name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.67M | allowCompletion bool 2
classes |
|---|---|---|---|
Nat.lcm.eq_1 | Mathlib.Algebra.Order.Antidiag.Nat | ∀ (m n : ℕ), m.lcm n = m * n / m.gcd n | true |
WithZero.exp_add | Mathlib.Algebra.GroupWithZero.WithZero | ∀ {M : Type u_4} [inst : AddMonoid M] (a b : M), WithZero.exp (a + b) = WithZero.exp a * WithZero.exp b | true |
PresheafOfModules.isColimitFreeYonedaCoproductsCokernelCofork._proof_6 | Mathlib.Algebra.Category.ModuleCat.Presheaf.Generator | ∀ {C : Type u_1} [inst : CategoryTheory.SmallCategory C] {R : CategoryTheory.Functor Cᵒᵖ RingCat}
(M : PresheafOfModules R),
CategoryTheory.Epi
{ X₁ := (CategoryTheory.Limits.kernel M.fromFreeYonedaCoproduct).freeYonedaCoproduct, X₂ := M.freeYonedaCoproduct,
X₃ := M, f := M.toFreeYonedaCoproduct, g :=... | false |
nonZeroDivisors.associated_coe._simp_1 | Mathlib.Algebra.GroupWithZero.NonZeroDivisors | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {a b : ↥(nonZeroDivisors M₀)}, Associated ↑a ↑b = Associated a b | false |
CompHaus.toCondensed | Mathlib.Condensed.Functors | CompHaus → CondensedSet | true |
IsCauSeq.cauchy₃ | Mathlib.Algebra.Order.CauSeq.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [inst_2 : IsStrictOrderedRing α]
[inst_3 : Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β},
IsCauSeq abv f → ∀ {ε : α}, 0 < ε → ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε | true |
Std.DHashMap.getKeyD_union_of_not_mem_right | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.DHashMap α β} [EquivBEq α]
[LawfulHashable α] {k fallback : α}, k ∉ m₂ → (m₁ ∪ m₂).getKeyD k fallback = m₁.getKeyD k fallback | true |
Lean.Parser.Term.macroArg | Lean.Parser.Term | Lean.Parser.Parser | true |
Std.TreeSet.Raw.get?_eq_some_get | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp],
t.WF → ∀ {a : α} (h' : a ∈ t), t.get? a = some (t.get a h') | true |
_private.Lean.Meta.Injective.0.Lean.Meta.mkInjectiveTheoremValue | Lean.Meta.Injective | Lean.Name → Lean.Expr → Lean.MetaM Lean.Expr | true |
Lean.Lsp.SignatureHelpParams.context? | Lean.Data.Lsp.LanguageFeatures | Lean.Lsp.SignatureHelpParams → Option Lean.Lsp.SignatureHelpContext | true |
Lean.IR.Expr.ctor.elim | Lean.Compiler.IR.Basic | {motive : Lean.IR.Expr → Sort u} →
(t : Lean.IR.Expr) →
t.ctorIdx = 0 → ((i : Lean.IR.CtorInfo) → (ys : Array Lean.IR.Arg) → motive (Lean.IR.Expr.ctor i ys)) → motive t | false |
USize.not_inj | Init.Data.UInt.Bitwise | ∀ {a b : USize}, ~~~a = ~~~b ↔ a = b | true |
_private.Mathlib.Data.Seq.Computation.0.Computation.LiftRelAux.match_1.splitter | Mathlib.Data.Seq.Computation | {α : Type u_1} →
{β : Type u_2} →
(motive : α ⊕ Computation α → β ⊕ Computation β → Sort u_3) →
(x : α ⊕ Computation α) →
(x_1 : β ⊕ Computation β) →
((a : α) → (b : β) → motive (Sum.inl a) (Sum.inl b)) →
((a : α) → (cb : Computation β) → motive (Sum.inl a) (Sum.inr cb)) →
... | true |
FreeAlgebra.star_ι | Mathlib.Algebra.Star.Free | ∀ {R : Type u_1} [inst : CommSemiring R] {X : Type u_2} (x : X), star (FreeAlgebra.ι R x) = FreeAlgebra.ι R x | true |
UniformSpace.ball | Mathlib.Topology.UniformSpace.Defs | {β : Type ub} → β → Set (β × β) → Set β | true |
CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_inv_app_app | Mathlib.CategoryTheory.Monoidal.DayConvolution | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {V : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} V]
[inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : CategoryTheory.MonoidalCategory V]
(F X : CategoryTheory.Functor C V)
(a :
(CategoryTheory.coyoneda.obj
(Opposite.op
... | true |
CategoryTheory.Precoverage.Saturate.below.pullback | Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] {J : CategoryTheory.Precoverage C}
{motive : (X : C) → (a : CategoryTheory.Sieve X) → J.Saturate X a → Prop} (X : C) (S : CategoryTheory.Sieve X)
(a : J.Saturate X S) (Y : C) (f : Y ⟶ X),
CategoryTheory.Precoverage.Saturate.below a → motive X S a → Ca... | true |
_private.Std.Data.DTreeMap.Internal.Operations.0.Std.DTreeMap.Internal.Impl.alter._proof_6 | Std.Data.DTreeMap.Internal.Operations | ∀ {α : Type u_1} {β : α → Type u_2}, Std.DTreeMap.Internal.Impl.leaf.size ≤ Std.DTreeMap.Internal.Impl.leaf.size + 1 | false |
Std.DTreeMap.Raw.containsThenInsertIfNew_snd | Std.Data.DTreeMap.Raw.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap.Raw α β cmp} [Std.TransCmp cmp],
t.WF → ∀ {k : α} {v : β k}, (t.containsThenInsertIfNew k v).2 = t.insertIfNew k v | true |
CategoryTheory.Under.pushout._proof_2 | Mathlib.CategoryTheory.Comma.Over.Pullback | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : CategoryTheory.Limits.HasPushoutsAlong f] (x : CategoryTheory.Under X) {x' : CategoryTheory.Under X}
{u : x ⟶ x'},
CategoryTheory.CategoryStruct.comp x.hom
(CategoryTheory.CategoryStruct.comp u.right (CategoryTheo... | false |
SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty | Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X : CategoryTheory.SimplicialObject C}
(s : SimplicialObject.Splitting X) [inst_1 : CategoryTheory.Preadditive C] (n : ℕ),
CategoryTheory.CategoryStruct.comp
(s.πSummand (SimplicialObject.Splitting.IndexSet.id (Opposite.op (SimplexCategory.mk n))... | true |
Aesop.handleNonfatalError | Aesop.Search.Main | {Q : Type} → [inst : Aesop.Queue Q] → Lean.MessageData → Aesop.SearchM Q (Array Lean.MVarId) | true |
Std.Rxo.IsAlwaysFinite.mk | Init.Data.Range.Polymorphic.PRange | ∀ {α : Type u} [inst : Std.PRange.UpwardEnumerable α] [inst_1 : LT α],
(∀ (init hi : α), ∃ n, (Std.PRange.succMany? n init).elim True fun x => ¬x < hi) → Std.Rxo.IsAlwaysFinite α | true |
Std.DHashMap.Internal.Raw₀.find?_toList_eq_some_iff_get?_eq_some | Std.Data.DHashMap.Internal.RawLemmas | ∀ {α : Type u} {β : α → Type v} (m : Std.DHashMap.Internal.Raw₀ α β) [inst : BEq α] [inst_1 : Hashable α]
[inst_2 : LawfulBEq α],
(↑m).WF → ∀ {k : α} {v : β k}, List.find? (fun x => x.fst == k) (↑m).toList = some ⟨k, v⟩ ↔ m.get? k = some v | true |
PartitionOfUnity.exists_finset_nhds | Mathlib.Topology.PartitionOfUnity | ∀ {ι : Type u} {X : Type v} [inst : TopologicalSpace X] (ρ : PartitionOfUnity ι X) (x₀ : X),
∃ I, ∀ᶠ (x : X) in nhds x₀, ∑ i ∈ I, (ρ i) x = 1 ∧ (Function.support fun x_1 => (ρ x_1) x) ⊆ ↑I | true |
SeparationQuotient.liftNormedAddGroupHomEquiv._proof_1 | Mathlib.Analysis.Normed.Group.SeparationQuotient | ∀ {M : Type u_1} [inst : SeminormedAddCommGroup M] {N : Type u_2} [inst_1 : SeminormedAddCommGroup N]
(g : NormedAddGroupHom (SeparationQuotient M) N) (x : M), ‖x‖ = 0 → (g.comp SeparationQuotient.normedMk) x = 0 | false |
CategoryTheory.BraidedCategory.ofBifunctor | Mathlib.CategoryTheory.Monoidal.Braided.Multifunctor | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
(β : CategoryTheory.MonoidalCategory.curriedTensor C ≅ (CategoryTheory.MonoidalCategory.curriedTensor C).flip) →
CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.curried... | true |
ArchimedeanClass.mk.eq_1 | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (a : M),
ArchimedeanClass.mk a = toAntisymmetrization (fun x1 x2 => x1 ≤ x2) (ArchimedeanOrder.of a) | true |
Std.ExtDTreeMap.forIn.congr_simp | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {δ : Type w} {m : Type w → Type w₂} [inst : Monad m]
[inst_1 : LawfulMonad m] [inst_2 : Std.TransCmp cmp] (f f_1 : (a : α) → β a → δ → m (ForInStep δ)),
f = f_1 →
∀ (init init_1 : δ),
init = init_1 →
∀ (t t_1 : Std.ExtDTreeMap α β cmp),
... | true |
isConjRoot_algHom_iff | Mathlib.FieldTheory.Minpoly.IsConjRoot | ∀ {R : Type u_1} {B : Type u_6} [inst : CommRing R] [inst_1 : Ring B] [inst_2 : Algebra R B] {A : Type u_7}
[inst_3 : DivisionRing A] [inst_4 : Algebra R A] [Nontrivial B] {x y : A} (f : A →ₐ[R] B),
IsConjRoot R (f x) (f y) ↔ IsConjRoot R x y | true |
Nat.instMeasurableSingletonClass | Mathlib.MeasureTheory.MeasurableSpace.Instances | MeasurableSingletonClass ℕ | true |
_private.Mathlib.Analysis.Convex.Deriv.0.StrictMonoOn.exists_slope_lt_deriv._simp_1_1 | Mathlib.Analysis.Convex.Deriv | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : PartialOrder G₀] [MulPosReflectLT G₀] {a b c : G₀},
0 < c → (b / c < a) = (b < a * c) | false |
List.findM?'.match_1 | Mathlib.Data.List.Defs | (motive : ULift.{u_1, 0} Bool → Sort u_2) →
(__discr : ULift.{u_1, 0} Bool) → ((px : Bool) → motive { down := px }) → motive __discr | false |
SeqCompactSpace.tendsto_subseq | Mathlib.Topology.Sequences | ∀ {X : Type u_1} [inst : TopologicalSpace X] [SeqCompactSpace X] (x : ℕ → X),
∃ a φ, StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a) | true |
Lean.Grind.Field.IsOrdered.mul_lt_mul_iff_of_pos_left | Init.Grind.Ordered.Field | ∀ {R : Type u} [inst : Lean.Grind.Field R] [inst_1 : LE R] [inst_2 : LT R] [Std.LawfulOrderLT R]
[inst_4 : Std.IsLinearOrder R] [Lean.Grind.OrderedRing R] {a b c : R}, 0 < c → (c * a < c * b ↔ a < b) | true |
Substring.Raw.ValidFor.atEnd | Batteries.Data.String.Lemmas | ∀ {l m r : List Char} {p : ℕ} {s : Substring.Raw},
Substring.Raw.ValidFor l m r s → (s.atEnd { byteIdx := p } = true ↔ p = String.utf8Len m) | true |
_private.Mathlib.GroupTheory.OrderOfElement.0.isMulTorsionFree_iff_not_isOfFinOrder._simp_1_1 | Mathlib.GroupTheory.OrderOfElement | ∀ {α : Type u_1} [inst : DivisionCommMonoid α] (a b : α) (n : ℕ), a ^ n / b ^ n = (a / b) ^ n | false |
List.Vector.scanl.eq_1 | Mathlib.Data.Vector.Basic | ∀ {α : Type u_1} {n : ℕ} {β : Type u_6} (f : β → α → β) (b : β) (v : List.Vector α n),
List.Vector.scanl f b v = ⟨List.scanl f b v.toList, ⋯⟩ | true |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic.0.WeierstrassCurve.Jacobian.X_eq_of_equiv._simp_1_2 | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | ∀ {R : Type r} [inst : CommRing R] (P : Fin 3 → R) (u : R), (u • P) 1 = u ^ 3 * P 1 | false |
instSelfSliceSubarrayDataSubarray | Init.Data.Array.Subarray | ∀ {α : Type u}, Std.Slice.Self (Std.Slice (Std.Slice.Internal.SubarrayData α)) (Subarray α) | true |
norm_div | Mathlib.Analysis.Normed.Field.Basic | ∀ {α : Type u_2} [inst : NormedDivisionRing α] (a b : α), ‖a / b‖ = ‖a‖ / ‖b‖ | true |
TietzeExtension.of_homeo | Mathlib.Topology.TietzeExtension | ∀ {Y : Type v} {Z : Type w} [inst : TopologicalSpace Y] [inst_1 : TopologicalSpace Z] [TietzeExtension Z] (e : Y ≃ₜ Z),
TietzeExtension Y | true |
CategoryTheory.Limits.Cotrident.mkHom._proof_1 | Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | ∀ {J : Type u_1} {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {X Y : C} {f : J → (X ⟶ Y)} [Nonempty J]
{s t : CategoryTheory.Limits.Cotrident f} (k : s.pt ⟶ t.pt),
CategoryTheory.CategoryStruct.comp s.π k = t.π →
∀ (j : CategoryTheory.Limits.WalkingParallelFamily J), CategoryTheory.CategoryStruc... | false |
CategoryTheory.Comonad.ComonadicityInternal.main_pair_coreflexive | Mathlib.CategoryTheory.Monad.Comonadicity | ∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₁, u₂} D]
{F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F ⊣ G) (A : adj.toComonad.Coalgebra),
CategoryTheory.IsCoreflexivePair (G.map A.a) (adj.unit.app (G.obj A.A)) | true |
Batteries.Tactic.Lint.LintVerbosity.low.elim | Batteries.Tactic.Lint.Frontend | {motive : Batteries.Tactic.Lint.LintVerbosity → Sort u} →
(t : Batteries.Tactic.Lint.LintVerbosity) → t.ctorIdx = 0 → motive Batteries.Tactic.Lint.LintVerbosity.low → motive t | false |
TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe._proof_1 | Mathlib.Topology.OpenPartialHomeomorph.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] (s : TopologicalSpace.Opens X), Topology.IsOpenEmbedding Subtype.val | false |
Equiv.Perm.sigmaCongrRightHom | Mathlib.Algebra.Group.End | {α : Type u_7} → (β : α → Type u_8) → ((a : α) → Equiv.Perm (β a)) →* Equiv.Perm ((a : α) × β a) | true |
ContMDiffMap.coeFnAlgHom._proof_7 | Mathlib.Geometry.Manifold.Algebra.SmoothFunctions | ∀ {𝕜 : Type u_3} [inst : NontriviallyNormedField 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_5} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_2}
[inst_4 : TopologicalSpace N] [inst_5 : ChartedSpace H N] {n : WithTop ℕ∞} {A : Type u_1} [inst_6... | false |
Lean.Lsp.CodeActionTriggerKind | Lean.Data.Lsp.CodeActions | Type | true |
_private.Lean.Meta.UnificationHint.0.Lean.Meta.tryUnificationHints.tryCandidate.match_3 | Lean.Meta.UnificationHint | (motive : Lean.LOption Lean.Expr → Sort u_1) →
(__do_lift : Lean.LOption Lean.Expr) →
((val : Lean.Expr) → motive (Lean.LOption.some val)) → ((x : Lean.LOption Lean.Expr) → motive x) → motive __do_lift | false |
StrictAntiOn.antitoneOn | Mathlib.Order.Monotone.Defs | ∀ {α : Type u} {β : Type v} [inst : PartialOrder α] [inst_1 : Preorder β] {f : α → β} {s : Set α},
StrictAntiOn f s → AntitoneOn f s | true |
Lean.Server.Watchdog.ServerContext.noConfusionType | Lean.Server.Watchdog | Sort u → Lean.Server.Watchdog.ServerContext → Lean.Server.Watchdog.ServerContext → Sort u | false |
Submodule.map_iInf | Mathlib.Algebra.Module.Submodule.Map | ∀ {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [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}
[inst_6 : RingHomSurjective σ₁₂] {ι : Sort u_9} [Nonempty ι] {p : ι → Submodule R M} (f : M... | true |
TensorialAt.add | Mathlib.Geometry.Manifold.VectorBundle.Tensoriality | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_5} [inst_6 : NormedAddCommG... | true |
StarSubalgebra.subtype._proof_6 | Mathlib.Algebra.Star.Subalgebra | ∀ {R : Type u_2} {A : Type u_1} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : Semiring A]
[inst_3 : StarRing A] [inst_4 : Algebra R A] [inst_5 : StarModule R A] (S : StarSubalgebra R A) (x : ↥S),
↑(star x) = ↑(star x) | false |
Matrix.toSquareBlockProp | Mathlib.Data.Matrix.Block | {m : Type u_2} → {α : Type u_12} → Matrix m m α → (p : m → Prop) → Matrix { a // p a } { a // p a } α | true |
AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace | Mathlib.Geometry.RingedSpace.LocallyRingedSpace | CategoryTheory.Functor AlgebraicGeometry.LocallyRingedSpace (AlgebraicGeometry.SheafedSpace CommRingCat) | true |
Lean.Elab.Do.ControlStack.restoreCont | Lean.Elab.Do.Control | Lean.Elab.Do.ControlStack → Lean.Elab.Do.DoElemCont → Lean.Elab.Do.DoElabM Lean.Elab.Do.DoElemCont | true |
instAssociativeMax_mathlib | Mathlib.Order.Lattice | ∀ {α : Type u} [inst : SemilatticeSup α], Std.Associative fun x1 x2 => x1 ⊔ x2 | true |
Lean.Parser.Command.declModifiers | Lean.Parser.Command | Bool → Lean.Parser.Parser | true |
_private.Mathlib.CategoryTheory.Bicategory.Free.0.CategoryTheory.FreeBicategory.«_aux_Mathlib_CategoryTheory_Bicategory_Free___macroRules__private_Mathlib_CategoryTheory_Bicategory_Free_0_CategoryTheory_FreeBicategory_termλ__1» | Mathlib.CategoryTheory.Bicategory.Free | Lean.Macro | false |
_private.Mathlib.Data.Finset.Sum.0.Finset.disjSum_subset._simp_1_1 | Mathlib.Data.Finset.Sum | ∀ {α : Type u_1} {s₁ s₂ : Finset α}, (s₁ ⊆ s₂) = ∀ ⦃x : α⦄, x ∈ s₁ → x ∈ s₂ | false |
CategoryTheory.CategoryOfElements.toStructuredArrow._proof_1 | Mathlib.CategoryTheory.Elements | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] (F : CategoryTheory.Functor C (Type u_1))
{X Y : F.Elements} (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp (CategoryTheory.StructuredArrow.mk fun x => X.snd).hom (F.map ↑f) =
(CategoryTheory.StructuredArrow.mk fun x => Y.snd).hom | false |
CategoryTheory.Functor.splitMonoEquiv._proof_4 | Mathlib.CategoryTheory.Functor.EpiMono | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_2, u_4} D] (F : CategoryTheory.Functor C D) {X Y : C} (f : X ⟶ Y)
[inst_2 : F.Full] [inst_3 : F.Faithful],
Function.LeftInverse (fun s => { retraction := F.preimage s.retraction, id := ⋯ }) fun f_1 =... | false |
AddCommGroup.modEq_iff_toIcoDiv_eq_toIocDiv_add_one | Mathlib.Algebra.Order.ToIntervalMod | ∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : LinearOrder α] [inst_2 : IsOrderedAddMonoid α] [hα : Archimedean α]
{p : α} (hp : 0 < p) {a b : α}, a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1 | true |
_private.Mathlib.CategoryTheory.WithTerminal.Basic.0.CategoryTheory.WithInitial.opEquiv.match_15.eq_3 | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C]
(motive : (x y : CategoryTheory.WithTerminal Cᵒᵖ) → (x ⟶ y) → Sort u_3)
(x : CategoryTheory.WithTerminal.star ⟶ CategoryTheory.WithTerminal.star)
(h_1 :
(x y : C) →
(f : CategoryTheory.WithTerminal.of (Opposite.op x) ⟶ CategoryTheory.WithTer... | true |
SetLike.smul_subset_self | Mathlib.GroupTheory.GroupAction.SubMulAction | ∀ {S : Type u_1} {R : Type u_2} {M : Type u_3} [inst : SetLike S M] [inst_1 : SMul R M] [SMulMemClass S R M] (r : R)
(s : S), r • ↑s ⊆ ↑s | true |
Std.Iterators.Types.StepSizeIterator.instProductive | Std.Data.Iterators.Combinators.Monadic.StepSize | ∀ {α : Type u_1} {m : Type u_1 → Type u_2} {β : Type u_1} [inst : Std.Iterator α m β] [inst_1 : Std.IteratorAccess α m]
[inst_2 : Monad m] [Std.Iterators.Productive α m],
Std.Iterators.Productive (Std.Iterators.Types.StepSizeIterator α m β) m | true |
Lean.Server.Watchdog.RequestQueueMap.queue | Lean.Server.Watchdog | Lean.Server.Watchdog.RequestQueueMap → Std.TreeMap ℕ (Lean.JsonRpc.RequestID × Lean.JsonRpc.Request Lean.Json) compare | true |
StarAlgEquiv.mk.inj | Mathlib.Algebra.Star.StarAlgHom | ∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {inst : Add A} {inst_1 : Add B} {inst_2 : Mul A} {inst_3 : Mul B}
{inst_4 : SMul R A} {inst_5 : SMul R B} {inst_6 : Star A} {inst_7 : Star B} {toStarRingEquiv : A ≃⋆+* B}
{map_smul' : ∀ (r : R) (a : A), toStarRingEquiv.toFun (r • a) = r • toStarRingEquiv.toFun a}
{to... | true |
_private.Mathlib.NumberTheory.LSeries.PrimesInAP.0.ArithmeticFunction.vonMangoldt.not_summable_residueClass_prime_div._simp_1_2 | Mathlib.NumberTheory.LSeries.PrimesInAP | ∀ {α : Type u} [inst : LinearOrder α] {a b c : α}, (a < min b c) = (a < b ∧ a < c) | false |
CategoryTheory.GradedObject.comapEq_trans | Mathlib.CategoryTheory.GradedObject | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] {β γ : Type w} {f g h : β → γ} (k : f = g) (l : g = h),
CategoryTheory.GradedObject.comapEq C ⋯ =
CategoryTheory.GradedObject.comapEq C k ≪≫ CategoryTheory.GradedObject.comapEq C l | true |
Lean.Doc.Inline.footnote.noConfusion | Lean.DocString.Types | {i : Type u} →
{P : Sort u_1} →
{name : String} →
{content : Array (Lean.Doc.Inline i)} →
{name' : String} →
{content' : Array (Lean.Doc.Inline i)} →
Lean.Doc.Inline.footnote name content = Lean.Doc.Inline.footnote name' content' →
(name = name' → content ≍ conten... | false |
Lean.initFn._@.Lean.Util.Trace.3737982518._hygCtx._hyg.4 | Lean.Util.Trace | IO (Lean.Option Bool) | false |
Lean.Grind.Linarith.imp_eq_cert | Init.Grind.Ordered.Linarith | Lean.Grind.Linarith.Poly → Lean.Grind.Linarith.Var → Lean.Grind.Linarith.Var → Bool | true |
ExistsAndEq.withExistsElimAlongPath | Mathlib.Tactic.Simproc.ExistsAndEq | {u : Lean.Level} →
{α : Q(Sort u)} →
{P goal : Q(Prop)} →
Q(«$P») →
{a a' : Q(«$α»)} →
List ExistsAndEq.VarQ →
ExistsAndEq.Path → (Q(«$a» = «$a'») → List ExistsAndEq.HypQ → Lean.MetaM Q(«$goal»)) → Lean.MetaM Q(«$goal») | true |
ProbabilityTheory.gaussianPDFReal_mul | Mathlib.Probability.Distributions.Gaussian.Real | ∀ {μ : ℝ} {v : NNReal} {c : ℝ},
c ≠ 0 →
∀ (x : ℝ),
ProbabilityTheory.gaussianPDFReal μ v (c * x) =
|c⁻¹| * ProbabilityTheory.gaussianPDFReal (c⁻¹ * μ) (⟨(c ^ 2)⁻¹, ⋯⟩ * v) x | true |
Ideal.IsTwoSided.mk | Mathlib.RingTheory.Ideal.Defs | ∀ {α : Type u} [inst : Semiring α] {I : Ideal α}, (∀ {a : α} (b : α), a ∈ I → a * b ∈ I) → I.IsTwoSided | true |
RestrictedProduct.mapAlongMonoidHom._proof_1 | Mathlib.Topology.Algebra.RestrictedProduct.Basic | ∀ {ι₁ : Type u_4} {ι₂ : Type u_1} (R₁ : ι₁ → Type u_5) (R₂ : ι₂ → Type u_2) {𝓕₁ : Filter ι₁} {𝓕₂ : Filter ι₂}
{S₁ : ι₁ → Type u_6} {S₂ : ι₂ → Type u_3} [inst : (i : ι₁) → SetLike (S₁ i) (R₁ i)]
[inst_1 : (j : ι₂) → SetLike (S₂ j) (R₂ j)] {B₁ : (i : ι₁) → S₁ i} {B₂ : (j : ι₂) → S₂ j} (f : ι₂ → ι₁)
(hf : Filter.T... | false |
CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_X₂ | Mathlib.Algebra.Homology.SpectralObject.Page | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ)
(hn₁ : autoParam (n₀ + 1 = n₁) Categ... | true |
CategoryTheory.Limits.MonoFactorisation.ext | Mathlib.CategoryTheory.Limits.Shapes.Images | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} {f : X ⟶ Y}
{F F' : CategoryTheory.Limits.MonoFactorisation f} (hI : F.I = F'.I),
F.m = CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom hI) F'.m → F = F' | true |
_private.Mathlib.Topology.MetricSpace.GromovHausdorffRealized.0.GromovHausdorff.candidates_le_maxVar | Mathlib.Topology.MetricSpace.GromovHausdorffRealized | ∀ {X : Type u} {Y : Type v} [inst : MetricSpace X] [inst_1 : MetricSpace Y] {f : GromovHausdorff.ProdSpaceFun✝ X Y}
{x y : X ⊕ Y}, f ∈ GromovHausdorff.candidates X Y → f (x, y) ≤ ↑(GromovHausdorff.maxVar✝ X Y) | true |
AffineSubspace.wSameSide_vadd_right_iff | Mathlib.Analysis.Convex.Side | ∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : CommRing R] [inst_1 : PartialOrder R]
[inst_2 : IsStrictOrderedRing R] [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : AddTorsor V P]
{s : AffineSubspace R P} {x y : P} {v : V}, v ∈ s.direction → (s.WSameSide x (v +ᵥ y) ↔ s.WSameSide x y) | true |
CategoryTheory.evaluationAdjunctionLeft | Mathlib.CategoryTheory.Adjunction.Evaluation | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
(D : Type u₂) →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
[inst_2 : ∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] →
(c : C) → (CategoryTheory.evaluation C D).obj c ⊣ CategoryTheory.evaluationRightAdjoint... | true |
Lean.findModuleOf? | Lean.MonadEnv | {m : Type → Type} → [Monad m] → [Lean.MonadEnv m] → [Lean.MonadError m] → Lean.Name → m (Option Lean.Name) | true |
_private.Init.Data.BitVec.Lemmas.0.BitVec.toNat_signExtend_of_le._proof_1_3 | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {v : ℕ} (i k : ℕ), ¬(i < w ∨ w ≤ i ∧ i < w + k ∨ w + k ≤ i) → False | false |
RestrictedProduct.mapAlongAddMonoidHom | Mathlib.Topology.Algebra.RestrictedProduct.Basic | {ι₁ : Type u_3} →
{ι₂ : Type u_4} →
(R₁ : ι₁ → Type u_5) →
(R₂ : ι₂ → Type u_6) →
{𝓕₁ : Filter ι₁} →
{𝓕₂ : Filter ι₂} →
{S₁ : ι₁ → Type u_7} →
{S₂ : ι₂ → Type u_8} →
[inst : (i : ι₁) → SetLike (S₁ i) (R₁ i)] →
[inst_1 : (j : ι₂)... | true |
OrderDual.instTrichotomousLt | Mathlib.Order.OrderDual | ∀ {α : Type u_1} [inst : LT α] [T : Std.Trichotomous LT.lt], Std.Trichotomous LT.lt | true |
_private.Mathlib.Combinatorics.Additive.AP.Three.Behrend.0.Behrend.le_sqrt_log._simp_1_3 | Mathlib.Combinatorics.Additive.AP.Three.Behrend | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 4] [NeZero 4], (4 = 0) = False | false |
Std.HashMap.Raw.getElem?_filter' | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} [inst : BEq α] [inst_1 : Hashable α] {m : Std.HashMap.Raw α β} [LawfulBEq α]
{f : α → β → Bool} {k : α}, m.WF → (Std.HashMap.Raw.filter f m)[k]? = Option.filter (f k) m[k]? | true |
Real.arcsin_nonpos | Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | ∀ {x : ℝ}, Real.arcsin x ≤ 0 ↔ x ≤ 0 | true |
MulOpposite.instSemifield._proof_3 | Mathlib.Algebra.Field.Opposite | ∀ {α : Type u_1} [inst : Semifield α] (n : ℕ) (a : αᵐᵒᵖ),
DivisionSemiring.zpow (↑n.succ) a = DivisionSemiring.zpow (↑n) a * a | false |
posMulReflectLT_iff | Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs | ∀ (α : Type u_1) [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α],
PosMulReflectLT α ↔ ContravariantClass { x // 0 ≤ x } α (fun x y => ↑x * y) fun x1 x2 => x1 < x2 | true |
SubalgebraClass.toAlgebra._proof_6 | Mathlib.Algebra.Algebra.Subalgebra.Basic | ∀ {S : Type u_2} {R : Type u_3} {A : Type u_1} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
[inst_3 : SetLike S A] [SubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S) (r : R), (algebraMap R A) r ∈ s | false |
Lean.Name.isAtomic | Lean.Data.Name | Lean.Name → Bool | true |
Lean.LibrarySuggestions.Suggestion.name | Lean.LibrarySuggestions.Basic | Lean.LibrarySuggestions.Suggestion → Lean.Name | true |
Set.addAntidiagonal_mono_left | Mathlib.Data.Set.MulAntidiagonal | ∀ {α : Type u_1} [inst : Add α] {s₁ s₂ t : Set α} {a : α}, s₁ ⊆ s₂ → s₁.addAntidiagonal t a ⊆ s₂.addAntidiagonal t a | true |
LinearEquiv.ofSubmodules_symm_apply | Mathlib.Algebra.Module.Submodule.Equiv | ∀ {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] {module_M : Module R M} {module_M₂ : Module R₂ M₂}
{σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (e : ... | true |
CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject.ctorIdx | Mathlib.CategoryTheory.Monoidal.Free.Coherence | {C : Type u} → CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C → ℕ | false |
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