name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Init.Data.Range.Polymorphic.Internal.SignedBitVec.0.BitVec.Signed.sle_iff_rotate_le_rotate._simp_1_15 | Init.Data.Range.Polymorphic.Internal.SignedBitVec | ∀ {a b : ℕ}, (¬a ≤ b) = (b < a) | null | false |
_private.Lean.Util.Recognizers.0.Lean.Expr.eqOrIff?._sparseCasesOn_1 | Lean.Util.Recognizers | {α : Type u} →
{motive : Option α → Sort u_1} →
(t : Option α) → ((val : α) → motive (some val)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
MeromorphicAt.neg_iff._simp_1 | Mathlib.Analysis.Meromorphic.Basic | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {x : 𝕜} {f : 𝕜 → E}, MeromorphicAt (-f) x = MeromorphicAt f x | null | false |
ProbabilityTheory.uniformOn_singleton | Mathlib.Probability.UniformOn | ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [MeasurableSingletonClass Ω] (ω : Ω) (t : Set Ω)
[inst_2 : Decidable (ω ∈ t)], (ProbabilityTheory.uniformOn {ω}) t = if ω ∈ t then 1 else 0 | null | true |
_private.Mathlib.Tactic.FBinop.0.FBinopElab.Tree.recOn | Mathlib.Tactic.FBinop | {motive : FBinopElab.Tree✝ → Sort u} →
(t : FBinopElab.Tree✝) →
((ref : Lean.Syntax) →
(infoTrees : Lean.PersistentArray Lean.Elab.InfoTree) →
(val : Lean.Expr) → motive (FBinopElab.Tree.term✝ ref infoTrees val)) →
((ref : Lean.Syntax) →
(f : Lean.Expr) →
(lhs rhs : F... | null | false |
Lean.mkLevelSuccEx | Lean.Level | Lean.Level → Lean.Level | null | true |
Vector.map_attachWith_eq_pmap | Init.Data.Vector.Attach | ∀ {α : Type u_1} {n : ℕ} {β : Type u_2} {xs : Vector α n} {P : α → Prop} {H : ∀ a ∈ xs, P a} {f : { x // P x } → β},
Vector.map f (xs.attachWith P H) = Vector.pmap (fun a h => f ⟨a, ⋯⟩) xs ⋯ | null | true |
Lean.Meta.Grind.DelayedTheoremInstance.mk.noConfusion | Lean.Meta.Tactic.Grind.Types | {P : Sort u} →
{thm : Lean.Meta.Grind.EMatchTheorem} →
{proof prop : Lean.Expr} →
{generation : ℕ} →
{guards : List Lean.Meta.Grind.TheoremGuard} →
{sources : List Lean.Meta.Grind.EMatchDiagNode} →
{thm' : Lean.Meta.Grind.EMatchTheorem} →
{proof' prop' : Lean.Expr... | null | false |
ContinuousAlgHom.prodEquiv._proof_1 | Mathlib.Topology.Algebra.Algebra | ∀ {R : Type u_4} [inst : CommSemiring R] {A : Type u_1} [inst_1 : Semiring A] [inst_2 : TopologicalSpace A]
{B : Type u_2} [inst_3 : Semiring B] [inst_4 : TopologicalSpace B] [inst_5 : Algebra R A] [inst_6 : Algebra R B]
{C : Type u_3} [inst_7 : Semiring C] [inst_8 : Algebra R C] [inst_9 : TopologicalSpace C],
Fu... | null | false |
BoundedContinuousFunction.extend._proof_1 | Mathlib.Topology.ContinuousMap.Bounded.Basic | ∀ {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst_1 : PseudoMetricSpace β] {δ : Type u_3}
[inst_2 : TopologicalSpace δ] (f : α ↪ δ) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β),
∃ C, ∀ (x y : δ), dist (Function.extend (⇑f) (⇑g) (⇑h) x) (Function.extend (⇑f) (⇑g) (⇑h) y) ≤ ... | null | false |
Equiv.Perm.two_le_length_toList_iff_mem_support | Mathlib.GroupTheory.Perm.Cycle.Concrete | ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] {p : Equiv.Perm α} {x : α},
2 ≤ (p.toList x).length ↔ x ∈ p.support | null | true |
ValuationSubring.valuation_le_one_iff | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] (A : ValuationSubring K) (x : K), A.valuation x ≤ 1 ↔ x ∈ A | null | true |
WeierstrassCurve.a₃_of_isCharTwoJNeZeroNF | Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | ∀ {R : Type u_1} [inst : CommRing R] (W : WeierstrassCurve R) [W.IsCharTwoJNeZeroNF], W.a₃ = 0 | null | true |
Lean.Meta.DiscrTree.Key.lit.injEq | Lean.Meta.DiscrTree.Types | ∀ (a a_1 : Lean.Literal), (Lean.Meta.DiscrTree.Key.lit a = Lean.Meta.DiscrTree.Key.lit a_1) = (a = a_1) | null | true |
Int8.instNeg | Init.Data.SInt.Basic | Neg Int8 | null | true |
AlgebraicIndependent.aevalEquiv | Mathlib.RingTheory.AlgebraicIndependent.Defs | {ι : Type u_1} →
{R : Type u_3} →
{A : Type u_4} →
{x : ι → A} →
[inst : CommRing R] →
[inst_1 : CommRing A] →
[inst_2 : Algebra R A] → AlgebraicIndependent R x → MvPolynomial ι R ≃ₐ[R] ↥(Algebra.adjoin R (Set.range x)) | Canonical isomorphism between polynomials and the subalgebra generated by
algebraically independent elements. | true |
Equiv.sigmaFiberEquiv.match_1 | Mathlib.Logic.Equiv.Sum | ∀ {α : Type u_2} {β : Type u_1} (f : α → β) (motive : (y : β) × { x // f x = y } → Prop)
(x : (y : β) × { x // f x = y }), (∀ (a : α), motive ⟨f a, ⟨a, ⋯⟩⟩) → motive x | null | false |
_private.Lean.Meta.LetToHave.0.Lean.Meta.LetToHave.visitApp.match_1 | Lean.Meta.LetToHave | (motive : Lean.Expr → Sort u_1) →
(fType : Lean.Expr) →
((binderName : Lean.Name) →
(d b : Lean.Expr) → (binderInfo : Lean.BinderInfo) → motive (Lean.Expr.forallE binderName d b binderInfo)) →
((x : Lean.Expr) → motive x) → motive fType | null | false |
FinBddDistLat.instBoundedOrderCarrier | Mathlib.Order.Category.FinBddDistLat | (X : FinBddDistLat) → BoundedOrder ↑X.toDistLat | null | true |
LaurentSeries.powerSeriesPart_eq_zero._simp_1 | Mathlib.RingTheory.LaurentSeries | ∀ {R : Type u_1} [inst : Semiring R] (x : LaurentSeries R), (x.powerSeriesPart = 0) = (x = 0) | null | false |
Std.Internal.List.self_le_maxKey_insertEntry | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
∀ {k : α} {v : β k}, (compare k (Std.Internal.List.maxKey (Std.Internal.List.insertEntry k v l) ⋯)).isLE = true | null | true |
BitVec.cpopLayer._unsafe_rec | Init.Data.BitVec.Bitblast | {len w iterNum : ℕ} → BitVec (len * w) → BitVec (iterNum * w) → 2 * (iterNum - 1) < len → BitVec ((len + 1) / 2 * w) | null | false |
CategoryTheory.Localization.HasProductsOfShapeAux.isLimitMapCone._proof_2 | Mathlib.CategoryTheory.Localization.FiniteProducts | ∀ {C : Type u_4} {D : Type u_2} [inst : CategoryTheory.Category.{u_3, u_4} C]
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] (L : CategoryTheory.Functor C D)
(W : CategoryTheory.MorphismProperty C) [inst_2 : L.IsLocalization W] (J : Type)
[inst_3 : CategoryTheory.Limits.HasProductsOfShape J C] [inst_4 : W.IsStab... | null | false |
MeasurableEquiv.symm_vadd | Mathlib.MeasureTheory.Group.MeasurableEquiv | ∀ {G : Type u_1} {α : Type u_3} [inst : MeasurableSpace α] [inst_1 : AddGroup G] [inst_2 : AddAction G α]
[inst_3 : MeasurableConstVAdd G α] (c : G), (MeasurableEquiv.vadd c).symm = MeasurableEquiv.vadd (-c) | null | true |
Lean.Server.ServerTask.BaseIO.mapTaskCostly | Lean.Server.ServerTask | {α : Type u_1} → {β : Type} → (α → BaseIO β) → Lean.Server.ServerTask α → BaseIO (Lean.Server.ServerTask β) | null | true |
Lean.Meta.LazyDiscrTree.buildPath._unsafe_rec | Lean.Meta.LazyDiscrTree | (Bool → Array Lean.Expr → Lean.Expr → Lean.MetaM (Lean.Meta.LazyDiscrTree.Key × Array Lean.Expr)) →
Bool → Array Lean.Expr → Array Lean.Meta.LazyDiscrTree.Key → Lean.MetaM (Array Lean.Meta.LazyDiscrTree.Key) | null | false |
Lean.Elab.Tactic.RCases.RCasesPatt.brecOn | Lean.Elab.Tactic.RCases | {motive_1 : Lean.Elab.Tactic.RCases.RCasesPatt → Sort u} →
{motive_2 : List Lean.Elab.Tactic.RCases.RCasesPatt → Sort u} →
(t : Lean.Elab.Tactic.RCases.RCasesPatt) →
((t : Lean.Elab.Tactic.RCases.RCasesPatt) → t.below → motive_1 t) →
((t : List Lean.Elab.Tactic.RCases.RCasesPatt) → Lean.Elab.Tactic.... | null | false |
AdjoinRoot.quotEquivQuotMap_symm_apply_mk | Mathlib.RingTheory.AdjoinRoot | ∀ {R : Type u_1} [inst : CommRing R] (f g : Polynomial R) (I : Ideal R),
(AdjoinRoot.quotEquivQuotMap f I).symm
((Ideal.Quotient.mk (Ideal.span {Polynomial.map (Ideal.Quotient.mk I) f}))
(Polynomial.map (Ideal.Quotient.mk I) g)) =
(Ideal.Quotient.mk (Ideal.map (AdjoinRoot.of f) I)) ((AdjoinRoot.mk f... | null | true |
StarMonoidHom.mk.noConfusion | Mathlib.Algebra.Star.MonoidHom | {A : Type u_6} →
{B : Type u_7} →
{inst : Monoid A} →
{inst_1 : Star A} →
{inst_2 : Monoid B} →
{inst_3 : Star B} →
{P : Sort u} →
{toMonoidHom : A →* B} →
{map_star' : ∀ (a : A), (↑toMonoidHom).toFun (star a) = star ((↑toMonoidHom).toFun a)} →
... | null | false |
AlgebraicGeometry.Scheme.Pullback.gluing_V | Mathlib.AlgebraicGeometry.Pullbacks | ∀ {X Y Z : AlgebraicGeometry.Scheme} (𝒰 : X.OpenCover) (f : X ⟶ Z) (g : Y ⟶ Z)
[inst : ∀ (i : 𝒰.I₀), CategoryTheory.Limits.HasPullback (CategoryTheory.CategoryStruct.comp (𝒰.f i) f) g]
(x : 𝒰.I₀ × 𝒰.I₀),
(AlgebraicGeometry.Scheme.Pullback.gluing 𝒰 f g).V x =
match x with
| (i, j) => AlgebraicGeometr... | null | true |
Set.Countable.of_sdiff | Mathlib.Data.Set.Countable | ∀ {α : Type u} {s t : Set α}, (s \ t).Countable → t.Countable → s.Countable | null | true |
TensorProduct.exists_finite_submodule_of_setFinite' | Mathlib.LinearAlgebra.TensorProduct.Finiteness | ∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
[inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] {M₁ : Submodule R M} {N₁ : Submodule R N}
(s : Set (TensorProduct R ↥M₁ ↥N₁)),
s.Finite →
∃ M' N',
∃ (hM : M' ≤ M₁) (hN : N' ≤ N₁),
... | Variation of `TensorProduct.exists_finite_submodule_of_setFinite` where `M` and `N` are
already submodules. | true |
IntermediateField.one_mem'._inherited_default | Mathlib.FieldTheory.IntermediateField.Basic | ∀ {K : Type u_1} {L : Type u_2} {inst : Field K} {inst_1 : Field L} {inst_2 : Algebra K L} (carrier : Set L),
(∀ (r : K), (algebraMap K L) r ∈ carrier) → 1 ∈ carrier | null | false |
Lean.Meta.Simp.ConfigCtx.unfoldPartialApp._inherited_default | Init.MetaTypes | Bool | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.Subgraph.0.SimpleGraph.Subgraph.comap_monotone._simp_1_2 | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
Std.DTreeMap.Internal.Impl.Const.entryAtIdx.match_1 | Std.Data.DTreeMap.Internal.Queries | {α : Type u_1} →
{β : Type u_2} →
(motive : (x : Std.DTreeMap.Internal.Impl α fun x => β) → x.Balanced → (x_2 : ℕ) → x_2 < x.size → Sort u_3) →
(x : Std.DTreeMap.Internal.Impl α fun x => β) →
(x_1 : x.Balanced) →
(x_2 : ℕ) →
(x_3 : x_2 < x.size) →
((size : ℕ) →
... | null | false |
_private.Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable.0.summable_jacobiTheta₂'_term_iff._simp_1_3 | Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | ∀ (n : ℤ), ↑n = ↑↑n | null | false |
_private.Mathlib.RingTheory.PowerSeries.Substitution.0.PowerSeries.substInvFun.match_1.eq_2 | Mathlib.RingTheory.PowerSeries.Substitution | ∀ (motive : ℕ → Sort u_1) (h_1 : Unit → motive 0) (h_2 : Unit → motive 1) (h_3 : (n : ℕ) → motive n.succ),
(match 1 with
| 0 => h_1 ()
| 1 => h_2 ()
| n.succ => h_3 n) =
h_2 () | null | true |
Field.toEuclideanDomain.match_1 | Mathlib.Algebra.EuclideanDomain.Field | ∀ {K : Type u_1} [inst : Field K] (x x_1 : K) (motive : x_1 = 0 ∧ x ≠ 0 → Prop) (x_2 : x_1 = 0 ∧ x ≠ 0),
(∀ (left : x_1 = 0) (hnb : x ≠ 0), motive ⋯) → motive x_2 | null | false |
_private.Mathlib.Tactic.ComputeDegree.0.Mathlib.Tactic.ComputeDegree.dispatchLemma.match_1 | Mathlib.Tactic.ComputeDegree | (motive : Lean.Name → Lean.Name → Sort u_1) →
(na1 na2 : Lean.Name) →
(Unit → motive `Polynomial.natDegree `LE.le) →
(Unit → motive `Polynomial.degree `LE.le) →
(Unit → motive `Polynomial.coeff `Eq) →
((x : Lean.Name) → motive x `LE.le) → ((x x_1 : Lean.Name) → motive x x_1) → motive na1 n... | null | false |
Order.Ideal.instLattice | Mathlib.Order.Ideal | {P : Type u_1} → [inst : SemilatticeSup P] → [IsCodirectedOrder P] → Lattice (Order.Ideal P) | null | true |
Submodule.existsUnique_add_of_isCompl | Mathlib.LinearAlgebra.Projection | ∀ {R : Type u_1} [inst : Ring R] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module R E] {p q : Submodule R E},
IsCompl p q → ∀ (x : E), ∃ u v, ↑u + ↑v = x ∧ ∀ (r : ↥p) (s : ↥q), ↑r + ↑s = x → r = u ∧ s = v | null | true |
MonoidHom.coe_ofClosureMEqTopRight | Mathlib.Algebra.Group.Submonoid.Basic | ∀ {M : Type u_1} {N : Type u_2} [inst : Monoid M] [inst_1 : Monoid N] {s : Set M} (f : M → N)
(hs : Submonoid.closure s = ⊤) (h1 : f 1 = 1) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y),
⇑(MonoidHom.ofClosureMEqTopRight f hs h1 hmul) = f | null | true |
Std.TreeSet.inter_eq | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeSet α cmp}, t₁.inter t₂ = t₁ ∩ t₂ | null | true |
Std.DHashMap.Internal.Raw.inter_eq | Std.Data.DHashMap.Internal.Raw | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] {m₁ m₂ : Std.DHashMap.Raw α β} (h₁ : m₁.WF)
(h₂ : m₂.WF), m₁.inter m₂ = ↑(Std.DHashMap.Internal.Raw₀.inter ⟨m₁, ⋯⟩ ⟨m₂, ⋯⟩) | null | true |
_private.Lean.Meta.InferType.0.Lean.Meta.isArrowProposition'.checkProp._sunfold | Lean.Meta.InferType | Lean.Expr → Lean.Meta.ArrowPropResult✝ | null | false |
Pointed.Hom.instInhabited | Mathlib.CategoryTheory.Category.Pointed | (X : Pointed) → Inhabited (X.Hom X) | null | true |
_private.Mathlib.Analysis.Calculus.ContDiff.Operations.0.contDiff_pi._simp_1_2 | Mathlib.Analysis.Calculus.ContDiff.Operations | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {s : Set E} {n : WithTop ℕ∞} {ι : Type u_3} [inst_3 : Fintype ι] {F' : ι → Type u_5}
[inst_4 : (i : ι) → NormedAddCommGroup (F' i)] [inst_5 : (i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i :... | null | false |
Std.HashSet.Raw.get?_of_isEmpty | Std.Data.HashSet.RawLemmas | ∀ {α : Type u} {m : Std.HashSet.Raw α} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α] [LawfulHashable α],
m.WF → ∀ {a : α}, m.isEmpty = true → m.get? a = none | null | true |
existsUnique_eq._simp_1 | Mathlib.Basic.ExistsUnique | ∀ {α : Sort u_1} {a' : α}, (∃! a, a = a') = True | null | false |
ModuleCat.semilinearMapAddEquiv | Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {R : Type u₁} →
{S : Type u₂} →
[inst : Ring R] →
[inst_1 : Ring S] →
(f : R →+* S) →
(M : ModuleCat R) → (N : ModuleCat S) → (↑M →ₛₗ[f] ↑N) ≃+ (M ⟶ (ModuleCat.restrictScalars f).obj N) | Semilinear maps `M →ₛₗ[f] N` identify to
morphisms `M ⟶ (ModuleCat.restrictScalars f).obj N`. | true |
CategoryTheory.IsKernelPair.lift' | Mathlib.CategoryTheory.Limits.Shapes.KernelPair | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{R X Y : C} →
{f : X ⟶ Y} →
{a b : R ⟶ X} →
{S : C} →
CategoryTheory.IsKernelPair f a b →
(p q : S ⟶ X) →
CategoryTheory.CategoryStruct.comp p f = CategoryTheory.CategoryStruct.comp q f →
... | Given a pair of morphisms `p`, `q` to `X` which factor through `f`, they factor through any kernel
pair of `f`.
| true |
QuaternionAlgebra.equivTuple._proof_2 | Mathlib.Algebra.Quaternion | ∀ {R : Type u_1} (c₁ c₂ c₃ : R) (x : Fin 4 → R),
(fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) x) = x | null | false |
MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE | Mathlib.MeasureTheory.VectorMeasure.Order | ∀ {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : MeasureTheory.IsFiniteMeasure μ],
μ.toSignedMeasure.toMeasureOfZeroLE Set.univ ⋯ ⋯ = μ | null | true |
MDifferentiableWithinAt.congr_nhds | Mathlib.Geometry.Manifold.MFDeriv.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] {E' : Type u_5} [inst_6 : NormedAddComm... | null | true |
WeakDual.CharacterSpace.continuousMapEval._proof_8 | Mathlib.Topology.ContinuousMap.Ideals | ∀ (X : Type u_1) (𝕜 : Type u_2) [inst : TopologicalSpace X] [inst_1 : CommRing 𝕜] [inst_2 : TopologicalSpace 𝕜]
[inst_3 : IsTopologicalRing 𝕜] (x : X) (x_1 : 𝕜) (x_2 : C(X, 𝕜)), (x_1 • x_2) x = (x_1 • x_2) x | null | false |
List.getLast_append_right._proof_1 | Init.Data.List.Lemmas | ∀ {α : Type u_1} {l' l : List α}, l' ≠ [] → l ++ l' = [] → False | null | false |
List.Vector.snoc_cons | Mathlib.Data.Vector.Snoc | ∀ {α : Type u_1} {n : ℕ} {x : α} (xs : List.Vector α n) {y : α}, (x ::ᵥ xs).snoc y = x ::ᵥ xs.snoc y | null | true |
BitVec.getMsbD_ofBoolListLE | Init.Data.BitVec.Lemmas | ∀ {bs : List Bool} {i : ℕ},
(BitVec.ofBoolListLE bs).getMsbD i = (decide (i < bs.length) && bs.getD (bs.length - 1 - i) false) | null | true |
HomologicalComplex₂.toGradedObject.match_1 | Mathlib.Algebra.Homology.HomologicalBicomplex | {I₁ : Type u_1} →
{I₂ : Type u_2} → (motive : I₁ × I₂ → Sort u_3) → (x : I₁ × I₂) → ((i₁ : I₁) → (i₂ : I₂) → motive (i₁, i₂)) → motive x | null | false |
ContinuousMultilinearMap.recOn | 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... | null | false |
GroupLike.instGroup._proof_5 | Mathlib.RingTheory.HopfAlgebra.GroupLike | ∀ {R : Type u_2} {A : Type u_1} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : HopfAlgebra R A]
(a : GroupLike R A), a⁻¹ * a = 1 | null | false |
Nat.multichoose_succ_succ | Mathlib.Data.Nat.Choose.Basic | ∀ (n k : ℕ), (n + 1).multichoose (k + 1) = n.multichoose (k + 1) + (n + 1).multichoose k | null | true |
ProbabilityTheory.Kernel.piecewise.congr_simp | Mathlib.Probability.Kernel.Basic | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {s s_1 : Set α} (e_s : s = s_1)
{inst : DecidablePred fun x => x ∈ s} [inst_1 : DecidablePred fun x => x ∈ s_1] (hs : MeasurableSet s)
(κ κ_1 : ProbabilityTheory.Kernel α β),
κ = κ_1 →
∀ (η η_1 : ProbabilityTheory.Kernel α β),
... | null | true |
_private.Lean.Elab.PreDefinition.Structural.FindRecArg.0.Lean.Elab.Structural.argsInGroup.match_5 | Lean.Elab.PreDefinition.Structural.FindRecArg | (motive : Option (ℕ × Std.Iter ℕ) → Sort u_1) →
(x : Option (ℕ × Std.Iter ℕ)) →
(Unit → motive none) → ((indIdx : ℕ) → (s' : Std.Iter ℕ) → motive (some (indIdx, s'))) → motive x | null | false |
instNontrivialFreeRing | Mathlib.RingTheory.FreeRing | ∀ (α : Type u_1), Nontrivial (FreeRing α) | null | true |
Lean.Elab.Term.Do.ToTerm.Context.casesOn | Lean.Elab.Do.Legacy | {motive : Lean.Elab.Term.Do.ToTerm.Context → Sort u} →
(t : Lean.Elab.Term.Do.ToTerm.Context) →
((m returnType : Lean.Syntax) →
(uvars : Array Lean.Elab.Term.Do.Var) →
(kind : Lean.Elab.Term.Do.ToTerm.Kind) →
motive { m := m, returnType := returnType, uvars := uvars, kind := kind }) ... | null | false |
Std.DHashMap.isEmpty_toArray | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} [EquivBEq α] [LawfulHashable α],
m.toArray.isEmpty = m.isEmpty | null | true |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.Const.alterKey.match_1.splitter | Std.Data.Internal.List.Associative | {β : Type u_1} →
(motive : Option β → Sort u_2) → (x : Option β) → (Unit → motive none) → ((v : β) → motive (some v)) → motive x | null | true |
CategoryTheory.Functor.mapGrp_obj_X | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{D : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.CartesianMonoidalCategory D]
(F : CategoryTheory.Functor C D) [inst_4 : F.Monoidal] (A : CategoryTheory.Grp C), (F.ma... | null | true |
AlgebraicGeometry.Scheme.Hom.coverPreserving_opensFunctor | Mathlib.AlgebraicGeometry.OpenImmersion | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f],
CategoryTheory.CoverPreserving (Opens.grothendieckTopology ↥X) (Opens.grothendieckTopology ↥Y)
(AlgebraicGeometry.Scheme.Hom.opensFunctor f) | null | true |
Lean.Meta.CasesSubgoal.toInductionSubgoal | Lean.Meta.Tactic.Cases | Lean.Meta.CasesSubgoal → Lean.Meta.InductionSubgoal | null | true |
_private.Mathlib.RingTheory.Finiteness.Cardinality.0.AddSubmonoid.fg_iff_exists_fin_addMonoidHom.match_1_3 | Mathlib.RingTheory.Finiteness.Cardinality | ∀ {M : Type u_1} [inst : AddCommMonoid M] {S : AddSubmonoid M} (n : ℕ)
(motive : (∃ f, AddMonoidHom.mrange f = (AddSubmonoid.toNatSubmodule S).toAddSubmonoid) → Prop)
(x : ∃ f, AddMonoidHom.mrange f = (AddSubmonoid.toNatSubmodule S).toAddSubmonoid),
(∀ (f : (Fin n → ℕ) →+ M) (hf : AddMonoidHom.mrange f = (AddSubm... | null | false |
contDiff_prod' | Mathlib.Analysis.Calculus.ContDiff.Operations | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {n : WithTop ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [inst_3 : NormedCommRing 𝔸']
[inst_4 : NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'},
(∀ i ∈ t, ContDiff 𝕜 n (f i)) → ContDiff... | null | true |
_private.Mathlib.Tactic.SuccessIfFailWithMsg.0.Mathlib.Tactic._aux_Mathlib_Tactic_SuccessIfFailWithMsg___elabRules_Mathlib_Tactic_successIfFailWithMsg_1.unsafe_impl_2 | Mathlib.Tactic.SuccessIfFailWithMsg | Lean.TSyntax `term → Lean.Elab.TermElabM String | null | false |
isBoundedBilinearMap_compMultilinear | Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_3} [inst_1 : SeminormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {G : Type u_4} [inst_3 : SeminormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G]
{ι : Type u_5} {E : ι → Type u_6} [inst_5 : Fintype ι] [inst_6 : (i : ι) → NormedAddCommGroup (E i)]
[... | The composition of a continuous linear map with a continuous multilinear map is a bounded
bilinear operation. | true |
LinearMap.mkContinuous₂OfExists._proof_7 | Mathlib.Analysis.Normed.Operator.Bilinear | ∀ {𝕜 : Type u_6} {𝕜₂ : Type u_4} {𝕜₃ : Type u_5} {E' : Type u_1} {F' : Type u_2} {G' : Type u_3}
[inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜₂] [inst_2 : NontriviallyNormedField 𝕜₃]
{σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [inst_3 : AddCommGroup E'] [inst_4 : Module 𝕜 E'] [inst_5 : To... | null | false |
CategoryTheory.Idempotents.karoubiChainComplexEquivalence | Mathlib.CategoryTheory.Idempotents.HomologicalComplex | (C : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
(α : Type u_3) →
[inst_2 : AddRightCancelSemigroup α] →
[inst_3 : One α] →
CategoryTheory.Idempotents.Karoubi (ChainComplex C α) ≌
ChainComplex (CategoryTheo... | The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. | true |
Lean.Meta.Simp.Diagnostics.mk.inj | Lean.Meta.Tactic.Simp.Types | ∀ {usedThmCounter triedThmCounter : Lean.PHashMap Lean.Meta.Origin ℕ} {congrThmCounter : Lean.PHashMap Lean.Name ℕ}
{thmsWithBadKeys : Lean.PArray Lean.Meta.SimpTheorem}
{usedThmCounter_1 triedThmCounter_1 : Lean.PHashMap Lean.Meta.Origin ℕ}
{congrThmCounter_1 : Lean.PHashMap Lean.Name ℕ} {thmsWithBadKeys_1 : Lea... | null | true |
Mathlib.Meta.Positivity.Strictness.ctorElim | Mathlib.Tactic.Positivity.Core | {u : Lean.Level} →
{α : Q(Type u)} →
{zα : Q(Zero «$α»)} →
{e : Q(«$α»)} →
{motive : (a : Option Q(PartialOrder «$α»)) → Mathlib.Meta.Positivity.Strictness zα e a → Sort u} →
(ctorIdx : ℕ) →
{a : Option Q(PartialOrder «$α»)} →
(t : Mathlib.Meta.Positivity.Strictne... | null | false |
_private.Lean.Parser.Term.0.Lean.Parser.Term.letrec._regBuiltin.Lean.Parser.Termination.partialFixpoint.formatter_35 | Lean.Parser.Term | IO Unit | null | false |
MaxTropical.List.trop_sum | Mathlib.Algebra.Tropical.BigOperators | ∀ {R : Type u_1} [inst : AddMonoid R] (l : List R), MaxTropical.trop l.sum = (List.map MaxTropical.trop l).prod | null | true |
Std.ExtDHashMap.instLawfulSingletonSigma | Std.Data.ExtDHashMap.Basic | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} [inst : EquivBEq α] [inst_1 : LawfulHashable α],
LawfulSingleton ((a : α) × β a) (Std.ExtDHashMap α β) | null | true |
WithZero.coeMonoidHom | Mathlib.Algebra.GroupWithZero.WithZero | {α : Type u_1} → [inst : MulOneClass α] → α →* WithZero α | Coercion as a monoid hom. | true |
tsirelson_inequality | Mathlib.Algebra.Star.CHSH | ∀ {R : Type u} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [StarOrderedRing R]
[inst_4 : Algebra ℝ R] [IsOrderedModule ℝ R] [StarModule ℝ R] (A₀ A₁ B₀ B₁ : R),
IsCHSHTuple A₀ A₁ B₀ B₁ → A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ √2 ^ 3 • 1 | In a noncommutative ordered \*-algebra over ℝ,
Tsirelson's bound for a CHSH tuple (A₀, A₁, B₀, B₁) is
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2^(3/2) • 1`.
We prove this by providing an explicit sum-of-squares decomposition
of the difference.
(We could work over `ℤ[2^(1/2), 2^(-1/2)]` if we really wanted to!)
| true |
Rat.mul_num_den' | Mathlib.Data.Rat.Lemmas | ∀ (q r : ℚ), (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den | null | true |
_private.Init.Data.String.Lemmas.Pattern.TakeDrop.Char.0.String.endsWith_char_eq_false_iff_get._simp_1_1 | Init.Data.String.Lemmas.Pattern.TakeDrop.Char | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.BackwardPattern pat] {s : String},
s.endsWith pat = s.toSlice.endsWith pat | null | false |
Nat.ascFactorial_zero | Mathlib.Data.Nat.Factorial.Basic | ∀ (n : ℕ), n.ascFactorial 0 = 1 | null | true |
«_aux_Mathlib_Algebra_Order_Hom_Monoid___macroRules_term_≃+o__1» | Mathlib.Algebra.Order.Hom.Monoid | Lean.Macro | null | false |
_private.Mathlib.LinearAlgebra.LinearIndependent.Lemmas.0.LinearIndependent.pair_symm_iff._simp_1_1 | Mathlib.LinearAlgebra.LinearIndependent.Lemmas | ∀ {R : Type u_2} {M : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {x y : M},
LinearIndependent R ![x, y] = ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0 | null | false |
_private.Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique.0.exists_isMIntegralCurveAt_of_contMDiffAt.match_1_3 | Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_3} [inst_2 : TopologicalSpace H]
{I : ModelWithCorners ℝ E H} {M : Type u_2} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
[inst_5 : IsManifold I 1 M] {v : (x : M) → TangentSpace I x} (t₀ : ℝ) {x₀ : M} (f : ℝ → E)
(m... | null | false |
Set.bijOn_empty | Mathlib.Data.Set.Function | ∀ {α : Type u_1} {β : Type u_2} (f : α → β), Set.BijOn f ∅ ∅ | null | true |
Std.Iter.atIdxSlow?_repeat | Std.Data.Iterators.Lemmas.Producers.Repeat | ∀ {α : Type w} {f : α → α} {init : α} {n : ℕ},
Std.Iter.atIdxSlow? n (Std.Iter.repeat f init) = some (Nat.repeat f n init) | null | true |
_private.Mathlib.NumberTheory.EllipticDivisibilitySequence.0.IsEllipticNet.atom_odd._simp_1_3 | Mathlib.NumberTheory.EllipticDivisibilitySequence | ∀ {R : Type v} [inst : Mul R] [inst_1 : Add R] [LeftDistribClass R] (a b c : R), a * b + a * c = a * (b + c) | null | false |
Lean.Meta.Simp.ConfigCtx.ground._inherited_default | Init.MetaTypes | Bool | null | false |
Set.Ico_union_Ico | Mathlib.Order.Interval.Set.LinearOrder | ∀ {α : Type u_1} [inst : LinearOrder α] {a b c d : α},
min a b ≤ max c d → min c d ≤ max a b → Set.Ico a b ∪ Set.Ico c d = Set.Ico (min a c) (max b d) | null | true |
TypeVec.typevecCasesCons₂._proof_1 | Mathlib.Data.TypeVec | ∀ (n : ℕ) (t : Type u_2) (t' : Type u_3) (v : TypeVec.{u_2} n) (v' : TypeVec.{u_3} n)
{β : (v ::: t).Arrow (v' ::: t') → Sort u_1} (fs : (v ::: t).Arrow (v' ::: t')),
β fs = β (TypeVec.splitFun (TypeVec.dropFun fs) (TypeVec.lastFun fs)) | null | false |
_private.Lean.Meta.Tactic.Grind.AC.Seq.0.Lean.Grind.AC.Seq.superpose?.go._sunfold | Lean.Meta.Tactic.Grind.AC.Seq | Lean.Grind.AC.Seq →
Lean.Grind.AC.Seq → Lean.Grind.AC.Seq → Option (Lean.Grind.AC.Seq × Lean.Grind.AC.Seq × Lean.Grind.AC.Seq) | null | false |
Lex.instMulActionWithZero._proof_1 | Mathlib.Algebra.Order.GroupWithZero.Action.Synonym | ∀ {G₀ : Type u_1} {M₀ : Type u_2} [inst : MonoidWithZero G₀] [inst_1 : AddMonoid M₀] [inst_2 : MulActionWithZero G₀ M₀]
(r : Lex G₀), r • 0 = 0 | null | false |
_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.Indep.union_isBasis_union_of_contract_isBasis._simp_1_3 | Mathlib.Combinatorics.Matroid.Minor.Contract | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
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