name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Monoid.CoprodI.NeWord.last.eq_def | Mathlib.GroupTheory.CoprodI | ∀ {ι : Type u_1} {M : ι → Type u_2} [inst : (i : ι) → Monoid (M i)] (x x_1 : ι) (x_2 : Monoid.CoprodI.NeWord M x x_1),
x_2.last =
match x, x_1, x_2 with
| x, .(x), Monoid.CoprodI.NeWord.singleton x_3 _hne1 => x_3
| x, x_3, _w₁.append _hne w₂ => w₂.last | null | true |
Finset.disjoint_val._simp_1 | Mathlib.Data.Finset.Disjoint | ∀ {α : Type u_2} {s t : Finset α}, Disjoint s.val t.val = Disjoint s t | null | false |
MeasureTheory.mem_fundamentalFrontier._simp_2 | Mathlib.MeasureTheory.Group.FundamentalDomain | ∀ {G : Type u_1} {α : Type u_3} [inst : Group G] [inst_1 : MulAction G α] {s : Set α} {x : α},
(x ∈ MeasureTheory.fundamentalFrontier G s) = (x ∈ s ∧ ∃ g, g ≠ 1 ∧ x ∈ g • s) | null | false |
_private.Mathlib.Combinatorics.Matroid.Map.0.Matroid.comap_isBasis_iff._simp_1_2 | Mathlib.Combinatorics.Matroid.Map | ∀ {α : Type u} {β : Type v} (f : α → β) (s : Set α) (y : β), (y ∈ f '' s) = ∃ x ∈ s, f x = y | null | false |
Lean.Expr.hasNonSyntheticSorry | Lean.Util.Sorry | Lean.Expr → Bool | null | true |
_private.Lean.Meta.Tactic.Grind.Types.0.Lean.Meta.Grind.Solvers.mergeTerms.go.match_3._arg_pusher | Lean.Meta.Tactic.Grind.Types | ∀ (motive : Lean.Meta.Grind.SolverTerms → Lean.Meta.Grind.SolverTerms → Sort u_1) (α : Sort u✝) (β : α → Sort v✝)
(f : (x : α) → β x) (rel : Lean.Meta.Grind.SolverTerms → Lean.Meta.Grind.SolverTerms → α → Prop)
(rhsTerms lhsTerms : Lean.Meta.Grind.SolverTerms)
(h_1 :
Unit →
((y : α) → rel Lean.Meta.Grin... | null | false |
CategoryTheory.Join.instUniqueHomLeftRight | Mathlib.CategoryTheory.Join.Basic | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
{X : C} → {Y : D} → Unique (CategoryTheory.Join.left X ⟶ CategoryTheory.Join.right Y) | null | true |
MemHolder.nsmul | Mathlib.Topology.MetricSpace.HolderNorm | ∀ {X : Type u_1} {Y : Type u_2} [inst : MetricSpace X] [inst_1 : NormedAddCommGroup Y] {r : NNReal} {f : X → Y}
[NormedSpace ℝ Y] (n : ℕ), MemHolder r f → MemHolder r (n • f) | null | true |
Fin.val_sub_one_of_ne_zero | Mathlib.Data.Fin.Basic | ∀ {n : ℕ} {i : Fin n}, i ≠ 0 → ↑(i - 1) = ↑i - 1 | null | true |
_private.Mathlib.Data.EReal.Basic.0.EReal.exists_rat_btwn_of_lt.match_1_3 | Mathlib.Data.EReal.Basic | ∀ (a : ℝ) (motive : (∃ q, ↑q < a) → Prop) (x : ∃ q, ↑q < a), (∀ (b : ℚ) (hab : ↑b < a), motive ⋯) → motive x | null | false |
CategoryTheory.Limits.IsLimit.liftConeMorphism | Mathlib.CategoryTheory.Limits.IsLimit | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
{C : Type u₃} →
[inst_1 : CategoryTheory.Category.{v₃, u₃} C] →
{F : CategoryTheory.Functor J C} →
{t : CategoryTheory.Limits.Cone F} →
CategoryTheory.Limits.IsLimit t → (s : CategoryTheory.Limits.Cone F) → s ⟶ t | The universal morphism from any other cone to a limit cone. | true |
Equiv.sumIsRight_apply | Mathlib.Logic.Equiv.Defs | ∀ {α : Type u_1} {β : Type u_2} (x : { x // x.isRight = true }), Equiv.sumIsRight x = (↑x).getRight ⋯ | null | true |
_private.Mathlib.Data.List.Cycle.0.List.next_eq_getElem._proof_1_10 | Mathlib.Data.List.Cycle | ∀ {α : Type u_1} {l : List α} (hl : l ≠ []), ¬[l.getLast ⋯].isEmpty = true → [l.getLast ⋯] ≠ [] | null | false |
CategoryTheory.Limits.pushoutPushoutRightIsPushout._proof_3 | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {X₁ X₂ X₃ Z₁ Z₂ : C} (g₁ : Z₁ ⟶ X₁) (g₂ : Z₁ ⟶ X₂)
(g₃ : Z₂ ⟶ X₂) (g₄ : Z₂ ⟶ X₃) [CategoryTheory.Limits.HasPushout g₁ g₂]
[inst_2 : CategoryTheory.Limits.HasPushout g₃ g₄]
[inst_3 :
CategoryTheory.Limits.HasPushout g₁
(CategoryTheory.Categor... | null | false |
CliffordAlgebra.reverse_involutive._simp_1 | Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation | ∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{Q : QuadraticForm R M}, Function.Involutive ⇑CliffordAlgebra.reverse = True | null | false |
AlgebraicGeometry.Scheme.Opens.partialIsoOfDense_iso | Mathlib.AlgebraicGeometry.Birational.Birational | ∀ {X : AlgebraicGeometry.Scheme} (U : X.Opens) (hU : Dense ↑U), (U.partialIsoOfDense hU).iso = (↑U).topIso | null | true |
Lean.Parser.numLitFn | Lean.Parser.Basic | Lean.Parser.ParserFn | null | true |
StarAlgebra.elemental.characterSpaceToSpectrum._proof_4 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | ContinuousAdd ℂ | null | false |
Algebra.normalizedTrace_algebraMap_apply | Mathlib.FieldTheory.NormalizedTrace | ∀ (F : Type u_3) (E : Type u_4) (K : Type u_5) [inst : Field F] [inst_1 : Field E] [inst_2 : Field K]
[inst_3 : Algebra F E] [inst_4 : Algebra E K] [inst_5 : Algebra F K] [IsScalarTower F E K]
[inst_7 : Algebra.IsIntegral F E] [inst_8 : Algebra.IsIntegral F K] [inst_9 : CharZero F] (a : E),
(Algebra.normalizedTra... | null | true |
sup_left_right_swap | Mathlib.Order.Defs.Lattice | ∀ {α : Type u} [inst : SemilatticeSup α] (a b c : α), a ⊔ b ⊔ c = c ⊔ b ⊔ a | null | true |
pythagoreanTriple_comm | Mathlib.NumberTheory.PythagoreanTriples | ∀ {x y z : ℤ}, PythagoreanTriple x y z ↔ PythagoreanTriple y x z | Pythagorean triples are interchangeable, i.e `x * x + y * y = y * y + x * x = z * z`.
This comes from additive commutativity. | true |
AlgebraicGeometry.IsOpenImmersion.ΓIsoTop._proof_1 | Mathlib.AlgebraicGeometry.OpenImmersion | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [inst : AlgebraicGeometry.IsOpenImmersion f],
AlgebraicGeometry.Scheme.Hom.opensRange f = (AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj ⊤ | null | false |
Asymptotics.isTheta_of_div_tendsto_nhds_ne_zero | Mathlib.Analysis.Asymptotics.Theta | ∀ {α : Type u_1} {𝕜 : Type u_11} [inst : NormedField 𝕜] {l : Filter α} {c : 𝕜} {f g : α → 𝕜},
Filter.Tendsto (fun x => g x / f x) l (nhds c) → c ≠ 0 → f =Θ[l] g | null | true |
_private.Init.Data.BitVec.Lemmas.0.BitVec.getMsbD_rotateLeft_of_lt._proof_1_2 | Init.Data.BitVec.Lemmas | ∀ {r n : ℕ} (w : ℕ), r < w + 1 → n < w + 1 - r → ¬n < w + 1 → False | null | false |
CategoryTheory.MorphismProperty.IsLocalAtSource.rec | Mathlib.CategoryTheory.MorphismProperty.Local | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{P : CategoryTheory.MorphismProperty C} →
{K : CategoryTheory.Precoverage C} →
{motive : P.IsLocalAtSource K → Sort u_1} →
([toRespects : P.Respects (CategoryTheory.MorphismProperty.isomorphisms C)] →
(comp :
... | null | false |
CategoryTheory.Functor.LaxMonoidal.right_unitality | Mathlib.CategoryTheory.Monoidal.Functor | ∀ {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}
(F : CategoryTheory.Functor C D) [self : F.LaxMonoidal] (X : C),
(CategoryTheory.MonoidalCategoryStruct.... | null | true |
_private.Mathlib.Analysis.SpecialFunctions.Log.Base.0.Real.logb_prod._simp_1_1 | Mathlib.Analysis.SpecialFunctions.Log.Base | ∀ {ι : Type u_1} {M₀ : Type u_4} [inst : CommMonoidWithZero M₀] {f : ι → M₀} {s : Finset ι} [Nontrivial M₀]
[NoZeroDivisors M₀], (∏ x ∈ s, f x ≠ 0) = ∀ a ∈ s, f a ≠ 0 | null | false |
CategoryTheory.FreeMonoidalCategory.Hom.ρ_hom.elim | Mathlib.CategoryTheory.Monoidal.Free.Basic | {C : Type u} →
{motive : (a a_1 : CategoryTheory.FreeMonoidalCategory C) → a.Hom a_1 → Sort u_1} →
{a a_1 : CategoryTheory.FreeMonoidalCategory C} →
(t : a.Hom a_1) →
t.ctorIdx = 5 →
((X : CategoryTheory.FreeMonoidalCategory C) →
motive (X.tensor CategoryTheory.FreeMonoidalCa... | null | false |
CategoryTheory.SimplicialObject.δ₀Iter.eq_1 | Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (X : CategoryTheory.SimplicialObject C) {n m : ℕ} (i : ℕ)
(hi : n + i = m), X.δ₀Iter i hi = X.map (SimplexCategory.δ₀Iter i hi).op | null | true |
UpperHalfPlane.dist_triangle | Mathlib.Analysis.Complex.UpperHalfPlane.Metric | ∀ (a b c : UpperHalfPlane), dist a c ≤ dist a b + dist b c | null | true |
MeromorphicOn.logDeriv_finprod_zpow_eventuallyEq | Mathlib.Analysis.Meromorphic.LogDeriv | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [inst_1 : NontriviallyNormedField 𝕜']
[inst_2 : NormedAlgebra 𝕜 𝕜'] {U : Set 𝕜} {ι : Type u_3} {F : ι → 𝕜 → 𝕜'} {d : ι → ℤ},
(Function.support d).Finite →
(∀ (i : ι), MeromorphicOn (F i) U) →
(∀ (i : ι), ∀ x ∈ U, meromorphicOrder... | The logarithmic derivative converts products into sums: away from a codiscrete subset of `U`, the
logarithmic derivative of a finite product of integer powers of meromorphic functions is the
corresponding weighted sum of logarithmic derivatives. This is the shape of statement used in the
differentiated Poisson–Jensen f... | true |
CategoryTheory.Functor.LaxMonoidal.prod'._aux_1 | Mathlib.CategoryTheory.Monoidal.Functor | {C : Type u_6} →
[inst : CategoryTheory.Category.{u_5, u_6} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{D : Type u_3} →
[inst_2 : CategoryTheory.Category.{u_1, u_3} D] →
[inst_3 : CategoryTheory.MonoidalCategory D] →
{E : Type u_4} →
[inst_4 : CategoryThe... | null | false |
_private.Mathlib.Combinatorics.Pigeonhole.0.Fintype.exists_card_fiber_lt_of_card_lt_nsmul.match_1_1 | Mathlib.Combinatorics.Pigeonhole | ∀ {α : Type u_3} {β : Type u_1} {M : Type u_2} [inst : DecidableEq β] [inst_1 : Fintype α] [inst_2 : Fintype β]
(f : α → β) {b : M} [inst_3 : CommSemiring M] [inst_4 : LinearOrder M]
(motive : (∃ y ∈ Finset.univ, ↑{x | f x = y}.card < b) → Prop) (x : ∃ y ∈ Finset.univ, ↑{x | f x = y}.card < b),
(∀ (y : β) (left :... | null | false |
MeasureTheory.average_const | Mathlib.MeasureTheory.Integral.Average | ∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
[CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [h : NeZero μ] (c : E),
⨍ (_x : α), c ∂μ = c | null | true |
Batteries.UnionFind.link | Batteries.Data.UnionFind.Basic | (self : Batteries.UnionFind) → Fin self.size → (y : Fin self.size) → self.parent ↑y = ↑y → Batteries.UnionFind | Link a union-find node to a root node. | true |
CategoryTheory.Iso.self_symm_conj | Mathlib.CategoryTheory.Conj | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X Y : C} (α : X ≅ Y) (f : CategoryTheory.End Y),
α.conj (α.symm.conj f) = f | null | true |
_private.Mathlib.Analysis.Complex.Poisson.0.le_re_herglotzRieszKernel_aux._simp_1_4 | Mathlib.Analysis.Complex.Poisson | ∀ {M₀ : Type u_1} [inst : Mul M₀] [inst_1 : Zero M₀] [NoZeroDivisors M₀] {a b : M₀}, a ≠ 0 → b ≠ 0 → (a * b = 0) = False | null | false |
Lean.PersistentHashMap.instIteratorLoop | Lean.Data.Iterators.Producers.PersistentHashMap | {α : Type u_1} →
{β : Type u_2} → {n : Type u_3 → Type u_4} → [Monad n] → Std.IteratorLoop (Lean.PersistentHashMap.Zipper α β) Id n | null | true |
EmbeddingLike.comp_injective._simp_1 | Mathlib.Basic.FunLike.Embedding | ∀ {α : Sort u_2} {β : Sort u_3} {γ : Sort u_4} {F : Sort u_5} [inst : FunLike F β γ] [EmbeddingLike F β γ] (f : α → β)
(e : F), Function.Injective (⇑e ∘ f) = Function.Injective f | null | false |
_private.Mathlib.Algebra.Homology.ShortComplex.Preadditive.0.CategoryTheory.ShortComplex.Homotopy.eq_add_nullHomotopic._abel_1_2 | Mathlib.Algebra.Homology.ShortComplex.Preadditive | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
{S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂),
CategoryTheory.CategoryStruct.comp S₁.g h.h₂ + CategoryTheory.CategoryStruct.comp h.h₁ S₂.f + φ₂.τ₂ =
... | null | false |
_private.Mathlib.Order.OrderIsoNat.0.exists_covBy_seq_of_wellFoundedLT_wellFoundedGT_of_le._simp_1_2 | Mathlib.Order.OrderIsoNat | ∀ {α : Sort u} {p : α → Prop} {a1 a2 : { x // p x }}, (a1 = a2) = (↑a1 = ↑a2) | null | false |
AddSubmonoid.matrix._proof_1 | Mathlib.Data.Matrix.Basic | ∀ {m : Type u_1} {n : Type u_2} {A : Type u_3} [inst : AddMonoid A] (S : AddSubmonoid A) {a b : Matrix m n A},
a ∈ (↑S).matrix → b ∈ (↑S).matrix → ∀ (i : m) (j : n), a i j + b i j ∈ S | null | false |
CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_hom_app_hom₃ | Mathlib.CategoryTheory.Triangulated.TriangleShift | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C),
((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).hom.app X).hom₃ =
(CategoryTheory.shiftFunctorZero C ℤ).hom.app X... | null | true |
IsCyclotomicExtension.Rat.ramificationIdxIn_eq | Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | ∀ (n : ℕ) {m p k : ℕ} [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]
[IsCyclotomicExtension {n} ℚ K],
n = p ^ (k + 1) * m → ¬p ∣ m → (Ideal.span {↑p}).ramificationIdxIn (NumberField.RingOfIntegers K) = p ^ k * (p - 1) | Write `n = p ^ (k + 1) * m` where the prime `p` does not divide `m`, then the ramification index
of `p` in `ℚ(ζₙ)` is `p ^ k * (p - 1)`.
| true |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.Option.map_dmap | Std.Data.Internal.List.Associative | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} (x : Option α) (f : (a : α) → x = some a → β) (g : β → γ),
Option.map g (Std.Internal.List.Option.dmap✝ x f) = Std.Internal.List.Option.dmap✝ x fun a h => g (f a h) | null | true |
Lean.Grind.CommRing.Mon.beq' | Init.Grind.Ring.CommSolver | Lean.Grind.CommRing.Mon → Lean.Grind.CommRing.Mon → Bool | null | true |
Polynomial.natDegree_multiset_prod_of_monic | Mathlib.Algebra.Polynomial.BigOperators | ∀ {R : Type u} [inst : CommSemiring R] (t : Multiset (Polynomial R)),
(∀ f ∈ t, f.Monic) → t.prod.natDegree = (Multiset.map Polynomial.natDegree t).sum | null | true |
Complex.UnitI | Mathlib.Analysis.InnerProductSpace.StandardSubspace | ℂˣ | The imaginary unit as an invertible element. | true |
CategoryTheory.Bicategory.rightUnitor_comp_inv | Mathlib.CategoryTheory.Bicategory.Basic | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
(CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).inv =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).inv)
(Ca... | null | true |
ISize.toBitVec_or | Init.Data.SInt.Bitwise | ∀ (a b : ISize), (a ||| b).toBitVec = a.toBitVec ||| b.toBitVec | null | true |
Mathlib.Tactic.BicategoryLike.HorizontalComp.cons.sizeOf_spec | Mathlib.Tactic.CategoryTheory.Coherence.Normalize | ∀ (e : Mathlib.Tactic.BicategoryLike.Mor₂) (η : Mathlib.Tactic.BicategoryLike.WhiskerRight)
(ηs : Mathlib.Tactic.BicategoryLike.HorizontalComp),
sizeOf (Mathlib.Tactic.BicategoryLike.HorizontalComp.cons e η ηs) = 1 + sizeOf e + sizeOf η + sizeOf ηs | null | true |
Std.DTreeMap.Raw.instInhabited | Std.Data.DTreeMap.Raw.Basic | {α : Type u} → {β : α → Type v} → {cmp : α → α → Ordering} → Inhabited (Std.DTreeMap.Raw α β cmp) | null | true |
AlgCat.instCategory._proof_1 | Mathlib.Algebra.Category.AlgCat.Basic | ∀ (R : Type u_2) [inst : CommRing R] {X Y : AlgCat R} (f : X.Hom Y),
{ hom' := f.hom'.comp { hom' := AlgHom.id R ↑X }.hom' } = f | null | false |
CategoryTheory.MonoidalCategory.MonoidalRightAction.actionOfMonoidalFunctorToEndofunctor._proof_14 | Mathlib.CategoryTheory.Monoidal.Action.End | ∀ {C : Type u_2} {D : Type u_4} [inst : CategoryTheory.Category.{u_1, u_2} C]
[inst_1 : CategoryTheory.MonoidalCategory C] [inst_2 : CategoryTheory.Category.{u_3, u_4} D]
(F : CategoryTheory.Functor C (CategoryTheory.Functor D D)) [inst_3 : F.Monoidal] (c : C) {c' c'' : C} (f : c' ⟶ c'')
(d : D),
(F.map (Catego... | null | false |
ContinuousMonoidHom.compLeft._proof_1 | Mathlib.Topology.Algebra.Group.CompactOpen | ∀ {A : Type u_1} {B : Type u_3} (E : Type u_2) [inst : Monoid A] [inst_1 : Monoid B] [inst_2 : CommGroup E]
[inst_3 : TopologicalSpace A] [inst_4 : TopologicalSpace B] [inst_5 : TopologicalSpace E]
[inst_6 : IsTopologicalGroup E] (f : A →ₜ* B), ContinuousMonoidHom.comp 1 f = ContinuousMonoidHom.comp 1 f | null | false |
Finsupp.Lex.wellFounded | Mathlib.Data.Finsupp.WellFounded | ∀ {α : Type u_1} {N : Type u_2} [inst : Zero N] {r : α → α → Prop} {s : N → N → Prop},
(∀ ⦃n : N⦄, ¬s n 0) → WellFounded s → WellFounded (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) → WellFounded (Finsupp.Lex r s) | null | true |
ENNReal.add_lt_add_iff_right | Mathlib.Basic.ENNReal.Operations | ∀ {a b c : ENNReal}, a ≠ ⊤ → (b + a < c + a ↔ b < c) | null | true |
PSigma.Lex.orderTop._proof_1 | Mathlib.Order.PSigma | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : PartialOrder ι] [inst_1 : OrderTop ι] [inst_2 : (i : ι) → Preorder (α i)]
[inst_3 : OrderTop (α ⊤)] (a : ι) (b : α a), ⟨a, b⟩ ≤ ⟨⊤, ⊤⟩ | null | false |
Std.DTreeMap.Internal.Unit.RioSliceData._sizeOf_inst | Std.Data.DTreeMap.Internal.Zipper | (α : Type u) → {inst : Ord α} → [SizeOf α] → SizeOf (Std.DTreeMap.Internal.Unit.RioSliceData α) | null | false |
Bundle.Trivialization.coordChangeL | Mathlib.Topology.VectorBundle.Basic | (R : Type u_1) →
{B : Type u_2} →
{F : Type u_3} →
{E : B → Type u_4} →
[inst : Semiring R] →
[inst_1 : TopologicalSpace F] →
[inst_2 : TopologicalSpace B] →
[inst_3 : TopologicalSpace (Bundle.TotalSpace F E)] →
[inst_4 : AddCommMonoid F] →
... | A coordinate change function between two trivializations, as a continuous linear equivalence.
Defined to be the identity when `b` does not lie in the base set of both trivializations. | true |
PresheafOfModulesOfCommRing.instPreservesColimitsOfSizeTensorRight | Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {R : CategoryTheory.Functor Cᵒᵖ CommRingCat}
(F : PresheafOfModulesOfCommRing R),
CategoryTheory.Limits.PreservesColimitsOfSize.{u, u, max u u_1, max u u_1, max (max (u + 1) u_1) v_1,
max (max (u + 1) u_1) v_1}
(CategoryTheory.MonoidalCategory... | null | true |
Std.ExtDTreeMap.maxKeyD_le | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.ExtDTreeMap α β cmp} [inst : Std.TransCmp cmp],
t ≠ ∅ → ∀ {k fallback : α}, (cmp (t.maxKeyD fallback) k).isLE = true ↔ ∀ k' ∈ t, (cmp k' k).isLE = true | null | true |
Set.subset_symmDiff_union_symmDiff_left | Mathlib.Data.Set.SymmDiff | ∀ {α : Type u} {s t u : Set α}, Disjoint s t → u ⊆ symmDiff s u ∪ symmDiff t u | null | true |
HomologicalComplex.extend.d_eq | Mathlib.Algebra.Homology.Embedding.Extend | ∀ {ι : Type u_1} {c : ComplexShape ι} {C : Type u_3} [inst : CategoryTheory.Category.{v_1, u_3} C]
[inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
(K : HomologicalComplex C c) {i j : Option ι} {a b : ι} (hi : i = some a) (hj : j = some b),
HomologicalComplex.ex... | null | true |
ContinuousMap.tendsto_iff_tendstoLocallyUniformly | Mathlib.Topology.UniformSpace.CompactConvergence | ∀ {α : Type u₁} {β : Type u₂} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] {f : C(α, β)} {ι : Type u₃}
{p : Filter ι} {F : ι → C(α, β)} [WeaklyLocallyCompactSpace α],
Filter.Tendsto F p (nhds f) ↔ TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p | In a weakly locally compact space,
convergence in the compact-open topology is the same as locally uniform convergence.
The right-to-left implication holds in any topological space,
see `ContinuousMap.tendsto_of_tendstoLocallyUniformly`. | true |
CompletelyRegularSpace.mk | Mathlib.Topology.Separation.CompletelyRegular | ∀ {X : Type u} [inst : TopologicalSpace X],
(∀ (x : X) (K : Set X), IsClosed K → x ∉ K → ∃ f, Continuous f ∧ f x = 0 ∧ Set.EqOn f 1 K) → CompletelyRegularSpace X | null | true |
BitVec.carry_extractLsb'_eq_carry | Init.Data.BitVec.Bitblast | ∀ {w i len : ℕ},
i < len →
∀ {x y : BitVec w} {b : Bool},
BitVec.carry i (BitVec.extractLsb' 0 len x) (BitVec.extractLsb' 0 len y) b = BitVec.carry i x y b | The value of `(carry i x y false)` can be computed by truncating `x` and `y`
to `len` bits where `len ≥ i`. | true |
CategoryTheory.Bicategory.InducedBicategory.bicategory._proof_2 | Mathlib.CategoryTheory.Bicategory.InducedBicategory | ∀ {B : Type u_1} {C : Type u_2} [inst : CategoryTheory.Bicategory C] {F : B → C}
{a b c : CategoryTheory.Bicategory.InducedBicategory C F} (f : a ⟶ b) (g : b ⟶ c),
CategoryTheory.Bicategory.InducedBicategory.mkHom₂
(CategoryTheory.Bicategory.whiskerLeft f.hom (CategoryTheory.CategoryStruct.id g).hom) =
Ca... | null | false |
Flag.ofIsMaxChain._proof_2 | Mathlib.Order.Preorder.Chain | ∀ {α : Type u_1} [inst : LE α] (c : Set α),
IsMaxChain (fun x1 x2 => x1 ≤ x2) c → ∀ ⦃t : Set α⦄, IsChain (fun x1 x2 => x1 ≤ x2) t → c ⊆ t → c = t | null | false |
_private.Mathlib.CategoryTheory.Presentable.Comma.0.CategoryTheory.Comma.isCardinalAccessibleCategory.J.homMk.congr_simp | Mathlib.CategoryTheory.Presentable.Comma | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C₁]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_3, u_3} D]
{F₁ : CategoryTheory.Functor C₁ D} {F₂ : CategoryTheory.Functor C₂ D} {κ : Cardinal.{w}} [inst_3 : Fact κ.IsRegular]
{f :... | null | true |
Mathlib.Tactic.IntervalCases.Bound._sizeOf_1 | Mathlib.Tactic.IntervalCases | Mathlib.Tactic.IntervalCases.Bound → ℕ | null | false |
TopCommRingCat.isCommRing | Mathlib.Topology.Category.TopCommRingCat | (self : TopCommRingCat) → CommRing self.α | null | true |
mulActionSphereClosedBall._proof_2 | Mathlib.Analysis.Normed.Module.Ball.Action | ∀ {𝕜 : Type u_2} {E : Type u_1} [inst : NormedField 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
{r : ℝ} (x : ↑(Metric.closedBall 0 r)), 1 • x = x | null | false |
Submonoid.map.congr_simp | Mathlib.Algebra.Group.Submonoid.Operations | ∀ {M : Type u_1} {N : Type u_2} [inst : MulOneClass M] [inst_1 : MulOneClass N] {F : Type u_4} [inst_2 : FunLike F M N]
[mc : MonoidHomClass F M N] (f f_1 : F),
f = f_1 → ∀ (S S_1 : Submonoid M), S = S_1 → Submonoid.map f S = Submonoid.map f_1 S_1 | null | true |
CategoryTheory.Limits.idZeroEquivIsoZero_apply_hom | Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C]
[inst_2 : CategoryTheory.Limits.HasZeroMorphisms C] (X : C) (h : CategoryTheory.CategoryStruct.id X = 0),
((CategoryTheory.Limits.idZeroEquivIsoZero X) h).hom = 0 | null | true |
CentroidHom.instFunLike._proof_1 | Mathlib.Algebra.Ring.CentroidHom | ∀ {α : Type u_1} [inst : NonUnitalNonAssocSemiring α] (f g : CentroidHom α),
(fun f => (↑f.toAddMonoidHom).toFun) f = (fun f => (↑f.toAddMonoidHom).toFun) g → f = g | null | false |
CommGroupWithZero.ctorIdx | Mathlib.Algebra.GroupWithZero.Defs | {G₀ : Type u_2} → CommGroupWithZero G₀ → ℕ | null | false |
HomologicalComplex.units_smul_f_apply | Mathlib.Algebra.Homology.Linear | ∀ {R : Type u_1} [inst : Semiring R] {C : Type u_2} [inst_1 : CategoryTheory.Category.{v_1, u_2} C]
[inst_2 : CategoryTheory.Preadditive C] [inst_3 : CategoryTheory.Linear R C] {ι : Type u_4} {c : ComplexShape ι}
{X Y : HomologicalComplex C c} (r : Rˣ) (f : X ⟶ Y) (n : ι), (r • f).f n = r • f.f n | null | true |
CliffordAlgebra.instRing._aux_3 | Mathlib.LinearAlgebra.CliffordAlgebra.Basic | {R : Type u_1} →
[inst : CommRing R] →
{M : Type u_2} →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] → (Q : QuadraticForm R M) → CliffordAlgebra Q → CliffordAlgebra Q → CliffordAlgebra Q | null | false |
CategoryTheory.Limits.IsImage.lift | Mathlib.CategoryTheory.Limits.Shapes.Images | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{X Y : C} →
{f : X ⟶ Y} →
{F : CategoryTheory.Limits.MonoFactorisation f} →
CategoryTheory.Limits.IsImage F → (F' : CategoryTheory.Limits.MonoFactorisation f) → F.I ⟶ F'.I | Data exhibiting that a given factorisation through a mono is initial. | true |
CategoryTheory.Limits.weakPullback.snd | Mathlib.CategoryTheory.Limits.WeakLimits.WeakPullbacks | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{X Y Z : C} →
(f : X ⟶ Z) →
(g : Y ⟶ Z) → [inst_1 : CategoryTheory.Limits.HasWeakPullback f g] → CategoryTheory.Limits.weakPullback f g ⟶ Y | The second projection of the weak pullback of `f` and `g`. | true |
UpperHalfPlane.cuspFunction.eq_1 | Mathlib.NumberTheory.ModularForms.QExpansion | ∀ (h : ℝ) (f : UpperHalfPlane → ℂ),
UpperHalfPlane.cuspFunction h f = Function.Periodic.cuspFunction h (f ∘ ↑UpperHalfPlane.ofComplex) | null | true |
Std.Sat.AIG.Entrypoint.ctorIdx | Std.Sat.AIG.Basic | {α : Type} → {inst : DecidableEq α} → {inst_1 : Hashable α} → Std.Sat.AIG.Entrypoint α → ℕ | null | false |
CategoryTheory.Adjunction.mapAddGrp | Mathlib.CategoryTheory.Monoidal.Grp | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.CartesianMonoidalCategory C] →
{D : Type u₂} →
[inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
[inst_3 : CategoryTheory.CartesianMonoidalCategory D] →
{F : CategoryTheory.Functor C D} →
... | An adjunction of monoidal functors lifts to an adjunction of their lifts
to additive group objects. | true |
CategoryTheory.Limits.Cone.equivalenceOfReindexing | Mathlib.CategoryTheory.Limits.Cones | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
{K : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} K] →
{C : Type u₃} →
[inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
{F : CategoryTheory.Functor J C} →
{G : CategoryTheory.Functor K C} →
... | The categories of cones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic
(possibly after changing the indexing category by an equivalence).
| true |
Lean.Unhygienic.Context.mk.inj | Lean.Hygiene | ∀ {ref : Lean.Syntax} {scope : Lean.MacroScope} {ref_1 : Lean.Syntax} {scope_1 : Lean.MacroScope},
{ ref := ref, scope := scope } = { ref := ref_1, scope := scope_1 } → ref = ref_1 ∧ scope = scope_1 | null | true |
Std.DHashMap.insert_eq_insert | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} {p : (a : α) × β a},
insert p m = m.insert p.fst p.snd | null | true |
CategoryTheory.Pseudofunctor.DescentData'.pullHom'_self'._auto_1 | Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | Lean.Syntax | null | false |
Std.DHashMap.Equiv.symm | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.DHashMap α β}, m₁.Equiv m₂ → m₂.Equiv m₁ | null | true |
UInt8.toFin_inj | Init.Data.UInt.Lemmas | ∀ {a b : UInt8}, a.toFin = b.toFin ↔ a = b | null | true |
Representation.IsIrreducible.finrank_intertwiningMap_self | Mathlib.RepresentationTheory.Irreducible | ∀ {G : Type u_1} {k : Type u_2} {V : Type u_3} [inst : Monoid G] [inst_1 : Field k] [inst_2 : AddCommGroup V]
[inst_3 : Module k V] (ρ : Representation k G V) [ρ.IsIrreducible] [FiniteDimensional k V] [IsAlgClosed k],
Module.finrank k (ρ.IntertwiningMap ρ) = 1 | null | true |
Set.encard_exchange' | Mathlib.Data.Set.Card | ∀ {α : Type u_1} {s : Set α} {a b : α}, a ∉ s → b ∈ s → (insert a s \ {b}).encard = s.encard | null | true |
Std.IterM.dropWhileWithPostcondition | Std.Data.Iterators.Combinators.Monadic.DropWhile | {α : Type w} →
{m : Type w → Type w'} →
{β : Type w} → (P : β → Std.Iterators.PostconditionT m (ULift.{w, 0} Bool)) → Std.IterM m β → Std.IterM m β | *Note: This is a very general combinator that requires an advanced understanding of monads,
dependent types and termination proofs. The variants `dropWhile` and `dropWhileM` are easier to use
and sufficient for most use cases.*
Given an iterator `it` and a monadic predicate `P`, `it.dropWhileWithPostcondition P` is an... | true |
IsMinOn.of_subset | Mathlib.Order.Filter.Extr | ∀ {α : Type u} {β : Type v} [inst : Preorder β] {f : α → β} {s : Set α} {a : α} {t : Set α},
IsMinOn f t a → s ⊆ t → IsMinOn f s a | null | true |
_private.Init.Data.FloatArray.Basic.0.FloatArray.forIn.loop._proof_3 | Init.Data.FloatArray.Basic | ∀ (as : FloatArray) (i : ℕ), as.size - 1 < as.size → as.size - 1 - i < as.size | null | false |
Matrix.self_mul_conjTranspose_mulVec_eq_zero | Mathlib.LinearAlgebra.Matrix.DotProduct | ∀ {m : Type u_1} {n : Type u_2} {R : Type u_4} [inst : Fintype m] [inst_1 : Fintype n] [inst_2 : PartialOrder R]
[inst_3 : NonUnitalRing R] [inst_4 : StarRing R] [StarOrderedRing R] [IsProperStar R] (A : Matrix m n R) (v : m → R),
(A * A.conjTranspose).mulVec v = 0 ↔ A.conjTranspose.mulVec v = 0 | null | true |
Rat.le_coe_toNNRat | Mathlib.Data.NNRat.Defs | ∀ (q : ℚ), q ≤ ↑q.toNNRat | null | true |
_private.Init.Grind.Ordered.Rat.0.Lean.Grind.instOrderedAddRat._simp_1 | Init.Grind.Ordered.Rat | ∀ {a b c : ℚ}, (c + a ≤ c + b) = (a ≤ b) | null | false |
SSet.Subcomplex.unionProd.pushoutObjObj._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.PushoutProduct | ∀ {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex),
CategoryTheory.CategoryStruct.comp (SSet.Subcomplex.unionProd.ι₁ S T) (S.unionProd T).ι =
((CategoryTheory.MonoidalCategory.curriedTensor SSet).obj X).map T.ι | null | false |
CategoryTheory.ObjectProperty.IsVerdierLeftLocalizing.recOn | Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{A B : CategoryTheory.ObjectProperty C} →
{motive : A.IsVerdierLeftLocalizing B → Sort u} →
(t : A.IsVerdierLeftLocalizing B) →
((fac :
∀ {X Y : C} (f : X ⟶ Y), A X → B Y → ∃ Z a b, A Z ∧ B Z ∧ CategoryTheory.... | null | false |
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