name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Substring.Raw.toString | Init.Data.String.Substring | Substring.Raw → String | {}
Copies the region of the underlying string pointed to by a substring into a fresh string.
| true |
Lean.Parser.antiquotExpr | Lean.Parser.Basic | Lean.Parser.Parser | null | true |
ZFSet.vonNeumann_subset_vonNeumann_iff | Mathlib.SetTheory.ZFC.VonNeumann | ∀ {a b : Ordinal.{u}}, ZFSet.vonNeumann a ⊆ ZFSet.vonNeumann b ↔ a ≤ b | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd.0.SSet.prodStdSimplex.weakRankFunction._proof_12 | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | ∀ {m : ℕ} (k : Fin (m + 1)) (n d : ℕ) (is : Fin (d + 1))
(s :
(CategoryTheory.MonoidalCategoryStruct.tensorObj (SSet.stdSimplex.obj { len := m + 1 })
(SSet.stdSimplex.obj { len := n })).obj
(Opposite.op { len := d + 1 }))
(hs₁ :
s ∈
(CategoryTheory.MonoidalCategoryStruct.tensorObj (SSe... | null | false |
Finpartition.ofSubset._proof_2 | Mathlib.Order.Partition.Finpartition | ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : OrderBot α] {a : α} (P : Finpartition a) {parts : Finset α},
parts ⊆ P.parts → ⊥ ∈ parts → False | null | false |
_private.Mathlib.Topology.Algebra.AsymptoticCone.0.asymptoticCone_union._simp_1_2 | 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] {v : V} {s : Set P},
(v ∈ asymptoticCone k s) = ∃ᶠ (p : P) in AffineSpace.asymptoticNhds k P v, p ∈ s | null | false |
String.Slice.Pattern.ToForwardSearcher.DefaultForwardSearcher.noConfusion | Init.Data.String.Pattern.Basic | {P : Sort u} →
{ρ : Type} →
{pat : ρ} →
{s : String.Slice} →
{t : String.Slice.Pattern.ToForwardSearcher.DefaultForwardSearcher pat s} →
{ρ' : Type} →
{pat' : ρ'} →
{s' : String.Slice} →
{t' : String.Slice.Pattern.ToForwardSearcher.DefaultForwardSe... | null | false |
_private.Lean.Server.FileWorker.RequestHandling.0.Lean.Server.FileWorker.NamespaceEntry.finish.match_1 | Lean.Server.FileWorker.RequestHandling | (motive : Option Lean.Syntax → Sort u_1) →
(endStx : Option Lean.Syntax) → ((endStx : Lean.Syntax) → motive (some endStx)) → (Unit → motive none) → motive endStx | null | false |
ContMDiffAt.eq_1 | Mathlib.Geometry.Manifold.ContMDiff.Defs | ∀ {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] {E' : Type u_5} [inst_6 : NormedAddComm... | null | true |
MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff | Mathlib.MeasureTheory.Measure.Real | ∀ {α : Type u_1} {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},
s₁ ⊆ s₂ →
s₂ ⊆ s₃ →
μ.real (s₃ \ s₁) = 0 →
autoParam (μ (s₃ \ s₁) ≠ ⊤) MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_sdiff._auto_1 →
μ.real s₁ = μ.real s₂ | **Alias** of `MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_sdiff`. | true |
Finsupp.instNonUnitalRing._proof_4 | Mathlib.Data.Finsupp.Pointwise | ∀ {α : Type u_1} {β : Type u_2} [inst : NonUnitalRing β] (g₁ g₂ : α →₀ β), ⇑(g₁ * g₂) = ⇑g₁ * ⇑g₂ | null | false |
CategoryTheory.Functor.Accessible.Limits.isColimitMapCocone.injective | Mathlib.CategoryTheory.Presentable.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {K : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} K]
{F : CategoryTheory.Functor K (CategoryTheory.Functor C (Type w'))} (c : CategoryTheory.Limits.Cone F)
(hc : (Y : C) → CategoryTheory.Limits.IsLimit (((CategoryTheory.evaluation C (Type w')).obj Y).m... | null | true |
KaehlerDifferential.map_D | Mathlib.RingTheory.Kaehler.Basic | ∀ (R : Type u) (S : Type v) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (A : Type u_2)
(B : Type u_3) [inst_3 : CommRing A] [inst_4 : CommRing B] [inst_5 : Algebra R A] [inst_6 : Algebra A B]
[inst_7 : Algebra S B] [inst_8 : Algebra R B] [inst_9 : IsScalarTower R A B] [inst_10 : IsScalarTower R... | null | true |
ArchimedeanClass.FiniteResidueField.ofArchimedean._proof_3 | Mathlib.Algebra.Order.Ring.StandardPart | ∀ {K : Type u_1} [inst : LinearOrder K] [inst_1 : Field K] [inst_2 : IsOrderedRing K] {R : Type u_2}
[inst_3 : LinearOrder R] [inst_4 : CommRing R] [inst_5 : IsStrictOrderedRing R] [inst_6 : Archimedean R]
(f : R →+*o K) (x y : R),
ArchimedeanClass.FiniteResidueField.mk (ArchimedeanClass.FiniteElement.mk (f (x * ... | null | false |
CategoryTheory.SymmetricCategory.toBraidedCategory | Mathlib.CategoryTheory.Monoidal.Braided.Basic | {C : Type u} →
{inst : CategoryTheory.Category.{v, u} C} →
{inst_1 : CategoryTheory.MonoidalCategory C} →
[self : CategoryTheory.SymmetricCategory C] → CategoryTheory.BraidedCategory C | null | true |
WittVector.fromPadicInt | Mathlib.RingTheory.WittVector.Compare | (p : ℕ) → [hp : Fact (Nat.Prime p)] → ℤ_[p] →+* WittVector p (ZMod p) | `fromPadicInt` uses `WittVector.lift` to lift `TruncatedWittVector.zmodEquivTrunc`
composed with `PadicInt.toZModPow` to a ring hom `ℤ_[p] →+* 𝕎 (ZMod p)`.
| true |
_private.Mathlib.MeasureTheory.Measure.Haar.OfBasis.0.Module.Basis.parallelepiped._simp_5 | Mathlib.MeasureTheory.Measure.Haar.OfBasis | ∀ (α : Sort u), (∀ (a : α), True) = True | null | false |
Option.toArray_eq_empty_iff._simp_1 | Init.Data.Option.Array | ∀ {α : Type u_1} {o : Option α}, (o.toArray = #[]) = (o = none) | null | false |
_private.Lean.Meta.InferType.0.Lean.Meta.typeFormerTypeLevel.go.match_1 | Lean.Meta.InferType | (motive : Lean.Expr → Sort u_1) →
(type : Lean.Expr) →
((l : Lean.Level) → motive (Lean.Expr.sort l)) →
((binderName : Lean.Name) →
(binderType body : Lean.Expr) →
(binderInfo : Lean.BinderInfo) → motive (Lean.Expr.forallE binderName binderType body binderInfo)) →
((x : Lean.Ex... | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.Trails.0.SimpleGraph.Walk.IsEulerian.card_filter_odd_degree._simp_1_3 | Mathlib.Combinatorics.SimpleGraph.Trails | ∀ {α : Sort u_1} {p : α → Prop}, (¬∀ (x : α), p x) = ∃ x, ¬p x | null | false |
TypeVec.toSubtype'._sunfold | Mathlib.Data.TypeVec | {n : ℕ} →
{α : TypeVec.{u} n} →
(p : (α.prod α).Arrow (TypeVec.repeat n Prop)) →
TypeVec.Arrow (fun i => { x // TypeVec.ofRepeat (p i (TypeVec.prod.mk i x.1 x.2)) }) (TypeVec.Subtype_ p) | null | false |
_private.Mathlib.Algebra.Lie.Weights.IsSimple.0.LieAlgebra.IsKilling.chi_not_in_q_aux | Mathlib.Algebra.Lie.Weights.IsSimple | ∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
[inst_4 : FiniteDimensional K L] [inst_5 : LieAlgebra.IsKilling K L] {H : LieSubalgebra K L}
[inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L] (q : Submodule K (Module... | null | true |
CategoryTheory.MorphismProperty.exists_isPushout_of_isFiltered | Mathlib.CategoryTheory.MorphismProperty.Ind | ∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {P : CategoryTheory.MorphismProperty C}
[self : P.PreIndSpreads] {J : Type w} [inst_1 : CategoryTheory.SmallCategory J] [CategoryTheory.IsFiltered J]
{D : CategoryTheory.Functor J C} {c : CategoryTheory.Limits.Cocone D} (x : CategoryTheory.Limits.IsColimit c)... | **Alias** of `CategoryTheory.MorphismProperty.PreIndSpreads.exists_isPushout`. | true |
PseudoMetricSpace.ofDistTopology._proof_3 | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u_1} [inst : TopologicalSpace α] (dist : α → α → ℝ) (dist_self : ∀ (x : α), dist x x = 0)
(dist_comm : ∀ (x y : α), dist x y = dist y x) (dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z)
(H : ∀ (s : Set α), IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : α), dist x y < ε → y ∈ s), uniformity α = unifo... | null | false |
Sum.LiftRel.isRight_right | Mathlib.Data.Sum.Basic | ∀ {α : Type u} {β : Type v} {γ : Type u_1} {δ : Type u_2} {r : α → γ → Prop} {s : β → δ → Prop} {y : γ ⊕ δ} {b : β},
Sum.LiftRel r s (Sum.inr b) y → y.isRight = true | null | true |
Stream'.Seq.set_cons_zero | Mathlib.Data.Seq.Basic | ∀ {α : Type u} (hd : α) (tl : Stream'.Seq α) (hd' : α), (Stream'.Seq.cons hd tl).set 0 hd' = Stream'.Seq.cons hd' tl | null | true |
Int8.neg_sub | Init.Data.SInt.Lemmas | ∀ {a b : Int8}, -(a - b) = b - a | null | true |
LinearMap.comap_prod_prod | Mathlib.LinearAlgebra.Prod | ∀ {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module R M₂]
[inst_6 : Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) (p : Submodule R M₂) (q : Submodule R M₃),
Submodule.... | null | true |
Matroid.loopyOn_isBasis_iff._simp_1 | Mathlib.Combinatorics.Matroid.Constructions | ∀ {α : Type u_1} {E I X : Set α}, (Matroid.loopyOn E).IsBasis I X = (I = ∅ ∧ X ⊆ E) | null | false |
Metric.thickening_thickening_subset | Mathlib.Topology.MetricSpace.Thickening | ∀ {α : Type u} [inst : PseudoEMetricSpace α] (ε δ : ℝ) (s : Set α),
Metric.thickening ε (Metric.thickening δ s) ⊆ Metric.thickening (ε + δ) s | For the equality, see `thickening_thickening`. | true |
ENNReal.inv_rpow | Mathlib.Analysis.SpecialFunctions.Pow.NNReal | ∀ (x : ENNReal) (y : ℝ), x⁻¹ ^ y = (x ^ y)⁻¹ | null | true |
_private.Lean.Elab.Tactic.Do.ProofMode.Exact.0.Lean.Elab.Tactic.Do.ProofMode.MGoal.exactPure._sparseCasesOn_1 | Lean.Elab.Tactic.Do.ProofMode.Exact | {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 |
MvPowerSeries.instNontrivial | Mathlib.RingTheory.MvPowerSeries.Basic | ∀ {σ : Type u_1} {R : Type u_2} [Nontrivial R], Nontrivial (MvPowerSeries σ R) | null | true |
CategoryTheory.NatTrans.IsMonoidal.instHomFunctorAssociator | Mathlib.CategoryTheory.Monoidal.NaturalTransformation | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C] {D : Type u₂}
[inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.MonoidalCategory D] {E : Type u₃}
[inst_4 : CategoryTheory.Category.{v₃, u₃} E] [inst_5 : CategoryTheory.MonoidalCategory E] {... | null | true |
AlgebraicGeometry.Scheme.Pullback.base_affine_hasPullback | Mathlib.AlgebraicGeometry.Pullbacks | ∀ {C : CommRingCat} {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ AlgebraicGeometry.Spec C)
(g : Y ⟶ AlgebraicGeometry.Spec C), CategoryTheory.Limits.HasPullback f g | null | true |
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.instCategory | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic | {A : Type u₁} →
{B : Type u₂} →
{C : Type u₃} →
[inst : CategoryTheory.Category.{v₁, u₁} A] →
[inst_1 : CategoryTheory.Category.{v₂, u₂} B] →
[inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
{F : CategoryTheory.Functor A B} →
{G : CategoryTheory.Functor C B} →
... | null | true |
_private.Mathlib.Data.Option.NAry.0.Option.map₂_assoc._proof_1_1 | Mathlib.Data.Option.NAry | ∀ {α : Type u_4} {β : Type u_5} {γ : Type u_3} {δ : Type u_2} {a : Option α} {b : Option β} {c : Option γ}
{ε : Type u_1} {ε' : Type u_6} {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'},
(∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)) →
Option.map₂ f (Option.map₂ g a b) c = Option.m... | null | false |
ZFSet.Subset | Mathlib.SetTheory.ZFC.Basic | ZFSet.{u} → ZFSet.{u} → Prop | `x ⊆ y` as ZFC sets means that all members of `x` are members of `y`. | true |
Lean.Meta.Grind.instInhabitedEMatchDiagNode.default | Lean.Meta.Tactic.Grind.Types | Lean.Meta.Grind.EMatchDiagNode | null | true |
String.apply_skipSuffixWhile_bool_eq_false | Init.Data.String.Lemmas.Pattern.TakeDrop.Pred | ∀ {p : Char → Bool} {s : String} {h : s.skipSuffixWhile p ≠ s.startPos},
p (((s.skipSuffixWhile p).prev h).get ⋯) = false | null | true |
_private.Lean.Meta.Sym.Canon.0.Lean.Meta.Sym.Canon.instReprShouldCanonResult | Lean.Meta.Sym.Canon | Repr Lean.Meta.Sym.Canon.ShouldCanonResult✝ | null | true |
CategoryTheory.Limits.sigmaConst_obj_map | Mathlib.CategoryTheory.Limits.Shapes.Products | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasCoproducts C] (X : C)
{X_1 Y : Type w} (f : X_1 ⟶ Y),
(CategoryTheory.Limits.sigmaConst.obj X).map f =
CategoryTheory.Limits.Sigma.map' ⇑(CategoryTheory.ConcreteCategory.hom f) fun x =>
CategoryTheory.CategoryStruc... | null | true |
LieSubalgebra.coe_incl' | Mathlib.Algebra.Lie.Subalgebra | ∀ {R : Type u} {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (L' : LieSubalgebra R L),
⇑L'.incl' = Subtype.val | null | true |
Std.ExtDTreeMap.Const.get_insertMany_list_of_mem | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : Std.ExtDTreeMap α (fun x => β) cmp} [inst : Std.TransCmp cmp]
{l : List (α × β)} {k k' : α},
cmp k k' = Ordering.eq →
∀ {v : β},
List.Pairwise (fun a b => ¬cmp a.1 b.1 = Ordering.eq) l →
(k, v) ∈ l →
∀ {h' : k' ∈ Std.ExtDTreeMap.C... | null | true |
_private.Mathlib.Topology.Algebra.Constructions.0.Units.isOpenMap_map.match_1_9 | Mathlib.Topology.Algebra.Constructions | ∀ {M : Type u_1} {N : Type u_2} [inst : Monoid M] [inst_1 : Monoid N] {f : M →* N} (U : Set (M × Mᵐᵒᵖ)) (y : Nˣ)
(motive : (∃ x, (↑x, MulOpposite.op ↑x⁻¹) ∈ U ∧ (Units.map f) x = y) → Prop)
(x : ∃ x, (↑x, MulOpposite.op ↑x⁻¹) ∈ U ∧ (Units.map f) x = y),
(∀ (x : Mˣ) (hxV : (↑x, MulOpposite.op ↑x⁻¹) ∈ U) (hx : (Uni... | null | false |
KaehlerDifferential.linearCombination_surjective | Mathlib.RingTheory.Kaehler.Basic | ∀ (R : Type u) (S : Type v) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],
Function.Surjective ⇑(Finsupp.linearCombination S ⇑(KaehlerDifferential.D R S)) | null | true |
UniformEquiv.refl_symm | Mathlib.Topology.UniformSpace.Equiv | ∀ {α : Type u} [inst : UniformSpace α], (UniformEquiv.refl α).symm = UniformEquiv.refl α | null | true |
_private.Init.Data.List.Nat.TakeDrop.0.List.mem_drop_iff_getElem._proof_1_2 | Init.Data.List.Nat.TakeDrop | ∀ {α : Type u_1} {i : ℕ} {l : List α} (i_1 : ℕ), i_1 + i < l.length → ¬i_1 < l.length - i → False | null | false |
Lean.Grind.OrderConfig.funCC._inherited_default | Init.Grind.Config | Bool | null | false |
Lean.Widget.RpcEncodablePacket._sizeOf_1._@.Lean.Widget.InteractiveDiagnostic.1765450820._hygCtx._hyg.1 | Lean.Widget.InteractiveDiagnostic | Lean.Widget.RpcEncodablePacket✝ → ℕ | null | false |
Turing.TM0.Stmt.move.elim | Mathlib.Computability.TuringMachine.PostTuringMachine | {Γ : Type u_1} →
{motive : Turing.TM0.Stmt Γ → Sort u} →
(t : Turing.TM0.Stmt Γ) → t.ctorIdx = 0 → ((a : Turing.Dir) → motive (Turing.TM0.Stmt.move a)) → motive t | null | false |
Option.instTransOrd | Init.Data.Order.Ord | ∀ {α : Type u_1} [inst : Ord α] [Std.TransOrd α], Std.TransOrd (Option α) | null | true |
HomologicalComplex.HomologySequence.snakeInput._proof_4 | Mathlib.Algebra.Homology.HomologySequence | ∀ {C : Type u_2} {ι : Type u_3} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
{c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (i j : ι),
CategoryTheory.CategoryStruct.comp (S.mapNatTrans (HomologicalComplex.natTransOpCyclesToCycles C c i j))
... | null | false |
MulOneClass.mk._flat_ctor | Mathlib.Algebra.Group.Defs | {M : Type u} → (one : M) → (mul : M → M → M) → (∀ (a : M), 1 * a = a) → (∀ (a : M), a * 1 = a) → MulOneClass M | null | false |
List.Nodup.count | Init.Data.List.Pairwise | ∀ {α : Type u_1} [inst : BEq α] [inst_1 : LawfulBEq α] {a : α} {l : List α},
l.Nodup → List.count a l = if a ∈ l then 1 else 0 | null | true |
Fin.coe_castPred | Mathlib.Data.Fin.SuccPred | ∀ {n : ℕ} (i : Fin (n + 1)) (h : i ≠ Fin.last n), ↑(i.castPred h) = ↑i | null | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.0.UInt8.reduceEq._regBuiltin.UInt8.reduceEq.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.781669616._hygCtx._hyg.3 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | IO Unit | null | false |
Ring.instIsScalarTowerFractionRingSubtypeAlgebraicClosureMemIntermediateFieldNormalClosure | Mathlib.RingTheory.NormalClosure | ∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : IsDomain S]
[inst_4 : Algebra R S] [inst_5 : Module.IsTorsionFree R S],
IsScalarTower S (FractionRing S)
↥(IntermediateField.normalClosure (FractionRing R) (FractionRing S) (AlgebraicClosure (FractionRing S... | null | true |
Fin.instSub | Init.Data.Fin.Basic | {n : ℕ} → Sub (Fin n) | null | true |
Std.TreeMap.alter | Std.Data.TreeMap.Basic | {α : Type u} →
{β : Type v} → {cmp : α → α → Ordering} → Std.TreeMap α β cmp → α → (Option β → Option β) → Std.TreeMap α β cmp | Modifies in place the value associated with a given key,
allowing creating new values and deleting values via an `Option` valued replacement function.
This function ensures that the value is used linearly.
| true |
HolderOnWith.ediam_image_le_of_subset_of_le | Mathlib.Topology.MetricSpace.Holder | ∀ {X : Type u_1} {Y : Type u_2} [inst : PseudoEMetricSpace X] [inst_1 : PseudoEMetricSpace Y] {C r : NNReal} {f : X → Y}
{s t : Set X},
HolderOnWith C r f s → t ⊆ s → ∀ {d : ENNReal}, Metric.ediam t ≤ d → Metric.ediam (f '' t) ≤ ↑C * d ^ ↑r | null | true |
CategoryTheory.Oplax.StrongTrans.isoMk_inv_as_app | Mathlib.CategoryTheory.Bicategory.Modification.Oplax | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
{F G : CategoryTheory.OplaxFunctor B C} {η θ : F ⟶ G} (app : (a : B) → η.app a ≅ θ.app a)
(naturality :
autoParam
(∀ {a b : B} (f : a ⟶ b),
CategoryTheory.CategoryStruct.comp (CategoryTheory.B... | null | true |
Bornology.IsBounded.disjoint_cobounded | Mathlib.Topology.Bornology.Basic | ∀ {α : Type u_2} [inst : Bornology α] {l : Filter α} {s : Set α},
Bornology.IsBounded s → s ∈ l → Disjoint l (Bornology.cobounded α) | null | true |
UniformSpace.hausdorff.isClosed_setOf_totallyBounded | Mathlib.Topology.UniformSpace.Closeds | ∀ {α : Type u_1} [inst : UniformSpace α], IsClosed {s | TotallyBounded s} | null | true |
Bornology.IsVonNBounded.image_multilinear | Mathlib.Topology.Algebra.Module.Multilinear.Bounded | ∀ {ι : Type u_1} {𝕜 : Type u_2} {F : Type u_3} {E : ι → Type u_4} [inst : NormedField 𝕜]
[inst_1 : (i : ι) → AddCommGroup (E i)] [inst_2 : (i : ι) → Module 𝕜 (E i)]
[inst_3 : (i : ι) → TopologicalSpace (E i)] [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
[inst_6 : TopologicalSpace F] [ContinuousSMul 𝕜 F] {... | The image of a von Neumann bounded set under a continuous multilinear map
is von Neumann bounded.
This version assumes that the codomain is a topological vector space.
| true |
CategoryTheory.Pseudofunctor.DescentData'.descentDataEquivalence._proof_11 | Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | ∀ {C : Type u_5} [inst : CategoryTheory.Category.{u_3, u_5} C]
(F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat) {ι : Type u_1} {S : C}
{X : ι → C} {f : (i : ι) → X i ⟶ S} (sq : (i j : ι) → CategoryTheory.Limits.ChosenPullback (f i) (f j))
(sq₃ : (i₁ i₂ i₃ : ι) → CategoryT... | null | false |
UInt64.toUInt16_lt._simp_1 | Init.Data.UInt.Lemmas | ∀ {a b : UInt64}, (a.toUInt16 < b.toUInt16) = (a % 65536 < b % 65536) | null | false |
CategoryTheory.Functor.leftOpRightOpEquiv._proof_1 | Mathlib.CategoryTheory.Opposites | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_4, u_2} C] (D : Type u_1)
[inst_1 : CategoryTheory.Category.{u_3, u_1} D] (X : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ),
CategoryTheory.NatTrans.rightOp (CategoryTheory.CategoryStruct.id X).unop =
CategoryTheory.CategoryStruct.id (Opposite.unop X).rightOp | null | false |
_private.Std.Data.DTreeMap.Internal.WF.Lemmas.0.Std.Internal.List.insertEntry.eq_1 | Std.Data.DTreeMap.Internal.WF.Lemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] (k : α) (v : β k) (l : List ((a : α) × β a)),
Std.Internal.List.insertEntry k v l =
bif Std.Internal.List.containsKey k l then Std.Internal.List.replaceEntry k v l else ⟨k, v⟩ :: l | null | true |
curveIntegral_symm | Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {a b : E} (ω : E → E →L[𝕜] F)
(γ : Path a b), ∫ᶜ (x : E) in γ.symm, ω x = -∫ᶜ (x : E) in γ, ω x | null | true |
LipschitzWith.iterate._f | Mathlib.Topology.EMetricSpace.Lipschitz | ∀ {α : Type u} [inst : PseudoEMetricSpace α] {K : NNReal} {f : α → α},
LipschitzWith K f → ∀ (x : ℕ) (f_1 : Nat.below x), LipschitzWith (K ^ x) f^[x] | null | false |
Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂ | Mathlib.LinearAlgebra.Matrix.SesquilinearForm | ∀ {R : Type u_1} {M₁ : Type u_6} {M₂ : Type u_7} {n : Type u_11} {m : Type u_12} [inst : CommRing R]
[inst_1 : DecidableEq m] [inst_2 : Fintype m] [inst_3 : DecidableEq n] [inst_4 : Fintype n] {M : Matrix m n R}
[inst_5 : AddCommMonoid M₁] [inst_6 : Module R M₁] [inst_7 : AddCommMonoid M₂] [inst_8 : Module R M₂]
... | null | true |
RingCon.instCommMagmaQuotient | Mathlib.RingTheory.Congruence.Defs | {R : Type u_1} → [inst : Add R] → [inst_1 : CommMagma R] → (c : RingCon R) → CommMagma c.Quotient | null | true |
_private.Mathlib.Analysis.Complex.CanonicalDecomposition.0.Complex.meromorphicNFOn_canonicalFactor._proof_1_1 | Mathlib.Analysis.Complex.CanonicalDecomposition | -1 < 0 | null | false |
ZFSet.image.match_1 | Mathlib.SetTheory.ZFC.Basic | ∀ (f : ZFSet.{u_1} → ZFSet.{u_1}) [inst : ZFSet.Definable₁ f] (x x_1 : PSet.{u_1})
(motive : (∃ z ∈ x, x_1.Equiv (ZFSet.Definable₁.out f z)) → Prop)
(x_2 : ∃ z ∈ x, x_1.Equiv (ZFSet.Definable₁.out f z)),
(∀ (w : PSet.{u_1}) (h1 : w ∈ x) (h2 : x_1.Equiv (ZFSet.Definable₁.out f w)), motive ⋯) → motive x_2 | null | false |
_private.Mathlib.MeasureTheory.Measure.AddContent.0.MeasureTheory.addContent_iUnion_eq_tsum_of_disjoint_of_addContent_iUnion_le._simp_1_5 | Mathlib.MeasureTheory.Measure.AddContent | ∀ {α : Sort u_2} {β : Sort u_1} {f : α → β} {p : α → Prop} {q : β → Prop},
(∀ (b : β) (a : α), p a → f a = b → q b) = ∀ (a : α), p a → q (f a) | null | false |
smul_mem_asymptoticCone_iff | 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 | true |
Matrix.transposeInvertibleEquivInvertible._proof_1 | Mathlib.LinearAlgebra.Matrix.Invertible | ∀ {n : Type u_1} {α : Type u_2} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommSemiring α] (A : Matrix n n α)
(x : Invertible A.transpose), A.invertibleTranspose = x | null | false |
_private.Mathlib.RingTheory.WittVector.InitTail.0.WittVector.init_sub._proof_1_6 | Mathlib.RingTheory.WittVector.InitTail | ∀ (n i : ℕ), i < n → ∀ k < i + 1, k < n | null | false |
_aux_Mathlib_Combinatorics_SimpleGraph_Basic___macroRules_aesop_graph_1 | Mathlib.Combinatorics.SimpleGraph.Basic | Lean.Macro | A variant of the `aesop` tactic for use in the graph library. Changes relative
to standard `aesop`:
- We use the `SimpleGraph` rule set in addition to the default rule sets.
- We instruct Aesop's `intro` rule to unfold with `default` transparency.
- We instruct Aesop to fail if it can't fully solve the goal. This allo... | false |
Std.Tactic.BVDecide.BVUnOp.eval_not | Std.Tactic.BVDecide.Bitblast.BVExpr.Basic | ∀ {w : ℕ}, Std.Tactic.BVDecide.BVUnOp.not.eval = fun x => ~~~x | null | true |
CategoryTheory.PreGaloisCategory.autMapHom_apply | Mathlib.CategoryTheory.Galois.GaloisObjects | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{u₂, u₁} C] [inst_1 : CategoryTheory.GaloisCategory C] {A B : C}
[inst_2 : CategoryTheory.PreGaloisCategory.IsConnected A] [inst_3 : CategoryTheory.PreGaloisCategory.IsGalois B]
(f : A ⟶ B) (σ : CategoryTheory.Aut A),
(CategoryTheory.PreGaloisCategory.autMapHom f) σ... | null | true |
PadicInt.unitCoeff_spec | Mathlib.NumberTheory.Padics.PadicIntegers | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] {x : ℤ_[p]} (hx : x ≠ 0), x = ↑(PadicInt.unitCoeff hx) * ↑p ^ x.valuation | null | true |
IsManifold.subset_maximalAtlas | Mathlib.Geometry.Manifold.IsManifold.Basic | ∀ {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} {n : WithTop ℕ∞}
{M : Type u_4} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [IsManifold I n M],
... | null | true |
List.consecutivePairs | Mathlib.Data.List.Defs | {α : Type u_1} → List α → List (α × α) | `consecutivePairs [a, b, c, d]` is `[(a, b), (b, c), (c, d)]`. | true |
Std.DHashMap.Raw.Equiv.mem_iff | Std.Data.DHashMap.RawLemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] {m₁ m₂ : Std.DHashMap.Raw α β} [EquivBEq α]
[LawfulHashable α] {k : α}, m₁.WF → m₂.WF → m₁.Equiv m₂ → (k ∈ m₁ ↔ k ∈ m₂) | null | true |
ULift.instLinearOrder | Mathlib.Order.Lattice | {α : Type u} → [LinearOrder α] → LinearOrder (ULift.{v, u} α) | null | true |
CategoryTheory.Grp.forget₂Mon_map_hom | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{A B : CategoryTheory.Grp C} (f : A ⟶ B), ((CategoryTheory.Grp.forget₂Mon C).map f).hom = f.hom.hom | null | true |
pos_of_right_mul_lt_le | Mathlib.Algebra.Order.Ring.Unbundled.Basic | ∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b c : R} [ExistsAddOfLE R] [PosMulMono R]
[AddRightMono R] [AddRightReflectLE R], a * b < a * c → b ≤ c → 0 < a | null | true |
Set.BijOn.union | Mathlib.Data.Set.Function | ∀ {α : Type u_1} {β : Type u_2} {s₁ s₂ : Set α} {t₁ t₂ : Set β} {f : α → β},
Set.BijOn f s₁ t₁ → Set.BijOn f s₂ t₂ → Set.InjOn f (s₁ ∪ s₂) → Set.BijOn f (s₁ ∪ s₂) (t₁ ∪ t₂) | null | true |
_private.Mathlib.NumberTheory.BernoulliPolynomials.0.Polynomial.sum_bernoulli._simp_1_4 | Mathlib.NumberTheory.BernoulliPolynomials | ∀ {M : Type u_4} [inst : AddMonoid M] [IsLeftCancelAdd M] {a b : M}, (a + b = a) = (b = 0) | null | false |
LieSubmodule.lowerCentralSeries_eq_lcs_comap | Mathlib.Algebra.Lie.Nilpotent | ∀ {R : Type u} {L : Type v} {M : Type w} [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] (k : ℕ) (N : LieSubmodule R L M)
[LieModule R L M], LieModule.lowerCentralSeries R L (↥N) k = LieSubmodule.comap N.incl (LieSubmodu... | null | true |
HopfAlgCat.noConfusionType | Mathlib.Algebra.Category.HopfAlgCat.Basic | Sort u_1 →
{R : Type u} → [inst : CommRing R] → HopfAlgCat R → {R' : Type u} → [inst' : CommRing R'] → HopfAlgCat R' → Sort u_1 | null | false |
String.Slice.isSome_skipPrefix? | Init.Data.String.Lemmas.Pattern.TakeDrop.Basic | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.ForwardPattern pat] [String.Slice.Pattern.LawfulForwardPattern pat]
{s : String.Slice}, (s.skipPrefix? pat).isSome = s.startsWith pat | null | true |
_private.Mathlib.NumberTheory.Transcendental.Liouville.Residual.0.setOf_liouville_eq_iInter_iUnion._simp_1_1 | Mathlib.NumberTheory.Transcendental.Liouville.Residual | ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋂ i, s i) = ∀ (i : ι), x ∈ s i | null | false |
CategoryTheory.ShortComplex.instPreservesLimitsOfShapeπ₂ | Mathlib.Algebra.Homology.ShortComplex.Limits | ∀ {J : Type u_1} {C : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} J]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[CategoryTheory.Limits.HasLimitsOfShape J C],
CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.ShortComplex.π₂ | null | true |
List.getElem_of_append | Init.Data.List.Lemmas | ∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α} {i : ℕ} {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i),
l[i] = a | null | true |
ZMod.χ₈'._proof_1 | Mathlib.NumberTheory.LegendreSymbol.ZModChar | (match 1 with
| 0 => 0
| 2 => 0
| 4 => 0
| 6 => 0
| 1 => 1
| 3 => 1
| 5 => -1
| 7 => -1) =
match 1 with
| 0 => 0
| 2 => 0
| 4 => 0
| 6 => 0
| 1 => 1
| 3 => 1
| 5 => -1
| 7 => -1 | null | false |
CategoryTheory.Idempotents.functorExtension₁._proof_1 | Mathlib.CategoryTheory.Idempotents.FunctorExtension | ∀ (C : Type u_1) (D : Type u_4) [inst : CategoryTheory.Category.{u_2, u_1} C]
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (F : CategoryTheory.Functor C (CategoryTheory.Idempotents.Karoubi D)),
CategoryTheory.Idempotents.FunctorExtension₁.map (CategoryTheory.CategoryStruct.id F) =
CategoryTheory.CategoryStru... | null | false |
ContinuousLinearMap.instSMul._proof_1 | Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | ∀ {R₁ : Type u_4} {R₂ : Type u_5} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_1}
[inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_2} [inst_4 : TopologicalSpace M₂]
[inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂] {S₂ : Type u_3}
[ins... | null | false |
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