name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
LinearMap.toMatrix._proof_1 | Mathlib.LinearAlgebra.Matrix.ToLin | ∀ {R : Type u_1} [inst : CommSemiring R] {M₂ : Type u_2} [inst_1 : AddCommMonoid M₂] [inst_2 : Module R M₂],
SMulCommClass R R M₂ |
continuous_iff_ultrafilter | Mathlib.Topology.Ultrafilter | ∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},
Continuous f ↔ ∀ (x : X) (g : Ultrafilter X), ↑g ≤ nhds x → Filter.Tendsto f (↑g) (nhds (f x)) |
_private.Lean.Elab.Tactic.BVDecide.Frontend.Attr.0.Lean.Elab.Tactic.BVDecide.Frontend.elabBVDecideConfig.match_1 | Lean.Elab.Tactic.BVDecide.Frontend.Attr | (motive :
DoResultPR Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig
PUnit.{1} →
Sort u_1) →
(r :
DoResultPR Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig
PUnit.{1}) →
((a : Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig) → (u : PUnit.{1}) → motive (DoResultPR.pure a u)) →
((b : Lean.Elab.Tactic.BVDecide.Frontend.BVDecideConfig) → (u : PUnit.{1}) → motive (DoResultPR.return b u)) →
motive r |
NumberField.Units.regOfFamily_div_regulator | Mathlib.NumberTheory.NumberField.Units.Regulator | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
(u : Fin (NumberField.Units.rank K) → (NumberField.RingOfIntegers K)ˣ),
NumberField.Units.regOfFamily u / NumberField.Units.regulator K =
↑(Subgroup.closure (Set.range u) ⊔ NumberField.Units.torsion K).index |
_private.Mathlib.CategoryTheory.WithTerminal.Basic.0.CategoryTheory.WithTerminal.comp.match_1.eq_4 | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ {C : Type u_1}
(motive : CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → Sort u_2)
(x : CategoryTheory.WithTerminal C) (_Y : C)
(h_1 :
(_X _Y _Z : C) →
motive (CategoryTheory.WithTerminal.of _X) (CategoryTheory.WithTerminal.of _Y)
(CategoryTheory.WithTerminal.of _Z))
(h_2 :
(_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive (CategoryTheory.WithTerminal.of _X) x CategoryTheory.WithTerminal.star)
(h_3 :
(_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _X) x)
(h_4 :
(x : CategoryTheory.WithTerminal C) →
(_Y : C) → motive x CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _Y))
(h_5 :
Unit → motive CategoryTheory.WithTerminal.star CategoryTheory.WithTerminal.star CategoryTheory.WithTerminal.star),
(match x, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _Y with
| CategoryTheory.WithTerminal.of _X, CategoryTheory.WithTerminal.of _Y, CategoryTheory.WithTerminal.of _Z =>
h_1 _X _Y _Z
| CategoryTheory.WithTerminal.of _X, x, CategoryTheory.WithTerminal.star => h_2 _X x
| CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _X, x => h_3 _X x
| x, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.of _Y => h_4 x _Y
| CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star => h_5 ()) =
h_4 x _Y |
NNReal.exists_pow_lt_of_lt_one | Mathlib.Data.NNReal.Defs | ∀ {a b : NNReal}, 0 < a → b < 1 → ∃ n, b ^ n < a |
IsFractionRing.isAlgebraic_iff | Mathlib.RingTheory.Localization.Integral | ∀ (A : Type u_3) (K : Type u_4) (C : Type u_5) [inst : CommRing A] [IsDomain A] [inst_2 : Field K]
[inst_3 : Algebra A K] [IsFractionRing A K] [inst_5 : CommRing C] [inst_6 : Algebra A C] [inst_7 : Algebra K C]
[IsScalarTower A K C] {x : C}, IsAlgebraic A x ↔ IsAlgebraic K x |
Quiver.Path.length_eq_zero_iff._simp_1 | Mathlib.Combinatorics.Quiver.Path.Vertices | ∀ {V : Type u_1} [inst : Quiver V] {a : V} (p : Quiver.Path a a), (p.length = 0) = (p = Quiver.Path.nil) |
Sym2.IsDiag._proof_1 | Mathlib.Data.Sym.Sym2 | ∀ {α : Type u_1} (x x_1 : α), (x = x_1) = (x_1 = x) |
idRestrGroupoid._proof_3 | Mathlib.Geometry.Manifold.StructureGroupoid | ∀ {H : Type u_1} [inst : TopologicalSpace H],
∃ s, ∃ (h : IsOpen s), OpenPartialHomeomorph.refl H ≈ OpenPartialHomeomorph.ofSet s h |
BitVec.ushiftRight_eq_zero | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {x : BitVec w} {n : ℕ}, w ≤ n → x >>> n = 0#w |
_private.Mathlib.Tactic.Abel.0.Mathlib.Tactic.Abel.eval._sparseCasesOn_5 | Mathlib.Tactic.Abel | {motive : Lean.Literal → Sort u} →
(t : Lean.Literal) →
((val : ℕ) → motive (Lean.Literal.natVal val)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t |
CategoryTheory.MorphismProperty.definition._proof_6._@.Mathlib.CategoryTheory.Localization.LocallySmall.822486777._hygCtx._hyg.2 | Mathlib.CategoryTheory.Localization.LocallySmall | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_4, u_2} C] {D : Type u_3}
[inst_1 : CategoryTheory.Category.{u_1, u_3} D] (L : CategoryTheory.Functor C D),
(CategoryTheory.inducedFunctor L.obj).Full |
isSaddlePointOn_value | Mathlib.Order.SaddlePoint | ∀ {E : Type u_1} {F : Type u_2} {β : Type u_3} {X : Set E} {Y : Set F} {f : E → F → β} [inst : CompleteLinearOrder β]
{a : E},
a ∈ X →
∀ {b : F}, b ∈ Y → IsSaddlePointOn X Y f a b → ⨅ x ∈ X, ⨆ y ∈ Y, f x y = f a b ∧ ⨆ y ∈ Y, ⨅ x ∈ X, f x y = f a b |
PrimeSpectrum.BasicConstructibleSetData.mk.sizeOf_spec | Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | ∀ {R : Type u_1} [inst : SizeOf R] (f : R) (n : ℕ) (g : Fin n → R),
sizeOf { f := f, n := n, g := g } = 1 + sizeOf f + sizeOf n |
Fin.image_succ_Ioc | Mathlib.Order.Interval.Set.Fin | ∀ {n : ℕ} (i j : Fin n), Fin.succ '' Set.Ioc i j = Set.Ioc i.succ j.succ |
_private.Init.Data.String.Slice.0.String.Slice.eqIgnoreAsciiCase.go._unary._proof_2 | Init.Data.String.Slice | ∀ (s1 s2 : String.Slice) (s1Curr s2Curr : String.Pos.Raw),
s1Curr < s1.rawEndPos ∧ s2Curr < s2.rawEndPos → s2Curr < s2.rawEndPos |
StieltjesFunction.instModuleNNReal._proof_1 | Mathlib.MeasureTheory.Measure.Stieltjes | ∀ {R : Type u_1} [inst : LinearOrder R] [inst_1 : TopologicalSpace R] (c : NNReal) (f : StieltjesFunction R),
Monotone fun x => ↑c * ↑f x |
_private.Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone.0.NumberField.mixedEmbedding.fundamentalCone.mem_integerSet._simp_1_2 | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) |
AddValuation.map_lt_sum | Mathlib.RingTheory.Valuation.Basic | ∀ {R : Type u_3} {Γ₀ : Type u_4} [inst : Ring R] [inst_1 : LinearOrderedAddCommMonoidWithTop Γ₀] (v : AddValuation R Γ₀)
{ι : Type u_6} {s : Finset ι} {f : ι → R} {g : Γ₀}, g ≠ ⊤ → (∀ i ∈ s, g < v (f i)) → g < v (∑ i ∈ s, f i) |
UInt16.toUInt8_ofNatTruncate_of_le | Init.Data.UInt.Lemmas | ∀ {n : ℕ},
UInt16.size ≤ n →
(UInt16.ofNatTruncate n).toUInt8 = UInt8.ofNatLT (UInt8.size - 1) UInt16.toUInt8_ofNatTruncate_of_le._proof_1 |
_private.Mathlib.Combinatorics.SimpleGraph.Basic.0.SimpleGraph.adj_incidenceSet_inter._simp_1_2 | Mathlib.Combinatorics.SimpleGraph.Basic | ∀ {α : Type u} {s : Set α} {p : α → Prop} {x : α}, (x ∈ {x | x ∈ s ∧ p x}) = (x ∈ s ∧ p x) |
_private.Mathlib.Analysis.InnerProductSpace.Adjoint.0.isStarProjection_iff_eq_starProjection_range._simp_1_2 | Mathlib.Analysis.InnerProductSpace.Adjoint | ∀ {R₁ : Type u_1} {R₂ : Type u_2} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4}
[inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_6} [inst_4 : TopologicalSpace M₂]
[inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂] {f g : M₁ →SL[σ₁₂] M₂},
(↑f = ↑g) = (f = g) |
Filter.tendsto_div_const_atBot_iff | Mathlib.Order.Filter.AtTopBot.Field | ∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {l : Filter β}
{f : β → α} {r : α} [l.NeBot],
Filter.Tendsto (fun x => f x / r) l Filter.atBot ↔
0 < r ∧ Filter.Tendsto f l Filter.atBot ∨ r < 0 ∧ Filter.Tendsto f l Filter.atTop |
Subgroup.ofUnitsEquivType._proof_3 | Mathlib.Algebra.Group.Submonoid.Units | ∀ {M : Type u_1} [inst : Monoid M] (S : Subgroup Mˣ) (x : ↥S.ofUnits), ↑x ∈ S.ofUnits |
Lean.Meta.Grind.SplitStatus.ready | Lean.Meta.Tactic.Grind.Split | ℕ → optParam Bool false → optParam Bool false → Lean.Meta.Grind.SplitStatus |
Lean.Export.Entry.ctorIdx | Mathlib.Util.Export | Lean.Export.Entry → ℕ |
tprod_setProd_singleton_right | Mathlib.Topology.Algebra.InfiniteSum.Constructions | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : CommMonoid α] [inst_1 : TopologicalSpace α] (s : Set β) (c : γ)
(f : β × γ → α), ∏' (x : ↑(s ×ˢ {c})), f ↑x = ∏' (b : ↑s), f (↑b, c) |
_private.Lean.Elab.DocString.Builtin.Keywords.0.Lean.Doc.Data.Atom.mk.injEq | Lean.Elab.DocString.Builtin.Keywords | ∀ (name category name_1 category_1 : Lean.Name),
({ name := name, category := category } = { name := name_1, category := category_1 }) =
(name = name_1 ∧ category = category_1) |
LieSubmodule.normalizer._proof_3 | Mathlib.Algebra.Lie.Normalizer | ∀ {R : Type u_2} {L : Type u_3} {M : Type u_1} [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]
(N : LieSubmodule R L M) (t : R), ∀ m ∈ {m | ∀ (x : L), ⁅x, m⁆ ∈ N}, ∀ (x : L), ⁅x, t • m⁆ ∈ N |
Lean.Meta.Grind.TopSort.State._sizeOf_1 | Lean.Meta.Tactic.Grind.EqResolution | Lean.Meta.Grind.TopSort.State → ℕ |
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 |
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 : IsTopologicalAddGroup F']
[inst_9 : ContinuousSMul 𝕜 F'] [inst_10 : CompleteSpace 𝕜] [inst_11 : T2Space E] [inst_12 : FiniteDimensional 𝕜 E],
LinearMap.toContinuousLinearMap = LinearMap.toContinuousLinearMap |
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' |
List.zipWithLeft'TR.go._unsafe_rec | Batteries.Data.List.Basic | {α : Type u_1} → {β : Type u_2} → {γ : Type u_3} → (α → Option β → γ) → List α → List β → Array γ → List γ × List β |
IO.FS.realPath | Init.System.IO | System.FilePath → IO System.FilePath |
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) →
(isOpen : ∀ (i : ι), IsOpen (toFun i)) →
(subset_iUnion : s ⊆ ⋃ i, toFun i) →
(closure_subset : ∀ {i : ι}, i ∈ carrier → closure (toFun i) ⊆ u i) →
(pred_of_mem : ∀ {i : ι}, i ∈ carrier → p (toFun i)) →
(apply_eq : ∀ {i : ι}, i ∉ carrier → toFun i = u i) →
motive
{ toFun := toFun, carrier := carrier, isOpen := isOpen, subset_iUnion := subset_iUnion,
closure_subset := closure_subset, pred_of_mem := pred_of_mem,
apply_eq := apply_eq }) →
(t : ShrinkingLemma.PartialRefinement u s p) → motive t |
SimpleGraph.IsMatchingFree | Mathlib.Combinatorics.SimpleGraph.Matching | {V : Type u_1} → SimpleGraph V → Prop |
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 |
invertibleSucc | Mathlib.Algebra.CharP.Invertible | {K : Type u_2} → [inst : DivisionSemiring K] → [CharZero K] → (n : ℕ) → Invertible ↑n.succ |
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 : Fintype α] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s)
[inst_7 : DecidablePred fun x => x ∈ s] (m₁ : (i : { a // a ∉ s }) → E₁ ↑i) (m₂ : α → (i : ι) → E₁ i),
‖((f.iteratedFDerivComponent e) m₁) m₂‖ ≤ (‖f‖ * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖ |
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 |
Lean.Doc.instFromDocArgMessageSeverity | Lean.Elab.DocString | Lean.Doc.FromDocArg Lean.MessageSeverity |
_private.Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital.0._auto_505 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | Lean.Syntax |
List.head_attach | Init.Data.List.Attach | ∀ {α : Type u_1} {xs : List α} (h : xs.attach ≠ []), xs.attach.head h = ⟨xs.head ⋯, ⋯⟩ |
_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 |
_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 |
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 |
FinEnum.PSigma.finEnumPropProp._proof_3 | Mathlib.Data.FinEnum | ∀ {α : Prop} {β : α → Prop}, (∃ (a : α), β a) → α |
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 → ENNReal.ofReal (Λ f) ≤ (RealRMK.rieszMeasure Λ) V |
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) |
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 |
Seminorm.comp_smul | Mathlib.Analysis.Seminorm | ∀ {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] E₂)
(c : 𝕜₂), p.comp (c • f) = ‖c‖₊ • p.comp f |
Lean.Widget.RpcEncodablePacket.leanTags?._@.Lean.Widget.InteractiveDiagnostic.2989700264._hygCtx._hyg.2 | Lean.Widget.InteractiveDiagnostic | Lean.Widget.RpcEncodablePacket✝ → Option Lean.Json |
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),
(AffineEquiv.ofBijective hφ) a = φ a |
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 } |
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 |
_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) |
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 = ⊤ |
Lean.instNonemptyKeyedDeclsAttribute | Lean.KeyedDeclsAttribute | ∀ {γ : Type}, Nonempty (Lean.KeyedDeclsAttribute γ) |
Lean.Server.Test.Runner.Client.instToJsonHighlightMatchesParams.toJson | Lean.Server.Test.Runner | Lean.Server.Test.Runner.Client.HighlightMatchesParams → Lean.Json |
_private.Mathlib.Topology.Order.Basic.0.exists_countable_generateFrom_Ioi_Iio._simp_1_2 | Mathlib.Topology.Order.Basic | ∀ {α : Sort u_1} {β : Sort u_2} {f : α → β} {p : α → Prop} {q : β → Prop},
(∃ b, (∃ a, p a ∧ f a = b) ∧ q b) = ∃ a, p a ∧ q (f a) |
_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 () |
instLawfulMonadContOptionT | Mathlib.Control.Monad.Cont | ∀ {m : Type u → Type v} [inst : Monad m] [inst_1 : MonadCont m] [LawfulMonadCont m], LawfulMonadCont (OptionT m) |
_private.Lean.Meta.Tactic.Grind.Order.StructId.0.Lean.Meta.Grind.Order.getInst? | Lean.Meta.Tactic.Grind.Order.StructId | Lean.Name → Lean.Level → Lean.Expr → Lean.Meta.Grind.GoalM (Option Lean.Expr) |
«term_ᵈᵃᵃ» | Mathlib.GroupTheory.GroupAction.DomAct.Basic | Lean.TrailingParserDescr |
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 => some ⟨k', v⟩
| base, Std.DTreeMap.Internal.Impl.ExplorationStep.eq a c r => (c.inner.or r.head?).or base
| base, Std.DTreeMap.Internal.Impl.ExplorationStep.gt a a_1 a_2 a_3 => base)
t |
_private.Lean.Elab.PreDefinition.WF.Preprocess.0._regBuiltin.Lean.Elab.WF.paramProj.declare_26._@.Lean.Elab.PreDefinition.WF.Preprocess.184633683._hygCtx._hyg.12 | Lean.Elab.PreDefinition.WF.Preprocess | IO Unit |
Lean.ExternEntry.adhoc.sizeOf_spec | Lean.Compiler.ExternAttr | ∀ (backend : Lean.Name), sizeOf (Lean.ExternEntry.adhoc backend) = 1 + sizeOf backend |
_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) |
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 |
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.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g) |
_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 |
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 |
_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_2 a) =
h_1 () |
Lean.PrettyPrinter.Delaborator.TopDownAnalyze.isTrivialBottomUp | Lean.PrettyPrinter.Delaborator.TopDownAnalyze | Lean.Expr → Lean.PrettyPrinter.Delaborator.TopDownAnalyze.AnalyzeM Bool |
RelUpperSet.isRelUpperSet' | Mathlib.Order.Defs.Unbundled | ∀ {α : Type u_1} [inst : LE α] {P : α → Prop} (self : RelUpperSet P), IsRelUpperSet self.carrier P |
_private.Mathlib.Data.List.Triplewise.0.List.triplewise_iff_getElem._proof_1_71 | Mathlib.Data.List.Triplewise | ∀ {α : Type u_1} (tail : List α) (i k : ℕ), k + 1 ≤ i → i + 1 ≤ k - 1 → k - 1 + 1 ≤ tail.length → k < tail.length |
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 |
_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) |
Nat.Linear.Expr.var.inj | Init.Data.Nat.Linear | ∀ {i i_1 : Nat.Linear.Var}, Nat.Linear.Expr.var i = Nat.Linear.Expr.var i_1 → i = i_1 |
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 |
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 := ⋯ } |
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 |
_private.Mathlib.Order.Interval.Set.Pi.0.Set.Icc_diff_pi_univ_Ioc_subset._simp_1_1 | Mathlib.Order.Interval.Set.Pi | ∀ {a b : Prop}, (¬(a ∧ b)) = (¬a ∨ ¬b) |
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)} →
{one_mem : ∀ (i : ι), 1 ∈ set i} →
{compl : ∀ (i : ι), Subgroup.IsComplement (↑(φ i).range) (set i)} →
{injective' : ∀ (i : ι), Function.Injective ⇑(φ i)} →
{set' : (i : ι) → Set (G i)} →
{one_mem' : ∀ (i : ι), 1 ∈ set' i} →
{compl' : ∀ (i : ι), Subgroup.IsComplement (↑(φ i).range) (set' i)} →
{ injective := injective, set := set, one_mem := one_mem, compl := compl } =
{ injective := injective', set := set', one_mem := one_mem', compl := compl' } →
(set ≍ set' → P) → P |
_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 |
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 |
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 |
Projectivization.Subspace.instCompleteLattice | Mathlib.LinearAlgebra.Projectivization.Subspace | {K : Type u_1} →
{V : Type u_2} →
[inst : Field K] →
[inst_1 : AddCommGroup V] → [inst_2 : Module K V] → CompleteLattice (Projectivization.Subspace K V) |
_aux_Mathlib_Algebra_Module_LinearMap_Defs___unexpand_LinearMap_1 | Mathlib.Algebra.Module.LinearMap.Defs | Lean.PrettyPrinter.Unexpander |
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 |
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 =>
have hV := ⋯;
have hr := ⋯;
let r := ⋯.choose;
have this := ⋯;
⟨this.unit, ⋯⟩,
invFun := SpecialLinearGroup.centerEquivRootsOfUnity_invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ } |
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) |
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 |
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 |
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 |
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 |
_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 |
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 |
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