name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Equiv.subtypeProdEquivProd._proof_3 | Mathlib.Logic.Equiv.Prod | ∀ {α : Type u_1} {β : Type u_2} {p : α → Prop} {q : β → Prop} (x : { c // p c.1 ∧ q c.2 }), p (↑x).1 | null | false |
LinearMap.toContinuousLinearMap.congr_simp | Mathlib.Topology.Algebra.Module.FiniteDimension | ∀ {𝕜 : Type u} [hnorm : NontriviallyNormedField 𝕜] {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E]
[inst_2 : TopologicalSpace E] [inst_3 : IsTopologicalAddGroup E] [inst_4 : ContinuousSMul 𝕜 E] {F' : Type x}
[inst_5 : AddCommGroup F'] [inst_6 : Module 𝕜 F'] [inst_7 : TopologicalSpace F'] [inst_8 : I... | null | true |
Lean.Compiler.LCNF.NormLevelParam.State.noConfusion | Lean.Compiler.LCNF.Level | {P : Sort u} →
{t t' : Lean.Compiler.LCNF.NormLevelParam.State} →
t = t' → Lean.Compiler.LCNF.NormLevelParam.State.noConfusionType P t t' | null | false |
IO.FS.realPath | Init.System.IO | System.FilePath → IO System.FilePath | Resolves a path to an absolute path that contains no '.', '..', or symbolic links.
This function coincides with the [POSIX `realpath`
function](https://pubs.opengroup.org/onlinepubs/9699919799/functions/realpath.html).
| true |
ShrinkingLemma.PartialRefinement.rec | Mathlib.Topology.ShrinkingLemma | {ι : Type u_1} →
{X : Type u_2} →
[inst : TopologicalSpace X] →
{u : ι → Set X} →
{s : Set X} →
{p : Set X → Prop} →
{motive : ShrinkingLemma.PartialRefinement u s p → Sort u} →
((toFun : ι → Set X) →
(carrier : Set ι) →
(isOp... | null | false |
SimpleGraph.IsMatchingFree | Mathlib.Combinatorics.SimpleGraph.Matching | {V : Type u_1} → SimpleGraph V → Prop | A graph is matching free if it has no perfect matching. It does not make much sense to
consider a graph being free of just matchings, because any non-trivial graph has those.
| true |
WeierstrassCurve.Jacobian.negY_eq | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | ∀ {R : Type r} [inst : CommRing R] {W' : WeierstrassCurve.Jacobian R} (X Y Z : R),
W'.negY ![X, Y, Z] = -Y - W'.a₁ * X * Z - W'.a₃ * Z ^ 3 | null | true |
Std.Http.Server.mk._flat_ctor | Std.Http.Server | Std.CancellationContext →
Std.Mutex ℕ →
Option Std.Semaphore → Std.Channel Unit → Std.Http.Config → Option Std.Net.SocketAddress → Std.Http.Server | null | false |
invertibleSucc | Mathlib.Algebra.CharP.Invertible | {K : Type u_2} → [inst : DivisionSemiring K] → [CharZero K] → (n : ℕ) → Invertible ↑n.succ | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex.0.SSet.prodStdSimplex.exists_nonDegenerate_max_dim_aux._simp_1_3 | Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex | ∀ {n : ℕ} {a b : Fin n}, (a < b) = (↑a < ↑b) | null | false |
ContinuousMultilinearMap.iteratedFDerivComponent._proof_3 | Mathlib.Analysis.Normed.Module.Multilinear.Basic | ∀ {𝕜 : Type u_5} {ι : Type u_1} {E₁ : ι → Type u_4} {G : Type u_2} [inst : NontriviallyNormedField 𝕜]
[inst_1 : (i : ι) → SeminormedAddCommGroup (E₁ i)] [inst_2 : (i : ι) → NormedSpace 𝕜 (E₁ i)]
[inst_3 : SeminormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] [inst_5 : Fintype ι] {α : Type u_3}
[inst_6 : Fintyp... | null | false |
List.getElem_modifyHead._proof_3 | Init.Data.List.Nat.Modify | ∀ {α : Type u_1} {l : List α} {f : α → α} {i : ℕ}, i < (List.modifyHead f l).length → i < l.length | null | false |
Lean.Doc.instFromDocArgMessageSeverity | Lean.Elab.DocString | Lean.Doc.FromDocArg Lean.MessageSeverity | null | true |
List.head_attach | Init.Data.List.Attach | ∀ {α : Type u_1} {xs : List α} (h : xs.attach ≠ []), xs.attach.head h = ⟨xs.head ⋯, ⋯⟩ | null | true |
_private.Mathlib.NumberTheory.LSeries.AbstractFuncEq.0.WeakFEPair.f_modif_aux2._simp_1_1 | Mathlib.NumberTheory.LSeries.AbstractFuncEq | ∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : SMulZeroClass R M] (s : Set α) (r : α → R)
(f : α → M), (fun a => r a • s.indicator f a) = s.indicator fun a => r a • f a | null | false |
_private.Mathlib.Order.KrullDimension.0.Order.exists_series_of_le_height._proof_1_1 | Mathlib.Order.KrullDimension | ∀ {α : Type u_1} [inst : Preorder α] {n : ℕ} (m : ℕ) (p : LTSeries α), p.length = m → n < m → m - n < p.length + 1 | null | false |
Subalgebra.coe_pi | Mathlib.Algebra.Algebra.Subalgebra.Pi | ∀ {ι : Type u_1} {R : Type u_2} {S : ι → Type u_3} [inst : CommSemiring R] [inst_1 : (i : ι) → Semiring (S i)]
[inst_2 : (i : ι) → Algebra R (S i)] (s : Set ι) (t : (i : ι) → Subalgebra R (S i)),
↑(Subalgebra.pi s t) = (Submodule.pi s fun i => Subalgebra.toSubmodule (t i)).carrier | null | true |
Int.abs_modEq_two._simp_1 | Mathlib.Data.Int.ModEq | ∀ {a : ℤ}, (|a| ≡ a [ZMOD 2]) = True | null | false |
FinEnum.PSigma.finEnumPropProp._proof_3 | Mathlib.Data.FinEnum | ∀ {α : Prop} {β : α → Prop}, (∃ (a : α), β a) → α | null | false |
RealRMK.le_rieszMeasure_tsupport_subset | Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | ∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : T2Space X] [inst_2 : MeasurableSpace X] [inst_3 : BorelSpace X]
(Λ : CompactlySupportedContinuousMap X ℝ →ₚ[ℝ] ℝ) [inst_4 : LocallyCompactSpace X]
{f : CompactlySupportedContinuousMap X ℝ},
(∀ (x : X), 0 ≤ f x ∧ f x ≤ 1) → ∀ {V : Set X}, tsupport ⇑f ⊆ V → ENN... | If `f` assumes values between `0` and `1` and the support is contained in `V`, then
`Λ f ≤ rieszMeasure V`. | true |
MeasureTheory.AEStronglyMeasurable.mono_set | Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α}
{f : α → β} {s t : Set α},
s ⊆ t → MeasureTheory.AEStronglyMeasurable f (μ.restrict t) → MeasureTheory.AEStronglyMeasurable f (μ.restrict s) | null | true |
Matrix.discr_fin_two | Mathlib.LinearAlgebra.Matrix.Charpoly.Disc | ∀ {R : Type u_1} [inst : CommRing R] (A : Matrix (Fin 2) (Fin 2) R), A.discr = A.trace ^ 2 - 4 * A.det | null | true |
Seminorm.comp_smul | Mathlib.Analysis.Normed.Module.Seminorm.Basic | ∀ {𝕜 : Type u_3} {𝕜₂ : Type u_4} {E : Type u_7} {E₂ : Type u_8} [inst : SeminormedRing 𝕜]
[inst_1 : SeminormedCommRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_2 : RingHomIsometric σ₁₂] [inst_3 : AddCommGroup E]
[inst_4 : AddCommGroup E₂] [inst_5 : Module 𝕜 E] [inst_6 : Module 𝕜₂ E₂] (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂]... | null | true |
Lean.Widget.RpcEncodablePacket.leanTags?._@.Lean.Widget.InteractiveDiagnostic.2989700264._hygCtx._hyg.2 | Lean.Widget.InteractiveDiagnostic | Lean.Widget.RpcEncodablePacket✝ → Option Lean.Json | null | false |
AffineEquiv.ofBijective_apply | Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | ∀ {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [inst : Ring k]
[inst_1 : AddCommGroup V₁] [inst_2 : AddCommGroup V₂] [inst_3 : Module k V₁] [inst_4 : Module k V₂]
[inst_5 : AddTorsor V₁ P₁] [inst_6 : AddTorsor V₂ P₂] {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Bijective ⇑φ) (a : P₁),
(Affin... | null | true |
Polynomial.powAddExpansion | Mathlib.Algebra.Polynomial.Identities | {R : Type u_1} →
[inst : CommSemiring R] → (x y : R) → (n : ℕ) → { k // (x + y) ^ n = x ^ n + ↑n * x ^ (n - 1) * y + k * y ^ 2 } | `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
| true |
Set.mulIndicator_le' | Mathlib.Algebra.Order.Group.Indicator | ∀ {α : Type u_2} {M : Type u_3} [inst : LE M] [inst_1 : One M] {s : Set α} {f g : α → M},
(∀ a ∈ s, f a ≤ g a) → (∀ a ∉ s, 1 ≤ g a) → s.mulIndicator f ≤ g | null | true |
_private.Mathlib.Dynamics.PeriodicPts.Defs.0.Function.periodicOrbit_eq_nil_iff_not_periodic_pt._simp_1_4 | Mathlib.Dynamics.PeriodicPts.Defs | ∀ {n : ℕ}, (List.range n = []) = (n = 0) | null | false |
Matrix.coeff_det_one_add_X_smul_eq_sum_minors | Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | ∀ {R : Type u} [inst : CommRing R] {n : Type v} [inst_1 : DecidableEq n] [inst_2 : Fintype n] (M : Matrix n n R)
(k : ℕ),
(1 + Polynomial.X • M.map ⇑Polynomial.C).det.coeff k =
∑ s ∈ Finset.powersetCard k Finset.univ, (M.submatrix Subtype.val Subtype.val).det | The k-th coefficient of `det (1 + X • M)` equals the sum of all k×k principal minors of M.
This generalizes `coeff_det_one_add_X_smul_one` (the k = 1 case, which gives the trace)
and `det_eq_sign_charpoly_coeff` (the k = n case, which gives the determinant). | true |
IsSimpleOrder.eq_bot_or_eq_top | Mathlib.Order.Atoms | ∀ {α : Type u_4} {inst : LE α} {inst_1 : BoundedOrder α} [self : IsSimpleOrder α] (a : α), a = ⊥ ∨ a = ⊤ | Every element is either `⊥` or `⊤` | true |
SSet.FundamentalGroupoid.rec_mk | Mathlib.AlgebraicTopology.SimplicialSet.FundamentalGroupoid.Basic | ∀ {X : SSet} {motive : X.FundamentalGroupoid → Sort u_1}
(mk : (x : X.obj (Opposite.op { len := 0 })) → motive (SSet.FundamentalGroupoid.mk x))
(x : X.obj (Opposite.op { len := 0 })), SSet.FundamentalGroupoid.rec mk (SSet.FundamentalGroupoid.mk x) = mk x | null | true |
Lean.instNonemptyKeyedDeclsAttribute | Lean.KeyedDeclsAttribute | ∀ {γ : Type}, Nonempty (Lean.KeyedDeclsAttribute γ) | null | true |
_private.Mathlib.Probability.Distributions.Poisson.Basic.0.ProbabilityTheory.map_cast_poissonMeasure_conv_real | Mathlib.Probability.Distributions.Poisson.Basic | ∀ (r₁ r₂ : NNReal),
(MeasureTheory.Measure.map Nat.cast (ProbabilityTheory.poissonMeasure r₁)).conv
(MeasureTheory.Measure.map Nat.cast (ProbabilityTheory.poissonMeasure r₂)) =
MeasureTheory.Measure.map Nat.cast (ProbabilityTheory.poissonMeasure (r₁ + r₂)) | null | true |
AddMonoidAlgebra.mapDomainOfBialgHom_comp | Mathlib.RingTheory.Bialgebra.MonoidAlgebra | ∀ {R : Type u_1} {G : Type u_5} {H : Type u_6} {I : Type u_7} [inst : CommRing R] [inst_1 : IsDomain R]
[inst_2 : AddGroup G] [inst_3 : AddGroup H] [inst_4 : AddGroup I]
(f : AddMonoidAlgebra R H →ₐc[R] AddMonoidAlgebra R I) (g : AddMonoidAlgebra R G →ₐc[R] AddMonoidAlgebra R H),
AddMonoidAlgebra.mapDomainOfBialg... | null | true |
_private.Init.Data.Iterators.Lemmas.Combinators.FilterMap.0.Std.Iter.val_step_filterMap.match_1.eq_1 | Init.Data.Iterators.Lemmas.Combinators.FilterMap | ∀ {γ : Type u_1} (motive : Option γ → Sort u_2) (h_1 : Unit → motive none) (h_2 : (out' : γ) → motive (some out')),
(match none with
| none => h_1 ()
| some out' => h_2 out') =
h_1 () | null | true |
instLawfulMonadContOptionT | Mathlib.Control.Monad.Cont | ∀ {m : Type u → Type v} [inst : Monad m] [inst_1 : MonadCont m] [LawfulMonadCont m], LawfulMonadCont (OptionT m) | null | true |
«term_ᵈᵃᵃ» | Mathlib.GroupTheory.GroupAction.DomAct.Basic | Lean.TrailingParserDescr | If `M` additively acts on `α`, then `DomAddAct M` acts on `α → β` as
well as some bundled maps from `α`. This is a type synonym for `AddOpposite M`, so this corresponds
to a right action of `M`. | true |
Std.DTreeMap.Internal.Impl.getEntryGE?ₘ'.eq_1 | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] (k : α) (t : Std.DTreeMap.Internal.Impl α β),
Std.DTreeMap.Internal.Impl.getEntryGE?ₘ' k t =
Std.DTreeMap.Internal.Impl.explore (compare k) none
(fun x x_1 =>
match x, x_1 with
| x, Std.DTreeMap.Internal.Impl.ExplorationStep.lt k' a v a_1 => som... | null | true |
_private.Lean.Compiler.LCNF.PhaseExt.0.Lean.Compiler.LCNF.initFn._@.Lean.Compiler.LCNF.PhaseExt.1453085006._hygCtx._hyg.2 | Lean.Compiler.LCNF.PhaseExt | IO (Lean.Compiler.LCNF.DeclExt Lean.Compiler.LCNF.Purity.pure) | null | false |
Lean.ExternEntry.adhoc.sizeOf_spec | Lean.Compiler.ExternAttr | ∀ (backend : Lean.Name), sizeOf (Lean.ExternEntry.adhoc backend) = 1 + sizeOf backend | null | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.get!_inter_of_contains_eq_false_left._simp_1_3 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {k : α},
(k ∈ t) = (Std.DTreeMap.Internal.Impl.contains k t = true) | null | false |
LocallyConstant.mulIndicator_apply | Mathlib.Topology.LocallyConstant.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] {R : Type u_5} [inst_1 : One R] {U : Set X} (f : LocallyConstant X R)
(hU : IsClopen U) (x : X), (f.mulIndicator hU) x = U.mulIndicator (⇑f) x | null | true |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap.0.WeierstrassCurve.addSubMapCoeff._proof_11 | Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap | (63 + 1).AtLeastTwo | null | false |
MeasureTheory.FiniteMeasure.map_prod_map | Mathlib.MeasureTheory.Measure.FiniteMeasureProd | ∀ {α : Type u_1} [inst : MeasurableSpace α] {β : Type u_2} [inst_1 : MeasurableSpace β]
(μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {α' : Type u_3} [inst_2 : MeasurableSpace α']
{β' : Type u_4} [inst_3 : MeasurableSpace β'] {f : α → α'} {g : β → β'},
Measurable f → Measurable g → (μ.ma... | null | true |
_private.Mathlib.Data.Fintype.Sets.0.Set.disjoint_toFinset._simp_1_1 | Mathlib.Data.Fintype.Sets | ∀ {α : Type u_2} {s t : Finset α}, Disjoint s t = Disjoint ↑s ↑t | null | false |
List.length_eraseIdx | Init.Data.List.Erase | ∀ {α : Type u_1} {l : List α} {i : ℕ}, (l.eraseIdx i).length = if i < l.length then l.length - 1 else l.length | null | true |
_private.Mathlib.CategoryTheory.WithTerminal.Cone.0.CategoryTheory.WithInitial.liftFromUnderComp.match_1.eq_1 | Mathlib.CategoryTheory.WithTerminal.Cone | ∀ {J : Type u_1} (motive : CategoryTheory.WithInitial J → Sort u_2)
(h_1 : Unit → motive CategoryTheory.WithInitial.star) (h_2 : (a : J) → motive (CategoryTheory.WithInitial.of a)),
(match CategoryTheory.WithInitial.star with
| CategoryTheory.WithInitial.star => h_1 ()
| CategoryTheory.WithInitial.of a => h... | null | true |
Lean.PrettyPrinter.Delaborator.TopDownAnalyze.isTrivialBottomUp | Lean.PrettyPrinter.Delaborator.TopDownAnalyze | Lean.Expr → Lean.PrettyPrinter.Delaborator.TopDownAnalyze.AnalyzeM Bool | null | true |
RelUpperSet.isRelUpperSet' | Mathlib.Order.Defs.Unbundled | ∀ {α : Type u_1} [inst : LE α] {P : α → Prop} (self : RelUpperSet P), IsRelUpperSet self.carrier P | The carrier of a `RelUpperSet` is an upper set relative to `P`.
Do NOT use directly. Please use `RelUpperSet.isRelUpperSet` instead. | true |
MeromorphicNFAt.meromorphicAt | Mathlib.Analysis.Meromorphic.NormalForm | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜}, MeromorphicNFAt f x → MeromorphicAt f x | If a function is meromorphic in normal form at `x`, then it is meromorphic at `x`. | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.getD_map_of_getKey?_eq_some._simp_1_3 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {k : α},
(k ∈ t) = (Std.DTreeMap.Internal.Impl.contains k t = true) | null | false |
AddMonoidHom.smul | Mathlib.Algebra.Module.Hom | {R : Type u_1} → {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [Module R M] → R →+ M →+ M | Scalar multiplication as a biadditive monoid homomorphism. We need `M` to be commutative
to have addition on `M →+ M`. | true |
Finmap.sigma_keys_lookup | Mathlib.Data.Finmap | ∀ {α : Type u} {β : α → Type v} [inst : DecidableEq α] (s : Finmap β),
(s.keys.sigma fun i => (Finmap.lookup i s).toFinset) = { val := s.entries, nodup := ⋯ } | null | true |
Filter.Tendsto.atBot_mul_eventuallyLE_one | Mathlib.Order.Filter.AtTopBot.Monoid | ∀ {α : Type u_1} {M : Type u_2} [inst : CommMonoid M] [inst_1 : Preorder M] [IsOrderedMonoid M] {l : Filter α}
{f g : α → M}, Filter.Tendsto f l Filter.atBot → g ≤ᶠ[l] 1 → Filter.Tendsto (fun x => f x * g x) l Filter.atBot | null | true |
Monoid.PushoutI.NormalWord.Transversal.mk.noConfusion | Mathlib.GroupTheory.PushoutI | {ι : Type u_1} →
{G : ι → Type u_2} →
{H : Type u_3} →
{inst : (i : ι) → Group (G i)} →
{inst_1 : Group H} →
{φ : (i : ι) → H →* G i} →
{P : Sort u} →
{injective : ∀ (i : ι), Function.Injective ⇑(φ i)} →
{set : (i : ι) → Set (G i)} →
... | null | false |
_private.Mathlib.Algebra.Ring.CentroidHom.0.CentroidHom._aux_Mathlib_Algebra_Ring_CentroidHom___macroRules__private_Mathlib_Algebra_Ring_CentroidHom_0_CentroidHom_termL_1 | Mathlib.Algebra.Ring.CentroidHom | Lean.Macro | null | false |
doublyStochastic.congr_simp | Mathlib.Analysis.Convex.DoublyStochasticMatrix | ∀ (R : Type u_3) (n : Type u_4) [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : Semiring R]
[inst_3 : PartialOrder R] [inst_4 : IsOrderedRing R], doublyStochastic R n = doublyStochastic R n | null | true |
_private.Mathlib.SetTheory.Cardinal.Cofinality.Club.0.IsClub.sInter._proof_1_4 | Mathlib.SetTheory.Cardinal.Cofinality.Club | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : WellFoundedLT α] (h : Nonempty α) (s : Set α),
¬BddAbove s → sSup s = sSup ∅ | null | false |
_private.Mathlib.Analysis.CStarAlgebra.CStarMatrix.0.CStarMatrix.instFinite._proof_1 | Mathlib.Analysis.CStarAlgebra.CStarMatrix | ∀ {n : Type u_2} {m : Type u_3} [Finite m] [Finite n] (α : Type u_1) [Finite α], Finite (CStarMatrix m n α) | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.0.Int64.reduceBoolPred._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.922157756._hygCtx._hyg.9 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt | Lean.Name → ℕ → (Int64 → Int64 → Bool) → Lean.Expr → Lean.Meta.SimpM Lean.Meta.Simp.DStep | null | false |
Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul | Mathlib.RingTheory.Finiteness.Nakayama | ∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (I : Ideal R)
(N : Submodule R M), N.FG → N ≤ I • N → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0 | **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2. | true |
Projectivization.Subspace.instCompleteLattice | Mathlib.LinearAlgebra.Projectivization.Subspace | {K : Type u_1} →
{V : Type u_2} →
[inst : DivisionRing K] →
[inst_1 : AddCommGroup V] → [inst_2 : Module K V] → CompleteLattice (Projectivization.Subspace K V) | The subspaces of a projective space form a complete lattice. | true |
_aux_Mathlib_Algebra_Module_LinearMap_Defs___unexpand_LinearMap_1 | Mathlib.Algebra.Module.LinearMap.Defs | Lean.PrettyPrinter.Unexpander | null | false |
Action.instIsIsoHomInv | Mathlib.CategoryTheory.Action.Basic | ∀ {V : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} V] {G : Type u_2} [inst_1 : Monoid G] {M N : Action V G}
(f : M ≅ N), CategoryTheory.IsIso f.inv.hom | null | true |
SpecialLinearGroup.centerEquivRootsOfUnity.eq_1 | Mathlib.LinearAlgebra.SpecialLinearGroup | ∀ {R : Type u_1} {V : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup V] [inst_2 : Module R V]
[inst_3 : Module.Free R V] [inst_4 : Module.Finite R V],
SpecialLinearGroup.centerEquivRootsOfUnity =
{
toFun := fun g =>
⋯.by_cases (fun x => 1) fun hR =>
⋯.by_cases (fun x => 1) fun hV =... | null | true |
Complex.UnitClosedDisc.instSMulCommClass_closedBall_circle | Mathlib.Analysis.Complex.UnitDisc.Basic | SMulCommClass Complex.UnitClosedDisc Circle Complex.UnitDisc | null | true |
Equiv.traverse.eq_1 | Mathlib.Control.Traversable.Equiv | ∀ {t t' : Type u → Type u} (eqv : (α : Type u) → t α ≃ t' α) [inst : Traversable t] {m : Type u → Type u}
[inst_1 : Applicative m] {α β : Type u} (f : α → m β) (x : t' α),
Equiv.traverse eqv f x = ⇑(eqv β) <$> traverse f ((eqv α).symm x) | null | true |
MvPolynomial.pderiv_X_of_ne | Mathlib.Algebra.MvPolynomial.PDeriv | ∀ {R : Type u} {σ : Type v} [inst : CommSemiring R] {i j : σ}, j ≠ i → (MvPolynomial.pderiv i) (MvPolynomial.X j) = 0 | null | true |
RelSeries.head_append | Mathlib.Order.RelSeries | ∀ {α : Type u_1} {r : SetRel α α} (p q : RelSeries r) (connect : (p.last, q.head) ∈ r),
(p.append q connect).head = p.head | null | true |
RestrictedProduct.instMonoidCoeOfSubmonoidClass._proof_4 | Mathlib.Topology.Algebra.RestrictedProduct.Basic | ∀ {ι : Type u_1} (R : ι → Type u_2) {𝓕 : Filter ι} {S : ι → Type u_3} [inst : (i : ι) → SetLike (S i) (R i)]
{B : (i : ι) → S i} [inst_1 : (i : ι) → Monoid (R i)] [inst_2 : ∀ (i : ι), SubmonoidClass (S i) (R i)], ⇑1 = ⇑1 | null | false |
Subsingleton.measurable | Mathlib.MeasureTheory.MeasurableSpace.Basic | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [Subsingleton α],
Measurable f | null | true |
_private.Mathlib.RingTheory.AdicCompletion.Exactness.0.AdicCompletion.mapPreimage._proof_2 | Mathlib.RingTheory.AdicCompletion.Exactness | ∀ {R : Type u_3} [inst : CommRing R] {I : Ideal R} {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{N : Type u_1} [inst_3 : AddCommGroup N] [inst_4 : Module R N] {f : M →ₗ[R] N} (hf : Function.Surjective ⇑f)
(x : AdicCompletion.AdicCauchySequence I N), f ⋯.choose = ↑x 0 | null | false |
_private.Mathlib.CategoryTheory.Presentable.Comma.0.CategoryTheory.Comma.isCardinalAccessibleCategory.π₂._proof_3 | Mathlib.CategoryTheory.Presentable.Comma | ∀ {C₁ : Type u_3} {C₂ : Type u_5} {D : Type u_7} [inst : CategoryTheory.Category.{u_2, u_3} C₁]
[inst_1 : CategoryTheory.Category.{u_4, u_5} C₂] [inst_2 : CategoryTheory.Category.{u_6, u_7} D]
{F₁ : CategoryTheory.Functor C₁ D} {F₂ : CategoryTheory.Functor C₂ D} (κ : Cardinal.{u_1}) [inst_3 : Fact κ.IsRegular]
(f... | null | false |
Ideal.isPrime_map_of_isLocalizationAtPrime | Mathlib.RingTheory.Localization.AtPrime.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (q : Ideal R) [inst_1 : q.IsPrime] {S : Type u_4} [inst_2 : CommSemiring S]
[inst_3 : Algebra R S] [IsLocalization.AtPrime S q] {p : Ideal R} [p.IsPrime],
p ≤ q → (Ideal.map (algebraMap R S) p).IsPrime | null | true |
TensorProduct.ext' | Mathlib.LinearAlgebra.TensorProduct.Basic | ∀ {R : Type u_1} {R₂ : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6}
{N : Type u_7} {P₂ : Type u_11} [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid N] [inst_4 : AddCommMonoid P₂]
[inst_5 : Module R M] [inst_6 : Module R N] [inst_7 : Module R₂ P₂] {g h : TensorProdu... | null | true |
CategoryTheory.Limits.hasFiniteProducts_of_hasFiniteLimits | Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteLimits C],
CategoryTheory.Limits.HasFiniteProducts C | If `C` has finite limits then it has finite products. | true |
_private.Mathlib.Topology.MetricSpace.Bounded.0.IsComplete.nonempty_iInter_of_nonempty_biInter._simp_1_1 | Mathlib.Topology.MetricSpace.Bounded | ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋂ i, s i) = ∀ (i : ι), x ∈ s i | null | false |
ModularForm.const_apply | Mathlib.NumberTheory.ModularForms.Basic | ∀ {Γ : Subgroup (GL (Fin 2) ℝ)} [inst : Γ.HasDetOne] (x : ℂ) (τ : UpperHalfPlane), (ModularForm.const x) τ = x | null | true |
LieIdeal.map_sup_ker_eq_map' | Mathlib.Algebra.Lie.Ideal | ∀ {R : Type u} {L : Type v} {L' : Type w₂} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieRing L']
[inst_3 : LieAlgebra R L'] [inst_4 : LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L},
LieIdeal.map f I ⊔ LieIdeal.map f f.ker = LieIdeal.map f I | null | true |
Set.vadd_empty | Mathlib.Algebra.Group.Pointwise.Set.Scalar | ∀ {α : Type u_1} {β : Type u_2} [inst : VAdd α β] {s : Set α}, s +ᵥ ∅ = ∅ | null | true |
ProofWidgets.CheckRequestResponse.ctorElimType | ProofWidgets.Cancellable | {motive : ProofWidgets.CheckRequestResponse → Sort u} → ℕ → Sort (max 1 u) | null | false |
CategoryTheory.MorphismProperty.HasQuotient.iff_of_eqvGen | Mathlib.CategoryTheory.MorphismProperty.Quotient | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (W : CategoryTheory.MorphismProperty C)
{homRel : HomRel C} [inst_1 : CategoryTheory.HomRel.IsStableUnderPrecomp homRel]
[inst_2 : CategoryTheory.HomRel.IsStableUnderPostcomp homRel] [W.HasQuotient homRel] {X Y : C} {f g : X ⟶ Y},
Relation.EqvGen homR... | null | true |
Lean.Lsp.instFromJsonDiagnosticCode.match_1 | Lean.Data.Lsp.Diagnostics | (motive : Lean.Json → Sort u_1) →
(x : Lean.Json) →
((i : ℤ) → motive (Lean.Json.num { mantissa := i, exponent := 0 })) →
((s : String) → motive (Lean.Json.str s)) → ((j : Lean.Json) → motive j) → motive x | null | false |
Std.ExtTreeSet.contains_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.contains km = true | null | true |
CategoryTheory.Functor.mapTriangleInvRotateIso_inv_app_hom₃ | Mathlib.CategoryTheory.Triangulated.Functor | ∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.HasShift C ℤ]
[inst_3 : CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [inst_4 : F.CommShift ℤ]
[inst_5 : CategoryTheory.Preadditive C] [inst_6 : Ca... | null | true |
instDecidableEqDihedralGroup.decEq._proof_1 | Mathlib.GroupTheory.SpecificGroups.Dihedral | ∀ {n : ℕ} (a : ZMod n), DihedralGroup.r a = DihedralGroup.r a | null | false |
_private.Mathlib.Topology.MetricSpace.HausdorffDimension.0.hausdorffMeasure_of_lt_dimH._simp_1_1 | Mathlib.Topology.MetricSpace.HausdorffDimension | ∀ {α : Type u_1} {ι : Sort u_3} [inst : CompleteLinearOrder α] {a : α} {f : ι → α}, (a < iSup f) = ∃ i, a < f i | null | false |
Int16.toInt_div_of_ne_left | Init.Data.SInt.Lemmas | ∀ (a b : Int16), a ≠ Int16.minValue → (a / b).toInt = a.toInt.tdiv b.toInt | null | true |
Filter.TendstoCofinite.mk | Mathlib.Order.Filter.TendstoCofinite | ∀ {α : Type u_1} {β : Type u_2} {f : α → β}, Filter.Tendsto f Filter.cofinite Filter.cofinite → Filter.TendstoCofinite f | null | true |
CategoryTheory.SimplicialObject.σ₀Iter_δ'._auto_5 | Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter | Lean.Syntax | null | false |
Lean.LLVM.DLLStorageClass.noConfusionType | Lean.Compiler.IR.LLVMBindings | Sort u → Lean.LLVM.DLLStorageClass → Lean.LLVM.DLLStorageClass → Sort u | null | false |
CategoryTheory.MonoidalCategory.InducedLawfulDayConvolutionMonoidalCategoryStructCore.ofHasDayConvolutions._proof_1 | Mathlib.CategoryTheory.Monoidal.DayConvolution | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] {V : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} V] [inst_2 : CategoryTheory.MonoidalCategory C]
[inst_3 : CategoryTheory.MonoidalCategory V] {D : Type u_6} [inst_4 : CategoryTheory.Category.{u_5, u_6} D]
(ι : CategoryTheory.Functor D (Cate... | null | false |
Std.Tactic.BVDecide.BVLogicalExpr.bitblast.go.match_3 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Substructure | (motive : Std.Tactic.BVDecide.Gate → Sort u_1) →
(g : Std.Tactic.BVDecide.Gate) →
(Unit → motive Std.Tactic.BVDecide.Gate.and) →
(Unit → motive Std.Tactic.BVDecide.Gate.xor) →
(Unit → motive Std.Tactic.BVDecide.Gate.beq) → (Unit → motive Std.Tactic.BVDecide.Gate.or) → motive g | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin.0.Fin.isValue._regBuiltin.Fin.isValue.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin.2142250532._hygCtx._hyg.28 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin | IO Unit | null | false |
_private.Init.Data.List.Sort.Lemmas.0.List.mergeSort_append._proof_1_8 | Init.Data.List.Sort.Lemmas | ∀ {α : Type u_1} (l₂ : List α), l₂.length + 1 + 1 ≤ 0 + 1 → False | null | false |
CategoryTheory.Functor.LaxLeftLinear.μₗ | Mathlib.CategoryTheory.Monoidal.Action.LinearFunctor | {D : Type u_1} →
{D' : Type u_2} →
{inst : CategoryTheory.Category.{v_1, u_1} D} →
{inst_1 : CategoryTheory.Category.{v_2, u_2} D'} →
(F : CategoryTheory.Functor D D') →
{C : Type u_3} →
{inst_2 : CategoryTheory.Category.{v_3, u_3} C} →
{inst_3 : CategoryTheory.Mo... | The "μₗ" morphism. | true |
Colex.instLeftCancelSemigroup | Mathlib.Algebra.Order.Group.Synonym | {α : Type u_1} → [LeftCancelSemigroup α] → LeftCancelSemigroup (Colex α) | null | true |
Function.Injective.groupWithZero | Mathlib.Algebra.GroupWithZero.InjSurj | {G₀ : Type u_2} →
{G₀' : Type u_4} →
[inst : GroupWithZero G₀] →
[inst_1 : Zero G₀'] →
[inst_2 : Mul G₀'] →
[inst_3 : One G₀'] →
[inst_4 : Inv G₀'] →
[inst_5 : Div G₀'] →
[inst_6 : Pow G₀' ℕ] →
[inst_7 : Pow G₀' ℤ] →
... | Pull back a `GroupWithZero` along an injective function.
See note [reducible non-instances]. | true |
ENat.floor_le_self | Mathlib.Algebra.Order.Floor.Extended | ∀ {r : ENNReal}, ↑⌊r⌋ₑ ≤ r | null | true |
_private.Mathlib.Combinatorics.SimpleGraph.Extremal.Turan.0.SimpleGraph.sum_ne_add_mod_eq_sub_one | Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | ∀ {n r c : ℕ}, (∑ w ∈ Finset.range r, if c % r ≠ (n + w) % r then 1 else 0) = r - 1 | null | true |
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