name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
CategoryTheory.Bicategory.rightUnitorNatIsoCat_hom_toNatTrans_app | Mathlib.CategoryTheory.Bicategory.Yoneda | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] (a b : B) (X : a ⟶ b),
(CategoryTheory.Bicategory.rightUnitorNatIsoCat a b).hom.toNatTrans.app X =
(CategoryTheory.Bicategory.rightUnitor X).hom |
_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion.0.iteratedDerivWithin_tsum_exp_aux_eq._simp_1_2 | Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | ∀ {G : Type u_3} [inst : InvolutiveNeg G] {a b : G}, (-a = -b) = (a = b) |
Configuration.Nondegenerate.exists_line | Mathlib.Combinatorics.Configuration | ∀ {P : Type u_1} {L : Type u_2} {inst : Membership P L} [self : Configuration.Nondegenerate P L] (p : P), ∃ l, p ∉ l |
Nat.EqResult.eq.injEq | Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat | ∀ (x y p x_1 y_1 p_1 : Lean.Expr), (Nat.EqResult.eq x y p = Nat.EqResult.eq x_1 y_1 p_1) = (x = x_1 ∧ y = y_1 ∧ p = p_1) |
ContMDiffMap.restrictMonoidHom._proof_1 | Mathlib.Geometry.Manifold.Algebra.SmoothFunctions | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {E' : Type u_5} [inst_3 : NormedAddCommGroup E'] [inst_4 : NormedSpace 𝕜 E'] {H : Type u_3}
[inst_5 : TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {H' : Type u_6} [inst_6 : TopologicalSpace H']
(I' : ModelWithCorners 𝕜 E' H') {N : Type u_4} [inst_7 : TopologicalSpace N] [inst_8 : ChartedSpace H N]
{n : WithTop ℕ∞} (G : Type u_7) [inst_9 : TopologicalSpace G] [inst_10 : ChartedSpace H' G]
{U V : TopologicalSpace.Opens N} (h : U ≤ V) (f : ContMDiffMap I I' (↥V) G n), ContMDiff I I' n (⇑f ∘ Set.inclusion h) |
mellinConvergent_of_isBigO_rpow_exp | Mathlib.Analysis.MellinTransform | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {a b : ℝ},
0 < a →
∀ {f : ℝ → E} {s : ℂ},
MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume →
(f =O[Filter.atTop] fun t => Real.exp (-a * t)) →
(f =O[nhdsWithin 0 (Set.Ioi 0)] fun x => x ^ (-b)) → b < s.re → MellinConvergent f s |
_private.Mathlib.NumberTheory.FLT.Three.0.FermatLastTheoremForThreeGen.Solution.w_spec | Mathlib.NumberTheory.FLT.Three | ∀ {K : Type u_1} [inst : Field K] {ζ : K} {hζ : IsPrimitiveRoot ζ 3} (S : FermatLastTheoremForThreeGen.Solution hζ),
S.c = (hζ.toInteger - 1) ^ S.multiplicity * FermatLastTheoremForThreeGen.Solution.w✝ S |
_private.Std.Sync.Channel.0.Std.CloseableChannel.Bounded.Consumer.ctorIdx | Std.Sync.Channel | {α : Type} → Std.CloseableChannel.Bounded.Consumer✝ α → ℕ |
_private.Batteries.CodeAction.Misc.0.Batteries.CodeAction.casesExpand.match_19 | Batteries.CodeAction.Misc | (motive : Option (Array (Lean.Name × Array Lean.Name)) → Sort u_1) →
(__discr : Option (Array (Lean.Name × Array Lean.Name))) →
((ctors : Array (Lean.Name × Array Lean.Name)) → motive (some ctors)) →
((x : Option (Array (Lean.Name × Array Lean.Name))) → motive x) → motive __discr |
AddAction.stabilizerEquivStabilizer_trans | Mathlib.GroupTheory.GroupAction.Basic | ∀ {G : Type u_1} {α : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] {g h k : G} {a b c : α} (hg : b = g +ᵥ a)
(hh : c = h +ᵥ b) (hk : c = k +ᵥ a),
k = h + g →
(AddAction.stabilizerEquivStabilizer hg).trans (AddAction.stabilizerEquivStabilizer hh) =
AddAction.stabilizerEquivStabilizer hk |
Lean.Meta.Grind.SymbolPriorityEntry.mk.noConfusion | Lean.Meta.Tactic.Grind.EMatchTheorem | {P : Sort u} →
{declName : Lean.Name} →
{prio : ℕ} →
{declName' : Lean.Name} →
{prio' : ℕ} →
{ declName := declName, prio := prio } = { declName := declName', prio := prio' } →
(declName = declName' → prio = prio' → P) → P |
BoundedOrderHomClass.toBotHomClass | Mathlib.Order.Hom.Bounded | ∀ {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : FunLike F α β] [inst_1 : LE α] [inst_2 : LE β]
[inst_3 : BoundedOrder α] [inst_4 : BoundedOrder β] [BoundedOrderHomClass F α β], BotHomClass F α β |
_private.Mathlib.Topology.UniformSpace.Defs.0.UniformSpace.hasBasis_nhds._simp_1_2 | Mathlib.Topology.UniformSpace.Defs | ∀ {a b c : Prop}, ((a ∧ b) ∧ c) = (a ∧ b ∧ c) |
Std.Iterators.Types.Zip.instFinite₂ | Std.Data.Iterators.Combinators.Monadic.Zip | ∀ {m : Type w → Type w'} {α₁ β₁ : Type w} [inst : Std.Iterator α₁ m β₁] {α₂ β₂ : Type w} [inst_1 : Std.Iterator α₂ m β₂]
[inst_2 : Monad m] [Std.Iterators.Productive α₁ m] [Std.Iterators.Finite α₂ m],
Std.Iterators.Finite (Std.Iterators.Types.Zip α₁ m α₂ β₂) m |
Std.TransCmp.lt_of_eq_of_lt | Init.Data.Order.Ord | ∀ {α : Type u} {cmp : α → α → Ordering} [Std.TransCmp cmp] {a b c : α},
cmp a b = Ordering.eq → cmp b c = Ordering.lt → cmp a c = Ordering.lt |
List.insert_of_not_mem | Init.Data.List.Lemmas | ∀ {α : Type u_1} [inst : BEq α] [LawfulBEq α] {a : α} {l : List α}, a ∉ l → List.insert a l = a :: l |
Real.denselyNormedField._proof_1 | Mathlib.Analysis.Normed.Field.Basic | ∀ (x x_1 : ℝ), 0 ≤ x → x < x_1 → ∃ a, x < ‖a‖ ∧ ‖a‖ < x_1 |
_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.at_least_as_long_as_coind._simp_1_8 | Mathlib.Data.Seq.Basic | ∀ {α : Type u} (s : Stream'.Seq α), (s = Stream'.Seq.nil) = (s.length' = 0) |
StrictMonoOn.mapsTo_Ioc | Mathlib.Order.Interval.Set.Image | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} [inst : PartialOrder α] [inst_1 : Preorder β] {a b : α},
StrictMonoOn f (Set.Icc a b) → Set.MapsTo f (Set.Ioc a b) (Set.Ioc (f a) (f b)) |
_private.Init.Data.String.Iterate.0.String.Slice.ByteIterator.finitenessRelation._proof_2 | Init.Data.String.Iterate | ∀ {m : Type → Type u_1},
WellFounded
(InvImage WellFoundedRelation.rel fun it => it.internalState.s.utf8ByteSize - it.internalState.offset.byteIdx) |
Std.DTreeMap.Internal.Impl.getKeyD_inter!_of_contains_eq_false_left | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {m₁ m₂ : Std.DTreeMap.Internal.Impl α β} [Std.TransOrd α],
m₁.WF →
m₂.WF →
∀ {k fallback : α},
Std.DTreeMap.Internal.Impl.contains k m₁ = false → (m₁.inter! m₂).getKeyD k fallback = fallback |
AddSubgroup.card_dvd_of_injective | Mathlib.GroupTheory.Coset.Card | ∀ {α : Type u_1} [inst : AddGroup α] {H : Type u_2} [inst_1 : AddGroup H] (f : α →+ H),
Function.Injective ⇑f → Nat.card α ∣ Nat.card H |
OrderHom.apply | Mathlib.Order.Hom.Basic | {α : Type u_2} → {β : Type u_3} → [inst : Preorder α] → [inst_1 : Preorder β] → α → (α →o β) →o β |
Manifold.«_aux_Mathlib_Geometry_Manifold_Instances_Real___macroRules_Manifold_term𝓡∂__1» | Mathlib.Geometry.Manifold.Instances.Real | Lean.Macro |
CompleteDistribLattice.toHNot | Mathlib.Order.CompleteBooleanAlgebra | {α : Type u_1} → [self : CompleteDistribLattice α] → HNot α |
MeasurableEquiv.piFinSuccAbove_apply | Mathlib.MeasureTheory.MeasurableSpace.Embedding | ∀ {n : ℕ} (α : Fin (n + 1) → Type u_8) [inst : (i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1)),
⇑(MeasurableEquiv.piFinSuccAbove α i) = ⇑(Fin.insertNthEquiv α i).symm |
LieAlgebra.radical_eq_top_of_isSolvable | Mathlib.Algebra.Lie.Solvable | ∀ (R : Type u) (L : Type v) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
[LieAlgebra.IsSolvable L], LieAlgebra.radical R L = ⊤ |
String.all_iff | Batteries.Data.String.Lemmas | ∀ (s : String) (p : Char → Bool), String.Legacy.all s p = true ↔ ∀ c ∈ s.toList, p c = true |
_private.Batteries.Data.List.Lemmas.0.List.getElem_findIdxs_lt._proof_1_5 | Batteries.Data.List.Lemmas | ∀ {i : ℕ} {α : Type u_1} {xs : List α} {p : α → Bool} {s : ℕ},
i < (List.findIdxs p xs s).length → 0 < (List.findIdxs p xs s).length |
Substring.Raw.toString | Init.Data.String.Substring | Substring.Raw → String |
Lean.Parser.antiquotExpr | Lean.Parser.Basic | Lean.Parser.Parser |
ZFSet.vonNeumann_subset_vonNeumann_iff | Mathlib.SetTheory.ZFC.VonNeumann | ∀ {a b : Ordinal.{u}}, ZFSet.vonNeumann a ⊆ ZFSet.vonNeumann b ↔ a ≤ b |
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 |
_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 |
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 |
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 S B]
[inst_11 : SMulCommClass S A B] (x : A),
(KaehlerDifferential.map R S A B) ((KaehlerDifferential.D R A) x) = (KaehlerDifferential.D S B) ((algebraMap A B) x) |
_private.Mathlib.MeasureTheory.Measure.Haar.OfBasis.0.Module.Basis.parallelepiped._simp_5 | Mathlib.MeasureTheory.Measure.Haar.OfBasis | ∀ (α : Sort u), (∀ (a : α), True) = True |
Option.toArray_eq_empty_iff._simp_1 | Init.Data.Option.Array | ∀ {α : Type u_1} {o : Option α}, (o.toArray = #[]) = (o = none) |
_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.Expr) → motive x) → motive type |
_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 |
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) {T : C}
(f : c.pt ⟶ T), P f → ∃ j T' f' g, CategoryTheory.IsPushout (c.ι.app j) f' f g ∧ P f' |
CategoryTheory.Mod_._sizeOf_1 | Mathlib.CategoryTheory.Monoidal.Mod_ | {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.MonoidalLeftAction C D} →
{A : C} → {inst_4 : CategoryTheory.MonObj A} → [SizeOf C] → [SizeOf D] → CategoryTheory.Mod_ D A → ℕ |
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 = uniformity α |
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 |
Int8.neg_sub | Init.Data.SInt.Lemmas | ∀ {a b : Int8}, -(a - b) = b - a |
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) |
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] {E' : Type u₄}
[inst_6 : CategoryTheory.Category.{v₄, u₄} E'] [inst_7 : CategoryTheory.MonoidalCategory E']
(F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) (H : CategoryTheory.Functor E E')
[inst_8 : F.LaxMonoidal] [inst_9 : G.LaxMonoidal] [inst_10 : H.LaxMonoidal],
CategoryTheory.NatTrans.IsMonoidal (F.associator G H).hom |
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 |
_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.map₂ f' a (Option.map₂ g' b c) |
ZFSet.Subset | Mathlib.SetTheory.ZFC.Basic | ZFSet.{u} → ZFSet.{u} → Prop |
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' f fun x => CategoryTheory.CategoryStruct.id X |
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.Const.insertMany t l},
Std.ExtDTreeMap.Const.get (Std.ExtDTreeMap.Const.insertMany t l) k' h' = v |
_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 : (Units.map f) x = y), motive ⋯) → motive x |
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)) |
UniformEquiv.refl_symm | Mathlib.Topology.UniformSpace.Equiv | ∀ {α : Type u} [inst : UniformSpace α], (UniformEquiv.refl α).symm = UniformEquiv.refl α |
_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 |
Lean.Widget.RpcEncodablePacket._sizeOf_1._@.Lean.Widget.InteractiveDiagnostic.1765450820._hygCtx._hyg.1 | Lean.Widget.InteractiveDiagnostic | Lean.Widget.RpcEncodablePacket✝ → ℕ |
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 |
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 |
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 |
Fin.coe_castPred | Mathlib.Data.Fin.SuccPred | ∀ {n : ℕ} (i : Fin (n + 1)) (h : i ≠ Fin.last n), ↑(i.castPred h) = ↑i |
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 |
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.Bicategory.whiskerLeft (F.map f) (app b).hom)
(θ.naturality f).hom =
CategoryTheory.CategoryStruct.comp (η.naturality f).hom
(CategoryTheory.Bicategory.whiskerRight (app a).hom (G.map f)))
CategoryTheory.Oplax.StrongTrans.isoMk._auto_1)
(a : B), (CategoryTheory.Oplax.StrongTrans.isoMk app naturality).inv.as.app a = (app a).inv |
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] {s : Set ((i : ι) → E i)},
Bornology.IsVonNBounded 𝕜 s → ∀ (f : ContinuousMultilinearMap 𝕜 E F), Bornology.IsVonNBounded 𝕜 (⇑f '' s) |
UInt64.toUInt16_lt._simp_1 | Init.Data.UInt.Lemmas | ∀ {a b : UInt64}, (a.toUInt16 < b.toUInt16) = (a % 65536 < b % 65536) |
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 |
_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 |
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 |
_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) |
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 : Set P} {c : k} {v : V},
0 < c → (c • v ∈ asymptoticCone k s ↔ v ∈ asymptoticCone k s) |
Matrix.transposeInvertibleEquivInvertible._proof_1 | Mathlib.Data.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 |
Std.Tactic.BVDecide.BVUnOp.eval_not | Std.Tactic.BVDecide.Bitblast.BVExpr.Basic | ∀ {w : ℕ}, Std.Tactic.BVDecide.BVUnOp.not.eval = fun x => ~~~x |
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 = CategoryTheory.PreGaloisCategory.autMap f σ |
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) |
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) |
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 (LieSubmodule.lcs k N) |
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 |
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 |
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.CategoryStruct.id (CategoryTheory.Idempotents.FunctorExtension₁.obj F) |
ContinuousLinearMap.instSMul._proof_1 | Mathlib.Topology.Algebra.Module.LinearMap | ∀ {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}
[inst_8 : DistribSMul S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂),
Continuous fun x => c • (↑f).toFun x |
_private.Mathlib.Combinatorics.Additive.VerySmallDoubling.0.Finset.doubling_lt_golden_ratio._simp_1_23 | Mathlib.Combinatorics.Additive.VerySmallDoubling | ∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p) |
NormedField.instRankLeOneNNRealValuation._proof_3 | Mathlib.Topology.Algebra.Valued.NormedValued | ∀ {K : Type u_1} [hK : NormedField K] [inst : IsUltrametricDist K], StrictMono ⇑MonoidWithZeroHom.ValueGroup₀.embedding |
Matrix.diagonal_mulVec_single | Mathlib.Data.Matrix.Mul | ∀ {n : Type u_3} {R : Type u_7} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : NonUnitalNonAssocSemiring R]
(v : n → R) (j : n) (x : R), (Matrix.diagonal v).mulVec (Pi.single j x) = Pi.single j (v j * x) |
_private.Mathlib.Analysis.Asymptotics.Theta.0.Asymptotics.isTheta_const_const_iff._simp_1_2 | Mathlib.Analysis.Asymptotics.Theta | ∀ {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [inst : NormedAddCommGroup E''] [inst_1 : NormedAddCommGroup F'']
{c : E''} {c' : F''} (l : Filter α) [l.NeBot], ((fun _x => c) =O[l] fun _x => c') = (c' = 0 → c = 0) |
Lean.Doc.instBEqListItem.beq | Lean.DocString.Types | {α : Type u_1} → [BEq α] → Lean.Doc.ListItem α → Lean.Doc.ListItem α → Bool |
Std.DTreeMap.isSome_maxKey?_iff_isEmpty_eq_false | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp],
t.maxKey?.isSome = true ↔ t.isEmpty = false |
Mathlib.Tactic.Monoidal.instMonadNormalizeNaturalityMonoidalM.match_1 | Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | (ctx : Mathlib.Tactic.Monoidal.Context) →
(motive : Option Q(CategoryTheory.MonoidalCategory unknown_1) → Sort u_1) →
(x : Option Q(CategoryTheory.MonoidalCategory unknown_1)) →
((_monoidal : Q(CategoryTheory.MonoidalCategory unknown_1)) → motive (some _monoidal)) →
((x : Option Q(CategoryTheory.MonoidalCategory unknown_1)) → motive x) → motive x |
ProbabilityTheory.preCDF_le_one | Mathlib.Probability.Kernel.Disintegration.CondCDF | ∀ {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ],
∀ᵐ (a : α) ∂ρ.fst, ∀ (r : ℚ), ProbabilityTheory.preCDF ρ r a ≤ 1 |
Metric.Snowflaking.preimage_toSnowflaking_emetricBall | Mathlib.Topology.MetricSpace.Snowflaking | ∀ {X : Type u_1} {α : ℝ} {hα₀ : 0 < α} {hα₁ : α ≤ 1} [inst : PseudoEMetricSpace X] (x : Metric.Snowflaking X α hα₀ hα₁)
(d : ENNReal),
⇑Metric.Snowflaking.toSnowflaking ⁻¹' Metric.eball x d = Metric.eball (Metric.Snowflaking.ofSnowflaking x) (d ^ α⁻¹) |
Finset.min_union | Mathlib.Data.Finset.Max | ∀ {α : Type u_2} [inst : LinearOrder α] {s t : Finset α}, (s ∪ t).min = min s.min t.min |
Lean.Compiler.LCNF.Code.collectUsed | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} → Lean.Compiler.LCNF.Code pu → optParam Lean.FVarIdHashSet ∅ → Lean.FVarIdHashSet |
CategoryTheory.Subgroupoid.instSetLikeSigmaHom | Mathlib.CategoryTheory.Groupoid.Subgroupoid | {C : Type u} → [inst : CategoryTheory.Groupoid C] → SetLike (CategoryTheory.Subgroupoid C) ((c : C) × (d : C) × (c ⟶ d)) |
ENat.one_lt_card._simp_1 | Mathlib.SetTheory.Cardinal.Finite | ∀ {α : Type u_1} [Nontrivial α], (1 < ENat.card α) = True |
Lean.Expr.isDIte | Lean.Util.Recognizers | Lean.Expr → Bool |
edist_lt_top | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u_3} [inst : PseudoMetricSpace α] (x y : α), edist x y < ⊤ |
CategoryTheory.Functor.PullbackObjObj.mk.inj | Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj | ∀ {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {inst : CategoryTheory.Category.{v₁, u₁} C₁}
{inst_1 : CategoryTheory.Category.{v₂, u₂} C₂} {inst_2 : CategoryTheory.Category.{v₃, u₃} C₃}
{G : CategoryTheory.Functor C₁ᵒᵖ (CategoryTheory.Functor C₃ C₂)} {X₁ Y₁ : C₁} {f₁ : X₁ ⟶ Y₁} {X₃ Y₃ : C₃}
{f₃ : X₃ ⟶ Y₃} {pt : C₂} {fst : pt ⟶ (G.obj (Opposite.op X₁)).obj X₃} {snd : pt ⟶ (G.obj (Opposite.op Y₁)).obj Y₃}
{isPullback : CategoryTheory.IsPullback fst snd ((G.obj (Opposite.op X₁)).map f₃) ((G.map f₁.op).app Y₃)} {pt_1 : C₂}
{fst_1 : pt_1 ⟶ (G.obj (Opposite.op X₁)).obj X₃} {snd_1 : pt_1 ⟶ (G.obj (Opposite.op Y₁)).obj Y₃}
{isPullback_1 : CategoryTheory.IsPullback fst_1 snd_1 ((G.obj (Opposite.op X₁)).map f₃) ((G.map f₁.op).app Y₃)},
{ pt := pt, fst := fst, snd := snd, isPullback := isPullback } =
{ pt := pt_1, fst := fst_1, snd := snd_1, isPullback := isPullback_1 } →
pt = pt_1 ∧ fst ≍ fst_1 ∧ snd ≍ snd_1 |
FirstOrder.Language.DefinableSet.coe_bot | Mathlib.ModelTheory.Definability | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {A : Set M} {α : Type u₁}, ↑⊥ = ∅ |
ArchimedeanClass.orderHom | Mathlib.Algebra.Order.Archimedean.Class | {M : Type u_1} →
[inst : AddCommGroup M] →
[inst_1 : LinearOrder M] →
[inst_2 : IsOrderedAddMonoid M] →
{N : Type u_2} →
[inst_3 : AddCommGroup N] →
[inst_4 : LinearOrder N] →
[inst_5 : IsOrderedAddMonoid N] → (M →+o N) → ArchimedeanClass M →o ArchimedeanClass N |
_private.Mathlib.CategoryTheory.Shift.Localization.0.CategoryTheory.Functor.commShiftOfLocalization._simp_1 | Mathlib.CategoryTheory.Shift.Localization | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C D} (self : CategoryTheory.NatTrans F G) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : D}
(h : G.obj Y ⟶ Z),
CategoryTheory.CategoryStruct.comp (self.app X) (CategoryTheory.CategoryStruct.comp (G.map f) h) =
CategoryTheory.CategoryStruct.comp (F.map f) (CategoryTheory.CategoryStruct.comp (self.app Y) h) |
ExteriorAlgebra.ιInv | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {R : Type u1} →
[inst : CommRing R] → {M : Type u2} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → ExteriorAlgebra R M →ₗ[R] M |
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