name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ContinuousLinearEquiv.toLinearEquiv | Mathlib.Topology.Algebra.Module.Equiv | {R : Type u_1} →
{S : Type u_2} →
[inst : Semiring R] →
[inst_1 : Semiring S] →
{σ : R →+* S} →
{σ' : S →+* R} →
[inst_2 : RingHomInvPair σ σ'] →
[inst_3 : RingHomInvPair σ' σ] →
{M : Type u_3} →
[inst_4 : TopologicalSpace M] →
... | null | true |
ProofWidgets.MarkdownDisplay.Props.mk | ProofWidgets.Component.Basic | String → ProofWidgets.MarkdownDisplay.Props | null | true |
Std.Http.Protocol.H1.Reader.State.complete.sizeOf_spec | Std.Http.Protocol.H1.Reader | ∀ {dir : Std.Http.Protocol.H1.Direction}, sizeOf Std.Http.Protocol.H1.Reader.State.complete = 1 | null | true |
CategoryTheory.Presieve.functorPushforward | Mathlib.CategoryTheory.Sites.Sieves | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
(F : CategoryTheory.Functor C D) → {X : C} → CategoryTheory.Presieve X → CategoryTheory.Presieve (F.obj X) | Given a presieve on `X`, we can define a presieve on `F(X)` (which is actually a sieve)
by taking the sieve generated by the image via `F`.
| true |
ENNReal.coe_iInf._simp_1 | Mathlib.Data.ENNReal.Basic | ∀ {ι : Sort u_3} [Nonempty ι] (f : ι → NNReal), ⨅ a, ↑(f a) = ↑(iInf f) | null | false |
SimplexCategory.Truncated.Hom.tr_id | Mathlib.AlgebraicTopology.SimplexCategory.Defs | ∀ {n : ℕ} (a : SimplexCategory) (ha : autoParam (a.len ≤ n) SimplexCategory.Truncated.Hom.tr_id._auto_1),
SimplexCategory.Truncated.Hom.tr (CategoryTheory.CategoryStruct.id a) ha ha =
CategoryTheory.CategoryStruct.id { obj := a, property := ha } | null | true |
_private.Mathlib.Algebra.Ring.Divisibility.Lemmas.0.dvd_smul_of_dvd.match_1_1 | Mathlib.Algebra.Ring.Divisibility.Lemmas | ∀ {R : Type u_1} [inst : Semigroup R] {x y : R} (motive : x ∣ y → Prop) (h : x ∣ y),
(∀ (k : R) (hk : y = x * k), motive ⋯) → motive h | null | false |
_private.Mathlib.Analysis.PSeries.0.summable_schlomilch_iff_of_nonneg._simp_1_1 | Mathlib.Analysis.PSeries | ∀ {r₁ r₂ : NNReal}, (↑r₁ ≤ ↑r₂) = (r₁ ≤ r₂) | null | false |
KaehlerDifferential.quotKerTotalEquiv._proof_2 | Mathlib.RingTheory.Kaehler.Basic | ∀ (R : Type u_2) (S : Type u_1) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],
Function.RightInverse (⇑(KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential)
((KaehlerDifferential.kerTotal R S).liftQ (Finsupp.linearCombination S ⇑(KaehlerDifferential.D R S)) ⋯).toFun | null | false |
ProbabilityTheory.Kernel.densityProcess_empty | Mathlib.Probability.Kernel.Disintegration.Density | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} [inst : MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β))
(ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) (x : γ), κ.densityProcess ν n a x ∅ = 0 | null | true |
Filter.lift_principal2 | Mathlib.Order.Filter.Lift | ∀ {α : Type u_1} {f : Filter α}, f.lift Filter.principal = f | null | true |
Lean.Elab.FVarAliasInfo.mk | Lean.Elab.InfoTree.Types | Lean.Name → Lean.FVarId → Lean.FVarId → Lean.Elab.FVarAliasInfo | null | true |
Monotone.seq_pos_lt_seq_of_le_of_lt | Mathlib.Order.Iterate | ∀ {α : Type u_1} [inst : Preorder α] {f : α → α} {x y : ℕ → α},
Monotone f →
∀ {n : ℕ}, 0 < n → x 0 ≤ y 0 → (∀ k < n, x (k + 1) ≤ f (x k)) → (∀ k < n, f (y k) < y (k + 1)) → x n < y n | null | true |
_private.Init.Data.Int.DivMod.Lemmas.0.Int.fmod_sub_cancel_left._simp_1_2 | Init.Data.Int.DivMod.Lemmas | ∀ {a b : ℤ}, (b.fmod a = 0) = (a ∣ b) | null | false |
Commute.pow_pow_self | Mathlib.Algebra.Group.Commute.Defs | ∀ {M : Type u_2} [inst : Monoid M] (a : M) (m n : ℕ), Commute (a ^ m) (a ^ n) | null | true |
Lean.Lsp.Registration.noConfusionType | Lean.Data.Lsp.Client | Sort u → Lean.Lsp.Registration → Lean.Lsp.Registration → Sort u | null | false |
Int.natAbs_eq_iff | Init.Data.Int.Order | ∀ {a : ℤ} {n : ℕ}, a.natAbs = n ↔ a = ↑n ∨ a = -↑n | null | true |
Mathlib.Tactic.BicategoryLike.Mor₂.whiskerLeft.inj | Mathlib.Tactic.CategoryTheory.Coherence.Datatypes | ∀ {e : Lean.Expr} {isoLift? : Option Mathlib.Tactic.BicategoryLike.IsoLift} {f g h : Mathlib.Tactic.BicategoryLike.Mor₁}
{η : Mathlib.Tactic.BicategoryLike.Mor₂} {e_1 : Lean.Expr} {isoLift?_1 : Option Mathlib.Tactic.BicategoryLike.IsoLift}
{f_1 g_1 h_1 : Mathlib.Tactic.BicategoryLike.Mor₁} {η_1 : Mathlib.Tactic.Bic... | null | true |
Std.DHashMap.Internal.Raw₀.filterₘ | Std.Data.DHashMap.Internal.Model | {α : Type u} →
{β : α → Type v} → Std.DHashMap.Internal.Raw₀ α β → ((a : α) → β a → Bool) → Std.DHashMap.Internal.Raw₀ α β | Internal implementation detail of the hash map | true |
Lean.Elab.Tactic.TacticParsedSnapshot.toSnapshot | Lean.Elab.Term.TermElabM | Lean.Elab.Tactic.TacticParsedSnapshot → Lean.Language.Snapshot | null | true |
MeasureTheory.L1.setToL1'.congr_simp | Mathlib.MeasureTheory.Integral.SetToL1 | ∀ {α : Type u_1} {E : Type u_2} {F : Type u_3} (𝕜 : Type u_6) [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
[inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
[inst_4 : NormedRing 𝕜] [inst_5 : Module 𝕜 E] [inst_6 : Module 𝕜 F] [inst_7 : Is... | null | true |
Lean.Lsp.InlayHintKind.casesOn | Lean.Data.Lsp.LanguageFeatures | {motive : Lean.Lsp.InlayHintKind → Sort u} →
(t : Lean.Lsp.InlayHintKind) → motive Lean.Lsp.InlayHintKind.type → motive Lean.Lsp.InlayHintKind.parameter → motive t | null | false |
MeasureTheory.lpTrimToLpMeasSubgroup._proof_1 | Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | ∀ {α : Type u_1} (F : Type u_2) (p : ENNReal) [inst : NormedAddCommGroup F] {m m0 : MeasurableSpace α}
(μ : MeasureTheory.Measure α) (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp F p (μ.trim hm))), MeasureTheory.MemLp (↑↑f) p μ | null | false |
_private.Mathlib.Tactic.Ring.Basic.0.Mathlib.Tactic.Ring.ExBase.evalIntCast.match_1 | Mathlib.Tactic.Ring.Basic | {u : Lean.Level} →
{α : Q(Type u)} →
{a : Q(ℤ)} →
(rα : Q(CommRing «$α»)) →
(motive : ℕ × { e' // ↑«$a» =Q «$e'» } → Sort u_1) →
(__discr : ℕ × { e' // ↑«$a» =Q «$e'» }) →
((i : ℕ) → (b' : Q(«$α»)) → (property : ↑«$a» =Q «$b'») → motive (i, ⟨b', property⟩)) → motive __discr | null | false |
RootPairing.Base.cartanMatrixIn_mul_diagonal_eq | Mathlib.LinearAlgebra.RootSystem.CartanMatrix | ∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
[inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (S : Type u_5) [inst_5 : CommRing S]
[inst_6 : Algebra S R] {P : RootPairing ι R M N} [inst_7 : P.IsRootSystem] [inst_8 : P.IsValuedIn S]
... | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex.0.SSet.stdSimplex.orderIsoOfNonDegenerate._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | ∀ {n d : ℕ} (x : ↑((SSet.stdSimplex.obj { len := n }).nonDegenerate d)) (i : Fin (d + 1)),
(SimplexCategory.Hom.toOrderHom (SSet.stdSimplex.objEquiv ↑x)) i ∈
Finset.image (⇑(SimplexCategory.Hom.toOrderHom (SSet.stdSimplex.objEquiv ↑x))) Finset.univ | null | false |
Aesop.LocalRuleSet.forwardRulePatternSubstsInLocalDecl | Aesop.RuleSet | Aesop.LocalRuleSet → Lean.LocalDecl → Aesop.BaseM (Array (Aesop.ForwardRule × Aesop.Substitution)) | null | true |
Set.Ioi_disjoint_Iio_iff._simp_2 | Mathlib.Order.Interval.Set.Disjoint | ∀ {α : Type v} [inst : Preorder α] {a b : α} [DenselyOrdered α], Disjoint (Set.Ioi a) (Set.Iio b) = ¬a < b | null | false |
Std.Iterators.Types.FilterMap.ctorIdx | Init.Data.Iterators.Combinators.Monadic.FilterMap | {α β γ : Type w} →
{m : Type w → Type w'} →
{n : Type w → Type w''} →
{lift : ⦃α : Type w⦄ → m α → n α} →
{f : β → Std.Iterators.PostconditionT n (Option γ)} → Std.Iterators.Types.FilterMap α m n lift f → ℕ | null | false |
RingHom.coe_inj | Mathlib.Algebra.Ring.Hom.Defs | ∀ {α : Type u_2} {β : Type u_3} {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} ⦃f g : α →+* β⦄, ⇑f = ⇑g → f = g | null | true |
OmegaCompletePartialOrder.Chain.ext | Mathlib.Order.OmegaCompletePartialOrder | ∀ {α : Type u_2} [inst : Preorder α] ⦃f g : OmegaCompletePartialOrder.Chain α⦄, ⇑f = ⇑g → f = g | See note [partially-applied ext lemmas]. | true |
onePointEquivSphereOfFinrankEq._proof_5 | Mathlib.Topology.Compactification.OnePoint.Sphere | ∀ {ι : Type u_1} {V : Type u_2} [inst : Fintype ι] [inst_1 : AddCommGroup V] [inst_2 : Module ℝ V],
Module.finrank ℝ V + 1 = Fintype.card ι → Nonempty ι | null | false |
Int.strongRec.eq_1 | Mathlib.Data.Int.Basic | ∀ {m : ℤ} {motive : ℤ → Sort u_1} (lt : (n : ℤ) → n < m → motive n)
(ge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → motive k) → motive n) (n : ℤ),
Int.strongRec lt ge n =
if hnm : n < m then lt n hnm
else
ge n ⋯
(Int.inductionOn' (motive := fun x => (k : ℤ) → k < x → motive k) n m lt
(fun... | null | true |
CovBy | Mathlib.Order.Defs.PartialOrder | {α : Type u_2} → [LT α] → α → α → Prop | `CovBy a b` means that `b` covers `a`. This means that `a < b` and there is no element in
between. This is denoted `a ⋖ b`. | true |
Nat.log2_eq_succ_log2_shiftRight | Mathlib.Data.Nat.BinaryRec | ∀ {n : ℕ}, n >>> 1 ≠ 0 → n.log2 = (n >>> 1).log2.succ | null | true |
_private.Mathlib.AlgebraicTopology.ExtraDegeneracy.0.CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_8 | Mathlib.AlgebraicTopology.ExtraDegeneracy | ∀ {n : ℕ} (j : Fin (n + 1)) (k : Fin (n + 1 + 1)), j.succ.rev = k → ↑k = ↑j.rev | null | false |
_private.Mathlib.Order.ConditionallyCompleteLattice.Indexed.0.cbiSup_of_not_bddAbove.match_1_3 | Mathlib.Order.ConditionallyCompleteLattice.Indexed | ∀ {α : Type u_1} {ι : Sort u_2} [inst : ConditionallyCompleteLinearOrder α] {p : ι → Prop} {f : (i : ι) → p i → α}
(motive : BddAbove (Set.range fun i => ⨆ (h : p i), f i h) → Prop)
(x : BddAbove (Set.range fun i => ⨆ (h : p i), f i h)),
(∀ (u : α) (hu : u ∈ upperBounds (Set.range fun i => ⨆ (h : p i), f i h)), m... | null | false |
CategoryTheory.PreOneHypercover.cylinderHomotopy._proof_2 | Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {S : C} {E : CategoryTheory.PreOneHypercover S}
{F : CategoryTheory.PreOneHypercover S} [inst_1 : CategoryTheory.Limits.HasPullbacks C] (f g : E.Hom F)
(p : (CategoryTheory.PreOneHypercover.cylinder f g).I₀),
CategoryTheory.CategoryStruct.comp
(... | null | false |
CategoryTheory.Abelian.Ext.covariant_sequence_exact₃' | Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : CategoryTheory.HasExt C] (X : C) {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) (n₀ n₁ : ℕ)
(h : n₀ + 1 = n₁),
{ X₁ := AddCommGrpCat.of (CategoryTheory.Abelian.Ext X S.X₂ n₀),
X₂ := AddCommGrpCat.... | Alternative formulation of `covariant_sequence_exact₃` | true |
Bimod.RightUnitorBimod.hom_right_act_hom' | Mathlib.CategoryTheory.Monoidal.Bimod | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.Limits.HasCoequalizers C] {R S : CategoryTheory.Mon C} (P : Bimod R S)
[inst_3 :
∀ (X : C),
CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}
(Catego... | null | true |
RingCon.instAddZeroClassQuotient | Mathlib.RingTheory.Congruence.Defs | {R : Type u_1} → [inst : AddZeroClass R] → [inst_1 : Mul R] → (c : RingCon R) → AddZeroClass c.Quotient | null | true |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.getValueCast?_filter_not_contains_of_contains_right._simp_1_1 | Std.Data.Internal.List.Associative | ∀ {α : Sort u_1} {p : Prop} [inst : Decidable p] {x y : α}, ((if p then x else y) = x) = (¬p → y = x) | null | false |
Filter.instDistribNeg._proof_2 | Mathlib.Order.Filter.Pointwise | ∀ {α : Type u_1} [inst : Mul α] [inst_1 : HasDistribNeg α] (x x_1 : Filter α),
Filter.map₂ (fun x1 x2 => x1 * x2) x (Filter.map Neg.neg x_1) =
Filter.map Neg.neg (Filter.map₂ (fun x1 x2 => x1 * x2) x x_1) | null | false |
CategoryTheory.LaxBraidedFunctor.hom_ext_iff | Mathlib.CategoryTheory.Monoidal.Braided.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] {D : Type u₂} [inst_3 : CategoryTheory.Category.{v₂, u₂} D]
[inst_4 : CategoryTheory.MonoidalCategory D] [inst_5 : CategoryTheory.BraidedCategory D]
{F G : CategoryT... | null | true |
instBooleanAlgebraAsBoolAlg._proof_4 | Mathlib.Algebra.Ring.BooleanRing | ∀ {α : Type u_1} [inst : BooleanRing α] (a b : AsBoolAlg α), a ≤ b → b ≤ a → a = b | null | false |
CategoryTheory.ShortComplex.RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork._proof_1 | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ {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.Internal.IsStrictCut.toIsCut | Std.Data.Internal.Cut | ∀ {α : Type u} {cmp : α → α → Ordering} {cut : α → Ordering} [self : Std.Internal.IsStrictCut cmp cut],
Std.Internal.IsCut cmp cut | null | true |
WellFoundedGT.toIsSuccArchimedean | Mathlib.Order.SuccPred.Archimedean | ∀ {α : Type u_1} [inst : PartialOrder α] [h : WellFoundedGT α] [inst_1 : SuccOrder α], IsSuccArchimedean α | null | true |
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose._proof_6 | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic | ∀ {A : Type u_6} {B : Type u_3} {C : Type u_8} [inst : CategoryTheory.Category.{u_5, u_6} A]
[inst_1 : CategoryTheory.Category.{u_2, u_3} B] [inst_2 : CategoryTheory.Category.{u_7, u_8} C]
(F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type u_1} {Y : Type u_9}
[inst_3 : CategoryTheory.Cate... | null | false |
AlgebraicGeometry.Scheme.IdealSheafData.support_bot | Mathlib.AlgebraicGeometry.IdealSheaf.Basic | ∀ {X : AlgebraicGeometry.Scheme}, ⊥.support = ⊤ | null | true |
PreAbstractSimplicialComplex.map._proof_2 | Mathlib.AlgebraicTopology.SimplicialComplex.Basic | ∀ {α : Type u_2} {β : Type u_1} [inst : DecidableEq β] (K : PreAbstractSimplicialComplex α) (f : α → β) {x : Finset β},
x ∈ (fun s => Finset.image f s) '' K.faces →
x.Nonempty ∧ ∀ ⦃b : Finset β⦄, b ≤ x → b.Nonempty → b ∈ (fun s => Finset.image f s) '' K.faces | null | false |
IntermediateField.intermediateFieldMap_apply_coe | Mathlib.FieldTheory.IntermediateField.Basic | ∀ {K : Type u_1} {L : Type u_2} {L' : Type u_3} [inst : Field K] [inst_1 : Field L] [inst_2 : Field L']
[inst_3 : Algebra K L] [inst_4 : Algebra K L'] (e : L ≃ₐ[K] L') (E : IntermediateField K L) (a : ↥E),
↑((IntermediateField.intermediateFieldMap e E) a) = e ↑a | null | true |
GradedObject.eulerChar_eq_sum_finSet_of_finrankSupport_subset | Mathlib.Algebra.Homology.EulerCharacteristic | ∀ {R : Type u_1} [inst : Ring R] {ι : Type u_2} (c : ComplexShape ι) [inst_1 : c.EulerCharSigns]
(X : CategoryTheory.GradedObject ι (ModuleCat R)) (indices : Finset ι),
GradedObject.finrankSupport X ⊆ ↑indices →
GradedObject.eulerChar c X = ∑ i ∈ indices, ↑(c.χ i) * ↑(Module.finrank R ↑(X i)) | If a graded object has finite rank support contained in a finite set,
the `finsum` Euler characteristic equals the finite sum over that set. | true |
Cardinal.powerlt_mono_left | Mathlib.SetTheory.Cardinal.Basic | ∀ (a : Cardinal.{u_1}), Monotone fun c => a ^< c | null | true |
Std.ExtHashMap.getElem!_union_of_not_mem_left | 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} [inst_2 : Inhabited β], k ∉ m₁ → (m₁ ∪ m₂)[k]! = m₂[k]! | null | true |
Lean.Expr.FindStep.done | Lean.Util.FindExpr | Lean.Expr.FindStep | Do not search subterms | true |
ContDiffMapSupportedIn.iteratedFDerivLM_eq_of_scalars | Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | ∀ (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace ℝ E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] [inst_5 : NormedSpace 𝕜 F]
[inst_6 : SMulCommClass ℝ 𝕜 F] {n k : ℕ∞} {K : TopologicalSpace.Compacts E} {i : ℕ} (... | null | true |
AddMonoidHom.le_map_tsub | Mathlib.Algebra.Order.Sub.Unbundled.Hom | ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : AddCommMonoid α] [inst_2 : Sub α] [OrderedSub α]
[inst_4 : Preorder β] [inst_5 : AddZeroClass β] [inst_6 : Sub β] [OrderedSub β] (f : α →+ β),
Monotone ⇑f → ∀ (a b : α), f a - f b ≤ f (a - b) | null | true |
AddSubgroup.finiteIndex_iInf' | Mathlib.GroupTheory.Index | ∀ {G : Type u_1} [inst : AddGroup G] {ι : Type u_3} {s : Finset ι} (f : ι → AddSubgroup G),
(∀ i ∈ s, (f i).FiniteIndex) → (⨅ i ∈ s, f i).FiniteIndex | null | true |
AlgebraicGeometry.Scheme.irreducibleComponentOpen._proof_1 | Mathlib.AlgebraicGeometry.IdealSheaf.IrreducibleComponent | ∀ (X : AlgebraicGeometry.Scheme) (Z : Set ↥X) [AlgebraicGeometry.IsNoetherian X],
IsOpen (⋃₀ (irreducibleComponents ↥X \ {Z}))ᶜ | null | false |
midpoint_eq_midpoint_iff_vsub_eq_vsub | Mathlib.LinearAlgebra.AffineSpace.Midpoint | ∀ (R : Type u_1) {V : Type u_2} {P : Type u_4} [inst : Ring R] [inst_1 : Invertible 2] [inst_2 : AddCommGroup V]
[inst_3 : Module R V] [inst_4 : AddTorsor V P] {x x' y y' : P}, midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y | null | true |
CategoryTheory.Limits.MultispanShape.ctorIdx | Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | CategoryTheory.Limits.MultispanShape → ℕ | null | false |
_private.Mathlib.Data.Int.CardIntervalMod.0.Nat.Ico_filter_modEq_cast._simp_1_4 | Mathlib.Data.Int.CardIntervalMod | ∀ {α : Sort u_1} {p : α → Prop} {q : (∃ x, p x) → Prop}, (∀ (h : ∃ x, p x), q h) = ∀ (x : α) (h : p x), q ⋯ | null | false |
IsLowerSet.vadd | Mathlib.Algebra.Order.UpperLower | ∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : Preorder α] [IsOrderedAddMonoid α] {s : Set α} {a : α},
IsLowerSet s → IsLowerSet (a +ᵥ s) | null | true |
PrimeSpectrum.noConfusion | Mathlib.RingTheory.Spectrum.Prime.Defs | {P : Sort u} →
{R : Type u_1} →
{inst : CommSemiring R} →
{t : PrimeSpectrum R} →
{R' : Type u_1} →
{inst' : CommSemiring R'} →
{t' : PrimeSpectrum R'} → R = R' → inst ≍ inst' → t ≍ t' → PrimeSpectrum.noConfusionType P t t' | null | false |
Finset.card_mul_singleton | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : Mul α] [IsRightCancelMul α] [inst_2 : DecidableEq α] (s : Finset α) (a : α),
(s * {a}).card = s.card | null | true |
Affine.Triangle.sbtw_touchpoint_empty | Mathlib.Geometry.Euclidean.Incenter | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3},
i₁ ≠ i₂ → i₁ ≠ i₃ → i₂ ≠ i₃ → Sbtw ℝ (t.points i₁) (Affine.Simplex.touchpoint t ∅ i₂) (t.points i₃) | null | true |
_private.Lean.Meta.Tactic.Grind.Arith.Linear.Reify.0.Lean.Meta.Grind.Arith.Linear.assertNatCastNonneg | Lean.Meta.Tactic.Grind.Arith.Linear.Reify | Lean.Expr → Lean.Meta.Grind.Arith.Linear.LinearM Unit | null | true |
CategoryTheory.ObjectProperty.shiftClosure_le_iff | Mathlib.CategoryTheory.ObjectProperty.Shift | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (P Q : CategoryTheory.ObjectProperty C) {A : Type u_2}
[inst_1 : AddMonoid A] [inst_2 : CategoryTheory.HasShift C A] [Q.IsClosedUnderIsomorphisms] [Q.IsStableUnderShift A],
P.shiftClosure A ≤ Q ↔ P ≤ Q | null | true |
_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent.0.tendsto_euler_sin_prod'._simp_1_6 | Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
_private.Mathlib.Analysis.InnerProductSpace.Positive.0.ContinuousLinearMap.isPositive_iff_eq_sum_rankOne._simp_1_4 | Mathlib.Analysis.InnerProductSpace.Positive | ∀ {K : Type u_1} [inst : RCLike K] (r s : ℝ), ↑r * ↑s = ↑(r * s) | null | false |
CategoryTheory.PreZeroHypercover.isLimitSigmaOfIsColimitEquiv._proof_7 | Mathlib.CategoryTheory.Sites.CoproductSheafCondition | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {S : C} (E : CategoryTheory.PreZeroHypercover S)
[inst_1 : E.HasPullbacks] {c : CategoryTheory.Limits.Cofan E.X} (hc : CategoryTheory.Limits.IsColimit c)
[inst_2 : (E.sigmaOfIsColimit hc).HasPullbacks] (i j : E.I₀),
CategoryTheory.CategoryStruct.comp
... | null | false |
_private.Mathlib.Algebra.Order.Star.Pi.0.Pi.instStarOrderedRing.match_6 | Mathlib.Algebra.Order.Star.Pi | ∀ {ι : Type u_1} {A : ι → Type u_2} [inst : (i : ι) → NonUnitalSemiring (A i)] [inst_1 : (i : ι) → StarRing (A i)]
(x : (i : ι) → A i) (motive : (∃ y, ∀ (x_1 : ι), star (y x_1) * y x_1 = x x_1) → Prop)
(x_1 : ∃ y, ∀ (x_1 : ι), star (y x_1) * y x_1 = x x_1),
(∀ (y : (i : ι) → A i) (hy : ∀ (x_2 : ι), star (y x_2) *... | null | false |
_private.Mathlib.LinearAlgebra.RootSystem.Finite.G2.0.RootPairing.zero_le_pairingIn_of_root_sub_mem._simp_1_1 | Mathlib.LinearAlgebra.RootSystem.Finite.G2 | ∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a + -b = a - b | null | false |
CommHopfAlgCat.instCategory | Mathlib.Algebra.Category.CommHopfAlgCat | {R : Type u} → [inst : CommRing R] → CategoryTheory.Category.{v, max (v + 1) u} (CommHopfAlgCat R) | null | true |
TendstoUniformlyOn.div | Mathlib.Topology.Algebra.IsUniformGroup.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : UniformSpace α] [inst_1 : Group α] [IsUniformGroup α] {ι : Type u_3}
{l : Filter ι} {f f' : ι → β → α} {g g' : β → α} {s : Set β},
TendstoUniformlyOn f g l s → TendstoUniformlyOn f' g' l s → TendstoUniformlyOn (f / f') (g / g') l s | null | true |
Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateRight._proof_12 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.RotateRight | ∀ {w : ℕ}, 0 ≤ w | null | false |
CategoryTheory.Limits.HasCokernels.mk._flat_ctor | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C],
autoParam (∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.Limits.HasCokernel f)
CategoryTheory.Limits.HasCokernels.has_colimit._autoParam →
CategoryTheory.Limits.HasCokernels C | null | false |
AddGroupSeminorm.smul_sup | Mathlib.Analysis.Normed.Group.Seminorm | ∀ {R : Type u_1} {E : Type u_3} [inst : AddGroup E] [inst_1 : SMul R ℝ] [inst_2 : SMul R NNReal]
[inst_3 : IsScalarTower R NNReal ℝ] (r : R) (p q : AddGroupSeminorm E), r • (p ⊔ q) = r • p ⊔ r • q | null | true |
_private.Mathlib.NumberTheory.Padics.MahlerBasis.0.PadicInt.mahlerSeries_apply_nat._proof_1_1 | Mathlib.NumberTheory.Padics.MahlerBasis | ∀ {m n : ℕ}, m ≤ n → ∀ (i : ℕ), m < i + (n + 1) | null | false |
PowerSeries.coe_rescaleAlgHom | Mathlib.RingTheory.PowerSeries.Substitution | ∀ {R : Type u_2} [inst : CommRing R] (r : R), ↑(PowerSeries.rescaleAlgHom r) = PowerSeries.rescale r | null | true |
Lean.Meta.RefinedDiscrTree.Trie.node._flat_ctor | Mathlib.Lean.Meta.RefinedDiscrTree.Basic | {α : Type} →
Array α →
Option Lean.Meta.RefinedDiscrTree.TrieIndex →
Std.HashMap ℕ Lean.Meta.RefinedDiscrTree.TrieIndex →
Std.HashMap Lean.Meta.RefinedDiscrTree.Key Lean.Meta.RefinedDiscrTree.TrieIndex →
Array (Lean.Meta.RefinedDiscrTree.LazyEntry × α) → Lean.Meta.RefinedDiscrTree.Trie α | null | false |
Equiv.lattice._proof_2 | Mathlib.Order.Lattice | ∀ {α : Type u_2} {β : Type u_1} (e : α ≃ β) [inst : Lattice β] {x y : α}, e x < e y ↔ e x < e y | null | false |
Lean.Lsp.WorkspaceSymbolParams.noConfusionType | Lean.Data.Lsp.LanguageFeatures | Sort u → Lean.Lsp.WorkspaceSymbolParams → Lean.Lsp.WorkspaceSymbolParams → Sort u | null | false |
NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace._proof_2 | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] (x x_1 : NumberField.mixedEmbedding.realSpace K),
(fun w => (x + x_1) ↑w - (↑(↑w).mult * ∑ w', (x + x_1) w') * (↑(Module.finrank ℚ K))⁻¹) =
(fun w => x ↑w - (↑(↑w).mult * ∑ w', x w') * (↑(Module.finrank ℚ K))⁻¹) + fun w =>
x_1 ↑w - (↑(↑w).mult * ∑ w... | null | false |
Int.lcm_mul_right_right | Init.Data.Int.Gcd | ∀ (a b : ℤ), a.lcm (a * b) = a.natAbs * b.natAbs | null | true |
UInt32.lt_add_one | Init.Data.UInt.Lemmas | ∀ {c : UInt32}, c ≠ -1 → c < c + 1 | null | true |
_private.Mathlib.Data.List.Triplewise.0.List.triplewise_iff_getElem._simp_1_2 | Mathlib.Data.List.Triplewise | ∀ {α : Type u_1} {R : α → α → Prop} {l : List α},
List.Pairwise R l = ∀ (i j : ℕ) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j] | null | false |
_private.Mathlib.Algebra.Algebra.Subalgebra.Basic.0.AlgHom.rangeRestrict_surjective.match_1_1 | Mathlib.Algebra.Algebra.Subalgebra.Basic | ∀ {R : Type u_2} {A : Type u_3} {B : Type u_1} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
[inst_3 : Semiring B] [inst_4 : Algebra R B] (f : A →ₐ[R] B) (_y : B) (motive : _y ∈ f.range → Prop)
(hy : _y ∈ f.range), (∀ (x : A) (hx : f.toRingHom x = _y), motive ⋯) → motive hy | null | false |
_private.Mathlib.Algebra.Order.Interval.Set.Instances.0.Set.Ioo.one_sub_mem._simp_1_2 | Mathlib.Algebra.Order.Interval.Set.Instances | ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LT α] [AddRightStrictMono α] {a b : α}, (0 < a - b) = (b < a) | null | false |
PrimeMultiset.prod | Mathlib.Data.PNat.Factors | PrimeMultiset → ℕ+ | The product of a `PrimeMultiset`, as a `ℕ+`. | true |
String.Slice.Pattern.ToForwardSearcher.mk | Init.Data.String.Pattern.Basic | {ρ : Type} →
{pat : ρ} →
{σ : outParam (String.Slice → Type)} →
((s : String.Slice) → Std.Iter (String.Slice.Pattern.SearchStep s)) → String.Slice.Pattern.ToForwardSearcher pat σ | null | true |
_private.Mathlib.Data.Nat.Choose.Multinomial.0.Nat.multinomial_congr_of_sdiff._proof_1_1 | Mathlib.Data.Nat.Choose.Multinomial | ∀ {α : Type u_1} [inst : DecidableEq α] {f : α → ℕ} {s t : Finset α}, s ⊆ t → ∀ a ∈ s \ t, f a = 0 | null | false |
Units.inv_eq_val_inv | Mathlib.Algebra.Group.Units.Defs | ∀ {α : Type u} [inst : Monoid α] (a : αˣ), a.inv = ↑a⁻¹ | null | true |
MeasureTheory.L2.instInnerSubtypeAEEqFunMemAddSubgroupLpOfNatENNReal | Mathlib.MeasureTheory.Function.L2Space | {α : Type u_1} →
{E : Type u_2} →
{𝕜 : Type u_4} →
[inst : RCLike 𝕜] →
{m : MeasurableSpace α} →
{μ : MeasureTheory.Measure α} →
[inst_1 : NormedAddCommGroup E] → [InnerProductSpace 𝕜 E] → Inner 𝕜 ↥(MeasureTheory.Lp E 2 μ) | null | true |
Nat.eq_two_pow_or_exists_odd_prime_and_dvd | Mathlib.Data.Nat.Factors | ∀ (n : ℕ), (∃ k, n = 2 ^ k) ∨ ∃ p, Nat.Prime p ∧ p ∣ n ∧ Odd p | null | true |
_private.Mathlib.Topology.EMetricSpace.BoundedVariation.0.BoundedVariationOn.tendsto_eVariationOn_Icc_zero_left._simp_1_2 | Mathlib.Topology.EMetricSpace.BoundedVariation | ∀ {α : Type u} [inst : LE α] [inst_1 : OrderBot α] {a : α}, (⊥ ≤ a) = True | null | false |
UpperHalfPlane.num._proof_1 | Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | NeZero (1 + 1) | null | false |
Lean.Elab.Tactic.BVDecide.Frontend.Normalize.getEqIffEnumToBitVecEqFor | Lean.Elab.Tactic.BVDecide.Frontend.Normalize.Enums | Lean.Name → Lean.MetaM Lean.Name | Assuming that `declName` is an enum inductive, construct a proof of
`∀ (x y : declName) : x = y ↔ x.enumToBitVec = y.enumToBitVec`.
| true |
_private.Mathlib.MeasureTheory.Group.Arithmetic.0.measurable_div_const'._simp_1_1 | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {inst : MeasurableSpace M} {inst_1 : Mul M} [self : MeasurableMul M] (c : M),
(Measurable fun x => x * c) = True | null | false |
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