name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Matroid.Indep.exists_insert_of_not_maximal | Mathlib.Combinatorics.Matroid.Basic | ∀ {α : Type u_1} (M : Matroid α) ⦃I B : Set α⦄,
M.Indep I → ¬Maximal M.Indep I → Maximal M.Indep B → ∃ x ∈ B \ I, M.Indep (insert x I) | This is the same as `Indep.exists_insert_of_not_isBase`, but phrased so that
it is defeq to the augmentation axiom for independent sets. | true |
and_imp_left_iff_true | Init.PropLemmas | ∀ {P Q : Prop}, P ∧ Q → P ↔ True | null | true |
LieModule.iSupIndep_genWeightSpace | Mathlib.Algebra.Lie.Weights.Basic | ∀ (R : Type u_2) (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]
[inst_7 : LieRing.IsNilpotent L] [IsDomain R] [Module.IsTorsionFree R M],
iSupIndep fun χ => Lie... | Lie module weight spaces are independent.
See also `LieModule.iSupIndep_genWeightSpace'`. | true |
Submodule.comapSubtypeEquivOfLe._proof_3 | Mathlib.Algebra.Module.Submodule.Map | ∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{p q : Submodule R M} (x : ↥(Submodule.comap q.subtype p)), ↑x ∈ Submodule.comap q.subtype p | null | false |
_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion.0.EisensteinSeries.summable_sigma_mul_cexp_pow._proof_1_1 | Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | ∀ {k : ℕ}, 1 ≤ k → ¬k - 1 + 1 = k → False | null | false |
Aesop.TreeM.Context.mk.noConfusion | Aesop.Tree.TreeM | {P : Sort u} →
{currentIteration : Aesop.Iteration} →
{ruleSet : Aesop.LocalRuleSet} →
{currentIteration' : Aesop.Iteration} →
{ruleSet' : Aesop.LocalRuleSet} →
{ currentIteration := currentIteration, ruleSet := ruleSet } =
{ currentIteration := currentIteration', ruleSet := ... | null | false |
_private.Lean.Compiler.LCNF.Simp.ConstantFold.0.Lean.Compiler.LCNF.Simp.ConstantFold.arithmeticFolders._proof_46 | Lean.Compiler.LCNF.Simp.ConstantFold | ∀ (a : UInt32), a + 0 = a | null | false |
Sym2.diagElemEquiv._proof_1 | Mathlib.Data.Sym.Sym2 | ∀ {α : Type u_1} (x : { a // a.IsDiag }), (↑x).IsDiag | null | false |
CategoryTheory.ShortComplex.LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork._proof_3 | Mathlib.Algebra.Homology.ShortComplex.LeftHomology | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) (hf : S.f = 0) (hg : S.g = 0) (c : CategoryTheory.Limits.CokernelCofork S.f)
(hc : CategoryTheory.Limits.IsColimit c),
CategoryTheory.CategoryStruct.comp (Category... | null | false |
Std.IterM.DefaultConsumers.forIn'_eq_forIn'._unary | Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop | ∀ {m : Type w → Type w'} {α β : Type w} [inst : Std.Iterator α m β] {n : Type x → Type x'} [inst_1 : Monad n]
[LawfulMonad n] {lift : (γ : Type w) → (δ : Type x) → (γ → n δ) → m γ → n δ} {γ : Type x} {P Q : β → Prop}
(Pl : β → γ → ForInStep γ → Prop) {f : (b : β) → P b → (c : γ) → n (Subtype (Pl b c))}
{g : (b : ... | null | false |
_private.Mathlib.Probability.Kernel.IonescuTulcea.Maps.0.IicProdIoc_preimage._simp_1_2 | Mathlib.Probability.Kernel.IonescuTulcea.Maps | ∀ {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {f : (i : ι) → α i},
(f ∈ s.pi t) = ∀ i ∈ s, f i ∈ t i | null | false |
FormalMultilinearSeries.applyComposition_ones | Mathlib.Analysis.Analytic.Composition | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : CommRing 𝕜] [inst_1 : AddCommGroup E] [inst_2 : AddCommGroup F]
[inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 F] [inst_5 : TopologicalSpace E] [inst_6 : TopologicalSpace F]
[inst_7 : IsTopologicalAddGroup E] [inst_8 : ContinuousConstSMul 𝕜 E] [inst_9 : IsTopolo... | null | true |
CategoryTheory.Profunctor.whiskerRight._proof_1 | Mathlib.CategoryTheory.Profunctor.Comp | ∀ {C D E : Type u_2} [inst : CategoryTheory.Category.{u_4, u_2} C] [inst_1 : CategoryTheory.Category.{u_1, u_2} D]
[inst_2 : CategoryTheory.Category.{u_5, u_2} E]
{P Q : CategoryTheory.Profunctor.{max u_2 u_3, u_4, u_1, u_2, u_2} C D}
(R : CategoryTheory.Profunctor.{max u_2 u_3, u_1, u_5, u_2, u_2} D E) (f : P ⟶ ... | null | false |
_private.Mathlib.Analysis.Convex.Integral.0.Convex.integral_mem._simp_1_3 | Mathlib.Analysis.Convex.Integral | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] {f : MeasureTheory.SimpleFunc α β} {b : β},
(b ∈ f.range) = (b ∈ Set.range ⇑f) | null | false |
CategoryTheory.Bicategory.LeftLift.whiskering | Mathlib.CategoryTheory.Bicategory.Extension | {B : Type u} →
[inst : CategoryTheory.Bicategory B] →
{a b c : B} →
{f : b ⟶ a} →
{g : c ⟶ a} →
{x : B} →
(h : x ⟶ c) →
CategoryTheory.Functor (CategoryTheory.Bicategory.LeftLift f g)
(CategoryTheory.Bicategory.LeftLift f (CategoryTheory.CategorySt... | Whiskering a 1-morphism is a functor. | true |
_private.Mathlib.RingTheory.Unramified.Dedekind.0.IsDedekindDomain.of_formallyUnramified._proof_1_1 | Mathlib.RingTheory.Unramified.Dedekind | 5 ≠ 0 | null | false |
Lean.Elab.Tactic.liftMetaTacticAux | Lean.Elab.Tactic.Basic | {α : Type} → (Lean.MVarId → Lean.MetaM (α × List Lean.MVarId)) → Lean.Elab.Tactic.TacticM α | null | true |
Bool.recOn._@.Mathlib.Util.CompileInductive.3618634379._hygCtx._hyg.5 | Mathlib.Util.CompileInductive | {motive : Bool → Sort u} → (t : Bool) → motive false → motive true → motive t | null | false |
Topology.IsInducing.isOpenMap | Mathlib.Topology.Maps.Basic | ∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
Topology.IsInducing f → IsOpen (Set.range f) → IsOpenMap f | An inducing map with an open range is an open map. | true |
Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_acChange!_1 | Mathlib.Tactic.Convert | Lean.Macro | null | false |
CategoryTheory.CartesianMonoidalCategory.whiskerRight_toUnit_comp_leftUnitor_hom_assoc | Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C] (X Y : C)
{Z : C} (h : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.SemiCartesianMonoidalCategory.toUnit X) Y)
(CategoryTheory.... | null | true |
CategoryTheory.Limits.ker._proof_5 | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] {X Y Z : CategoryTheory.Arrow C} (f : X ⟶ Y) (g : Y ⟶ Z),
CategoryTheory.Limits.kernel.lift Z.hom
(CategoryTheory.CategoryStruct.comp (CategoryTheory... | null | false |
MeasureTheory.Measure.ext_of_generateFrom_of_iUnion | Mathlib.MeasureTheory.Measure.Restrict | ∀ {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (C : Set (Set α)) (B : ℕ → Set α),
m0 = MeasurableSpace.generateFrom C →
IsPiSystem C → ⋃ i, B i = Set.univ → (∀ (i : ℕ), B i ∈ C) → (∀ (i : ℕ), μ (B i) ≠ ⊤) → (∀ s ∈ C, μ s = ν s) → μ = ν | Two measures are equal if they are equal on the π-system generating the σ-algebra,
and they are both finite on an increasing spanning sequence of sets in the π-system.
This lemma is formulated using `iUnion`.
`FiniteSpanningSetsIn.ext` is a reformulation of this lemma. | true |
CategoryTheory.MorphismProperty.FunctorialFactorizationData.fac_app | Mathlib.CategoryTheory.MorphismProperty.Factorization | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {W₁ W₂ : CategoryTheory.MorphismProperty C}
(data : W₁.FunctorialFactorizationData W₂) {f : CategoryTheory.Arrow C},
CategoryTheory.CategoryStruct.comp (data.i.app f) (data.p.app f) = f.hom | null | true |
Submonoid.unitsTypeEquivIsUnitSubmonoid._proof_1 | Mathlib.Algebra.Group.Submonoid.Operations | ∀ {M : Type u_1} [inst : Monoid M] (x : ↥(IsUnit.submonoid M)), (fun x => ⟨↑x, ⋯⟩) ((fun x => IsUnit.unit ⋯) x) = x | null | false |
TrivSqZeroExt.kerIdeal._proof_2 | Mathlib.Algebra.TrivSqZeroExt.Ideal | ∀ (R : Type u_1) (M : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
IsScalarTower R R M | null | false |
Std.HashSet.erase_emptyWithCapacity | Std.Data.HashSet.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {a : α} {c : ℕ},
(Std.HashSet.emptyWithCapacity c).erase a = Std.HashSet.emptyWithCapacity c | null | true |
Set.vadd_set_univ_pi | Mathlib.Algebra.Group.Pointwise.Set.Scalar | ∀ {M : Type u_4} {ι : Type u_5} {π : ι → Type u_6} [inst : (i : ι) → VAdd M (π i)] (c : M) (s : (i : ι) → Set (π i)),
c +ᵥ Set.univ.pi s = Set.univ.pi (c +ᵥ s) | null | true |
HomologicalComplex.cyclesFunctor_map | Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{ι : Type u_2} (c : ComplexShape ι) (i : ι) [inst_2 : CategoryTheory.CategoryWithHomology C]
{X Y : HomologicalComplex C c} (f : X ⟶ Y),
(HomologicalComplex.cyclesFunctor C c i).map f = HomologicalCo... | null | true |
Nat.Partrec.Code.zero.elim | Mathlib.Computability.PartrecCode | {motive : Nat.Partrec.Code → Sort u} → (t : Nat.Partrec.Code) → t.ctorIdx = 0 → motive Nat.Partrec.Code.zero → motive t | null | false |
AlgebraicTopology.NormalizedMooreComplex.objD._proof_1 | Mathlib.AlgebraicTopology.MooreComplex | ∀ (n : ℕ) (i : Fin (n + 1)), i.succ ∈ Finset.univ | null | false |
ComplexShape.Embedding.not_boundaryGE_next' | Mathlib.Algebra.Homology.Embedding.Boundary | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff]
{j k : ι}, ¬e.BoundaryGE j → c.next j = k → ¬e.BoundaryGE k | null | true |
_private.Batteries.Data.MLList.Basic.0.MLList.Spec.noConfusion | Batteries.Data.MLList.Basic | {P : Sort u_1} →
{m : Type u → Type u} →
{t : MLList.Spec✝ m} →
{m' : Type u → Type u} → {t' : MLList.Spec✝ m'} → m = m' → t ≍ t' → MLList.Spec.noConfusionType✝ P t t' | null | false |
Rat.cast_min | Mathlib.Data.Rat.Cast.Order | ∀ {K : Type u_2} [inst : Field K] [inst_1 : LinearOrder K] [IsStrictOrderedRing K] (p q : ℚ), ↑(min p q) = min ↑p ↑q | null | true |
Mathlib.Meta.FunProp.Config.mk | Mathlib.Tactic.FunProp.Types | ℕ → ℕ → Mathlib.Meta.FunProp.Config | null | true |
ZMod.val_le | Mathlib.Data.ZMod.Basic | ∀ {n : ℕ} [NeZero n] (a : ZMod n), a.val ≤ n | null | true |
_private.Lean.Data.KVMap.0.Lean.KVMap.insert.match_1 | Lean.Data.KVMap | (motive : Lean.KVMap → Lean.Name → Lean.DataValue → Sort u_1) →
(x : Lean.KVMap) →
(x_1 : Lean.Name) →
(x_2 : Lean.DataValue) →
((m : List (Lean.Name × Lean.DataValue)) →
(k : Lean.Name) → (v : Lean.DataValue) → motive { entries := m } k v) →
motive x x_1 x_2 | null | false |
HomologicalComplex.truncLE'ToRestriction_naturality | Mathlib.Algebra.Homology.Embedding.TruncLE | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
[inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{K L : HomologicalComplex C c'} (φ : K ⟶ L) (e : c.Embedding c') [inst_2 : e.IsTruncLE]
[inst_3 : ∀ (i' : ι'), K.HasHomology... | null | true |
_private.Mathlib.Order.Interval.Finset.Fin.0.Fin.finsetImage_val_Iio._simp_1_1 | Mathlib.Order.Interval.Finset.Fin | ∀ {α : Type u_1} {s₁ s₂ : Finset α}, (s₁ = s₂) = (↑s₁ = ↑s₂) | null | false |
_private.Mathlib.RingTheory.Extension.Presentation.Submersive.0.Algebra.PreSubmersivePresentation.jacobian_reindex._simp_1_4 | Mathlib.RingTheory.Extension.Presentation.Submersive | ∀ {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : Semiring B] [inst_3 : Semiring C] [inst_4 : Algebra R A] [inst_5 : Algebra R B] [inst_6 : Algebra R C]
(φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B), ⇑φ₁ ∘ ⇑φ₂ = ⇑(φ₁.comp φ₂) | null | false |
Lean.AttributeImpl.applicationTime._inherited_default | Lean.Attributes | Lean.AttributeApplicationTime | null | false |
Tactic.ComputeAsymptotics.MultiseriesExpansion.mk_eq_mk_iff_iff | Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | ∀ {basis_hd : ℝ → ℝ} {basis_tl : List (ℝ → ℝ)}
(ms : Tactic.ComputeAsymptotics.MultiseriesExpansion (basis_hd :: basis_tl))
(s : Tactic.ComputeAsymptotics.MultiseriesExpansion.Multiseries basis_hd basis_tl) (f : ℝ → ℝ),
Tactic.ComputeAsymptotics.MultiseriesExpansion.mk s f = ms ↔ ms.seq = s ∧ ms.toFun = f | null | true |
instCoeSortProfiniteGrpType | Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic | CoeSort ProfiniteGrp.{u} (Type u) | null | true |
Topology.IsQuotientMap.trivializationOfSMulDisjoint._proof_9 | Mathlib.Topology.Covering.Quotient | ∀ {E : Type u_2} {G : Type u_1} [inst : Group G] [inst_1 : MulAction G E] (U : Set E),
(∀ (g : G), ((fun x => g • x) '' U ∩ U).Nonempty → g = 1) →
∀ {g₁ g₂ : G}, g₁ ≠ g₂ → Disjoint ((fun x => g₁ • x) ⁻¹' U) ((fun x => g₂ • x) ⁻¹' U) | null | false |
Order.krullDim_int | Mathlib.Order.KrullDimension | Order.krullDim ℤ = ⊤ | null | true |
MvPolynomial.support_rename_killCompl_subset | Mathlib.Algebra.MvPolynomial.Rename | ∀ {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [inst : CommSemiring R] {p : MvPolynomial τ R} {f : σ → τ}
(hf : Function.Injective f), ((MvPolynomial.rename f) ((MvPolynomial.killCompl hf) p)).support ⊆ p.support | null | true |
MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left | Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | ∀ {α : Type u_1} {m : MeasurableSpace α} {f : α → ENNReal} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
[MeasureTheory.SigmaFinite ν],
(∀ᵐ (x : α) ∂μ, f x ≠ ⊤) → ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν | Auxiliary lemma for `rnDeriv_withDensity_left`. | true |
AdicCompletion.instCommRingAdicCauchySequence | Mathlib.RingTheory.AdicCompletion.Algebra | {R : Type u_1} → [inst : CommRing R] → (I : Ideal R) → CommRing (AdicCompletion.AdicCauchySequence I R) | null | true |
CochainComplex.homOfDegreewiseSplit | Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {C : Type u_1} →
[inst : CategoryTheory.Category.{v, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
(S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) →
((n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) →
(S.X₃ ⟶ (CategoryTheory.shiftFunctor (CochainC... | The canonical morphism `S.X₃ ⟶ S.X₁⟦(1 : ℤ)⟧` attached to a degreewise split
short exact sequence of cochain complexes. | true |
_private.Mathlib.Tactic.NormNum.ModEq.0.Mathlib.Meta.NormNum.evalNatModEq.match_1 | Mathlib.Tactic.NormNum.ModEq | (αP : Q(Type)) →
(e : Q(«$αP»)) →
(motive : (b : Bool) × Mathlib.Meta.NormNum.BoolResult e b → Sort u_1) →
(x : (b : Bool) × Mathlib.Meta.NormNum.BoolResult e b) →
((b : Bool) → (pb : Mathlib.Meta.NormNum.BoolResult e b) → motive ⟨b, pb⟩) → motive x | null | false |
upperSemicontinuous_biInf | Mathlib.Topology.Semicontinuity.Basic | ∀ {α : Type u_1} [inst : TopologicalSpace α] {ι : Sort u_4} {δ : Type u_5} [inst_1 : CompleteLinearOrder δ]
{p : ι → Prop} {f : (i : ι) → p i → α → δ},
(∀ (i : ι) (hi : p i), UpperSemicontinuous (f i hi)) → UpperSemicontinuous fun x' => ⨅ i, ⨅ (hi : p i), f i hi x' | null | true |
AddMonoidAlgebra.coeff_mapDomainLinearEquiv | Mathlib.Algebra.MonoidAlgebra.Module | ∀ {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Semiring R] [inst_1 : Semiring S]
[inst_2 : Module R S] (e : M ≃ N) (x : AddMonoidAlgebra S M),
((AddMonoidAlgebra.mapDomainLinearEquiv R S e) x).coeff = Finsupp.equivMapDomain e x.coeff | null | true |
_private.Std.Time.Time.HourMarker.0.Std.Time.HourMarker.toAbsolute._proof_1 | Std.Time.Time.HourMarker | 12 ≤ 23 | null | false |
Equiv.Perm.one_lt_of_mem_cycleType | Mathlib.GroupTheory.Perm.Cycle.Type | ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] {σ : Equiv.Perm α} {n : ℕ}, n ∈ σ.cycleType → 1 < n | null | true |
Nat.two_pow_pred_mul_two | Init.Data.Nat.Lemmas | ∀ {w : ℕ}, 0 < w → 2 ^ (w - 1) * 2 = 2 ^ w | null | true |
Lean.Meta.Config.ctxApprox._default | Lean.Meta.Basic | Bool | null | false |
AddGroupNormClass.toAddGroupSeminormClass | Mathlib.Algebra.Order.Hom.GroupNorm | ∀ {F : Type u_4} {α : outParam (Type u_5)} {β : outParam (Type u_6)} {inst : AddGroup α} {inst_1 : AddCommMonoid β}
{inst_2 : PartialOrder β} {inst_3 : FunLike F α β} [self : AddGroupNormClass F α β], AddGroupSeminormClass F α β | null | true |
AlgebraicGeometry.Scheme.Cover.RelativeGluingData.instCategoryI₀Cover._proof_7 | Mathlib.AlgebraicGeometry.RelativeGluing | ∀ {S : AlgebraicGeometry.Scheme} {𝒰 : S.OpenCover} [inst : CategoryTheory.Category.{u_3, u_1} 𝒰.I₀]
[inst_1 : AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.RelativeGluingData 𝒰)
[inst_2 : Small.{u_2, u_1} 𝒰.I₀] [inst_3 : Quiver.IsThin 𝒰.I₀],
autoParam
(∀ {X Y : d.... | null | false |
_private.Mathlib.Analysis.Convex.Deriv.0.bddBelow_slope_lt_of_mem_interior.match_1_1 | Mathlib.Analysis.Convex.Deriv | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} {x : 𝕜} (y' : 𝕜)
(motive : y' ∈ slope f x '' {y | y ∈ s ∧ x < y} → Prop) (x_1 : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}),
(∀ (z : 𝕜) (hz : z ∈ {y | y ∈ s ∧ x < y}) (hz' : slope f x z = y'), motive ⋯) → motive x_1 | null | false |
NonUnitalSubring.instSetLike | Mathlib.RingTheory.NonUnitalSubring.Defs | {R : Type u} → [inst : NonUnitalNonAssocRing R] → SetLike (NonUnitalSubring R) R | null | true |
Filter.Germ.instDivisionRing._proof_6 | Mathlib.Order.Filter.FilterProduct | ∀ {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [DivisionRing β], Nontrivial ((↑φ).Germ β) | null | false |
LinearMap.shortComplexKer._proof_1 | Mathlib.Algebra.Homology.ShortComplex.ModuleCat | ∀ {R : Type u_2} [inst : Ring R] {M : Type u_1} [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),
CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom f.ker.subtype) (ModuleCat.ofHom f) = 0 | null | false |
MulOpposite.instSemifield._proof_2 | Mathlib.Algebra.Field.Opposite | ∀ {α : Type u_1} [inst : Semifield α] (a : αᵐᵒᵖ), a ^ 0 = 1 | null | false |
StarMulEquiv.coe_trans | Mathlib.Algebra.Star.MonoidHom | ∀ {A : Type u_2} {B : Type u_3} {C : Type u_4} [inst : Mul A] [inst_1 : Mul B] [inst_2 : Mul C] [inst_3 : Star A]
[inst_4 : Star B] [inst_5 : Star C] (e₁ : A ≃⋆* B) (e₂ : B ≃⋆* C), ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁ | null | true |
AlgebraicClosure.instGroupWithZero._proof_8 | Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | ∀ (k : Type u_1) [inst : Field k], autoParam (∀ (a : AlgebraicClosure k), a ^ 0 = 1) DivInvMonoid.zpow_zero'._autoParam | null | false |
TensorProduct.quotientTensorQuotientEquiv._proof_9 | Mathlib.LinearAlgebra.TensorProduct.Quotient | ∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[inst_3 : AddCommGroup N] [inst_4 : Module R N], IsScalarTower R R (TensorProduct R M N) | null | false |
CategoryTheory.Functor.whiskeringLeft₃ObjObjMap_app | Mathlib.CategoryTheory.Whiskering | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} {D₃ : Type u_7}
[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} C₃] [inst_3 : CategoryTheory.Category.{v_5, u_5} D₁]
[inst_4 : CategoryTheo... | null | true |
PolynomialLaw.instZero._proof_2 | Mathlib.RingTheory.PolynomialLaw.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Type u_3}
[inst_3 : AddCommMonoid N] [inst_4 : Module R N] {S : Type u_1} [inst_5 : CommSemiring S] [inst_6 : Algebra R S]
{S' : Type u_1} [inst_7 : CommSemiring S'] [inst_8 : Algebra R S'] (φ : S →ₐ[R] S')... | null | false |
FirstOrder.Language.isFraisse_finite_linear_order | Mathlib.ModelTheory.Order | FirstOrder.Language.IsFraisse {M | Finite ↑M ∧ ↑M ⊨ FirstOrder.Language.order.linearOrderTheory} | The class of finite models of the theory of linear orders is Fraïssé. | true |
Colex.instAddMonoid.eq_1 | Mathlib.Algebra.Order.Group.Synonym | ∀ {α : Type u_1} [inst : AddMonoid α],
Colex.instAddMonoid =
{ toZero := Colex.instZero, toAdd := Colex.instAdd, add_assoc := ⋯, zero_add := ⋯, add_zero := ⋯,
toNSMul := NSMul.ofSMul, nsmul_zero := ⋯, nsmul_succ := ⋯ } | null | true |
TestFunction.monoCLM._proof_4 | Mathlib.Analysis.Distribution.TestFunction | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {Ω₁ Ω₂ : TopologicalSpace.Opens E} {n₁ n₂ : ℕ∞}
(K : TopologicalSpace.Compacts E), ↑K ⊆ ↑Ω₁ → n₂ ≤ n₁ ∧ Ω₁ ≤ Ω₂ → ↑K ⊆ ↑Ω₂ | null | false |
CategoryTheory.Cat.freeMapIdIso | Mathlib.CategoryTheory.Category.Quiv | (V : Type u_1) →
[inst : Quiver V] → CategoryTheory.Cat.freeMap (𝟭q V) ≅ CategoryTheory.Functor.id (CategoryTheory.Paths V) | The functor `free : Quiv ⥤ Cat` preserves identities up to natural isomorphism and in fact up
to equality. | true |
ULift.addGroup._proof_4 | Mathlib.Algebra.Group.ULift | ∀ {α : Type u_2} [inst : AddGroup α] (x x_1 : ULift.{u_1, u_2} α), Equiv.ulift (x - x_1) = Equiv.ulift (x - x_1) | null | false |
CategoryTheory.Limits.Types.colimitCocone | Mathlib.CategoryTheory.Limits.Types.Colimits | {J : Type v} →
[inst : CategoryTheory.Category.{w, v} J] →
(F : CategoryTheory.Functor J (Type u)) → [Small.{u, max u v} F.ColimitType] → CategoryTheory.Limits.Cocone F | (internal implementation) the colimit cocone of a functor,
implemented as a quotient of a sigma type
| true |
Nat.recOnPrimePow._proof_2 | Mathlib.Data.Nat.Factorization.Induction | ∀ (k : ℕ), Nat.Prime (k + 2).minFac | null | false |
Lean.Meta.instInhabitedCaseArraySizesSubgoal.default | Lean.Meta.Match.CaseArraySizes | Lean.Meta.CaseArraySizesSubgoal | null | true |
FiniteField.instFieldExtension._proof_68 | Mathlib.FieldTheory.Finite.Extension | ∀ (k : Type u_1) [inst : Field k] (p : ℕ) [inst_1 : Fact (Nat.Prime p)] [inst_2 : CharP k p] (n : ℕ),
autoParam (∀ (q : ℚ≥0) (a : FiniteField.Extension k p n), FiniteField.instFieldExtension._aux_66 k p n q a = ↑q * a)
DivisionRing.nnqsmul_def._autoParam | null | false |
CategoryTheory.Limits.SequentialProduct.functorObj_eq_neg | Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | ∀ {C : Type u_1} {M N : ℕ → C} {n m : ℕ}, ¬m < n → (fun i => if x : i < n then M i else N i) m = N m | null | true |
Lean.Grind.Linarith.Expr.toPolyN._f | Init.Grind.Module.NatModuleNorm | (x : Lean.Grind.Linarith.Expr) → Lean.Grind.Linarith.Expr.below x → Lean.Grind.Linarith.Poly | null | false |
AddAction.vadd_mem_fixedBy_iff_mem_fixedBy | Mathlib.GroupTheory.GroupAction.FixedPoints | ∀ {α : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] {a : α} {g : G},
g +ᵥ a ∈ AddAction.fixedBy α g ↔ a ∈ AddAction.fixedBy α g | null | true |
_private.Lean.PrettyPrinter.Delaborator.Options.0.Lean.initFn._@.Lean.PrettyPrinter.Delaborator.Options.2604662143._hygCtx._hyg.4 | Lean.PrettyPrinter.Delaborator.Options | IO (Lean.Option Bool) | null | false |
_private.Mathlib.RingTheory.LocalIso.0.Algebra.IsLocalIso.trans._simp_1_4 | Mathlib.RingTheory.LocalIso | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
_private.Mathlib.NumberTheory.LSeries.ZMod.0.ZMod.LFunction_apply_zero_of_even._simp_1_2 | Mathlib.NumberTheory.LSeries.ZMod | ∀ {α : Type u_1} [inst : Fintype α] (x : α), (x ∈ Finset.univ) = True | null | false |
_private.Plausible.Gen.0.Plausible.Gen.runUntil.repeatGen.match_1 | Plausible.Gen | (motive : Option ℕ → Sort u_1) →
(attempts : Option ℕ) → (Unit → motive (some 0)) → ((x : Option ℕ) → motive x) → motive attempts | null | false |
LieModule.Weight.mk | Mathlib.Algebra.Lie.Weights.Basic | {R : Type u_2} →
{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... | null | true |
MeasureTheory.FiniteMeasure.restrict_apply | Mathlib.MeasureTheory.Measure.FiniteMeasure | ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (A : Set Ω) {s : Set Ω},
MeasurableSet s → (μ.restrict A) s = μ (s ∩ A) | null | true |
Rep.FiniteCyclicGroup.groupCohomologyIsoEven._proof_4 | Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic | ∀ (i : ℕ) [h₀ : NeZero i], Even i → (ComplexShape.up ℕ).Rel ((ComplexShape.up ℕ).prev i) i | null | false |
ContinuousLinearMap.instAddMonoid._proof_5 | Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | ∀ {R₁ : Type u_1} {R₂ : Type u_2} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_3}
[inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_4} [inst_4 : TopologicalSpace M₂]
[inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂] [inst_8 : ContinuousAd... | null | false |
_private.Mathlib.CategoryTheory.Limits.Shapes.Countable.0.CategoryTheory.Limits.IsFiltered.sequentialFunctor_final._simp_3 | Mathlib.CategoryTheory.Limits.Shapes.Countable | ∀ {α : Type u_1} {R : α → α → Prop} (a : α), List.IsChain R [a] = True | null | false |
NumberField.mixedEmbedding.negAt_apply_isComplex | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | ∀ {K : Type u_1} [inst : Field K] {s : Set { w // w.IsReal }} (x : NumberField.mixedEmbedding.mixedSpace K)
(w : { w // w.IsComplex }), ((NumberField.mixedEmbedding.negAt s) x).2 w = x.2 w | null | true |
_private.Mathlib.NumberTheory.ArithmeticFunction.Misc.0.ArithmeticFunction.sum_Ioc_sigma0_eq_sum_div._proof_1_1 | Mathlib.NumberTheory.ArithmeticFunction.Misc | ∀ (N x : ℕ), x ∈ Finset.Ioc 0 N → N / x = if x = 0 then 0 else N / x | null | false |
_private.Lean.Meta.Tactic.Grind.Types.0.Lean.Meta.Grind.Goal.getEqc.go._unsafe_rec | Lean.Meta.Tactic.Grind.Types | Lean.Meta.Grind.Goal → Lean.Expr → Lean.Expr → Array Lean.Expr → Array Lean.Expr | null | false |
LocalSubring.instPartialOrder._proof_2 | Mathlib.RingTheory.LocalRing.LocalSubring | ∀ {R : Type u_1} [inst : CommRing R], SubringClass (Subring R) R | null | false |
DerivedCategory.Plus.instIsHomologicalHomologyFunctor._proof_1 | Mathlib.Algebra.Homology.DerivedCategory.Plus | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_3, u_2} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] (n : ℤ), (DerivedCategory.Plus.homologyFunctor C n).IsHomological | null | false |
Lean.Compiler.LCNF.MonadFVarSubstState.recOn | Lean.Compiler.LCNF.CompilerM | {m : Type → Type} →
{pu : Lean.Compiler.LCNF.Purity} →
{motive : Lean.Compiler.LCNF.MonadFVarSubstState m pu → Sort u} →
(t : Lean.Compiler.LCNF.MonadFVarSubstState m pu) →
((modifySubst : (Lean.Compiler.LCNF.FVarSubst pu → Lean.Compiler.LCNF.FVarSubst pu) → m Unit) →
motive { modifySubs... | null | false |
Submodule.projection_eq_id_sub_projection | Mathlib.LinearAlgebra.Projection | ∀ {R : Type u_1} [inst : Ring R] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module R E] {p q : Submodule R E}
(hpq : IsCompl p q), q.projection p ⋯ = LinearMap.id - p.projection q hpq | null | true |
_private.Mathlib.Data.Finset.Image.0.Finset.coe_map_subset_range._proof_1_1 | Mathlib.Data.Finset.Image | ∀ {α : Type u_2} {β : Type u_1} (f : α ↪ β) (s : Finset α), ↑(Finset.map f s) ⊆ Set.range ⇑f | null | false |
_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform.0.EisensteinSeries.G2_slash_action._simp_1_2 | Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | ∀ {α : Type u_1} {a b : α}, (a ∈ {b}) = (a = b) | null | false |
_private.Init.Data.String.Lemmas.Pattern.Char.0.String.Slice.Pattern.Model.Char.revMatchesAt_iff_revMatchesAt_beq._simp_1_1 | Init.Data.String.Lemmas.Pattern.Char | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.Model.PatternModel pat] {s : String.Slice} {pos : s.Pos},
String.Slice.Pattern.Model.RevMatchesAt pat pos =
∃ startPos, String.Slice.Pattern.Model.IsLongestRevMatchAt pat startPos pos | null | false |
List.Vector.ofFn | Mathlib.Data.Vector.Defs | {α : Type u_1} → {n : ℕ} → (Fin n → α) → List.Vector α n | Vector of length `n` from a function on `Fin n`. | true |
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