name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Std.Tactic.BVDecide.BVExpr.Cache.Key.w | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Expr | Std.Tactic.BVDecide.BVExpr.Cache.Key → ℕ | null | true |
CategoryTheory.MorphismProperty.Over.pullbackComp._proof_2 | Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | ∀ {T : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} T] {P Q : CategoryTheory.MorphismProperty T}
[inst_1 : Q.IsMultiplicative] {X Y Z : T} (f : X ⟶ Y) (g : Y ⟶ Z) [inst_2 : P.IsStableUnderBaseChangeAlong f]
[inst_3 : P.IsStableUnderBaseChangeAlong g] [inst_4 : P.HasPullbacksAlong f] [inst_5 : P.HasPullbacks... | null | false |
Set.MapsTo.vadd_set | Mathlib.GroupTheory.GroupAction.Pointwise | ∀ {M : Type u_1} {α : Type u_2} {β : Type u_3} {F : Type u_4} [inst : VAdd M α] [inst_1 : VAdd M β]
[inst_2 : FunLike F α β] [AddActionHomClass F M α β] {f : F} {s : Set α} {t : Set β},
Set.MapsTo (⇑f) s t → ∀ (c : M), Set.MapsTo (⇑f) (c +ᵥ s) (c +ᵥ t) | null | true |
Codisjoint.map | Mathlib.Order.Hom.BoundedLattice | ∀ {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Lattice α] [inst_1 : Lattice β] [inst_2 : FunLike F α β]
[inst_3 : OrderTop α] [inst_4 : OrderTop β] [TopHomClass F α β] [SupHomClass F α β] {a b : α} (f : F),
Codisjoint a b → Codisjoint (f a) (f b) | null | true |
NonUnitalStarSubalgebra.map | Mathlib.Algebra.Star.NonUnitalSubalgebra | {R : Type u} →
{A : Type v} →
{B : Type w} →
[inst : CommSemiring R] →
[inst_1 : NonUnitalNonAssocSemiring A] →
[inst_2 : Module R A] →
[inst_3 : Star A] →
[inst_4 : NonUnitalNonAssocSemiring B] →
[inst_5 : Module R B] →
[inst_6 :... | Transport a non-unital star subalgebra via a non-unital star algebra homomorphism. | true |
PrimeSpectrum.instNonemptyOfNontrivial | Mathlib.RingTheory.Spectrum.Prime.Basic | ∀ {R : Type u} [inst : CommSemiring R] [Nontrivial R], Nonempty (PrimeSpectrum R) | null | true |
Real.Angle.sign_two_nsmul_eq_neg_sign_iff | Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | ∀ {θ : Real.Angle}, (2 • θ).sign = -θ.sign ↔ θ = 0 ∨ Real.pi / 2 < |θ.toReal| | null | true |
List.Perm.eq_reverse_of_sortedLE_of_sortedGE | Mathlib.Data.List.Sort | ∀ {α : Type u_1} [inst : PartialOrder α] {l₁ l₂ : List α}, l₁.Perm l₂ → l₁.SortedLE → l₂.SortedGE → l₁ = l₂.reverse | null | true |
nonempty_linearEquiv_of_lift_rank_eq | Mathlib.LinearAlgebra.Dimension.Free | ∀ {R : Type u} {M : Type v} {M' : Type v'} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[Module.Free R M] [inst_4 : AddCommMonoid M'] [inst_5 : Module R M'] [Module.Free R M'] [StrongRankCondition R],
Cardinal.lift.{v', v} (Module.rank R M) = Cardinal.lift.{v, v'} (Module.rank R M') → Nonemp... | Two vector spaces are isomorphic if they have the same dimension. | true |
Matrix.isSymm_transpose_iff | Mathlib.LinearAlgebra.Matrix.Symmetric | ∀ {α : Type u_1} {n : Type u_3} {A : Matrix n n α}, A.transpose.IsSymm ↔ A.IsSymm | null | true |
PointwiseConvergenceCLM.coeLMₛₗ | Mathlib.Topology.Algebra.Module.Spaces.PointwiseConvergenceCLM | {𝕜₁ : Type u_4} →
{𝕜₂ : Type u_5} →
[inst : NormedField 𝕜₁] →
[inst_1 : NormedField 𝕜₂] →
(σ : 𝕜₁ →+* 𝕜₂) →
(E : Type u_7) →
(F : Type u_8) →
[inst_2 : AddCommGroup E] →
[inst_3 : TopologicalSpace E] →
[inst_4 : AddCommGroup... | Coercion from `E →SLₚₜ[σ] F` to `E →ₛₗ[σ] F` as a `𝕜₂`-linear map. | true |
CategoryTheory.Adjunction.CoreHomEquivUnitCounit.homEquiv_unit._autoParam | Mathlib.CategoryTheory.Adjunction.Basic | Lean.Syntax | null | false |
Real.cosPartialEquiv._proof_5 | Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | ∀ θ ∈ Set.Icc 0 Real.pi, θ ≤ Real.pi | null | false |
AddSubgroup.relIndex_eq_one._simp_1 | Mathlib.GroupTheory.Index | ∀ {G : Type u_1} [inst : AddGroup G] {H K : AddSubgroup G}, (H.relIndex K = 1) = (K ≤ H) | null | false |
AlgebraicGeometry.Scheme.Hom | Mathlib.AlgebraicGeometry.Scheme | AlgebraicGeometry.Scheme → AlgebraicGeometry.Scheme → Type u_1 | A morphism between schemes is a morphism between the underlying locally ringed spaces. | true |
AdicCompletion.map_of | Mathlib.RingTheory.AdicCompletion.Functoriality | ∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R) {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{N : Type u_3} [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M →ₗ[R] N) (x : M),
((AdicCompletion.map I) f) ((AdicCompletion.of I M) x) = (AdicCompletion.of I N) (f x) | null | true |
Std.DTreeMap.toList_rci | Std.Data.DTreeMap.Slice | ∀ {α : Type u} {β : α → Type v} (cmp : autoParam (α → α → Ordering) Std.DTreeMap.toList_rci._auto_1) [Std.TransCmp cmp]
{t : Std.DTreeMap α β cmp} {bound : α},
Std.Slice.toList (Std.Rci.Sliceable.mkSlice t (bound...*)) = List.filter (fun e => (cmp e.fst bound).isGE) t.toList | null | true |
CategoryTheory.NatTrans.hcomp_app._to_dual_1 | Mathlib.CategoryTheory.Functor.Category | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] {F G : CategoryTheory.Functor C D}
{H I : CategoryTheory.Functor D E} (α : G ⟶ F) (β : I ⟶ H) (X : C),
(α ◫ β).app X = CategoryTheory.... | null | false |
CategoryTheory.ReflexiveRelation.map | Mathlib.CategoryTheory.EquivalenceRelation | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{D : Type u_2} →
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
{R X : C} →
{p₁ p₂ : R ⟶ X} →
CategoryTheory.ReflexiveRelation p₁ p₂ →
(F : CategoryTheory.Functor C D) →
[CategoryT... | Given a functor `F : C ⥤ D`, if `F.map p₁` and `F.map p₂` form a jointly monic pair of
morphisms, then `F` preserves reflexive relations. | true |
ContMDiff.smul | Mathlib.Geometry.Manifold.Algebra.SMul | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
[inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {H' : Type u_4}
[inst_4 : TopologicalSpace H'] {E' : Type u_5} [inst_5 : NormedAddCommGroup E'] [inst_6 : Normed... | null | true |
ProofWidgets.RefreshComponent.RpcEncodablePacket.mk._flat_ctor._@.ProofWidgets.Component.RefreshComponent.311896448._hygCtx._hyg.1 | ProofWidgets.Component.RefreshComponent | Lean.Json → Lean.Json → ProofWidgets.RefreshComponent.RpcEncodablePacket✝ | null | false |
LinearMap.lift_rank_le_of_surjective | Mathlib.LinearAlgebra.Dimension.Basic | ∀ {R : Type u} {M : Type v} {M' : Type v'} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M'),
Function.Surjective ⇑f → Cardinal.lift.{v, v'} (Module.rank R M') ≤ Cardinal.lift.{v', v} (Module.rank R M) | null | true |
NumberField.mixedEmbedding.fundamentalCone.integerSetQuotEquivAssociates._proof_3 | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | ∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K],
Function.Injective (Quotient.lift (NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates K) ⋯) ∧
Function.Surjective (Quotient.lift (NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates K) ⋯) | null | false |
fwdDiff_aux.fwdDiffₗ_apply | Mathlib.Algebra.Group.ForwardDiff | ∀ (M : Type u_1) (G : Type u_2) [inst : AddCommMonoid M] [inst_1 : AddCommGroup G] (h : M) (f : M → G) (a : M),
(fwdDiff_aux.fwdDiffₗ M G h) f a = fwdDiff h f a | null | true |
_private.Mathlib.Topology.ContinuousOn.0.continuous_prod_of_discrete_left._simp_1_1 | Mathlib.Topology.ContinuousOn | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β},
Continuous f = ContinuousOn f Set.univ | null | false |
_private.Mathlib.MeasureTheory.VectorMeasure.BoundedVariation.0.BoundedVariationOn.variation_vectorMeasure_Ioo_right._proof_1_5 | Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, a < b → b ∈ Set.Ioc a b | null | false |
ENat.lt_top_of_sum_ne_top | Mathlib.Data.ENat.BigOperators | ∀ {α : Type u_1} {s : Finset α} {f : α → ℕ∞}, ∑ x ∈ s, f x ≠ ⊤ → ∀ {a : α}, a ∈ s → f a < ⊤ | null | true |
Graph.Simple.anti | Mathlib.Combinatorics.Graph.Simple | ∀ {α : Type u_1} {β : Type u_2} {G H : Graph α β} [G.Simple], H ≤ G → H.Simple | null | true |
UnitalShelf.act_self_act_eq | Mathlib.Algebra.Quandle | ∀ {S : Type u_1} [inst : UnitalShelf S] (x y : S), Shelf.act x (Shelf.act x y) = Shelf.act x y | null | true |
IsDedekindDomain.HeightOneSpectrum.instCoeIdeal | Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {R : Type u_1} → [inst : CommRing R] → Coe (IsDedekindDomain.HeightOneSpectrum R) (Ideal R) | null | true |
_private.Mathlib.MeasureTheory.SpecificCodomains.WithLp.0.MeasureTheory.memLp_piLp_iff._simp_1_1 | Mathlib.MeasureTheory.SpecificCodomains.WithLp | ∀ {X : Type u_1} {mX : MeasurableSpace X} {μ : MeasureTheory.Measure X} {p : ENNReal} {ι : Type u_2} [inst : Fintype ι]
{E : ι → Type u_3} [inst_1 : (i : ι) → NormedAddCommGroup (E i)] {f : X → (i : ι) → E i},
(∀ (i : ι), MeasureTheory.MemLp (fun x => f x i) p μ) = MeasureTheory.MemLp f p μ | null | false |
SemiNormedGrp₁.of | Mathlib.Analysis.Normed.Group.SemiNormedGrp | (carrier : Type u) → [str : SeminormedAddCommGroup carrier] → SemiNormedGrp₁ | Construct a bundled `SemiNormedGrp₁` from the underlying type and typeclass. | true |
MvPowerSeries.map._proof_7 | Mathlib.RingTheory.MvPowerSeries.Basic | ∀ {σ : Type u_1} {R : Type u_2} {S : Type u_3} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S)
(φ ψ : MvPowerSeries σ R),
(fun n => f ((MvPowerSeries.coeff n) (φ + ψ))) =
(fun n => f ((MvPowerSeries.coeff n) φ)) + fun n => f ((MvPowerSeries.coeff n) ψ) | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.StronglyRegular.0.SimpleGraph.IsSRGWith.param_eq._simp_1_3 | Mathlib.Combinatorics.SimpleGraph.StronglyRegular | ∀ {α : Type u_1} (s t : Set α) [inst : DecidableEq α] [inst_1 : Fintype ↑s] [inst_2 : Fintype ↑t]
[inst_3 : Fintype ↑(s ∩ t)], s.toFinset ∩ t.toFinset = (s ∩ t).toFinset | null | false |
CompositionAsSet.card_boundaries_pos | Mathlib.Combinatorics.Enumerative.Composition | ∀ {n : ℕ} (c : CompositionAsSet n), 0 < c.boundaries.card | null | true |
Polynomial.roots_C_mul | Mathlib.Algebra.Polynomial.Roots | ∀ {R : Type u} {a : R} [inst : CommRing R] [inst_1 : IsDomain R] (p : Polynomial R),
a ≠ 0 → (Polynomial.C a * p).roots = p.roots | null | true |
SMulWithZero.compHom | Mathlib.Algebra.GroupWithZero.Action.Defs | {M₀ : Type u_2} →
{M₀' : Type u_3} →
(A : Type u_7) →
[inst : Zero M₀] →
[inst_1 : Zero A] → [SMulWithZero M₀ A] → [inst_3 : Zero M₀'] → ZeroHom M₀' M₀ → SMulWithZero M₀' A | Compose a `SMulWithZero` with a `ZeroHom`, with action `f r' • m` | true |
_private.Mathlib.Analysis.SpecialFunctions.Complex.Log.0.Complex.expOpenPartialHomeomorph._simp_3 | Mathlib.Analysis.SpecialFunctions.Complex.Log | ∀ {α : Type u} {a : α} {p : α → Prop}, (a ∈ {x | p x}) = p a | null | false |
ImportGraph.Lake.instReprBaseWorkspace | ImportGraph.WorkspaceModel.Base | Repr ImportGraph.Lake.BaseWorkspace | null | true |
Con.instCompleteLattice._proof_2 | Mathlib.GroupTheory.Congruence.Defs | ∀ {M : Type u_1} [inst : Mul M] (s : Set (Con M)), sInf s ∈ lowerBounds s ∧ sInf s ∈ upperBounds (lowerBounds s) | null | false |
NumberField.InfinitePlace.liesOver_conjugate_embedding_of_mem_ramifiedPlacesOver | Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | ∀ {K : Type u_4} {L : Type u_5} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]
{v : NumberField.InfinitePlace K} {w : NumberField.InfinitePlace L},
w ∈ NumberField.InfinitePlace.ramifiedPlacesOver L v →
NumberField.ComplexEmbedding.LiesOver (NumberField.ComplexEmbedding.conjugate w.embedding) v.embe... | null | true |
CategoryTheory.StrictPseudofunctor.comp_obj | Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C] {D : Type u₃}
[inst_2 : CategoryTheory.Bicategory D] (F : CategoryTheory.StrictPseudofunctor B C)
(G : CategoryTheory.StrictPseudofunctor C D) (X : B), (F.comp G).obj X = G.obj (F.obj X) | null | true |
IsTopologicalAddGroup.toHSpace._proof_5 | Mathlib.Topology.Homotopy.HSpaces | ∀ (M : Type u_1) [inst : AddZeroClass M] [inst_1 : TopologicalSpace M] [inst_2 : ContinuousAdd M],
{ toFun := Function.uncurry Add.add, continuous_toFun := ⋯ }.comp
((ContinuousMap.const M 0).prodMk (ContinuousMap.id M)) =
{ toFun := Function.uncurry Add.add, continuous_toFun := ⋯ }.comp
((ContinuousM... | null | false |
Submodule.IsMinimalPrimaryDecomposition.comap_localized₀_eq_ite | Mathlib.RingTheory.Lasker | ∀ {R : Type u_3} {M : Type u_4} [inst : CommRing R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Submodule R M}
(s₀ : Finset ↑N.associatedPrimes),
IsLowerSet ↑s₀ →
∀ (q : Submodule R M),
q.IsPrimary →
∀ (p : ↑N.associatedPrimes),
(q.colon Set.univ).radical = ↑p →
Sub... | null | true |
Finmap.keys_singleton | Mathlib.Data.Finmap | ∀ {α : Type u} {β : α → Type v} (a : α) (b : β a), (Finmap.singleton a b).keys = {a} | null | true |
Std.TreeMap.mk.injEq | Std.Data.TreeMap.Basic | ∀ {α : Type u} {β : Type v} {cmp : autoParam (α → α → Ordering) Std.TreeMap._auto_1}
(inner inner_1 : Std.DTreeMap α (fun x => β) cmp), ({ inner := inner } = { inner := inner_1 }) = (inner = inner_1) | null | true |
OpenPartialHomeomorph.extend_preimage_mem_nhdsWithin | Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | ∀ {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : TopologicalSpace H] [inst_4 : TopologicalSpace M]
(f : OpenPartialHomeomorph M H) {I : ModelWithCorners 𝕜 E H} {s t : Set M} {x : M},
x ∈ f.sour... | Technical lemma ensuring that the preimage under an extended chart of a neighborhood of a point
in the source is a neighborhood of the preimage, within a set. | true |
_private.Mathlib.Algebra.Order.Archimedean.Basic.0.WithBot.instArchimedean._simp_1 | Mathlib.Algebra.Order.Archimedean.Basic | ∀ {α : Type u} [inst : AddMonoid α] (a : α) (n : ℕ), n • ↑a = ↑(n • a) | null | false |
RBTree.RBNode.upperBound?_ge | Batteries.Recycling.RBTree.Lemmas | ∀ {α : Type u_1} {cut : α → Ordering} {x : α} {t : RBTree.RBNode α},
RBTree.RBNode.upperBound? cut t = some x → cut x ≠ Ordering.gt | The value `x` returned by `upperBound?` is greater or equal to the `cut`. | true |
LinearIsometry.map_sub | Mathlib.Analysis.Normed.Operator.LinearIsometry | ∀ {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [inst : Semiring R] [inst_1 : Semiring R₂]
{σ₁₂ : R →+* R₂} [inst_2 : SeminormedAddCommGroup E] [inst_3 : SeminormedAddCommGroup E₂] [inst_4 : Module R E]
[inst_5 : Module R₂ E₂] (f : E →ₛₗᵢ[σ₁₂] E₂) (x y : E), f (x - y) = f x - f y | null | true |
TwoSidedIdeal.rightModule | Mathlib.RingTheory.TwoSidedIdeal.Operations | {R : Type u_1} → [inst : Ring R] → (I : TwoSidedIdeal R) → Module Rᵐᵒᵖ ↥I | null | true |
Fintype.linearCombination._proof_3 | Mathlib.LinearAlgebra.Finsupp.LinearCombination | ∀ {α : Type u_2} {M : Type u_1} (R : Type u_3) [inst : Fintype α] [inst_1 : Semiring R] [inst_2 : AddCommMonoid M]
[inst_3 : Module R M] (v : α → M) (f g : α → R), ∑ i, (f + g) i • v i = ∑ i, f i • v i + ∑ i, g i • v i | null | false |
Homeomorph.toOpenPartialHomeomorphOfImageEq._proof_2 | Mathlib.Topology.OpenPartialHomeomorph.Defs | ∀ {X : Type u_2} {Y : Type u_1} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X),
IsOpen s → ∀ (t : Set Y) (h : ⇑e '' s = t), IsOpen (e.toPartialHomeomorphOfImageEq s t h).target | null | false |
TensorAlgebra.instSemiring._proof_17 | Mathlib.LinearAlgebra.TensorAlgebra.Basic | ∀ (R : Type u_1) [inst : CommSemiring R] (M : Type u_2) [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(a : TensorAlgebra R M), a * 1 = a | null | false |
Projectivization.independent_iff_iSupIndep | Mathlib.LinearAlgebra.Projectivization.Independence | ∀ {ι : Type u_1} {K : Type u_2} {V : Type u_3} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V]
{f : ι → Projectivization K V}, Projectivization.Independent f ↔ iSupIndep fun i => (f i).submodule | A family of points in projective space is independent if and only if the family of
submodules which the points determine is independent in the lattice-theoretic sense. | true |
_private.Mathlib.Data.Nat.Hyperoperation.0.hyperoperation_ge_two_eq_self._proof_1_2 | Mathlib.Data.Nat.Hyperoperation | ∀ (m n : ℕ), hyperoperation (n + 2) m 1 = m → hyperoperation (n + 1 + 2) m 1 = m | null | false |
norm_eq_zero._simp_1 | Mathlib.Analysis.Normed.Group.Basic | ∀ {E : Type u_4} [inst : NormedAddGroup E] {a : E}, (‖a‖ = 0) = (a = 0) | null | false |
MvPolynomial.rTensorAlgEquiv.eq_1 | Mathlib.RingTheory.TensorProduct.MvPolynomial | ∀ {R : Type u} {N : Type v} [inst : CommSemiring R] {σ : Type u_1} {S : Type u_3} [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [inst_3 : CommSemiring N] [inst_4 : Algebra R N],
MvPolynomial.rTensorAlgEquiv = AddMonoidAlgebra.rTensorEquivAlgEquiv R S S N | null | true |
ContinuousAlternatingMap.instTopologicalSpace | Mathlib.Topology.Algebra.Module.Alternating.Topology | {𝕜 : Type u_1} →
{ι : Type u_2} →
{E : Type u_3} →
{F : Type u_4} →
[inst : NormedField 𝕜] →
[inst_1 : AddCommGroup E] →
[inst_2 : Module 𝕜 E] →
[inst_3 : TopologicalSpace E] →
[inst_4 : AddCommGroup F] →
[inst_5 : Module 𝕜 F]... | null | true |
LeanSearchClient.unicode_turnstile | LeanSearchClient.LoogleSyntax | Lean.Parser.Parser | null | true |
DomMulAct.instMulDistribMulActionMonoidHom._proof_2 | Mathlib.GroupTheory.GroupAction.DomAct.Basic | ∀ {A : Type u_2} {M : Type u_1} {B : Type u_3} [inst : Monoid M] [inst_1 : Monoid A] [inst_2 : MulDistribMulAction M A]
[inst_3 : CommMonoid B] (x : Mᵈᵐᵃ) (x_1 : A →* B), (MonoidHom.coeFn A B) (x • x_1) = (MonoidHom.coeFn A B) (x • x_1) | null | false |
Disjoint.of_preimage | Mathlib.Data.Set.Image | ∀ {α : Type u_1} {β : Type u_2} {f : α → β},
Function.Surjective f → ∀ {s t : Set β}, Disjoint (f ⁻¹' s) (f ⁻¹' t) → Disjoint s t | null | true |
CommRingCat.coyonedaUnique_inv_app_hom_apply | Mathlib.Algebra.Category.Ring.Adjunctions | ∀ {n : Type v} [inst : Unique n] (X : CommRingCat) (a : ↑X) (a_1 : Opposite.unop (Opposite.op n)),
(CommRingCat.Hom.hom (CommRingCat.coyonedaUnique.inv.app X)) a a_1 = a | null | true |
Int.emod_eq_zero_of_dvd | Init.Data.Int.DivMod.Bootstrap | ∀ {a b : ℤ}, a ∣ b → b % a = 0 | null | true |
Lean.Meta.SynthInstance.TableEntry.mk.injEq | Lean.Meta.SynthInstance | ∀ (waiters : Array Lean.Meta.SynthInstance.Waiter) (answers : Array Lean.Meta.SynthInstance.Answer)
(waiters_1 : Array Lean.Meta.SynthInstance.Waiter) (answers_1 : Array Lean.Meta.SynthInstance.Answer),
({ waiters := waiters, answers := answers } = { waiters := waiters_1, answers := answers_1 }) =
(waiters = wa... | null | true |
FirstOrder.Language.BoundedFormula.realize_iAlls._simp_1 | Mathlib.ModelTheory.Semantics | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {α : Type u'} {β : Type v'} [inst_1 : Finite β]
{φ : L.Formula (α ⊕ β)} {v : α → M} {v' : Fin 0 → M},
FirstOrder.Language.BoundedFormula.Realize (FirstOrder.Language.Formula.iAlls β φ) v v' =
∀ (i : β → M), φ.Realize fun a => Sum.elim v i a | null | false |
_private.Mathlib.Data.Nat.Digits.Defs.0.Nat.digitsAux0.match_1.eq_1 | Mathlib.Data.Nat.Digits.Defs | ∀ (motive : ℕ → Sort u_1) (h_1 : Unit → motive 0) (h_2 : (n : ℕ) → motive n.succ),
(match 0 with
| 0 => h_1 ()
| n.succ => h_2 n) =
h_1 () | null | true |
SemimoduleCat.coe_of | Mathlib.Algebra.Category.ModuleCat.Semi | ∀ (R : Type u) [inst : Semiring R] (X : Type v) [inst_1 : Semiring X] [inst_2 : Module R X], ↑↧X = X | null | true |
Localization.lt._proof_2 | Mathlib.GroupTheory.MonoidLocalization.Order | ∀ {α : Type u_1} [inst : CommMonoid α] [inst_1 : PartialOrder α] [IsOrderedCancelMonoid α] {s : Submonoid α} {a₁ b₁ : α}
{a₂ b₂ : ↥s} {c₁ d₁ : α} {c₂ d₂ : ↥s},
(Localization.r s) (a₁, a₂) (b₁, b₂) →
(Localization.r s) (c₁, c₂) (d₁, d₂) → (↑c₂ * a₁ < ↑a₂ * c₁) = (↑d₂ * b₁ < ↑b₂ * d₁) | null | false |
vsub_left_injective | Mathlib.Algebra.Torsor.Basic | ∀ {G : Type u_1} {P : Type u_2} [inst : AddGroup G] [T : AddTorsor G P] (p : P), Function.Injective fun x => x -ᵥ p | Subtracting the point `p` is an injective function. | true |
CategoryTheory.Limits.Types.Image.ι | Mathlib.CategoryTheory.Limits.Types.Images | {α β : Type u} → (f : α ⟶ β) → CategoryTheory.Limits.Types.Image f ⟶ β | the inclusion of `Image f` into the target | true |
WeierstrassCurve.natDegree_Ψ₂Sq_le | Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | ∀ {R : Type u} [inst : CommRing R] (W : WeierstrassCurve R), W.Ψ₂Sq.natDegree ≤ 3 | null | true |
Prod.seminormedRing._proof_14 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : SeminormedRing α] [inst_1 : SeminormedRing β] (a : α × β), 0 • a = 0 | null | false |
Complex.log_mul_ofReal | Mathlib.Analysis.SpecialFunctions.Complex.Log | ∀ (r : ℝ), 0 < r → ∀ (x : ℂ), x ≠ 0 → Complex.log (x * ↑r) = ↑(Real.log r) + Complex.log x | null | true |
CategoryTheory.TwoSquare.GuitartExact.quotient_of_nonempty_leftHomotopy | Mathlib.CategoryTheory.GuitartExact.Quotient | ∀ {C₀ : Type u_1} {C : Type u_2} {H₀ : Type u_3} {H : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} C₀]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C] [inst_2 : CategoryTheory.Category.{v_3, u_3} H₀]
[inst_3 : CategoryTheory.Category.{v_4, u_4} H] {T : CategoryTheory.Functor C₀ H₀} {L : CategoryTheory.Funct... | null | true |
_private.Init.Data.BitVec.Bitblast.0.BitVec.fastUmulOverflow._proof_1_1 | Init.Data.BitVec.Bitblast | ∀ (x : BitVec (0 + 1)), ¬x.toNat ≤ 1 → False | null | false |
Lean.Elab.Do.observingPostpone | Lean.Elab.Do.Basic | {α : Type} → Lean.Elab.Do.DoElabM α → Lean.Elab.Do.DoElabM (Option α) | null | true |
CategoryTheory.Limits.HasImageMaps.has_image_map | Mathlib.CategoryTheory.Limits.Shapes.Images | ∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {inst_1 : CategoryTheory.Limits.HasImages C}
[self : CategoryTheory.Limits.HasImageMaps C] {f g : CategoryTheory.Arrow C} (st : f ⟶ g),
CategoryTheory.Limits.HasImageMap st | null | true |
_private.Mathlib.Analysis.SpecialFunctions.Integrals.PosLog.0.circleAverage_log_norm_sub_const_eq_log_radius_add_posLog._simp_1_4 | Mathlib.Analysis.SpecialFunctions.Integrals.PosLog | ∀ {M₀ : Type u_1} [inst : Mul M₀] [inst_1 : Zero M₀] [NoZeroDivisors M₀] {a b : M₀}, a ≠ 0 → b ≠ 0 → (a * b = 0) = False | null | false |
IncidenceAlgebra.coe_add._simp_1 | Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | ∀ {𝕜 : Type u_1} {α : Type u_4} [inst : AddZeroClass 𝕜] [inst_1 : LE α] (f g : IncidenceAlgebra 𝕜 α), ⇑f + ⇑g = ⇑(f + g) | null | false |
CategoryTheory.CommGrp.forget_obj | Mathlib.CategoryTheory.Monoidal.CommGrp_ | ∀ (C : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] (X : CategoryTheory.CommGrp C),
(CategoryTheory.CommGrp.forget C).obj X = X.X | null | true |
AlgebraicGeometry.Scheme.Hom.resLE_appLE | Mathlib.AlgebraicGeometry.Restrict | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens}
(e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (O : (↑U).Opens) (W : (↑V).Opens)
(e' : W ≤ (TopologicalSpace.Opens.map (AlgebraicGeometry.Scheme.Hom.resLE f U V e).base).obj O),
AlgebraicGeometry.Scheme.Hom.appLE (AlgebraicGeometry.Sc... | null | true |
Lean.NamePart.num.noConfusion | Lean.Data.NameTrie | {P : Sort u} → {n n' : ℕ} → Lean.NamePart.num n = Lean.NamePart.num n' → (n = n' → P) → P | null | false |
CategoryTheory.Sieve.pushforward_monotone | Mathlib.CategoryTheory.Sites.Sieves.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : Y ⟶ X),
Monotone (CategoryTheory.Sieve.pushforward f) | null | true |
CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.DiagramWithUniqueTerminal.mk.congr_simp | Mathlib.CategoryTheory.Presentable.Directed | ∀ {J : Type w} [inst : CategoryTheory.SmallCategory J] {κ : Cardinal.{w}}
(toDiagram : CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram J κ) (top : J)
(isTerminal isTerminal_1 : toDiagram.IsTerminal top),
isTerminal = isTerminal_1 →
∀ (uniq_terminal : ∀ (j : J) (hj : toDiagram.IsTerminal j),... | null | true |
QuotientGroup.Quotient.group._proof_8 | Mathlib.GroupTheory.QuotientGroup.Defs | ∀ {G : Type u_1} [inst : Group G] (N : Subgroup G) [nN : N.Normal] (x x_1 : G),
x ≈ x_1 → ∀ (x_2 x_3 : G), x_2 ≈ x_3 → { toSetoid := QuotientGroup.leftRel N, mul' := ⋯ } (x / x_2) (x_1 / x_3) | null | false |
Group.exponent_dvd_iff_forall_zpow_eq_one | Mathlib.GroupTheory.Exponent | ∀ {G : Type u} [inst : Group G] {n : ℤ}, ↑(Monoid.exponent G) ∣ n ↔ ∀ (g : G), g ^ n = 1 | null | true |
affineCombination_mem_affineSpan | Mathlib.LinearAlgebra.AffineSpace.Combination | ∀ {ι : Type u_1} {k : Type u_2} {V : Type u_3} {P : Type u_4} [inst : Ring k] [inst_1 : AddCommGroup V]
[inst_2 : Module k V] [inst_3 : AddTorsor V P] [Nontrivial k] {s : Finset ι} {w : ι → k},
∑ i ∈ s, w i = 1 → ∀ (p : ι → P), (Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p) | An `affineCombination` with sum of weights 1 is in the
`affineSpan` of an indexed family, if the underlying ring is
nontrivial. | true |
le_gauge_of_subset_closedBall | Mathlib.Analysis.Convex.Gauge | ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E},
Absorbent ℝ s → 0 ≤ r → s ⊆ Metric.closedBall 0 r → ‖x‖ / r ≤ gauge s x | null | true |
CochainComplex.mappingCocone.liftCochain.congr_simp | Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
{K L : CochainComplex C ℤ} (φ : K ⟶ L) [inst_2 : HomologicalComplex.HasHomotopyCofiber φ] {M : CochainComplex C ℤ}
{n m : ℤ} (α α_1 : CochainComplex.HomComplex.Cochain M K n),
α = α_1 →
∀ (β β_1 : CochainCom... | null | true |
MonoidAlgebra.comapDomainAddMonoidHom._proof_2 | Mathlib.Algebra.MonoidAlgebra.MapDomain | ∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Semiring R] (f : M → N) (hf : Function.Injective f),
MonoidAlgebra.comapDomain f hf 0 = 0 | null | false |
_private.Mathlib.Combinatorics.Enumerative.Partition.GenFun.0.Nat.Partition.aux_dvd_of_coeff_ne_zero | Mathlib.Combinatorics.Enumerative.Partition.GenFun | ∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : TopologicalSpace R] [T2Space R] {f : ℕ → ℕ → R} {d : ℕ}
{s : Finset ℕ},
0 ∉ s →
∀ {g : ℕ →₀ ℕ},
g ∈ s.finsuppAntidiag d →
(∀ i ∈ s, (PowerSeries.coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • PowerSeries.X ^ (i * (j + 1))) ≠ 0) →
∀ (x : ℕ),... | null | true |
FirstOrder.Field.FieldAxiom.mulAssoc | Mathlib.ModelTheory.Algebra.Field.Basic | FirstOrder.Field.FieldAxiom | null | true |
convexJoin_singleton_segment | Mathlib.Analysis.Convex.Join | ∀ {𝕜 : Type u_2} {E : Type u_3} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[inst_3 : AddCommGroup E] [inst_4 : Module 𝕜 E] (a b c : E),
convexJoin 𝕜 {a} (segment 𝕜 b c) = (convexHull 𝕜) {a, b, c} | null | true |
Int16.lt_of_lt_of_le | Init.Data.SInt.Lemmas | ∀ {a b c : Int16}, a < b → b ≤ c → a < c | null | true |
Vector.setIfInBounds_setIfInBounds | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n i : ℕ} (a : α) {b : α} {xs : Vector α n},
(xs.setIfInBounds i a).setIfInBounds i b = xs.setIfInBounds i b | null | true |
CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionRight_op | Mathlib.CategoryTheory.Monoidal.Action.Opposites | ∀ (C : Type u_1) (D : Type u_2) [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Category.{v_2, u_2} D]
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (c : C) {d d' : D} (f : d ⟶ d'),
CategoryTheory.MonoidalCategory.MonoidalLeft... | null | true |
CategoryTheory.Functor.FullyFaithful.ofFullyFaithful | Mathlib.CategoryTheory.Functor.FullyFaithful | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(F : CategoryTheory.Functor C D) → [F.Full] → [F.Faithful] → F.FullyFaithful | A `FullyFaithful` structure can be obtained from the assumption the `F` is both
full and faithful. | true |
TwoP.concreteCategory._proof_6 | Mathlib.CategoryTheory.Category.TwoP | autoParam (∀ {X Y : TwoP} (f : X ⟶ Y), TwoP.concreteCategory._aux_3 (TwoP.concreteCategory._aux_1 f) = f)
CategoryTheory.ConcreteCategory.ofHom_hom._autoParam | null | false |
Prod.gameAdd_mk_iff | Mathlib.Order.GameAdd | ∀ {α : Type u_1} {β : Type u_2} {rα : α → α → Prop} {rβ : β → β → Prop} {a₁ a₂ : α} {b₁ b₂ : β},
Prod.GameAdd rα rβ (a₁, b₁) (a₂, b₂) ↔ rα a₁ a₂ ∧ b₁ = b₂ ∨ rβ b₁ b₂ ∧ a₁ = a₂ | null | true |
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