name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
IsLocalHomeomorphOn.of_comp_left | Mathlib.Topology.IsLocalHomeomorph | ∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
[inst_2 : TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X},
IsLocalHomeomorphOn (g ∘ f) s → IsLocalHomeomorphOn g (f '' s) → (∀ x ∈ s, ContinuousAt f x) → IsLocalHomeomorphOn f s | null | true |
AddChar.PrimitiveAddChar.char | Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {R : Type u} →
[inst : CommRing R] →
{R' : Type v} →
[inst_1 : Field R'] → (self : AddChar.PrimitiveAddChar R R') → AddChar R (CyclotomicField (↑self.n) R') | The second projection from `PrimitiveAddChar`, giving the character. | true |
_private.Mathlib.LinearAlgebra.Contraction.0.dualTensorHom_bijective._simp_1_1 | Mathlib.LinearAlgebra.Contraction | ∀ {R : Type u_1} [inst : CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7}
[inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : AddCommMonoid Q]
[inst_5 : Module R M] [inst_6 : Module R N] [inst_7 : Module R P] [inst_8 : Module R Q] (g : P →ₗ[R] Q)
(... | null | false |
skyscraperSheafForgetAdjunction | Mathlib.Topology.Sheaves.Skyscraper | {X : TopCat} →
(p₀ : ↑X) →
[inst : (U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] →
{C : Type v} →
[inst_1 : CategoryTheory.Category.{u, v} C] →
[inst_2 : CategoryTheory.Limits.HasTerminal C] →
[inst_3 : CategoryTheory.Limits.HasColimits C] →
TopCat.Preshea... | Taking stalks is the left adjoint of `skyscraperSheafFunctor ⋙ Sheaf.forget`. Useful
only when the fact that `skyscraperPresheafFunctor` factors through `Sheaf C X` is relevant. | true |
MulEquiv.coe_subgroupMap_apply | Mathlib.Algebra.Group.Subgroup.Map | ∀ {G : Type u_1} {G' : Type u_2} [inst : Group G] [inst_1 : Group G'] (e : G ≃* G') (H : Subgroup G) (g : ↥H),
↑((e.subgroupMap H) g) = e ↑g | null | true |
Polynomial.expand_eq_zero | Mathlib.Algebra.Polynomial.Expand | ∀ {R : Type u} [inst : CommSemiring R] {p : ℕ}, 0 < p → ∀ {f : Polynomial R}, (Polynomial.expand R p) f = 0 ↔ f = 0 | null | true |
MeasureTheory.measureReal_univ_pos | Mathlib.MeasureTheory.Measure.Real | ∀ {α : Type u_1} {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ],
0 < μ.real Set.univ | null | true |
LieModuleHom.codRestrict.congr_simp | Mathlib.Algebra.Lie.Weights.Cartan | ∀ {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [inst : CommRing R] [inst_1 : LieRing L]
[inst_2 : AddCommGroup M] [inst_3 : Module R M] [inst_4 : LieRingModule L M] [inst_5 : AddCommGroup N]
[inst_6 : Module R N] [inst_7 : LieRingModule L N] (P : LieSubmodule R L N) (f f_1 : M →ₗ⁅R,L⁆ N) (e_f : f = f_1)
(... | null | true |
List.SortedGT.reverse | Mathlib.Data.List.Sort | ∀ {α : Type u_1} {l : List α} [inst : Preorder α], l.SortedGT → l.reverse.SortedLT | **Alias** of the reverse direction of `List.sortedLT_reverse`. | true |
Std.ExtDTreeMap.Const.isSome_get?_iff_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]
{a : α}, (Std.ExtDTreeMap.Const.get? t a).isSome = true ↔ a ∈ t | null | true |
CategoryTheory.End.smul_left | Mathlib.CategoryTheory.Endomorphism | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} {r : (CategoryTheory.End X)ᵐᵒᵖ} {f : X ⟶ Y},
r • f = CategoryTheory.CategoryStruct.comp (MulOpposite.unop r).asHom f | null | true |
StarAddMonoid.mk | Mathlib.Algebra.Star.Basic | {R : Type u} →
[inst : AddMonoid R] →
[toInvolutiveStar : InvolutiveStar R] → (∀ (r s : R), star (r + s) = star r + star s) → StarAddMonoid R | null | true |
CategoryTheory.ComposableArrows.IsComplex.opcyclesToCycles._proof_6 | Mathlib.Algebra.Homology.ExactSequenceFour | ∀ {n : ℕ} (k : ℕ),
autoParam (k ≤ n) CategoryTheory.ComposableArrows.IsComplex.opcyclesToCycles._auto_1 → k + 1 + 2 ≤ n + 3 | null | false |
CategoryTheory.MonoOver.supLe._proof_1 | Mathlib.CategoryTheory.Subobject.Lattice | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [CategoryTheory.Limits.HasBinaryCoproducts C] {A : C}
(f g : CategoryTheory.MonoOver A), CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.pair f.obj.left g.obj.left) | null | false |
CategoryTheory.Functor.instIsTriangulatedPlusMapHomotopyCategoryPlus | Mathlib.Algebra.Homology.HomotopyCategory.Plus | ∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Preadditive C]
[inst_3 : CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D) [inst_4 : F.Additive]
[inst_5 : CategoryTheory.Limits.HasZeroObject C] [ins... | null | true |
Num.pos.injEq | Mathlib.Data.Num.Basic | ∀ (a a_1 : PosNum), (Num.pos a = Num.pos a_1) = (a = a_1) | null | true |
Function.FromTypes | Mathlib.Logic.Function.FromTypes | {n : ℕ} → (Fin n → Type u) → Type u → Type u | The type of `n`-ary functions `p 0 → p 1 → ... → p (n - 1) → τ`. | true |
_private.Lean.Server.FileWorker.ExampleHover.0.Lean.Server.FileWorker.Hover.RWState.output.injEq | Lean.Server.FileWorker.ExampleHover | ∀ (indent ticks indent_1 ticks_1 : ℕ),
(Lean.Server.FileWorker.Hover.RWState.output✝ indent ticks =
Lean.Server.FileWorker.Hover.RWState.output✝ indent_1 ticks_1) =
(indent = indent_1 ∧ ticks = ticks_1) | null | true |
Lean.Elab.Term.elabStateRefT | Lean.Elab.BuiltinNotation | Lean.Elab.Term.TermElab | null | true |
Real.instFloorRing | Mathlib.Algebra.Order.Archimedean.Real.Basic | FloorRing ℝ | null | true |
Std.DTreeMap.Equiv.foldrM_eq | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.DTreeMap α β cmp} {δ : Type w}
{m : Type w → Type w'} [Std.TransCmp cmp] [inst : Monad m] [LawfulMonad m] {f : (a : α) → β a → δ → m δ} {init : δ},
t₁.Equiv t₂ → Std.DTreeMap.foldrM f init t₁ = Std.DTreeMap.foldrM f init t₂ | null | true |
Subsemiring.mk'._proof_4 | Mathlib.Algebra.Ring.Subsemiring.Defs | ∀ {R : Type u_1} [inst : NonAssocSemiring R] (s : Set R) (sa : AddSubmonoid R), ↑sa = s → 0 ∈ s | null | false |
deriv_fun_pow | Mathlib.Analysis.Calculus.Deriv.Pow | ∀ {𝕜 : Type u_1} {𝔸 : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedCommRing 𝔸]
[inst_2 : NormedAlgebra 𝕜 𝔸] {f : 𝕜 → 𝔸} {x : 𝕜},
DifferentiableAt 𝕜 f x → ∀ (n : ℕ), deriv (fun i => f i ^ n) x = ↑n * f x ^ (n - 1) * deriv f x | Eta-expanded form of `deriv_pow` | true |
_private.Mathlib.Data.Nat.Prime.Defs.0.Nat.minFac_has_prop | Mathlib.Data.Nat.Prime.Defs | ∀ {n : ℕ}, n ≠ 1 → Nat.minFacProp✝ n n.minFac | null | true |
Int.Cooper.mul_resolve_left_inv_le | Init.Data.Int.Cooper | ∀ {b q : ℤ} (a p k : ℤ), 0 < a → b * k + b * p ≤ a * q → a ∣ k + p → b * Int.Cooper.resolve_left_inv a p k ≤ q | null | true |
CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj | Mathlib.CategoryTheory.Abelian.LeftDerived | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{D : Type u_1} →
[inst_1 : CategoryTheory.Category.{v_1, u_1} D] →
[inst_2 : CategoryTheory.Abelian C] →
[inst_3 : CategoryTheory.HasProjectiveResolutions C] →
[inst_4 : CategoryTheory.Abelian D] →
{X : C}... | If `P : ProjectiveResolution Z` and `F : C ⥤ D` is an additive functor, this is
an isomorphism between `F.leftDerivedToHomotopyCategory.obj X` and the complex
obtained by applying `F` to `P.complex`. | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.mem_toList_iff_getKey?_eq_some_and_get?_eq_some._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {x : Ord α} {x_1 : BEq α} [Std.LawfulBEqOrd α] {a b : α}, (compare a b = Ordering.eq) = ((a == b) = true) | null | false |
_private.Mathlib.Order.Cover.0.Set.Ioo_eq_singleton_iff._to_dual_1.match_1_6 | Mathlib.Order.Cover | ∀ {α : Type u_1} [inst : LinearOrder α] {a b c : α} (motive : c < a ∧ b < c → Prop) (x : c < a ∧ b < c),
(∀ (hac : c < a) (hcb : b < c), motive ⋯) → motive x | null | false |
monotoneOn_univ._simp_1 | Mathlib.Order.Monotone.Defs | ∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] {f : α → β}, MonotoneOn f Set.univ = Monotone f | null | false |
CategoryTheory.Functor.IsDenseSubsite.sheafEquiv | Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{D : Type u_2} →
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
(J : CategoryTheory.GrothendieckTopology C) →
(K : CategoryTheory.GrothendieckTopology D) →
(G : CategoryTheory.Functor C D) →
(A : ... | If `G : C ⥤ D` exhibits `(C, J)` as a dense subsite of `(D, K)`, and the
pushforward functor `Sheaf K A ⥤ Sheaf J A` is an equivalence, then this
is the equivalence `Sheaf K A ≌ Sheaf J A`. | true |
Std.DHashMap.mem_toArray_iff_get?_eq_some | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} [inst : LawfulBEq α] {k : α}
{v : β k}, ⟨k, v⟩ ∈ m.toArray ↔ m.get? k = some v | null | true |
_private.Mathlib.Tactic.Linter.FlexibleLinter.0.Mathlib.Linter.Flexible.TacticData.mk.sizeOf_spec | Mathlib.Tactic.Linter.FlexibleLinter | ∀ (stx : Lean.Syntax) (ci : Lean.Elab.ContextInfo) (mctxBefore mctxAfter : Lean.MetavarContext)
(goalsTargetedBy goalsCreatedBy : List Lean.MVarId),
sizeOf
{ stx := stx, ci := ci, mctxBefore := mctxBefore, mctxAfter := mctxAfter, goalsTargetedBy := goalsTargetedBy,
goalsCreatedBy := goalsCreatedBy } =... | null | true |
smul_add_smul_le_smul_add_smul | Mathlib.Algebra.Order.Module.Defs | ∀ {α : Type u_1} {β : Type u_2} [inst : Semiring α] [inst_1 : PartialOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
[inst_4 : AddCommMonoid β] [inst_5 : PartialOrder β] [IsOrderedCancelAddMonoid β] [inst_7 : Module α β]
[PosSMulMono α β] {a₁ a₂ : α} {b₁ b₂ : β}, a₁ ≤ a₂ → b₁ ≤ b₂ → a₁ • b₂ + a₂ • b₁ ≤ a₁ • b₁ +... | Binary **rearrangement inequality**. | true |
_private.Lean.Meta.Tactic.Grind.MatchCond.0.Lean.Meta.Grind.collectMatchCondLhss._sparseCasesOn_1 | Lean.Meta.Tactic.Grind.MatchCond | {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 |
Quaternion.coe_real_complex_mul | Mathlib.Analysis.Quaternion | ∀ (r : ℝ) (z : ℂ), r • ↑z = ↑r * ↑z | null | true |
IsLocalization.instForallPiUniv | Mathlib.RingTheory.Localization.Pi | ∀ {ι : Type u_1} (R : ι → Type u_2) (S : ι → Type u_3) [inst : (i : ι) → CommSemiring (R i)]
[inst_1 : (i : ι) → CommSemiring (S i)] [inst_2 : (i : ι) → Algebra (R i) (S i)] (M : (i : ι) → Submonoid (R i))
[∀ (i : ι), IsLocalization (M i) (S i)], IsLocalization (Submonoid.pi Set.univ M) ((i : ι) → S i) | If `S i` is a localization of `R i` at the submonoid `M i` for each `i`,
then `Π i, S i` is a localization of `Π i, R i` at the product submonoid. | true |
Homotopy.nullHomotopicMap'_f_eq_zero | Mathlib.Algebra.Homology.Homotopy | ∀ {ι : Type u_1} {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Preadditive V]
{c : ComplexShape ι} {C D : HomologicalComplex V c} {k₀ : ι},
(∀ (l : ι), ¬c.Rel k₀ l) →
(∀ (l : ι), ¬c.Rel l k₀) → ∀ (h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)), (Homotopy.nullHomotopicMap' h).f k₀... | null | true |
IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe | Mathlib.RingTheory.Localization.AtPrime.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [inst_3 : IsLocalization.AtPrime S I] (a : { p // p.IsPrime ∧ p ≤ I }),
↑↑((RelIso.symm (IsLocalization.AtPrime.orderIsoOfPrime S I)) a) =
⋂ s,
⋂ (_ :
↑↑((Set.ord... | null | true |
List.subset_dedup | Mathlib.Data.List.Dedup | ∀ {α : Type u_1} [inst : DecidableEq α] (l : List α), l ⊆ l.dedup | null | true |
Subring.topologicalClosure._proof_3 | Mathlib.Topology.Algebra.Ring.Basic | ∀ {R : Type u_1} [inst : TopologicalSpace R] [inst_1 : Ring R] [inst_2 : IsSemitopologicalRing R] (S : Subring R)
{a b : R}, a ∈ S.topologicalClosure.carrier → b ∈ S.topologicalClosure.carrier → a * b ∈ S.topologicalClosure.carrier | null | false |
USize.ofFin_mk | Init.Data.UInt.Lemmas | ∀ {n : ℕ} (hn : n < USize.size), USize.ofFin ⟨n, hn⟩ = USize.ofNatLT n hn | null | true |
LawfulBitraversable.binaturality' | Mathlib.Control.Bitraversable.Basic | ∀ {t : Type u → Type u → Type u} {inst : Bitraversable t} [self : LawfulBitraversable t] {F G : Type u → Type u}
[inst_1 : Applicative F] [inst_2 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
(η : ApplicativeTransformation F G) {α α' β β' : Type u} (f : α → F β) (f' : α' → F β'),
(fun {α} => η.app ... | null | true |
Finsupp.fst_sumFinsuppAddEquivProdFinsupp | Mathlib.Data.Finsupp.Basic | ∀ {M : Type u_5} [inst : AddMonoid M] {α : Type u_12} {β : Type u_13} (f : α ⊕ β →₀ M) (x : α),
(Finsupp.sumFinsuppAddEquivProdFinsupp f).1 x = f (Sum.inl x) | null | true |
List.scanrM_map | Init.Data.List.Scan.Lemmas | ∀ {m : Type u_1 → Type u_2} {α₁ : Type u_3} {α₂ : Type u_4} {β : Type u_1} {init : β} [inst : Monad m] [LawfulMonad m]
{f : α₁ → α₂} {g : α₂ → β → m β} {as : List α₁},
List.scanrM g init (List.map f as) = List.scanrM (fun a b => g (f a) b) init as | null | true |
Representation.linearize._proof_2 | Mathlib.RepresentationTheory.Action | ∀ (k : Type u_1) (G : Type u_3) [inst : Monoid G] [inst_1 : Semiring k] (X : Action (Type u_2) G) (x x_1 : G),
MonoidAlgebra.mapDomainLinearMap k k ⇑(CategoryTheory.ConcreteCategory.hom (X.ρ (x * x_1)).asHom) =
MonoidAlgebra.mapDomainLinearMap k k ⇑(CategoryTheory.ConcreteCategory.hom (X.ρ x).asHom) *
Monoi... | null | false |
_private.Init.Data.String.Lemmas.Pattern.Memcmp.0.String.Slice.Pattern.Internal.memcmpStr_eq_true_iff._proof_1_8 | Init.Data.String.Lemmas.Pattern.Memcmp | ∀ {lhs rhs : String} {lstart rstart : String.Pos.Raw} {len : String.Pos.Raw} (curr : String.Pos.Raw),
¬len.byteIdx < curr.byteIdx → ¬len.byteIdx = curr.byteIdx + (len.byteIdx - curr.byteIdx) → False | null | false |
SeminormedAddCommGroup.toIsTopologicalAddGroup | Mathlib.Analysis.Normed.Group.Uniform | ∀ {E : Type u_2} [inst : SeminormedAddCommGroup E], IsTopologicalAddGroup E | null | true |
FirstOrder.Language.BoundedFormula.realize_relabel._simp_1 | Mathlib.ModelTheory.Semantics | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {α : Type u'} {β : Type v'} {m n : ℕ}
{φ : L.BoundedFormula α n} {g : α → β ⊕ Fin m} {v : β → M} {xs : Fin (m + n) → M},
(FirstOrder.Language.BoundedFormula.relabel g φ).Realize v xs =
φ.Realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAd... | null | false |
Float.Model.rec.eq._@.Mathlib.Util.CompileInductive.3310122970._hygCtx._hyg.315 | Mathlib.Util.CompileInductive | @Float.Model.rec = @Float.Model.rec✝ | null | false |
List.map_zip_eq_zipWith | Init.Data.List.Zip | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α × β → γ} {l : List α} {l' : List β},
List.map f (l.zip l') = List.zipWith (Function.curry f) l l' | null | true |
Matrix.kroneckerTMulAlgEquiv_symm_apply | Mathlib.RingTheory.MatrixAlgebra | ∀ {m : Type u_2} {n : Type u_3} (R : Type u_5) (S : Type u_6) {A : Type u_7} {B : Type u_8} [inst : CommSemiring R]
[inst_1 : Semiring A] [inst_2 : Semiring B] [inst_3 : Algebra R A] [inst_4 : Algebra R B] [inst_5 : Fintype n]
[inst_6 : DecidableEq n] [inst_7 : CommSemiring S] [inst_8 : Algebra R S] [inst_9 : Algeb... | null | true |
tendsto_fib_succ_div_fib_atTop | Mathlib.Analysis.SpecificLimits.Fibonacci | Filter.Tendsto (fun n => ↑(Nat.fib (n + 1)) / ↑(Nat.fib n)) Filter.atTop (nhds Real.goldenRatio) | The limit of `fib (n + 1) / fib n` as `n → ∞` is the golden ratio. | true |
CommGroup.subgroupOrderIsoSubgroupMonoidHom.eq_1 | Mathlib.GroupTheory.FiniteAbelian.Duality | ∀ (G : Type u_1) (M : Type u_2) [inst : CommGroup G] [inst_1 : Finite G] [inst_2 : CommMonoid M]
[hM : HasEnoughRootsOfUnity M (Monoid.exponent G)],
CommGroup.subgroupOrderIsoSubgroupMonoidHom G M =
{ toFun := fun H => OrderDual.toDual (MonoidHom.domRestrictHom H Mˣ).ker,
invFun := fun Φ =>
(CommG... | null | true |
Nat.lcm_eq_mul_iff | Init.Data.Nat.Lcm | ∀ {m n : ℕ}, m.lcm n = m * n ↔ m = 0 ∨ n = 0 ∨ m.gcd n = 1 | null | true |
CategoryTheory.Limits.WidePushoutShape.struct._proof_5 | Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | ∀ {J : Type u_1} {Z : CategoryTheory.Limits.WidePushoutShape J}, Z = Z | null | false |
CategoryTheory.Functor.copyObj.eq_1 | Mathlib.CategoryTheory.NatIso | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(F : CategoryTheory.Functor C D) (obj : C → D) (e : (X : C) → F.obj X ≅ obj X),
F.copyObj obj e =
{ obj := obj,
map := fun {X Y} f =>
CategoryTheory.CategoryStruct.comp (e X).inv... | null | true |
Basis.le_span'' | Mathlib.LinearAlgebra.Dimension.StrongRankCondition | ∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [RankCondition R]
{ι : Type u_2} [inst_4 : Fintype ι] (b : Module.Basis ι R M) {w : Set M} [inst_5 : Fintype ↑w],
Submodule.span R w = ⊤ → Fintype.card ι ≤ Fintype.card ↑w | An auxiliary lemma for `Basis.le_span`.
If `R` satisfies the rank condition,
then for any finite basis `b : Basis ι R M`,
and any finite spanning set `w : Set M`,
the cardinality of `ι` is bounded by the cardinality of `w`.
| true |
Module.Baer.extensionOfMaxAdjoin._proof_2 | Mathlib.Algebra.Module.Injective | ∀ {R : Type u_2} [inst : Ring R] {Q : Type u_1} [inst_1 : AddCommGroup Q] [inst_2 : Module R Q] {M : Type u_3}
{N : Type u_4} [inst_3 : AddCommGroup M] [inst_4 : AddCommGroup N] [inst_5 : Module R M] [inst_6 : Module R N]
(i : M →ₗ[R] N) (f : M →ₗ[R] Q) [inst_7 : Fact (Function.Injective ⇑i)] (h : Module.Baer R Q) ... | null | false |
List.Vector.get_replicate | Mathlib.Data.Vector.Basic | ∀ {α : Type u_1} {n : ℕ} (a : α) (i : Fin n), (List.Vector.replicate n a).get i = a | null | true |
Int.sign_eq_neg_one_iff_neg | Init.Data.Int.Order | ∀ {a : ℤ}, a.sign = -1 ↔ a < 0 | null | true |
AddCircle.equivAddCircle.congr_simp | Mathlib.Topology.Instances.AddCircle.Defs | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0),
AddCircle.equivAddCircle p q hp hq = AddCircle.equivAddCircle p q hp hq | null | true |
_private.Mathlib.Tactic.HigherOrder.0.Mathlib.Tactic.higherOrderGetParam.match_1 | Mathlib.Tactic.HigherOrder | (motive : Lean.FVarId × Lean.MVarId → Sort u_1) →
(x : Lean.FVarId × Lean.MVarId) → ((fst : Lean.FVarId) → (mvarId : Lean.MVarId) → motive (fst, mvarId)) → motive x | null | false |
ISize.toInt_inj | Init.Data.SInt.Lemmas | ∀ {x y : ISize}, x.toInt = y.toInt ↔ x = y | null | true |
CategoryTheory.ChosenPullbacksAlong.isoInv_pullback_obj_left | Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (Z : CategoryTheory.Over Y),
((CategoryTheory.ChosenPullbacksAlong.pullback f.inv).obj Z).left = Z.left | null | true |
Lean.Elab.Term.wrapAsyncAsSnapshot | Lean.Elab.Term.TermElabM | {α : Type} →
(α → Lean.Elab.TermElabM Unit) →
Option IO.CancelToken →
autoParam String Lean.Elab.Term.wrapAsyncAsSnapshot._auto_1 →
Lean.Elab.TermElabM (α → BaseIO Lean.Language.SnapshotTree) | Wraps the given action for use in `BaseIO.asTask` etc., discarding its final state except for
`logSnapshotTask` tasks, which are reported as part of the returned tree. The given cancellation
token, if any, should be stored in a `SnapshotTask` for the server to trigger it when the result is
no longer needed.
| true |
Mathlib.Tactic.Sat.Clause.expr | Mathlib.Tactic.Sat.FromLRAT | Mathlib.Tactic.Sat.Clause → Lean.Expr | The clause expression of type `Clause` | true |
_private.Mathlib.Algebra.ContinuedFractions.Computation.Translations.0.Option.getD.match_1.eq_2 | Mathlib.Algebra.ContinuedFractions.Computation.Translations | ∀ {α : Type u_1} (motive : Option α → Sort u_2) (h_1 : (x : α) → motive (some x)) (h_2 : Unit → motive none),
(match none with
| some x => h_1 x
| none => h_2 ()) =
h_2 () | null | true |
Real.Angle.abs_toReal_coe_eq_self_iff | Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | ∀ {θ : ℝ}, |(↑θ).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ Real.pi | null | true |
_private.Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy.0.HomotopicalAlgebra.FibrantObject.instHasQuotientWeakEquivalencesHomRel._simp_1 | Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C],
HomotopicalAlgebra.weakEquivalences C f = HomotopicalAlgebra.WeakEquivalence f | null | false |
Std.TreeSet.getD_max? | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp] {km fallback : α},
t.max? = some km → t.getD km fallback = km | null | true |
Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Init.Tactics | Lean.Macro | `letI` behaves like `let`, but inlines the value instead of producing a `let` term. | false |
CategoryTheory.Under.instHasColimitsOfSizeOfHasWidePushouts | Mathlib.CategoryTheory.Limits.Constructions.Over.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {B : C} [CategoryTheory.Limits.HasWidePushouts C],
CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, max u v} (CategoryTheory.Under B) | null | true |
_private.Mathlib.Probability.Kernel.IonescuTulcea.Maps.0.IocProdIoc_preimage._proof_1_15 | Mathlib.Probability.Kernel.IonescuTulcea.Maps | ∀ {ι : Type u_1} [inst : LinearOrder ι] [inst_1 : LocallyFiniteOrder ι] {a b c : ι} (hab : a ≤ b) (w : ι) (w_1 : b < w)
(w_2 : w ≤ c), ↑⟨w, ⋯⟩ ∈ Finset.Ioc b c | null | false |
summable_of_hasFiniteSupport | Mathlib.Topology.Algebra.InfiniteSum.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f : β → α}
{L : SummationFilter β} [L.HasSupport], Function.HasFiniteSupport f → Summable f L | null | true |
Std.Time.Modifier.ctorElimType | Std.Time.Format.Modifier | {motive : Std.Time.Modifier → Sort u} → ℕ → Sort (max 1 u) | null | false |
Lean.IR.Decl.ctorElim | Lean.Compiler.IR.Basic | {motive : Lean.IR.Decl → Sort u} →
(ctorIdx : ℕ) → (t : Lean.IR.Decl) → ctorIdx = t.ctorIdx → Lean.IR.Decl.ctorElimType ctorIdx → motive t | null | false |
instIsPredArchimedeanOrderDualOfIsSuccArchimedean | Mathlib.Order.SuccPred.Archimedean | ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : SuccOrder α] [IsSuccArchimedean α], IsPredArchimedean αᵒᵈ | null | true |
MulEquiv.val_piUnits_apply | Mathlib.Algebra.Group.Pi.Units | ∀ {ι : Type u_1} {M : ι → Type u_2} [inst : (i : ι) → Monoid (M i)] (f : ((i : ι) → M i)ˣ) (i : ι),
↑(MulEquiv.piUnits f i) = ↑f i | null | true |
Equiv.isCancelAdd | Mathlib.Algebra.Group.TransferInstance | ∀ {α : Type u_2} {β : Type u_3} (e : α ≃ β) [inst : Add β] [IsCancelAdd β], IsCancelAdd α | Transfer `IsCancelAdd` across an `Equiv` | true |
_private.Mathlib.Topology.UniformSpace.Closeds.0.UniformSpace.hausdorff.uniformContinuous_union.match_1_1 | Mathlib.Topology.UniformSpace.Closeds | ∀ {α : Type u_1} (U : Set (α × α)) (a : (Set α × Set α) × Set α × Set α)
(motive : a ∈ entourageProd (hausdorffEntourage U) (hausdorffEntourage U) → Prop)
(h : a ∈ entourageProd (hausdorffEntourage U) (hausdorffEntourage U)),
(∀ (h₁ : (a.1.1, a.2.1) ∈ hausdorffEntourage U) (h₂ : (a.1.2, a.2.2) ∈ hausdorffEntourag... | null | false |
_private.Batteries.Data.Array.Scan.0.Array.getElem_succ_scanl._proof_1_4 | Batteries.Data.Array.Scan | ∀ {β : Type u_1} {α : Type u_2} {i : ℕ} {b : β} {as : Array α} {f : β → α → β} (h : i + 1 < (Array.scanl f b as).size),
(List.scanl f b as.toList)[i + 1] = f (List.scanl f b as.toList)[i] as[i] | null | false |
MinTropical.instDecidableLT.eq_1 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u} [inst : LT R] [inst_1 : DecidableLT R], MinTropical.instDecidableLT = inferInstance | null | true |
RootPairing.RootPositiveForm.posForm._proof_5 | Mathlib.LinearAlgebra.RootSystem.RootPositive | ∀ {S : Type u_1} [inst : CommRing S], RingHomSurjective (RingHom.id S) | null | false |
_private.Init.Data.String.Basic.0.String.Pos.ofSliceTo_sliceTo._simp_1_1 | Init.Data.String.Basic | ∀ {s : String} {x y : s.Pos}, (x = y) = (x.offset = y.offset) | null | false |
Std.DHashMap.Const.get!_inter_of_not_mem_right | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m₁ m₂ : Std.DHashMap α fun x => β} [EquivBEq α]
[LawfulHashable α] [inst : Inhabited β] {k : α}, k ∉ m₂ → Std.DHashMap.Const.get! (m₁.inter m₂) k = default | null | true |
Std.TreeMap.minKey_le_minKey_erase | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
{he : (t.erase k).isEmpty = false}, (cmp (t.minKey ⋯) ((t.erase k).minKey he)).isLE = true | null | true |
contDiffGroupoid_prod | Mathlib.Geometry.Manifold.IsManifold.Basic | ∀ {n : WithTop ℕ∞} {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] {E' : Type u_5} {H' : Type u_6}
[inst_4 : NormedAddCommGroup E'] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : TopologicalSpace H']
... | The product of two `C^n` open partial homeomorphisms is `C^n`. | true |
LinOrd.hom_comp | Mathlib.Order.Category.LinOrd | ∀ {X Y Z : LinOrd} (f : X ⟶ Y) (g : Y ⟶ Z),
LinOrd.Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (LinOrd.Hom.hom g).comp (LinOrd.Hom.hom f) | null | true |
isOpen_setOfPred_eventually_nhdsWithin | Mathlib.Topology.Separation.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {p : X → Prop},
IsOpen {x | ∀ᶠ (y : X) in nhdsWithin x {x}ᶜ, p y} | null | true |
_private.Mathlib.Data.Num.Basic.0.PosNum.sqrtAux._unsafe_rec | Mathlib.Data.Num.Basic | PosNum → Num → Num → Num | null | false |
_private.Mathlib.Util.WhatsNew.0.Mathlib.WhatsNew.whatsNew.match_1 | Mathlib.Util.WhatsNew | (motive : Lean.Name × Lean.ConstantInfo → Sort u_1) →
(x : Lean.Name × Lean.ConstantInfo) → ((c : Lean.Name) → (i : Lean.ConstantInfo) → motive (c, i)) → motive x | null | false |
Set.piecewise_preimage | Mathlib.Data.Set.Piecewise | ∀ {α : Type u_1} {β : Type u_2} (s : Set α) [inst : (j : α) → Decidable (j ∈ s)] (f g : α → β) (t : Set β),
s.piecewise f g ⁻¹' t = s.ite (f ⁻¹' t) (g ⁻¹' t) | null | true |
Submodule.hasDistribPointwiseNeg._proof_3 | Mathlib.Algebra.Algebra.Operations | ∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra R A] (S : Submodule R A),
(-S).toAddSubmonoid = -S.toAddSubmonoid | null | false |
Polynomial.normUnit_X | Mathlib.Algebra.Polynomial.FieldDivision | ∀ {R : Type u} [inst : CommRing R] [inst_1 : NoZeroDivisors R] [inst_2 : NormalizationMonoid R],
normUnit Polynomial.X = 1 | null | true |
ProofWidgets.FilterDetailsProps.recOn | ProofWidgets.Component.FilterDetails | {motive : ProofWidgets.FilterDetailsProps → Sort u} →
(t : ProofWidgets.FilterDetailsProps) →
((summary filtered all : ProofWidgets.Html) →
(initiallyFiltered : Bool) →
motive { summary := summary, filtered := filtered, all := all, initiallyFiltered := initiallyFiltered }) →
motive t | null | false |
CategoryTheory.Limits.functorCategoryHasLimitsOfShape | Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type u₁} [inst_1 : CategoryTheory.Category.{v₁, u₁} J]
{K : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C],
CategoryTheory.Limits.HasLimitsOfShape J (CategoryTheory.Functor K C) | null | true |
AddSubsemigroup.prod_eq_top_iff | Mathlib.Algebra.Group.Subsemigroup.Operations | ∀ {M : Type u_1} {N : Type u_2} [inst : Add M] [inst_1 : Add N] [Nonempty M] [Nonempty N] {s : AddSubsemigroup M}
{t : AddSubsemigroup N}, s.prod t = ⊤ ↔ s = ⊤ ∧ t = ⊤ | null | true |
AlgebraicClosure.instCommRing._aux_18 | Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | (k : Type u_1) → [inst : Field k] → ℕ → AlgebraicClosure k → AlgebraicClosure k | null | false |
WeierstrassCurve.Jacobian.Point.toAffineAddEquiv_symm_apply | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | ∀ {F : Type u} [inst : Field F] (W : WeierstrassCurve.Jacobian F) [inst_1 : DecidableEq F] (a : W.toAffine.Point),
(WeierstrassCurve.Jacobian.Point.toAffineAddEquiv W).symm a = WeierstrassCurve.Jacobian.Point.fromAffine a | null | true |
_private.Lean.Language.Lean.Util.0.Lean.Language.SnapshotTree.findInfoTreeAtPos.match_1 | Lean.Language.Lean.Util | (motive : Option Lean.Elab.InfoTree → Sort u_1) →
(x : Option Lean.Elab.InfoTree) →
((infoTree : Lean.Elab.InfoTree) → motive (some infoTree)) → ((x : Option Lean.Elab.InfoTree) → motive x) → motive x | null | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.