name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ArchimedeanClass.liftOrderHom_mk | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {α : Type u_2}
[inst_3 : PartialOrder α] (f : M → α) (h : ∀ (a b : M), ArchimedeanClass.mk a ≤ ArchimedeanClass.mk b → f a ≤ f b)
(a : M), (ArchimedeanClass.liftOrderHom f h) (ArchimedeanClass.mk a) = f a | null | true |
Mathlib.Tactic.Monoidal.evalWhiskerLeft_comp | Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.MonoidalCategory C] {f g h i : C}
{η : h ⟶ i}
{η₁ : CategoryTheory.MonoidalCategoryStruct.tensorObj g h ⟶ CategoryTheory.MonoidalCategoryStruct.tensorObj g i}
{η₂ :
CategoryTheory.MonoidalCategoryStruct.tensorObj f (CategoryTheo... | null | true |
CategoryTheory.GrothendieckTopology.diagramPullback._proof_1 | Mathlib.CategoryTheory.Sites.Plus | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_4, u_3} C] (J : CategoryTheory.GrothendieckTopology C)
{D : Type u_2} [inst_1 : CategoryTheory.Category.{u_1, u_2} D]
[inst_2 :
∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X), CategoryTheory.Limits.HasMultiequalizer (S.index P)]
(P : CategoryTh... | null | false |
Subsemiring.closure_singleton_one | Mathlib.Algebra.Ring.Subsemiring.Basic | ∀ {R : Type u} [inst : NonAssocSemiring R], Subsemiring.closure {1} = ⊥ | null | true |
Std.ExtHashMap.getD_inter | Std.Data.ExtHashMap.Lemmas | ∀ {α : Type u} {β : Type v} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.ExtHashMap α β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] {k : α} {fallback : β},
(m₁ ∩ m₂).getD k fallback = if k ∈ m₂ then m₁.getD k fallback else fallback | null | true |
normalizationMonoidOfMonoidHomRightInverse._proof_5 | Mathlib.Algebra.GCDMonoid.Basic | ∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : DecidableEq α] (f : Associates α →* α)
(hinv : Function.RightInverse (⇑f) Associates.mk), (if 0 = 0 then 1 else Classical.choose ⋯) = 1 | null | false |
Submonoid.LocalizationMap.map_surjective_of_surjOn | Mathlib.GroupTheory.MonoidLocalization.Maps | ∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {P : Type u_3}
[inst_2 : CommMonoid P] (f : S.LocalizationMap N) {g : M →* P} {T : Submonoid P} (hy : ∀ (y : ↥S), g ↑y ∈ T)
{Q : Type u_4} [inst_3 : CommMonoid Q] {k : T.LocalizationMap Q},
Set.SurjOn ⇑g ↑S ↑T → Functi... | null | true |
Lean.Elab.Term.tryPostponeIfHasMVars | Lean.Elab.Term.TermElabM | Option Lean.Expr → String → Lean.Elab.TermElabM Lean.Expr | Throws `Exception.postpone`, if `expectedType?` contains unassigned metavariables.
If `mayPostpone == false`, it throws error `msg`.
| true |
Lean.Meta.Grind.AC.EqCnstrProof.ctorIdx | Lean.Meta.Tactic.Grind.AC.Types | Lean.Meta.Grind.AC.EqCnstrProof → ℕ | null | false |
_private.Init.Data.String.Pattern.String.0.String.Slice.Pattern.ForwardSliceSearcher.buildTable._proof_11 | Init.Data.String.Pattern.String | ∀ (pat : String.Slice) (table : Array ℕ) (guess : ℕ) (hg : guess < table.size) (this : table[guess - 1] < guess),
¬table[guess - 1] < table.size → False | null | false |
Primrec.list_flatten | Mathlib.Computability.Primrec.List | ∀ {α : Type u_1} [inst : Primcodable α], Primrec List.flatten | null | true |
Turing.ToPartrec.Code.succ | Mathlib.Computability.TuringMachine.Config | Turing.ToPartrec.Code | null | true |
CategoryTheory.IsHomLift.eq_of_isHomLift | Mathlib.CategoryTheory.FiberedCategory.HomLift | ∀ {𝒮 : Type u₁} {𝒳 : Type u₂} [inst : CategoryTheory.Category.{v₁, u₂} 𝒳] [inst_1 : CategoryTheory.Category.{v₂, u₁} 𝒮]
(p : CategoryTheory.Functor 𝒳 𝒮) {a b : 𝒳} (f : p.obj a ⟶ p.obj b) (φ : a ⟶ b) [p.IsHomLift f φ], f = p.map φ | null | true |
RootPairing.InvariantForm.isOrthogonal_reflection | Mathlib.LinearAlgebra.RootSystem.RootPositive | ∀ {ι : Type u_1} {R : Type u_2} {M : Type u_4} {N : Type u_5} [inst : CommRing R] [inst_1 : AddCommGroup M]
[inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N}
(self : P.InvariantForm) (i : ι), LinearMap.IsOrthogonal self.form ⇑(P.reflection i) | null | true |
PresheafOfModules.presheaf.eq_1 | Mathlib.Algebra.Category.ModuleCat.Presheaf.Free | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat}
(M : PresheafOfModules R),
M.presheaf =
{ obj := fun X => (CategoryTheory.forget₂ (ModuleCat ↑(R.obj X)) Ab).obj (M.obj X),
map := fun {X Y} f => AddCommGrpCat.ofHom (AddMonoidHom.mk' ⇑(CategoryTheory.Conc... | null | true |
Subfield.rank_comap | Mathlib.FieldTheory.Relrank | ∀ {E : Type v} [inst : Field E] (A : Subfield E) {L : Type v} [inst_1 : Field L] (f : L →+* E),
Module.rank (↥(Subfield.comap f A)) L = A.relrank f.fieldRange | null | true |
ContinuousLinearMap.ebound | Mathlib.Analysis.Normed.Operator.Basic | ∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : SeminormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : ... | null | true |
List.rotate_eq_nil_iff | Mathlib.Data.List.Rotate | ∀ {α : Type u} {l : List α} {n : ℕ}, l.rotate n = [] ↔ l = [] | null | true |
SubMulAction.ofFixingSubgroup_equivariantMap.eq_1 | Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | ∀ (M : Type u_1) {α : Type u_2} [inst : Group M] [inst_1 : MulAction M α] (s : Set α),
SubMulAction.ofFixingSubgroup_equivariantMap M s = { toFun := fun x => ↑x, map_smul' := ⋯ } | null | true |
Ideal.comap._proof_6 | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u_1} {S : Type u_2} {F : Type u_3} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S]
(f : F) [RingHomClass F R S] (I : Ideal S) (c x : R), x ∈ ⇑f ⁻¹' ↑I → c • x ∈ ⇑f ⁻¹' ↑I | null | false |
Int64.toInt_maxValue | Init.Data.SInt.Lemmas | Int64.maxValue.toInt = 2 ^ 63 - 1 | null | true |
GenContFract.of_s_head | Mathlib.Algebra.ContinuedFractions.Computation.Translations | ∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] {v : K},
Int.fract v ≠ 0 → (GenContFract.of v).s.head = some { a := 1, b := ↑⌊(Int.fract v)⁻¹⌋ } | This gives the first pair of coefficients of the continued fraction of a non-integer `v`.
| true |
Associates.is_pow_of_dvd_count | Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | ∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : UniqueFactorizationMonoid α]
[inst_2 : DecidableEq (Associates α)] [inst_3 : (p : Associates α) → Decidable (Irreducible p)] {a : Associates α},
a ≠ 0 → ∀ {k : ℕ}, (∀ (p : Associates α), Irreducible p → k ∣ p.count a.factors) → ∃ b, a = b ^ k | null | true |
Quaternion.dualNumberEquiv._proof_2 | Mathlib.Algebra.DualQuaternion | ∀ {R : Type u_1} [inst : CommRing R],
Function.RightInverse
(fun d =>
{ re := ((TrivSqZeroExt.fst d).re, (TrivSqZeroExt.snd d).re),
imI := ((TrivSqZeroExt.fst d).imI, (TrivSqZeroExt.snd d).imI),
imJ := ((TrivSqZeroExt.fst d).imJ, (TrivSqZeroExt.snd d).imJ),
imK := ((TrivSqZeroExt.fst... | null | false |
CategoryTheory.IsFilteredOrEmpty | Mathlib.CategoryTheory.Filtered.Basic | (C : Type u) → [CategoryTheory.Category.{v, u} C] → Prop | A category `IsFilteredOrEmpty` if
1. for every pair of objects there exists another object "to the right", and
2. for every pair of parallel morphisms there exists a morphism to the right so the compositions
are equal.
| true |
Std.TreeMap.isEmpty_inter_iff | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap α β cmp} [Std.TransCmp cmp],
(t₁ ∩ t₂).isEmpty = true ↔ ∀ k ∈ t₁, k ∉ t₂ | null | true |
_private.Mathlib.Util.CompileInductive.0.Mathlib.Util.replaceConst.match_1 | Mathlib.Util.CompileInductive | (motive : Lean.Expr → Sort u_1) →
(x : Lean.Expr) →
((n : Lean.Name) → (us : List Lean.Level) → motive (Lean.Expr.const n us)) → ((x : Lean.Expr) → motive x) → motive x | null | false |
_private.Init.Meta.Defs.0.Lean.Name.hasNum._sunfold | Init.Meta.Defs | Lean.Name → Bool | null | false |
CategoryTheory.Bicategory.associator_naturality_middle_assoc | Mathlib.CategoryTheory.Bicategory.Basic | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d)
{Z : a ⟶ d} (h_1 : CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g' h) ⟶ Z),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Bicategory.whiskerRight (CategoryTheo... | null | true |
_private.Lean.Meta.Tactic.Grind.Arith.Cutsat.Proof.0.Lean.Meta.Grind.Arith.Cutsat.mkMulEqProof.go | Lean.Meta.Tactic.Grind.Arith.Cutsat.Proof | Option Lean.Expr →
Array (Lean.Expr × ℤ × Lean.Meta.Grind.Arith.Cutsat.EqCnstr) →
Lean.Expr → Lean.Meta.Grind.Arith.Cutsat.ProofM✝ Lean.Meta.Grind.Arith.Cutsat.MulEqProof✝ | null | true |
ZeroHom.zero_comp | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} {P : Type u_6} [inst : Zero M] [inst_1 : Zero N] [inst_2 : Zero P] (f : ZeroHom M N),
ZeroHom.comp 0 f = 0 | null | true |
differentiableOn_pi'' | Mathlib.Analysis.Calculus.FDeriv.Prod | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {F' : ι → Type u_7}
[inst_3 : (i : ι) → NormedAddCommGroup (F' i)] [inst_4 : (i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i},
(∀ (i : ι), Differenti... | null | true |
_private.Mathlib.CategoryTheory.Functor.TypeValuedFlat.0.CategoryTheory.FunctorToTypes.fromOverFunctorElementsEquivalence._proof_5 | Mathlib.CategoryTheory.Functor.TypeValuedFlat | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (F : CategoryTheory.Functor C (Type u_3)) {X : C}
(x : F.obj X) (X_1 : (CategoryTheory.FunctorToTypes.fromOverFunctor F x).Elements),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.CategoryOfElements.homMk
(CategoryTheory.Over.mk
... | null | false |
_private.Mathlib.SetTheory.ZFC.Rank.0.PSet.rank_powerset._simp_1_2 | Mathlib.SetTheory.ZFC.Rank | ∀ {x y : PSet.{u_1}}, (y ∈ x.powerset) = (y ⊆ x) | null | false |
Monoid.End.instInhabited | Mathlib.Algebra.Group.Hom.Defs | (M : Type u_4) → [inst : MulOne M] → Inhabited (Monoid.End M) | null | true |
DFinsupp.subtypeSupportEqEquiv._proof_5 | Mathlib.Data.DFinsupp.Defs | ∀ {ι : Type u_1} {β : ι → Type u_2} [inst : DecidableEq ι] [inst_1 : (i : ι) → Zero (β i)]
[inst_2 : (i : ι) → (x : β i) → Decidable (x ≠ 0)] (s : Finset ι) (f : (i : ↥s) → { x // x ≠ 0 }) (i : ι),
i ∈ (DFinsupp.mk s fun i => ↑(f i)).support ↔ i ∈ s | null | false |
CategoryTheory.Grp.instMonoidalMonForget₂Mon._proof_6 | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] (X Y Z : CategoryTheory.Grp C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight
(CategoryTheory.Catego... | null | false |
Lean.ProjectionFunctionInfo.mk | Lean.ProjFns | Lean.Name → ℕ → ℕ → Bool → Lean.ProjectionFunctionInfo | null | true |
CFC.rpow_sqrt_nnreal | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | ∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A]
[inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [inst_6 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
[inst_7 : NonnegSpectrumClass ℝ A] [IsSemitopologicalRing A] [T2Space A] {a : A} {x : NNReal},
... | null | true |
ne_zero_and_ne_zero_of_mul | Mathlib.Algebra.GroupWithZero.Basic | ∀ {M₀ : Type u_1} [inst : MulZeroClass M₀] {a b : M₀}, a * b ≠ 0 → a ≠ 0 ∧ b ≠ 0 | null | true |
_private.Mathlib.Analysis.MellinInversion.0.mellin_eq_fourier._simp_1_6 | Mathlib.Analysis.MellinInversion | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
Real.mk_add | Mathlib.Data.Real.Basic | ∀ {f g : CauSeq ℚ abs}, Real.mk (f + g) = Real.mk f + Real.mk g | null | true |
DivisionSemiring.mk.noConfusion | Mathlib.Algebra.Field.Defs | {K : Type u_2} →
{P : Sort u} →
{toSemiring : Semiring K} →
{toInv : Inv K} →
{toDiv : Div K} →
{div_eq_mul_inv : autoParam (∀ (a b : K), a / b = a * b⁻¹) DivInvMonoid.div_eq_mul_inv._autoParam} →
{zpow : ℤ → K → K} →
{zpow_zero' : autoParam (∀ (a : K), zpow 0 a =... | null | false |
LieModule.isLieAbelian_of_ker_traceForm_eq_bot | Mathlib.Algebra.Lie.TraceForm | ∀ (R : Type u_1) (L : Type u_3) (M : Type u_4) [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] [inst_6 : LieModule R L M]
[LieRing.IsNilpotent L] [IsDomain R] [Module.Free R M] [Module.Finite R M],
LinearMap.ker (LieMo... | A nilpotent Lie algebra with a representation whose trace form is non-singular is Abelian. | true |
_private.Mathlib.Probability.Distributions.Gaussian.Real.0.ProbabilityTheory.gaussianPDFReal_inv_mul._simp_1_4 | Mathlib.Probability.Distributions.Gaussian.Real | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False | null | false |
SubMulAction.instOne | Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise | {R : Type u_1} → {M : Type u_2} → [inst : Monoid R] → [inst_1 : MulAction R M] → [One M] → One (SubMulAction R M) | null | true |
Function.const_inv | Mathlib.Algebra.Notation.Pi.Defs | ∀ {ι : Type u_1} {G : Type u_7} [inst : Inv G] (a : G), (Function.const ι a)⁻¹ = Function.const ι a⁻¹ | null | true |
instOrderBotSubtypeIsIdempotentElem._proof_1 | Mathlib.Algebra.Order.Ring.Idempotent | ∀ {M₀ : Type u_1} [inst : CommMonoidWithZero M₀], IsIdempotentElem 0 | null | false |
getElem?_eq_none_iff._simp_1 | Init.GetElem | ∀ {cont : Type u_1} {idx : Type u_2} {elem : Type u_3} {dom : cont → idx → Prop} [inst : GetElem? cont idx elem dom]
[LawfulGetElem cont idx elem dom] (c : cont) (i : idx) [Decidable (dom c i)], (c[i]? = none) = ¬dom c i | null | false |
_private.Std.Sat.AIG.RefVecOperator.Map.0.Std.Sat.AIG.RefVec.denote_map._proof_1_1 | Std.Sat.AIG.RefVecOperator.Map | ∀ {len : ℕ} (idx : ℕ), ¬0 ≤ idx → False | null | false |
lowerCentralSeries_pi_of_finite | Mathlib.GroupTheory.Nilpotent | ∀ {η : Type u_2} {Gs : η → Type u_3} [inst : (i : η) → Group (Gs i)] [Finite η] (Ss : (i : η) → Subgroup (Gs i))
(n : ℕ), (Subgroup.pi Set.univ Ss).lowerCentralSeries n = Subgroup.pi Set.univ fun i => (Ss i).lowerCentralSeries n | **Alias** of `Subgroup.lowerCentralSeries_pi_of_finite`. | true |
OpenSubgroup.instOrderTop._proof_1 | Mathlib.Topology.Algebra.OpenSubgroup | ∀ {G : Type u_1} [inst : Group G] [inst_1 : TopologicalSpace G] (x : OpenSubgroup G), ↑x ⊆ Set.univ | null | false |
_private.Init.Data.String.Decode.0.Char.utf8Size_eq_four_iff._proof_1_5 | Init.Data.String.Decode | ∀ {c : Char}, 127 < c.toNat → c.toNat ≤ 2047 → ¬c.toNat ≤ 65535 → False | null | false |
Std.PRange.UpwardEnumerable.succMany?_succ?_eq_succ?_bind_succMany? | Init.Data.Range.Polymorphic.UpwardEnumerable | ∀ {α : Type u_1} [inst : Std.PRange.UpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerable α] (n : ℕ) (a : α),
Std.PRange.succMany? (n + 1) a = (Std.PRange.succ? a).bind fun x => Std.PRange.succMany? n x | null | true |
Nat.add_le_add_iff_left._simp_1 | Init.Data.Nat.Basic | ∀ {m k n : ℕ}, (n + m ≤ n + k) = (m ≤ k) | null | false |
Subfield.mk.congr_simp | Mathlib.Algebra.Field.Subfield.Basic | ∀ {K : Type u} [inst : DivisionRing K] (toSubring toSubring_1 : Subring K) (e_toSubring : toSubring = toSubring_1)
(inv_mem' : ∀ x ∈ toSubring.carrier, x⁻¹ ∈ toSubring.carrier),
{ toSubring := toSubring, inv_mem' := inv_mem' } = { toSubring := toSubring_1, inv_mem' := ⋯ } | null | true |
BddDistLat.Hom.Simps.hom | Mathlib.Order.Category.BddDistLat | (X Y : BddDistLat) → X.Hom Y → BoundedLatticeHom ↑X.toDistLat ↑Y.toDistLat | Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. | true |
Polynomial.smeval_assoc_X_pow | Mathlib.Algebra.Polynomial.Smeval | ∀ (R : Type u_1) [inst : Semiring R] (p : Polynomial R) {S : Type u_2} [inst_1 : NonAssocSemiring S]
[inst_2 : Module R S] [inst_3 : Pow S ℕ] (x : S) [NatPowAssoc S] [IsScalarTower R S S] (m n : ℕ),
p.smeval x * x ^ m * x ^ n = p.smeval x * (x ^ m * x ^ n) | null | true |
smul_mem_asymptoticCone_iff._simp_1 | Mathlib.Topology.Algebra.AsymptoticCone | ∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Field k] [inst_1 : LinearOrder k] [inst_2 : AddCommGroup V]
[inst_3 : Module k V] [inst_4 : AddTorsor V P] [inst_5 : TopologicalSpace V] [inst_6 : TopologicalSpace k]
[OrderTopology k] [IsStrictOrderedRing k] [IsTopologicalAddGroup V] [ContinuousSMul k V] {s : ... | null | false |
CategoryTheory.Limits.CokernelCofork.isColimitMapBifunctor.exists_desc | Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms C₁] [inst_4 : CategoryTheory.Limits.HasZeroMorphisms C₂]
[inst_5 : Categ... | null | true |
Lean.Server.RequestContext | Lean.Server.Requests | Type | null | true |
AlgebraicGeometry.Scheme.Modules.Hom | Mathlib.AlgebraicGeometry.Modules.Sheaf | {X : AlgebraicGeometry.Scheme} → X.Modules → X.Modules → Type u | Morphisms between `𝒪ₓ`-modules. Use `Hom.app` to act on sections. | true |
Quaternion.imJ_coe | Mathlib.Algebra.Quaternion | ∀ {R : Type u_3} [inst : CommRing R] (x : R), (↑x).imJ = 0 | null | true |
ENNReal.instSemilatticeSup._aux_1 | Mathlib.Data.ENNReal.Basic | ENNReal → ENNReal → ENNReal | null | false |
Polynomial.instCommRingUniversalFactorizationRing._proof_25 | Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | ∀ {R : Type u_1} [inst : CommRing R] {n : ℕ} (m k : ℕ) (hn : n = m + k) (p : Polynomial.MonicDegreeEq R n)
(a : Polynomial.UniversalFactorizationRing m k hn p), a * 0 = 0 | null | false |
CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker | Mathlib.RingTheory.Idempotents | ∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S) {I : Type u_3} [inst_2 : Fintype I],
(∀ x ∈ RingHom.ker f, IsNilpotent x) →
∀ {e : I → S},
CompleteOrthogonalIdempotents e →
(∀ (i : I), e i ∈ f.range) → ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e | A system of complete orthogonal idempotents lift along nil ideals. | true |
_private.Lean.Meta.Tactic.Grind.CasesMatch.0.Lean.Meta.Grind.casesMatch.match_6 | Lean.Meta.Tactic.Grind.CasesMatch | (motive : Lean.Expr × Array Lean.Expr → Sort u_1) →
(x : Lean.Expr × Array Lean.Expr) →
((motive_1 : Lean.Expr) → (eqRefls : Array Lean.Expr) → motive (motive_1, eqRefls)) → motive x | null | false |
AddAction.instElemOrbit_1._proof_1 | Mathlib.GroupTheory.GroupAction.Defs | ∀ {G : Type u_2} {α : Type u_1} [inst : AddGroup G] [inst_1 : AddAction G α] (x : AddAction.orbitRel.Quotient G α)
(g g' : G) (a' : ↑x.orbit), (g + g') +ᵥ a' = g +ᵥ g' +ᵥ a' | null | false |
ContractingWith.fixedPoint_unique | Mathlib.Topology.MetricSpace.Contracting | ∀ {α : Type u_1} [inst : MetricSpace α] {K : NNReal} {f : α → α} (hf : ContractingWith K f) [inst_1 : Nonempty α]
[inst_2 : CompleteSpace α] {x : α}, Function.IsFixedPt f x → x = ContractingWith.fixedPoint f hf | null | true |
_private.Lean.Parser.Extension.0.Lean.Parser.compileParserDescr.visit.match_3 | Lean.Parser.Extension | (motive : Option Lean.Parser.ParserCategory → Sort u_1) →
(x : Option Lean.Parser.ParserCategory) →
((val : Lean.Parser.ParserCategory) → motive (some val)) → (Unit → motive none) → motive x | null | false |
Lean.PrettyPrinter.Delaborator.State.mk._flat_ctor | Lean.PrettyPrinter.Delaborator.Basic | ℕ →
Lean.SubExpr.PosMap Lean.Elab.Info →
Lean.PrettyPrinter.Delaborator.SubExpr.HoleIterator → Lean.PrettyPrinter.Delaborator.State | null | false |
LinearEquiv.extendScalarsOfSurjective | Mathlib.Algebra.Algebra.Basic | {R : Type u_1} →
{S : Type u_2} →
[inst : CommSemiring R] →
[inst_1 : Semiring S] →
[inst_2 : Algebra R S] →
{M : Type u_3} →
{N : Type u_4} →
[inst_3 : AddCommMonoid M] →
[inst_4 : AddCommMonoid N] →
[inst_5 : Module R M] →
... | If `R →+* S` is surjective, then `R`-linear isomorphisms are also `S`-linear. | true |
Localization.AtPrime.mapPiEvalRingHom_comp_algebraMap | Mathlib.RingTheory.Localization.AtPrime.Basic | ∀ {ι : Type u_4} {R : ι → Type u_5} [inst : (i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i))
[inst_1 : I.IsPrime],
(Localization.AtPrime.mapPiEvalRingHom I).comp
(algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I))) =
(algebraMap (R i) (Localization.AtPrime I)).co... | null | true |
notation_class | Mathlib.Tactic.Simps.NotationClass | Lean.ParserDescr | The `@[notation_class]` attribute specifies that this is a notation class,
and this notation should be used instead of projections by `@[simps]`.
* This is only important if the projection is written differently using notation, e.g.
`+` uses `HAdd.hAdd`, not `Add.add` and `0` uses `OfNat.ofNat` not `Zero.zero`.
... | true |
FreeRing.coe_sub | Mathlib.RingTheory.FreeCommRing | ∀ {α : Type u} (x y : FreeRing α), ↑(x - y) = ↑x - ↑y | null | true |
LocalizedModule.instRing._proof_9 | Mathlib.Algebra.Module.LocalizedModule.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Submonoid R},
autoParam (∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)) AddGroupWithOne.intCast_negSucc._autoParam | null | false |
_private.Lean.Meta.ExprDefEq.0.Lean.Meta.isEtaUnassignedMVar._sparseCasesOn_1 | Lean.Meta.ExprDefEq | {α : Type u} →
{motive : Option α → Sort u_1} →
(t : Option α) → ((val : α) → motive (some val)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
CategoryTheory.ShortComplex.homologyπ | Mathlib.Algebra.Homology.ShortComplex.Homology | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
(S : CategoryTheory.ShortComplex C) → [inst_2 : S.HasHomology] → S.cycles ⟶ S.homology | The canonical morphism `S.cycles ⟶ S.homology` for a short complex `S` that has homology. | true |
Topology.IsLowerSet.closure_eq_upperClosure | Mathlib.Topology.Order.UpperLowerSetTopology | ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [Topology.IsLowerSet α] {s : Set α},
closure s = ↑(upperClosure s) | null | true |
CategoryTheory.OplaxFunctor.PseudoCore | Mathlib.CategoryTheory.Bicategory.Functor.Oplax | {B : Type u₁} →
[inst : CategoryTheory.Bicategory B] →
{C : Type u₂} → [inst_1 : CategoryTheory.Bicategory C] → CategoryTheory.OplaxFunctor B C → Type (max (max u₁ v₁) w₂) | A structure on an oplax functor that promotes an oplax functor to a pseudofunctor.
See `Pseudofunctor.mkOfOplax`. | true |
Unitary.spectrum_subset_slitPlane_iff_norm_lt_two | Mathlib.Analysis.CStarAlgebra.Unitary.Connected | ∀ {A : Type u_1} [inst : CStarAlgebra A] {u : A}, u ∈ unitary A → (spectrum ℂ u ⊆ Complex.slitPlane ↔ ‖u - 1‖ < 2) | null | true |
Std.TreeSet.Raw.le_min! | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp] [inst : Inhabited α],
t.WF → t.isEmpty = false → ∀ {k : α}, (cmp k t.min!).isLE = true ↔ ∀ k' ∈ t, (cmp k k').isLE = true | null | true |
MeasureTheory.isMulLeftInvariant_map | Mathlib.MeasureTheory.Group.Measure | ∀ {G : Type u_1} [inst : MeasurableSpace G] [inst_1 : Mul G] {μ : MeasureTheory.Measure G} [MeasurableMul G]
{H : Type u_3} [inst_3 : MeasurableSpace H] [inst_4 : Mul H] [MeasurableMul H] [μ.IsMulLeftInvariant] (f : G →ₙ* H),
Measurable ⇑f → Function.Surjective ⇑f → (MeasureTheory.Measure.map (⇑f) μ).IsMulLeftInvar... | null | true |
CategoryTheory.Limits.binaryBiconeOfIsSplitEpiOfKernel._proof_7 | Mathlib.CategoryTheory.Preadditive.Biproducts | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {X Y : C} {f : X ⟶ Y}
[inst_1 : CategoryTheory.IsSplitEpi f],
CategoryTheory.CategoryStruct.comp (CategoryTheory.section_ f) f = CategoryTheory.CategoryStruct.id Y | null | false |
MeasureTheory.Measure.rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular | Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν]
[μ.HaveLebesgueDecomposition (ν + ν')] [MeasureTheory.SigmaFinite ν],
μ.AbsolutelyContinuous ν → ν.MutuallySingular ν' → μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν | Auxiliary lemma for `rnDeriv_add_right_of_mutuallySingular`. | true |
AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen | Mathlib.AlgebraicGeometry.AffineScheme | ∀ {X : AlgebraicGeometry.Scheme} {U : X.Opens},
AlgebraicGeometry.IsAffineOpen U →
∀ (f : ↑(X.presheaf.obj (Opposite.op U))) (g : ↑(X.presheaf.obj (Opposite.op (X.basicOpen f)))),
∃ f', X.basicOpen f' = X.basicOpen g | null | true |
Multiset.foldl_zero | Mathlib.Data.Multiset.MapFold | ∀ {α : Type u_1} {β : Type v} (f : β → α → β) [inst : RightCommutative f] (b : β), Multiset.foldl f b 0 = b | null | true |
DistribLattice.ofInfSupLe | Mathlib.Order.Lattice | {α : Type u} → [inst : Lattice α] → (∀ (a b c : α), a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) → DistribLattice α | Prove distributivity of an existing lattice from the dual distributive law. | true |
Finset.bipartiteBelow.eq_1 | Mathlib.Combinatorics.Enumerative.DoubleCounting | ∀ {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (s : Finset α) (b : β) [inst : (a : α) → Decidable (r a b)],
Finset.bipartiteBelow r s b = {a ∈ s | r a b} | null | true |
Lean.Meta.Sym.DSimp.Cache | Lean.Meta.Sym.DSimp.DSimpM | Type | Cache mapping expressions (by pointer equality) to their simplified results. | true |
KaehlerDifferential.moduleBaseChange._proof_3 | Mathlib.RingTheory.Kaehler.TensorProduct | ∀ (R : Type u_3) (S : Type u_1) (A : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[inst_3 : CommRing A] [inst_4 : Algebra R A] (r s : A) (x : TensorProduct R S Ω[A⁄R]), (r + s) • x = r • x + s • x | null | false |
instContinuousAddWeakSpace | Mathlib.Topology.Algebra.Module.Spaces.WeakDual | ∀ (𝕜 : Type u_2) (E : Type u_1) [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],
ContinuousAdd (WeakSpace 𝕜 E) | null | true |
Batteries.AssocList.findEntryP?._unsafe_rec | Batteries.Data.AssocList | {α : Type u_1} → {β : Type u_2} → (α → β → Bool) → Batteries.AssocList α β → Option (α × β) | null | false |
nndist_inv_inv₀ | Mathlib.Analysis.Normed.Field.Basic | ∀ {α : Type u_2} [inst : NormedDivisionRing α] {z w : α}, z ≠ 0 → w ≠ 0 → nndist z⁻¹ w⁻¹ = nndist z w / (‖z‖₊ * ‖w‖₊) | null | true |
not_fermatLastTheoremFor_two | Mathlib.NumberTheory.FLT.Basic | ¬FermatLastTheoremFor 2 | null | true |
Sum.isLeft_iff | Init.Data.Sum.Lemmas | ∀ {α : Type u_1} {β : Type u_2} {x : α ⊕ β}, x.isLeft = true ↔ ∃ y, x = Sum.inl y | null | true |
boolAlg_dual_comp_forget_to_bddDistLat | Mathlib.Order.Category.BoolAlg | BoolAlg.dual.comp (CategoryTheory.forget₂ BoolAlg BddDistLat) =
(CategoryTheory.forget₂ BoolAlg BddDistLat).comp BddDistLat.dual | null | true |
NumberField.InfinitePlace.mkReal._proof_1 | Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | ∀ {K : Type u_1} [inst : Field K] (w : { w // w.IsReal }),
(fun φ => ⟨NumberField.InfinitePlace.mk ↑φ, ⋯⟩) ⟨(↑w).embedding, ⋯⟩ = w | null | false |
FreeLieAlgebra.liftAux_map_mul | Mathlib.Algebra.Lie.Free | ∀ (R : Type u) {X : Type v} [inst : CommRing R] {L : Type w} [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (f : X → L)
(a b : FreeNonUnitalNonAssocAlgebra R X),
(FreeLieAlgebra.liftAux R f) (a * b) = ⁅(FreeLieAlgebra.liftAux R f) a, (FreeLieAlgebra.liftAux R f) b⁆ | null | true |
TypeVec.toSubtype._f | Mathlib.Data.TypeVec | (x : ℕ) →
(x_1 : Fin2 x) →
Fin2.below (motive := fun x x_2 =>
(x_3 : TypeVec.{u} x) →
(x_4 : x_3.Arrow (TypeVec.repeat x Prop)) →
{ x_5 // TypeVec.ofRepeat (x_4 x_2 x_5) } → TypeVec.Subtype_ x_4 x_2)
x_1 →
(x_2 : TypeVec.{u} x) →
(x_3 : x_2.Arrow (TypeVec.repeat... | null | false |
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