w →₀ R) (g : (Fin w →₀ R) →ₗ[R] M) (eq : LinearMap.id = g ∘ₗ f), motive ⋯) → motive x | null | false |
Module.finitePresentation_iff_finite | Mathlib.Algebra.Module.FinitePresentation | ∀ (R : Type u_1) (M : Type u_2) [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsNoetherianRing R],
Module.FinitePresentation R M ↔ Module.Finite R M | null | true |
AddMonoidAlgebra.ofMagma_apply | Mathlib.Algebra.MonoidAlgebra.Defs | ∀ (R : Type u_8) (M : Type u_9) [inst : Semiring R] [inst_1 : Add M] (a : Multiplicative M),
(AddMonoidAlgebra.ofMagma R M) a = AddMonoidAlgebra.single (Multiplicative.toAdd a) 1 | null | true |
FilterBasis | Mathlib.Order.Filter.Bases.Basic | Type u_5 → Type u_5 | A filter basis `B` on a type `α` is a nonempty collection of sets of `α`
such that the intersection of two elements of this collection contains some element
of the collection. | true |
List.le_max_of_mem | Init.Data.List.MinMax | ∀ {α : Type u_1} [inst : Max α] [inst_1 : LE α] [Std.IsLinearOrder α] [Std.LawfulOrderMax α] {l : List α} {a : α}
(ha : a ∈ l), a ≤ l.max ⋯ | null | true |
bind_pure_unit | Init.Control.Lawful.Basic | ∀ {m : Type u_1 → Type u_2} [inst : Monad m] [LawfulMonad m] {x : m PUnit.{u_1 + 1}},
(do
x
pure PUnit.unit) =
x | null | true |
rank_eq_card_basis | Mathlib.LinearAlgebra.Dimension.StrongRankCondition | ∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
{ι : Type w} [inst_4 : Fintype ι] (h : Module.Basis ι R M), Module.rank R M = ↑(Fintype.card ι) | If a vector space has a finite basis, then its dimension (seen as a cardinal) is equal to the
cardinality of the basis. | true |
MultilinearMap.currySum._proof_7 | Mathlib.LinearAlgebra.Multilinear.Curry | ∀ {R : Type u_5} {ι : Type u_2} {ι' : Type u_1} {M₂ : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M₂]
[inst_2 : Module R M₂] {N : ι ⊕ ι' → Type u_4} [inst_3 : (i : ι ⊕ ι') → AddCommMonoid (N i)]
[inst_4 : (i : ι ⊕ ι') → Module R (N i)] (f : MultilinearMap R N M₂) [inst_5 : DecidableEq ι]
(u : (i : ι... | null | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.