name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
UInt32.mul_def | Init.Data.UInt.Lemmas | ∀ (a b : UInt32), a * b = { toBitVec := a.toBitVec * b.toBitVec } |
List.findIdx_map | Init.Data.List.Find | ∀ {α : Type u_1} {β : Type u_2} (xs : List α) (f : α → β) (p : β → Bool),
List.findIdx p (List.map f xs) = List.findIdx (p ∘ f) xs |
Lean.Meta.LazyDiscrTree.InitEntry._sizeOf_1 | Lean.Meta.LazyDiscrTree | {α : Type} → [SizeOf α] → Lean.Meta.LazyDiscrTree.InitEntry α → ℕ |
Lean.MonadStateCacheT | Lean.Util.MonadCache | (α : Type) → Type → (Type → Type) → [BEq α] → [Hashable α] → Type → Type |
MulHom.coe_ofDense | Mathlib.Algebra.Group.Subsemigroup.Basic | ∀ {M : Type u_1} {N : Type u_2} [inst : Semigroup M] [inst_1 : Semigroup N] {s : Set M} (f : M → N)
(hs : Subsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y), ⇑(MulHom.ofDense f hs hmul) = f |
_private.Init.Data.ByteArray.Lemmas.0.ByteArray.data_extract._proof_1_4 | Init.Data.ByteArray.Lemmas | ∀ {a : ByteArray} {b e : ℕ}, ¬b ≤ e → ¬min e a.data.size ≤ b → False |
Submonoid.mem_divPairs | Mathlib.GroupTheory.MonoidLocalization.DivPairs | ∀ {M : Type u_1} {G : Type u_2} [inst : CommMonoid M] [inst_1 : CommGroup G] {f : ⊤.LocalizationMap G} {s : Submonoid G}
{x : M × M}, x ∈ Submonoid.divPairs f s ↔ f x.1 / f x.2 ∈ s |
_private.Mathlib.Algebra.Module.FinitePresentation.0.Module.finitePresentation_of_free_of_surjective._simp_1_7 | Mathlib.Algebra.Module.FinitePresentation | ∀ {α : Type u_1} {M : Type u_2} (R : Type u_5) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{M' : Type u_8} [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] (f : M →ₗ[R] M') (v : α → M) (l : α →₀ R),
(Finsupp.linearCombination R (⇑f ∘ v)) l = f ((Finsupp.linearCombination R v) l) |
_private.Lean.Level.0.Lean.Level.normLtAux._unary._proof_3 | Lean.Level | ∀ (l₁ : Lean.Level) (k₁ : ℕ) (l₂ : Lean.Level) (k₂ : ℕ),
(invImage
(fun x =>
PSigma.casesOn x fun a a_1 => PSigma.casesOn a_1 fun a_2 a_3 => PSigma.casesOn a_3 fun a_4 a_5 => (a, a_4))
Prod.instWellFoundedRelation).1
⟨l₁, ⟨k₁, ⟨l₂, k₂ + 1⟩⟩⟩ ⟨l₁, ⟨k₁, ⟨l₂.succ, k₂⟩⟩⟩ |
MeasureTheory.FiniteMeasure.coeFn_def | Mathlib.MeasureTheory.Measure.FiniteMeasure | ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω), ⇑μ = fun s => (↑μ s).toNNReal |
_private.Mathlib.RingTheory.Extension.Cotangent.Basis.0.Algebra.Generators.exists_presentation_of_basis_cotangent._simp_1_3 | Mathlib.RingTheory.Extension.Cotangent.Basis | ∀ {α : Type u_1} {β : Type u_2} {ι : Sort u_4} (g : α → β) (f : ι → α), g '' Set.range f = Set.range (g ∘ f) |
Quotient.finChoice._proof_2 | Mathlib.Data.Fintype.Quotient | ∀ {ι : Type u_1} (x x_1 : { l // ∀ (i : ι), i ∈ l }),
(Subtype.instSetoid_mathlib fun l => ∀ (i : ι), i ∈ l) x x_1 ↔
(Subtype.instSetoid_mathlib fun l => ∀ (i : ι), i ∈ l) x x_1 |
_private.Mathlib.RingTheory.Polynomial.UniversalFactorizationRing.0.MvPolynomial.universalFactorizationMapPresentation_jacobian._simp_1_2 | Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | ∀ {R : Type u_1} [inst : CommRing R] (f g : Polynomial R) (m n : ℕ), (f.sylvester g m n).det = f.resultant g m n |
GrpWithZero.carrier | Mathlib.Algebra.Category.GrpWithZero | GrpWithZero → Type u_1 |
Std.Do.SPred.exists.match_1 | Std.Do.SPred.SPred | {α : Sort u_3} →
(motive : (σs : List (Type u_1)) → (α → Std.Do.SPred σs) → Sort u_2) →
(σs : List (Type u_1)) →
(P : α → Std.Do.SPred σs) →
((P : α → Std.Do.SPred []) → motive [] P) →
((σ : Type u_1) → (tail : List (Type u_1)) → (P : α → Std.Do.SPred (σ :: tail)) → motive (σ :: tail) P) →
motive σs P |
IsPrimitiveRoot.toRootsOfUnity | Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {M : Type u_1} → [inst : CommMonoid M] → {μ : M} → {n : ℕ} → [NeZero n] → IsPrimitiveRoot μ n → ↥(rootsOfUnity n M) |
Mathlib.Meta.NormNum.not_isSquare_of_isNNRat_rat_of_num | Mathlib.Tactic.NormNum.IsSquare | ∀ (a : ℚ) (n d : ℕ), ¬IsSquare n → n.Coprime d → Mathlib.Meta.NormNum.IsNNRat a n d → ¬IsSquare a |
DFinsupp.lsum_single | Mathlib.LinearAlgebra.DFinsupp | ∀ {ι : Type u_1} {R : Type u_3} (S : Type u_4) {M : ι → Type u_5} {N : Type u_6} [inst : Semiring R]
[inst_1 : (i : ι) → AddCommMonoid (M i)] [inst_2 : (i : ι) → Module R (M i)] [inst_3 : AddCommMonoid N]
[inst_4 : Module R N] [inst_5 : DecidableEq ι] [inst_6 : Semiring S] [inst_7 : Module S N]
[inst_8 : SMulCommClass R S N] (F : (i : ι) → M i →ₗ[R] N) (i : ι) (x : M i),
(((DFinsupp.lsum S) F) fun₀ | i => x) = (F i) x |
Lean.Meta.Grind.AC.MonadGetStruct.noConfusionType | Lean.Meta.Tactic.Grind.AC.Util | Sort u →
{m : Type → Type} →
Lean.Meta.Grind.AC.MonadGetStruct m → {m' : Type → Type} → Lean.Meta.Grind.AC.MonadGetStruct m' → Sort u |
_private.Init.Meta.Defs.0.Lean.Name.replacePrefix.match_1 | Init.Meta.Defs | (motive : Lean.Name → Lean.Name → Lean.Name → Sort u_1) →
(x x_1 x_2 : Lean.Name) →
((newP : Lean.Name) → motive Lean.Name.anonymous Lean.Name.anonymous newP) →
((x x_3 : Lean.Name) → motive Lean.Name.anonymous x x_3) →
((n p : Lean.Name) →
(s : String) →
(h : n = p.str s) → (queryP newP : Lean.Name) → motive (namedPattern n (p.str s) h) queryP newP) →
((n p : Lean.Name) →
(s : ℕ) →
(h : n = p.num s) → (queryP newP : Lean.Name) → motive (namedPattern n (p.num s) h) queryP newP) →
motive x x_1 x_2 |
Lean.Expr.replace | Lean.Util.ReplaceExpr | (Lean.Expr → Option Lean.Expr) → Lean.Expr → Lean.Expr |
Lean.instToJsonPrintImportResult.toJson | Lean.Elab.ParseImportsFast | Lean.PrintImportResult → Lean.Json |
SemiNormedGrp.of.injEq | Mathlib.Analysis.Normed.Group.SemiNormedGrp | ∀ (carrier : Type u) [str : SeminormedAddCommGroup carrier] (carrier_1 : Type u)
(str_1 : SeminormedAddCommGroup carrier_1),
({ carrier := carrier, str := str } = { carrier := carrier_1, str := str_1 }) = (carrier = carrier_1 ∧ str ≍ str_1) |
Finset.Ioo_subset_Ioi_self | Mathlib.Order.Interval.Finset.Basic | ∀ {α : Type u_2} {a b : α} [inst : Preorder α] [inst_1 : LocallyFiniteOrderTop α] [inst_2 : LocallyFiniteOrder α],
Finset.Ioo a b ⊆ Finset.Ioi a |
Lean.Elab.Attribute.name | Lean.Elab.Attributes | Lean.Elab.Attribute → Lean.Name |
Std.DHashMap.Internal.Raw₀.Const.any_toList | Std.Data.DHashMap.Internal.RawLemmas | ∀ {α : Type u} {β : Type v} (m : Std.DHashMap.Internal.Raw₀ α fun x => β) {p : α → β → Bool},
((Std.DHashMap.Raw.Const.toList ↑m).any fun x => p x.1 x.2) = (↑m).any p |
Std.Time.Timestamp.ofPlainDateTimeAssumingUTC | Std.Time.DateTime | Std.Time.PlainDateTime → Std.Time.Timestamp |
CuspFormClass.rec | Mathlib.NumberTheory.ModularForms.Basic | {F : Type u_2} →
{Γ : Subgroup (GL (Fin 2) ℝ)} →
{k : ℤ} →
[inst : FunLike F UpperHalfPlane ℂ] →
{motive : CuspFormClass F Γ k → Sort u} →
([toSlashInvariantFormClass : SlashInvariantFormClass F Γ k] →
(holo : ∀ (f : F), MDifferentiable (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ⇑f) →
(zero_at_cusps : ∀ (f : F) {c : OnePoint ℝ}, IsCusp c Γ → c.IsZeroAt (⇑f) k) → motive ⋯) →
(t : CuspFormClass F Γ k) → motive t |
Turing.TM1to1.trCfg.eq_1 | Mathlib.Computability.PostTuringMachine | ∀ {Γ : Type u_1} {Λ : Type u_2} {σ : Type u_3} {n : ℕ} (enc : Γ → List.Vector Bool n) [inst : Inhabited Γ]
(enc0 : enc default = List.Vector.replicate n false) (l : Option Λ) (v : σ) (T : Turing.Tape Γ),
Turing.TM1to1.trCfg enc enc0 { l := l, var := v, Tape := T } =
{ l := Option.map Turing.TM1to1.Λ'.normal l, var := v, Tape := Turing.TM1to1.trTape enc0 T } |
GroupExtension.Splitting.conjAct | Mathlib.GroupTheory.GroupExtension.Basic | {N : Type u_1} →
{G : Type u_2} →
[inst : Group N] →
[inst_1 : Group G] →
{E : Type u_3} → [inst_2 : Group E] → {S : GroupExtension N E G} → S.Splitting → G →* MulAut N |
Std.HashMap.Raw.getElem_congr | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} {m : Std.HashMap.Raw α β} [inst : BEq α] [inst_1 : Hashable α] [inst_2 : EquivBEq α]
[inst_3 : LawfulHashable α] (h : m.WF) {a b : α} (hab : (a == b) = true) {h' : a ∈ m}, m[a] = m[b] |
WeakSpace.instModule' | Mathlib.Topology.Algebra.Module.WeakDual | {𝕜 : Type u_2} →
{𝕝 : Type u_3} →
{E : Type u_4} →
[inst : CommSemiring 𝕜] →
[inst_1 : TopologicalSpace 𝕜] →
[inst_2 : ContinuousAdd 𝕜] →
[inst_3 : ContinuousConstSMul 𝕜 𝕜] →
[inst_4 : AddCommMonoid E] →
[inst_5 : Module 𝕜 E] →
[inst_6 : TopologicalSpace E] → [inst_7 : CommSemiring 𝕝] → [Module 𝕝 E] → Module 𝕝 (WeakSpace 𝕜 E) |
CategoryTheory.Subfunctor.range | Mathlib.CategoryTheory.Subfunctor.Image | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{F F' : CategoryTheory.Functor C (Type w)} → (F' ⟶ F) → CategoryTheory.Subfunctor F |
QuaternionAlgebra.equivProd | Mathlib.Algebra.Quaternion | {R : Type u_1} → (c₁ c₂ c₃ : R) → QuaternionAlgebra R c₁ c₂ c₃ ≃ R × R × R × R |
Vector.set_mk._proof_3 | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n : ℕ} {xs : Array α} (h : xs.size = n) {i : ℕ} {x : α} (w : i < n), (xs.set i x ⋯).size = n |
ContDiffOn.pow | 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∞} {𝔸 : Type u_3} [inst_3 : NormedRing 𝔸]
[inst_4 : NormedAlgebra 𝕜 𝔸] {f : E → 𝔸}, ContDiffOn 𝕜 n f s → ∀ (m : ℕ), ContDiffOn 𝕜 n (fun y => f y ^ m) s |
Std.Internal.IO.Async.UDP.Socket.getPeerName | Std.Internal.Async.UDP | Std.Internal.IO.Async.UDP.Socket → IO Std.Net.SocketAddress |
Aesop.GoalOrigin | Aesop.Tree.Data | Type |
_private.Mathlib.Probability.Distributions.Gaussian.Real.0.ProbabilityTheory.complexMGF_id_gaussianReal._simp_1_5 | Mathlib.Probability.Distributions.Gaussian.Real | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False |
Submodule.span_smul | Mathlib.Algebra.Module.Submodule.Pointwise | ∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),
Submodule.span R (a • s) = a • Submodule.span R s |
Lean.Meta.Grind.InjectiveInfo.inv? | Lean.Meta.Tactic.Grind.Types | Lean.Meta.Grind.InjectiveInfo → Option (Lean.Expr × Lean.Expr) |
_private.Mathlib.CategoryTheory.Triangulated.Opposite.OpOp.0.CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app._proof_3 | Mathlib.CategoryTheory.Triangulated.Opposite.OpOp | ∀ (n m : ℤ), autoParam (n + m = 0) CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app._auto_1 → m + n = 0 |
Valuation.val_le_one_or_val_inv_lt_one | Mathlib.RingTheory.Valuation.Basic | ∀ {K : Type u_1} [inst : DivisionRing K] {Γ₀ : Type u_4} [inst_1 : LinearOrderedCommGroupWithZero Γ₀]
(v : Valuation K Γ₀) (x : K), v x ≤ 1 ∨ v x⁻¹ < 1 |
Lean.Grind.IntInterval.lo?.eq_4 | Init.Grind.ToIntLemmas | Lean.Grind.IntInterval.ii.lo? = none |
AddAction.stabilizer.eq_1 | Mathlib.GroupTheory.GroupAction.Defs | ∀ (G : Type u_1) {α : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] (a : α),
AddAction.stabilizer G a = { toAddSubmonoid := AddAction.stabilizerAddSubmonoid G a, neg_mem' := ⋯ } |
ContinuousMultilinearMap.nnnorm_constOfIsEmpty | Mathlib.Analysis.Normed.Module.Multilinear.Basic | ∀ (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) {G : Type wG} [inst : NontriviallyNormedField 𝕜]
[inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)]
[inst_3 : SeminormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] [inst_5 : Fintype ι] [inst_6 : IsEmpty ι] (x : G),
‖ContinuousMultilinearMap.constOfIsEmpty 𝕜 E x‖₊ = ‖x‖₊ |
groupCohomology.Hilbert90.aux.eq_1 | Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | ∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]
[inst_3 : FiniteDimensional K L] (f : Gal(L/K) → Lˣ),
groupCohomology.Hilbert90.aux f =
(Finsupp.linearCombination L fun φ => ⇑φ) (Finsupp.equivFunOnFinite.symm fun φ => ↑(f φ)) |
Polynomial.Monic.degree_pos | Mathlib.Algebra.Polynomial.Degree.Operations | ∀ {R : Type u} [inst : CommSemiring R] {p : Polynomial R}, p.Monic → (0 < p.degree ↔ p ≠ 1) |
SeparationQuotient.instDistrib._proof_1 | Mathlib.Topology.Algebra.SeparationQuotient.Basic | ∀ {R : Type u_1} [inst : TopologicalSpace R] [inst_1 : Distrib R] [inst_2 : ContinuousAdd R] (a b : R),
SeparationQuotient.mk (a + b) = SeparationQuotient.mk a + SeparationQuotient.mk b |
_private.Mathlib.Data.List.TakeDrop.0.List.takeD_eq_take.match_1_1 | Mathlib.Data.List.TakeDrop | ∀ {α : Type u_1} (motive : (x : ℕ) → (x_1 : List α) → α → x ≤ x_1.length → Prop) (x : ℕ) (x_1 : List α) (x_2 : α)
(x_3 : x ≤ x_1.length),
(∀ (x : List α) (x_4 : α) (x_5 : 0 ≤ x.length), motive 0 x x_4 x_5) →
(∀ (n : ℕ) (head : α) (tail : List α) (a : α) (h : n + 1 ≤ (head :: tail).length),
motive n.succ (head :: tail) a h) →
motive x x_1 x_2 x_3 |
Lean.instInhabitedTheoremVal | Lean.Declaration | Inhabited Lean.TheoremVal |
t1Space_iff_specializes_imp_eq | Mathlib.Topology.Separation.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X], T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y |
Submodule.IsOrtho.le | Mathlib.Analysis.InnerProductSpace.Orthogonal | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{U V : Submodule 𝕜 E}, U ⟂ V → U ≤ Vᗮ |
AdicCompletion.of_injective | Mathlib.RingTheory.AdicCompletion.Basic | ∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R) (M : Type u_4) [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[IsHausdorff I M], Function.Injective ⇑(AdicCompletion.of I M) |
MeasureTheory.Measure.pi | Mathlib.MeasureTheory.Constructions.Pi | {ι : Type u_4} →
{α : ι → Type u_5} →
[Fintype ι] →
[inst : (i : ι) → MeasurableSpace (α i)] →
((i : ι) → MeasureTheory.Measure (α i)) → MeasureTheory.Measure ((i : ι) → α i) |
Std.Internal.IO.Async.Signal.sigxfsz.elim | Std.Internal.Async.Signal | {motive : Std.Internal.IO.Async.Signal → Sort u} →
(t : Std.Internal.IO.Async.Signal) → t.ctorIdx = 17 → motive Std.Internal.IO.Async.Signal.sigxfsz → motive t |
MonoidWithZeroHom | Mathlib.Algebra.GroupWithZero.Hom | (α : Type u_7) → (β : Type u_8) → [MulZeroOneClass α] → [MulZeroOneClass β] → Type (max u_7 u_8) |
_private.Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise.0.SubMulAction.instMulOneClass._simp_1 | Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise | ∀ {R : Type u_1} {M : Type u_2} [inst : Monoid R] [inst_1 : MulAction R M] [inst_2 : Mul M]
[inst_3 : IsScalarTower R M M] {p q : SubMulAction R M} {x : M}, (x ∈ p * q) = ∃ y ∈ p, ∃ z ∈ q, y * z = x |
Std.DTreeMap.Internal.Impl.Const.get!_insertIfNew! | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {β : Type v} {t : Std.DTreeMap.Internal.Impl α fun x => β} [Std.TransOrd α]
[inst : Inhabited β],
t.WF →
∀ {k a : α} {v : β},
Std.DTreeMap.Internal.Impl.Const.get! (Std.DTreeMap.Internal.Impl.insertIfNew! k v t) a =
if compare k a = Ordering.eq ∧ k ∉ t then v else Std.DTreeMap.Internal.Impl.Const.get! t a |
Lean.Elab.Tactic.BVDecide.Frontend.ReifiedBVExpr.mkBVConst | Lean.Elab.Tactic.BVDecide.Frontend.BVDecide.ReifiedBVExpr | {w : ℕ} → BitVec w → Lean.Elab.Tactic.BVDecide.Frontend.M Lean.Elab.Tactic.BVDecide.Frontend.ReifiedBVExpr |
ContDiffOn.continuousOn_fderivWithin_apply | Mathlib.Analysis.Calculus.ContDiff.Comp | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F}
{n : WithTop ℕ∞},
ContDiffOn 𝕜 n f s → UniqueDiffOn 𝕜 s → 1 ≤ n → ContinuousOn (fun p => (fderivWithin 𝕜 f s p.1) p.2) (s ×ˢ Set.univ) |
MultiplierAlgebra.«_aux_Mathlib_Analysis_CStarAlgebra_Multiplier___macroRules_MultiplierAlgebra_term𝓜(_,_)_1» | Mathlib.Analysis.CStarAlgebra.Multiplier | Lean.Macro |
iteratedFDerivWithin_insert | Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
{f : E → F} {x : E} {n : ℕ} {y : E}, iteratedFDerivWithin 𝕜 n f (insert x s) y = iteratedFDerivWithin 𝕜 n f s y |
AddActionHom.inverse._proof_2 | Mathlib.GroupTheory.GroupAction.Hom | ∀ {M : Type u_2} {X : Type u_1} [inst : VAdd M X] {Y₁ : Type u_3} [inst_1 : VAdd M Y₁] (f : X →ₑ[id] Y₁) (g : Y₁ → X)
(m : M) (x : Y₁), g (m +ᵥ f (g x)) = g (f (m +ᵥ g x)) |
Lean.Meta.Simp.debug.simp.check.have | Lean.Meta.Tactic.Simp.Main | Lean.Option Bool |
_private.Aesop.Script.Step.0.Aesop.Script.LazyStep.runFirstSuccessfulTacticBuilder.tryTacticBuilder.match_4 | Aesop.Script.Step | (motive : Option (Lean.Meta.SavedState × List Lean.MVarId) → Sort u_1) →
(tacticResult : Option (Lean.Meta.SavedState × List Lean.MVarId)) →
((actualPostState : Lean.Meta.SavedState) →
(actualPostGoals : List Lean.MVarId) → motive (some (actualPostState, actualPostGoals))) →
((x : Option (Lean.Meta.SavedState × List Lean.MVarId)) → motive x) → motive tacticResult |
_private.Mathlib.Order.Cover.0.LT.lt.exists_disjoint_Iio_Ioi._proof_1_1 | Mathlib.Order.Cover | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, a < b → ∃ a', a < a' ∧ ∃ b' < b, ∀ x < a', ∀ (y : α), b' < y → x < y |
String.Slice.getUTF8Byte.eq_1 | Init.Data.String.Basic | ∀ (s : String.Slice) (p : String.Pos.Raw) (h : p < s.rawEndPos),
s.getUTF8Byte p h = s.str.getUTF8Byte (p.offsetBy s.startInclusive.offset) ⋯ |
Mathlib.Tactic.UnfoldBoundary.UnfoldBoundaries.recOn | Mathlib.Tactic.Translate.UnfoldBoundary | {motive : Mathlib.Tactic.UnfoldBoundary.UnfoldBoundaries → Sort u} →
(t : Mathlib.Tactic.UnfoldBoundary.UnfoldBoundaries) →
((unfolds : Lean.NameMap Lean.Meta.SimpTheorem) →
(casts : Lean.NameMap (Lean.Name × Lean.Name)) →
(insertionFuns : Lean.NameSet) →
motive { unfolds := unfolds, casts := casts, insertionFuns := insertionFuns }) →
motive t |
ContinuousMap.HomotopicRel.symm | Mathlib.Topology.Homotopy.Basic | ∀ {X : Type u} {Y : Type v} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {S : Set X} ⦃f g : C(X, Y)⦄,
f.HomotopicRel g S → g.HomotopicRel f S |
Polynomial.roots_map_of_injective_of_card_eq_natDegree | Mathlib.Algebra.Polynomial.Roots | ∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : IsDomain A] [inst_3 : IsDomain B]
{p : Polynomial A} {f : A →+* B},
Function.Injective ⇑f → p.roots.card = p.natDegree → Multiset.map (⇑f) p.roots = (Polynomial.map f p).roots |
Int8.ofUInt8.sizeOf_spec | Init.Data.SInt.Basic | ∀ (toUInt8 : UInt8), sizeOf { toUInt8 := toUInt8 } = 1 + sizeOf toUInt8 |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.String.0._regBuiltin.String.reducePush.declare_28._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.String.1574800046._hygCtx._hyg.14 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.String | IO Unit |
_private.Lean.Meta.Tactic.Grind.Arith.Cutsat.ReorderVars.0.Lean.Meta.Grind.Arith.Cutsat.updateVarCoeff | Lean.Meta.Tactic.Grind.Arith.Cutsat.ReorderVars | ℤ → Int.Linear.Var → Lean.Meta.Grind.Arith.Cutsat.CollectM✝ Unit |
Matrix.IsAdjMatrix.apply_diag | Mathlib.Combinatorics.SimpleGraph.AdjMatrix | ∀ {V : Type u_1} {α : Type u_2} [inst : Zero α] [inst_1 : One α] {A : Matrix V V α},
A.IsAdjMatrix → ∀ (i : V), A i i = 0 |
Real.sin_arctan_lt_zero | Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | ∀ {x : ℝ}, Real.sin (Real.arctan x) < 0 ↔ x < 0 |
SSet.N.eq_iff | Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | ∀ {X : SSet} {x y : X.N}, x = y ↔ x.subcomplex = y.subcomplex |
Aesop.NormalizationState.isNormal | Aesop.Tree.Data | Aesop.NormalizationState → Bool |
String.Slice.posGT_le_iff | Init.Data.String.Lemmas.FindPos | ∀ {s : String.Slice} {p : String.Pos.Raw} {h : p < s.rawEndPos} {q : s.Pos}, s.posGT p h ≤ q ↔ p < q.offset |
Zero.ctorIdx | Init.Prelude | {α : Type u} → Zero α → ℕ |
Matrix.toLpLinAlgEquiv_symm_apply | Mathlib.Analysis.Normed.Lp.Matrix | ∀ {n : Type u_2} {R : Type u_4} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing R] (p : ENNReal)
(a : Module.End R (WithLp p (n → R))), (Matrix.toLpLinAlgEquiv p).symm a = (Matrix.toLpLin p p).symm a |
Lean.Elab.TerminationBy.synthetic | Lean.Elab.PreDefinition.TerminationHint | Lean.Elab.TerminationBy → Bool |
Mathlib.Tactic.BicategoryLike.Mor₂.comp.sizeOf_spec | Mathlib.Tactic.CategoryTheory.Coherence.Datatypes | ∀ (e : Lean.Expr) (isoLift? : Option Mathlib.Tactic.BicategoryLike.IsoLift) (f g h : Mathlib.Tactic.BicategoryLike.Mor₁)
(η θ : Mathlib.Tactic.BicategoryLike.Mor₂),
sizeOf (Mathlib.Tactic.BicategoryLike.Mor₂.comp e isoLift? f g h η θ) =
1 + sizeOf e + sizeOf isoLift? + sizeOf f + sizeOf g + sizeOf h + sizeOf η + sizeOf θ |
_private.Lean.Meta.Sym.InstantiateS.0.Lean.Meta.Sym.instantiateRevRangeS._proof_1 | Lean.Meta.Sym.InstantiateS | ∀ (beginIdx endIdx : ℕ) (subst : Array Lean.Expr),
¬beginIdx > endIdx →
¬endIdx > subst.size →
∀ (offset idx : ℕ),
idx ≥ beginIdx + offset →
idx < offset + (endIdx - beginIdx) → endIdx - beginIdx - (idx - offset) - 1 < subst.size |
Manifold.«_aux_Mathlib_Geometry_Manifold_ContMDiffMap___macroRules_Manifold_termC^_⟮_,_;_⟯_1» | Mathlib.Geometry.Manifold.ContMDiffMap | Lean.Macro |
GradedMonoid.GSMul.rec | Mathlib.Algebra.GradedMulAction | {ιA : Type u_1} →
{ιM : Type u_3} →
{A : ιA → Type u_4} →
{M : ιM → Type u_5} →
[inst : VAdd ιA ιM] →
{motive : GradedMonoid.GSMul A M → Sort u} →
((smul : {i : ιA} → {j : ιM} → A i → M j → M (i +ᵥ j)) → motive { smul := smul }) →
(t : GradedMonoid.GSMul A M) → motive t |
Std.Time.TimeZone.TZif.TZifV2._sizeOf_inst | Std.Time.Zoned.Database.TzIf | SizeOf Std.Time.TimeZone.TZif.TZifV2 |
PresheafOfModules.evaluationJointlyReflectsLimits | Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{R : CategoryTheory.Functor Cᵒᵖ RingCat} →
{J : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} J] →
(F : CategoryTheory.Functor J (PresheafOfModules R)) →
[∀ (X : Cᵒᵖ),
Small.{v, max u₂ v}
↑((F.comp (PresheafOfModules.evaluation R X)).comp
(CategoryTheory.forget (ModuleCat ↑(R.obj X)))).sections] →
(c : CategoryTheory.Limits.Cone F) →
((X : Cᵒᵖ) → CategoryTheory.Limits.IsLimit ((PresheafOfModules.evaluation R X).mapCone c)) →
CategoryTheory.Limits.IsLimit c |
EstimatorData.ctorIdx | Mathlib.Deprecated.Estimator | {α : Type u_1} → {a : Thunk α} → {ε : Type u_3} → EstimatorData a ε → ℕ |
_private.Mathlib.RingTheory.Spectrum.Prime.Topology.0.PrimeSpectrum.localization_away_comap_range._simp_1_3 | Mathlib.RingTheory.Spectrum.Prime.Topology | ∀ {α : Type u} {a : α} {s : Set α}, ({a} ⊆ s) = (a ∈ s) |
Submodule.equivMapOfInjective._proof_4 | Mathlib.Algebra.Module.Submodule.Map | ∀ {R : Type u_3} {R₂ : Type u_4} {M : Type u_1} {M₂ : Type u_2} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂}
(f : M →ₛₗ[σ₁₂] M₂) (i : Function.Injective ⇑f) (p : Submodule R M),
Function.RightInverse (Equiv.Set.image (⇑f) (↑p) i).invFun (Equiv.Set.image (⇑f) (↑p) i).toFun |
Mathlib.PrintSorries.State.mk.inj | Mathlib.Util.PrintSorries | ∀ {visited : Lean.NameSet} {sorries : Std.HashSet Lean.Expr} {sorryMsgs : Array Lean.MessageData}
{visited_1 : Lean.NameSet} {sorries_1 : Std.HashSet Lean.Expr} {sorryMsgs_1 : Array Lean.MessageData},
{ visited := visited, sorries := sorries, sorryMsgs := sorryMsgs } =
{ visited := visited_1, sorries := sorries_1, sorryMsgs := sorryMsgs_1 } →
visited = visited_1 ∧ sorries = sorries_1 ∧ sorryMsgs = sorryMsgs_1 |
NonUnitalStarSubalgebra.toNonUnitalSubring_injective | Mathlib.Algebra.Star.NonUnitalSubalgebra | ∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : NonUnitalRing A] [inst_2 : Module R A] [inst_3 : Star A],
Function.Injective NonUnitalStarSubalgebra.toNonUnitalSubring |
Lean.ErrorExplanation.Metadata.removedVersion? | Lean.ErrorExplanation | Lean.ErrorExplanation.Metadata → Option String |
CategoryTheory.sheafificationNatIso | Mathlib.CategoryTheory.Sites.Sheafification | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
(J : CategoryTheory.GrothendieckTopology C) →
(D : Type u_1) →
[inst_1 : CategoryTheory.Category.{v_1, u_1} D] →
[inst_2 : CategoryTheory.HasWeakSheafify J D] →
CategoryTheory.Functor.id (CategoryTheory.Sheaf J D) ≅
(CategoryTheory.sheafToPresheaf J D).comp (CategoryTheory.presheafToSheaf J D) |
RelIso.sumLexComplRight_symm_apply | Mathlib.Order.Hom.Lex | ∀ {α : Type u_1} {r : α → α → Prop} {x : α} [inst : IsTrans α r] [inst_1 : Std.Trichotomous r] [inst_2 : DecidableRel r]
(a : { x_1 // ¬r x x_1 } ⊕ Subtype (r x)), (RelIso.sumLexComplRight r x) a = (Equiv.sumCompl (r x)) a.swap |
CategoryTheory.ShortComplex.RightHomologyMapData.comp_φQ | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ S₃ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {φ' : S₂ ⟶ S₃} {h₁ : S₁.RightHomologyData}
{h₂ : S₂.RightHomologyData} {h₃ : S₃.RightHomologyData} (ψ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂)
(ψ' : CategoryTheory.ShortComplex.RightHomologyMapData φ' h₂ h₃),
(ψ.comp ψ').φQ = CategoryTheory.CategoryStruct.comp ψ.φQ ψ'.φQ |
WithSeminorms.hasBasis_ball | Mathlib.Analysis.LocallyConvex.WithSeminorms | ∀ {𝕜 : Type u_2} {E : Type u_6} {ι : Type u_9} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] {p : SeminormFamily 𝕜 E ι},
WithSeminorms p → ∀ {x : E}, (nhds x).HasBasis (fun sr => 0 < sr.2) fun sr => (sr.1.sup p).ball x sr.2 |
List.sortedLE_ofFn_iff | Mathlib.Data.List.Sort | ∀ {α : Type u_1} [inst : Preorder α] {n : ℕ} {f : Fin n → α}, (List.ofFn f).SortedLE ↔ Monotone f |
Rel.edgeDensity.congr_simp | Mathlib.Combinatorics.SimpleGraph.Density | ∀ {α : Type u_4} {β : Type u_5} (r r_1 : α → β → Prop),
r = r_1 →
∀ {inst : (a : α) → DecidablePred (r a)} [inst_1 : (a : α) → DecidablePred (r_1 a)] (s s_1 : Finset α),
s = s_1 → ∀ (t t_1 : Finset β), t = t_1 → Rel.edgeDensity r s t = Rel.edgeDensity r_1 s_1 t_1 |
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