name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
OrderIso.mapSetOfPredMinimal | Mathlib.Order.Minimal | {α : Type u_2} →
{β : Type u_3} →
[inst : Preorder α] →
[inst_1 : Preorder β] →
{s : Set α} → {t : Set β} → ↑s ≃o ↑t → ↑{x | Minimal (fun x => x ∈ s) x} ≃o ↑{x | Minimal (fun x => x ∈ t) x} | If two sets are order isomorphic, their minimals are also order isomorphic. | true |
MLList.filterMap | Batteries.Data.MLList.Basic | {m : Type u_1 → Type u_1} → {α β : Type u_1} → [Monad m] → (α → Option β) → MLList m α → MLList m β | Filter and transform a `MLList` using an `Option` valued function. | true |
Std.Http.Method.uncheckout.sizeOf_spec | Std.Http.Data.Method | sizeOf Std.Http.Method.uncheckout = 1 | null | true |
CFC.log_one | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | ∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : StarRing A] [inst_2 : NormedAlgebra ℝ A]
[inst_3 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint], CFC.log 1 = 0 | null | true |
_private.Init.Data.Range.Polymorphic.SInt.0.Int32.instUpwardEnumerable._proof_1 | Init.Data.Range.Polymorphic.SInt | ∀ (n : ℕ) (i : Int32), Int32.minValue.toInt ≤ i.toInt → ¬Int32.minValue.toInt ≤ i.toInt + ↑n → False | null | false |
FirstOrder.Language.HomClass.mk._flat_ctor | Mathlib.ModelTheory.Basic | ∀ {L : outParam FirstOrder.Language} {F : Type u_3} {M : outParam (Type u_4)} {N : outParam (Type u_5)}
[inst : FunLike F M N] [inst_1 : L.Structure M] [inst_2 : L.Structure N],
(∀ (φ : F) {n : ℕ} (f : L.Functions n) (x : Fin n → M),
φ (FirstOrder.Language.Structure.funMap f x) = FirstOrder.Language.Structure... | null | false |
Std.Time.DayPeriodSymbols.am | Std.Time.Format.DateFormat | Std.Time.DayPeriodSymbols → String | AM string (e.g., "AM"). | true |
TendstoLocallyUniformlyOn.tendsto_at | Mathlib.Topology.UniformSpace.LocallyUniformConvergence | ∀ {α : Type u_1} {β : Type u_2} {ι : Type u_4} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] {F : ι → α → β}
{f : α → β} {s : Set α} {p : Filter ι},
TendstoLocallyUniformlyOn F f p s → ∀ {a : α}, a ∈ s → Filter.Tendsto (fun i => F i a) p (nhds (f a)) | null | true |
CategoryTheory.StrictlyUnitaryLaxFunctor.mk.injEq | Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
(toLaxFunctor : CategoryTheory.LaxFunctor B C)
(map_id :
autoParam
(∀ (X : B),
toLaxFunctor.map (CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id (toLaxFunctor.obj X))
... | null | true |
padicValInt | Mathlib.NumberTheory.Padics.PadicVal.Basic | ℕ → ℤ → ℕ | For `p ≠ 1`, the `p`-adic valuation of an integer `z ≠ 0` is the largest natural number `k` such
that `p^k` divides `z`. If `x = 0` or `p = 1`, then `padicValInt p q` defaults to `0`. | true |
Partition.mem_removeBot | Mathlib.Order.Partition.Basic | ∀ {α : Type u_1} {s x : α} [inst : CompleteLattice α] (P : Set α) (indep : sSupIndep P) (hsSup : sSup P = s),
x ∈ Partition.removeBot P indep hsSup ↔ x ∈ P ∧ x ≠ ⊥ | null | true |
Std.ExtDHashMap.getD_ofList_of_contains_eq_false | Std.Data.ExtDHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : α → Type v} [inst : LawfulBEq α] {l : List ((a : α) × β a)} {k : α}
{fallback : β k}, (List.map Sigma.fst l).contains k = false → (Std.ExtDHashMap.ofList l).getD k fallback = fallback | null | true |
Std.ExtDTreeMap.get_getKey? | 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] {a : α}
{h : (t.getKey? a).isSome = true}, (t.getKey? a).get h = t.getKey a ⋯ | null | true |
MeasureTheory.Measure.singularPart_eq_zero | Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | ∀ {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν],
μ.singularPart ν = 0 ↔ μ.AbsolutelyContinuous ν | null | true |
CliffordAlgebra.ofBaseChangeAux._proof_5 | Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | ∀ {R : Type u_1} [inst : CommRing R], RingHomCompTriple (RingHom.id R) (RingHom.id R) (RingHom.id R) | null | false |
_private.Mathlib.Analysis.Analytic.Order.0.AnalyticAt.analyticOrderAt_deriv_add_one._simp_1_2 | Mathlib.Analysis.Analytic.Order | ∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a - b = 0) = (a = b) | null | false |
LinearMap.coe_equivOfIsUnitDet | Mathlib.LinearAlgebra.Determinant | ∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[inst_3 : Module.Free R M] [inst_4 : Module.Finite R M] {f : M →ₗ[R] M} (h : IsUnit (LinearMap.det f)),
↑(LinearMap.equivOfIsUnitDet h) = f | null | true |
Std.Http.URI.Query.instEmptyCollection | Std.Http.Data.URI.Basic | EmptyCollection Std.Http.URI.Query | null | true |
Function.Surjective.comp_left | Mathlib.Logic.Function.Basic | ∀ {α : Sort u} {β : Sort v} {γ : Sort w} {g : β → γ}, Function.Surjective g → Function.Surjective fun x => g ∘ x | Composition by a surjective function on the left is itself surjective. | true |
Lean.Widget.instInhabitedStrictOrLazy | Lean.Widget.InteractiveDiagnostic | {a : Type} → [Inhabited a] → {a_1 : Type} → Inhabited (Lean.Widget.StrictOrLazy a a_1) | null | true |
List.headD.eq_2 | Init.Data.List.Lemmas | ∀ {α : Type u} (x a : α) (as : List α), (a :: as).headD x = a | null | true |
Set.biUnion_union | Mathlib.Data.Set.Lattice.Bounded | ∀ {α : Type u_1} {β : Type u_2} (s t : Set α) (u : α → Set β), ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x | null | true |
_private.Lean.Meta.LetToHave.0.Lean.Meta.LetToHave.State.mk | Lean.Meta.LetToHave | ℕ → Std.HashMap Lean.ExprStructEq Lean.Meta.LetToHave.Result✝ → Lean.Meta.LetToHave.State✝ | null | true |
MaximalSpectrum.recOn | Mathlib.RingTheory.Spectrum.Maximal.Defs | {R : Type u_1} →
[inst : CommSemiring R] →
{motive : MaximalSpectrum R → Sort u} →
(t : MaximalSpectrum R) →
((asIdeal : Ideal R) →
(isMaximal : asIdeal.IsMaximal) → motive { asIdeal := asIdeal, isMaximal := isMaximal }) →
motive t | null | false |
CategoryTheory.PreGaloisCategory.PointedGaloisObject.incl.match_1 | Mathlib.CategoryTheory.Galois.Prorepresentability | {C : Type u_2} →
[inst : CategoryTheory.Category.{u_1, u_2} C] →
[inst_1 : CategoryTheory.GaloisCategory C] →
(F : CategoryTheory.Functor C FintypeCat) →
{X Y : CategoryTheory.PreGaloisCategory.PointedGaloisObject F} →
(motive : (X ⟶ Y) → Sort u_4) →
(x : X ⟶ Y) →
... | null | false |
Lean.DeclNameGenerator.mk.inj | Lean.CoreM | ∀ {namePrefix : Lean.Name} {idx : ℕ} {parentIdxs : List ℕ} {namePrefix_1 : Lean.Name} {idx_1 : ℕ}
{parentIdxs_1 : List ℕ},
{ namePrefix := namePrefix, idx := idx, parentIdxs := parentIdxs } =
{ namePrefix := namePrefix_1, idx := idx_1, parentIdxs := parentIdxs_1 } →
namePrefix = namePrefix_1 ∧ idx = idx_1... | null | true |
SSet.horn₂₀.ι₀₂._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.HornColimits | 1 ≠ 0 | null | false |
ProperCone.toPointedCone_bot | Mathlib.Analysis.Convex.Cone.Basic | ∀ {R : Type u_2} {E : Type u_3} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R]
[inst_3 : AddCommMonoid E] [inst_4 : TopologicalSpace E] [inst_5 : Module R E] [inst_6 : T1Space E], ↑⊥ = ⊥ | null | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.0.Int16.reduceToNatClampNeg._regBuiltin.Int16.reduceToNatClampNeg.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt.1259268813._hygCtx._hyg.439 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt | IO Unit | null | false |
Equiv.decidableEq | Mathlib.Logic.Equiv.Defs | {α : Sort u} → {β : Sort v} → α ≃ β → [DecidableEq β] → DecidableEq α | Transfer `DecidableEq` across an equivalence. | true |
MulOpposite.instNonUnitalCommCStarAlgebra._proof_2 | Mathlib.Analysis.CStarAlgebra.Classes | ∀ {A : Type u_1} [inst : NonUnitalCommCStarAlgebra A], CStarRing Aᵐᵒᵖ | null | false |
Lean.Parser.Command.namespace.parenthesizer | Lean.Parser.Command | Lean.PrettyPrinter.Parenthesizer | null | true |
RelIso.apply_eq_iff_eq | Mathlib.Order.RelIso.Basic | ∀ {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) {x y : α}, f x = f y ↔ x = y | null | true |
MonoidAlgebra.instCoalgebra | Mathlib.RingTheory.Coalgebra.MonoidAlgebra | (R : Type u_1) →
[inst : CommSemiring R] →
(A : Type u_2) →
[inst_1 : Semiring A] → (X : Type u_3) → [inst_2 : Module R A] → [Coalgebra R A] → Coalgebra R (MonoidAlgebra A X) | null | true |
HilbertBasis.toUnconditionalSchauderBasis | Mathlib.Analysis.InnerProductSpace.l2Space | {ι : Type u_1} →
{𝕜 : Type u_2} →
[inst : RCLike 𝕜] →
{E : Type u_3} →
[inst_1 : NormedAddCommGroup E] →
[inst_2 : InnerProductSpace 𝕜 E] → HilbertBasis ι 𝕜 E → UnconditionalSchauderBasis ι 𝕜 E | A Hilbert basis of is an unconditional Schauder basis (`UnconditionalSchauderBasis`),
with coordinate functionals `x ↦ ⟪b i, x⟫`. The basis expansion `x = ∑' i, ⟪b i, x⟫ • b i`
converges unconditionally. | true |
CategoryTheory.Functor.RightExtension.isPointwiseRightKanExtensionOfCompTwoSquare | Mathlib.CategoryTheory.GuitartExact.KanExtension | {C₁ : Type u₁} →
{C₂ : Type u₂} →
{C₃ : Type u₃} →
{C₄ : Type u₄} →
{D : Type u₅} →
[inst : CategoryTheory.Category.{v₁, u₁} C₁] →
[inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] →
[inst_2 : CategoryTheory.Category.{v₃, u₃} C₃] →
[inst_3 : CategoryT... | If `w : TwoSquare T L R B` is a Guitart exact square, with `R` essentially surjective,
and `E` is a right extension of `F` along `B`, then `E` is a pointwise
right Kan extension of `F` along `R` provided `E.compTwoSquare w` is a pointwise right
Kan extension of `L ⋙ F` along `T`. | true |
RCLike.instPosMulReflectLE | Mathlib.Analysis.RCLike.Basic | ∀ {K : Type u_1} [inst : RCLike K], PosMulReflectLE K | null | true |
IsOpen.exists_eq_add_of_deriv_eq | Mathlib.Analysis.Calculus.MeanValue | ∀ {𝕜 : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup G] [inst_2 : NormedSpace 𝕜 G] {s : Set 𝕜}
{f g : 𝕜 → G},
IsOpen s →
IsPreconnected s →
DifferentiableOn 𝕜 f s →
DifferentiableOn 𝕜 g s → Set.EqOn (deriv f) (deriv g) s → ∃ a, Set.EqOn f (fun x => g x + a) s | null | true |
_private.Batteries.Data.List.Lemmas.0.List.take_succ_drop._proof_1 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {l : List α} {n stop : ℕ}, n < l.length - stop → ¬stop + n < l.length → False | null | false |
AlgebraicGeometry.Scheme.Modules.pseudofunctor._proof_7 | Mathlib.AlgebraicGeometry.Modules.Sheaf | ∀ {b₀ b₁ b₂ b₃ : AlgebraicGeometry.Schemeᵒᵖ} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (h : b₂ ⟶ b₃),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Bicategory.Adj.iso₂Mk
(CategoryTheory.Cat.Hom.isoMk
(AlgebraicGeometry.Scheme.Modules.pullbackComp h.unop (CategoryTheory.CategoryStruct.comp f g).unop... | null | false |
instCircularOrderZMod._proof_8 | Mathlib.Order.Circular.ZMod | ∀ {a b c d : ZMod 0}, sbtw a b c → sbtw b d c → sbtw a d c | null | false |
NonUnitalSubsemiring.map_equiv_eq_comap_symm | Mathlib.RingTheory.NonUnitalSubsemiring.Basic | ∀ {R : Type u} {S : Type v} [inst : NonUnitalNonAssocSemiring R] [inst_1 : NonUnitalNonAssocSemiring S] (f : R ≃+* S)
(K : NonUnitalSubsemiring R), NonUnitalSubsemiring.map (↑f) K = NonUnitalSubsemiring.comap (↑f.symm) K | null | true |
Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos | Mathlib.Algebra.Polynomial.Degree.TrailingDegree | ∀ {R : Type u} [inst : Semiring R] {p : Polynomial R} {n : ℕ}, n ≠ 0 → (p.trailingDegree = ↑n ↔ p.natTrailingDegree = n) | null | true |
ContinuousLinearMap.IsPositive.inner_nonneg_right | Mathlib.Analysis.InnerProductSpace.Positive | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{T : E →L[𝕜] E}, T.IsPositive → ∀ (x : E), 0 ≤ inner 𝕜 x (T x) | null | true |
_private.Mathlib.GroupTheory.Perm.Cycle.Basic.0.Equiv.Perm.IsCycle.of_pow._simp_1_1 | Mathlib.GroupTheory.Perm.Cycle.Basic | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : Fintype α] {f : Equiv.Perm α} {x : α}, (f x ≠ x) = (x ∈ f.support) | null | false |
_private.Mathlib.Data.Set.Finite.Basic.0.Set.finite_of_forall_not_lt_lt._simp_1_1 | Mathlib.Data.Set.Finite.Basic | ∀ {α : Type u} {s : Set α} {p : (x : α) → x ∈ s → Prop}, (∀ (x : α) (h : x ∈ s), p x h) = ∀ (x : ↑s), p ↑x ⋯ | null | false |
Batteries.Tactic.Lint.SimpTheoremInfo.rec | Batteries.Tactic.Lint.Simp | {motive : Batteries.Tactic.Lint.SimpTheoremInfo → Sort u} →
((hyps : Array Lean.Expr) → (lhs rhs : Lean.Expr) → motive { hyps := hyps, lhs := lhs, rhs := rhs }) →
(t : Batteries.Tactic.Lint.SimpTheoremInfo) → motive t | null | false |
Equiv.sigmaCongrRight_trans | Mathlib.Logic.Equiv.Defs | ∀ {α : Type u_4} {β₁ : α → Type u_1} {β₂ : α → Type u_2} {β₃ : α → Type u_3} (F : (a : α) → β₁ a ≃ β₂ a)
(G : (a : α) → β₂ a ≃ β₃ a),
(Equiv.sigmaCongrRight F).trans (Equiv.sigmaCongrRight G) = Equiv.sigmaCongrRight fun a => (F a).trans (G a) | null | true |
ImportGraph.Shake.Bitset.recOn | ImportGraph.Shake.Core | {motive : ImportGraph.Shake.Bitset → Sort u} →
(t : ImportGraph.Shake.Bitset) → ((toNat : ℕ) → motive { toNat := toNat }) → motive t | null | false |
Std.TreeMap.getElem!_diff_of_mem_right | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap α β cmp} [Std.TransCmp cmp] {k : α}
[inst : Inhabited β], k ∈ t₂ → (t₁ \ t₂)[k]! = default | null | true |
Lean.Grind.CommSemiring.casesOn | Init.Grind.Ring.Basic | {α : Type u} →
{motive : Lean.Grind.CommSemiring α → Sort u_1} →
(t : Lean.Grind.CommSemiring α) →
([toSemiring : Lean.Grind.Semiring α] →
(mul_comm : ∀ (a b : α), a * b = b * a) → motive { toSemiring := toSemiring, mul_comm := mul_comm }) →
motive t | null | false |
pointedToBipointedSndBipointedToPointedSndAdjunction | Mathlib.CategoryTheory.Category.Bipointed | pointedToBipointedSnd ⊣ bipointedToPointedSnd | The free/forgetful adjunction between `PointedToBipointed_snd` and `BipointedToPointed_snd`.
| true |
RatFunc.instDiv | Mathlib.FieldTheory.RatFunc.Basic | {K : Type u} → [inst : CommRing K] → [IsDomain K] → Div (RatFunc K) | null | true |
_private.Mathlib.Data.List.Basic.0.List.dropLast_append_getLast.match_1_1 | Mathlib.Data.List.Basic | ∀ {α : Type u_1} (motive : (x : List α) → x ≠ [] → Prop) (x : List α) (x_1 : x ≠ []),
(∀ (h : [] ≠ []), motive [] h) →
(∀ (head : α) (x : [head] ≠ []), motive [head] x) →
(∀ (a b : α) (l : List α) (h : a :: b :: l ≠ []), motive (a :: b :: l) h) → motive x x_1 | null | false |
_private.Lean.Parser.Term.0.Lean.Parser.Term.withDeclName._regBuiltin.Lean.Parser.Term.withDeclName_1 | Lean.Parser.Term | IO Unit | null | false |
_private.Mathlib.Data.Finsupp.Basic.0.Finsupp.support_mapDomain_embedding._simp_1_1 | Mathlib.Data.Finsupp.Basic | ∀ {α : Type u_1} {M : Type u_4} [inst : Zero M] {f : α →₀ M} {a : α}, (a ∈ f.support) = (f a ≠ 0) | null | false |
HomotopicalAlgebra.cofibration_iff | Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : HomotopicalAlgebra.CategoryWithCofibrations C],
HomotopicalAlgebra.Cofibration f ↔ HomotopicalAlgebra.cofibrations C f | null | true |
Function.mulSupport_fun_curry | Mathlib.Algebra.Notation.Support | ∀ {ι : Type u_1} {κ : Type u_2} {M : Type u_3} [inst : One M] (f : ι × κ → M),
(Function.mulSupport fun i j => f (i, j)) = Prod.fst '' Function.mulSupport f | null | true |
Lean.MonadRecDepth.getRecDepth | Lean.Exception | {m : Type → Type} → [self : Lean.MonadRecDepth m] → m ℕ | null | true |
List.le_sum_of_subadditive_on_pred | Mathlib.Algebra.Order.BigOperators.Group.List | ∀ {α : Type u_5} {β : Type u_6} [inst : AddMonoid α] [inst_1 : AddCommMonoid β] [inst_2 : Preorder β]
[IsOrderedAddMonoid β] (f : α → β) (p : α → Prop),
f 0 ≤ 0 →
p 0 →
(∀ (a b : α), p a → p b → f (a + b) ≤ f a + f b) →
(∀ (a b : α), p a → p b → p (a + b)) → ∀ (l : List α), (∀ a ∈ l, p a) → f l.su... | null | true |
Units.ofPow._proof_1 | Mathlib.Algebra.Group.Commute.Units | ∀ {M : Type u_1} [inst : Monoid M] (u : Mˣ) (x : M) {n : ℕ}, n ≠ 0 → x ^ n = ↑u → x * x ^ (n - 1) = ↑u | null | false |
Array.exists_mem_empty | Init.Data.Array.Lemmas | ∀ {α : Type u_1} (p : α → Prop), ¬∃ x, ∃ (_ : x ∈ #[]), p x | null | true |
Function.Injective.unique | Mathlib.Basic.Unique | {α : Sort u_1} → {β : Sort u_2} → {f : α → β} → [Inhabited α] → [Subsingleton β] → Function.Injective f → Unique α | If `α` is inhabited and admits an injective map to a subsingleton type, then `α` is `Unique`. | true |
InnerProductSpace.Core.inner_smul_left | Mathlib.Analysis.InnerProductSpace.Defs | ∀ {𝕜 : Type u_1} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : AddCommGroup F] [inst_2 : Module 𝕜 F]
[c : PreInnerProductSpace.Core 𝕜 F] (x y : F) {r : 𝕜}, inner 𝕜 (r • x) y = (starRingEnd 𝕜) r * inner 𝕜 x y | null | true |
_private.Mathlib.RingTheory.PrincipalIdealDomain.0.Ideal.nonPrincipals_eq_empty_iff._simp_1_1 | Mathlib.RingTheory.PrincipalIdealDomain | ∀ {α : Type u} {s : Set α}, (s = ∅) = ∀ (x : α), x ∉ s | null | false |
Lean.Lsp.DeclInfo.mk | Lean.Data.Lsp.Internal | ℕ → ℕ → ℕ → ℕ → ℕ → ℕ → ℕ → ℕ → Lean.Lsp.DeclInfo | null | true |
AddSubgroup.le_normalClosure | Mathlib.Algebra.Group.Subgroup.Basic | ∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G}, H ≤ AddSubgroup.normalClosure ↑H | null | true |
AddCommute.op | Mathlib.Algebra.Group.Opposite | ∀ {α : Type u_1} [inst : Add α] {x y : α}, AddCommute x y → AddCommute (AddOpposite.op x) (AddOpposite.op y) | null | true |
MeasureTheory.Lp.instModule._proof_5 | Mathlib.MeasureTheory.Function.LpSpace.Basic | ∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α}
[inst : NormedAddCommGroup E] [inst_1 : NormedRing 𝕜] [inst_2 : Module 𝕜 E] [inst_3 : IsBoundedSMul 𝕜 E] (r s : 𝕜)
(x : ↥(MeasureTheory.Lp E p μ)), (r + s) • x = r • x + s • x | null | false |
CategoryTheory.Limits.Cone.fromStructuredArrow._proof_2 | Mathlib.CategoryTheory.Limits.ConeCategory | ∀ {J : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} J] {C : Type u_6}
[inst_1 : CategoryTheory.Category.{u_5, u_6} C] {D : Type u_4} [inst_2 : CategoryTheory.Category.{u_3, u_4} D]
(F : CategoryTheory.Functor C D) {X : D} (G : CategoryTheory.Functor J (CategoryTheory.StructuredArrow X F))
⦃X_1 Y : J⦄ (f :... | null | false |
_private.Mathlib.Order.Disjoint.0.disjoint_assoc._proof_1_1 | Mathlib.Order.Disjoint | ∀ {α : Type u_1} [inst : SemilatticeInf α] [inst_1 : OrderBot α] {a b c : α}, Disjoint (a ⊓ b) c ↔ Disjoint a (b ⊓ c) | null | false |
AlgebraicGeometry.IsFinite.rec | Mathlib.AlgebraicGeometry.Morphisms.Finite | {X Y : AlgebraicGeometry.Scheme} →
{f : X ⟶ Y} →
{motive : AlgebraicGeometry.IsFinite f → Sort u} →
([toIsAffineHom : AlgebraicGeometry.IsAffineHom f] →
(finite_app :
∀ (U : Y.Opens),
AlgebraicGeometry.IsAffineOpen U →
(CommRingCat.Hom.hom (Algebraic... | null | false |
Num.toZNum_inj | Mathlib.Data.Num.Lemmas | ∀ {m n : Num}, m.toZNum = n.toZNum ↔ m = n | null | true |
Std.DTreeMap.Internal.Impl.getEntryLT | Std.Data.DTreeMap.Internal.Queries | {α : Type u} →
{β : α → Type v} →
[inst : Ord α] →
[Std.TransOrd α] →
(k : α) →
(t : Std.DTreeMap.Internal.Impl α β) → t.Ordered → (∃ a ∈ t, compare a k = Ordering.lt) → (a : α) × β a | Implementation detail of the tree map | true |
Lean.Meta.Grind.Arith.Cutsat.LeCnstr.brecOn.eq | Lean.Meta.Tactic.Grind.Arith.Cutsat.Types | ∀ {motive_1 : Lean.Meta.Grind.Arith.Cutsat.EqCnstr → Sort u}
{motive_2 : Lean.Meta.Grind.Arith.Cutsat.EqCnstrProof → Sort u}
{motive_3 : Lean.Meta.Grind.Arith.Cutsat.DvdCnstr → Sort u}
{motive_4 : Lean.Meta.Grind.Arith.Cutsat.CooperSplitPred → Sort u}
{motive_5 : Lean.Meta.Grind.Arith.Cutsat.CooperSplit → Sort ... | null | true |
CategoryTheory.SimplicialObject.Splitting.PInfty_comp_πSummand_id_assoc | Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : X.Splitting)
[inst_1 : CategoryTheory.Preadditive C] (n : ℕ) {Z : C}
(h :
s.N (Opposite.unop (CategoryTheory.SimplicialObject.Splitting.IndexSet.id (Opposite.op { len := n })).fst).len ⟶ Z),
CategoryThe... | null | true |
_private.Mathlib.Data.Seq.Parallel.0.Computation.parallel.aux2.match_1.congr_eq_2 | Mathlib.Data.Seq.Parallel | ∀ {α : Type u_1} (motive : α ⊕ List (Computation α) → Sort u_2) (o : α ⊕ List (Computation α))
(h_1 : (a : α) → motive (Sum.inl a)) (h_2 : (ls : List (Computation α)) → motive (Sum.inr ls))
(ls : List (Computation α)),
o = Sum.inr ls →
(match o with
| Sum.inl a => h_1 a
| Sum.inr ls => h_2 ls) ≍
... | null | true |
Metric.diam_cthickening_le | Mathlib.Topology.MetricSpace.Thickening | ∀ {ε : ℝ} {α : Type u_2} [inst : PseudoMetricSpace α] (s : Set α),
0 ≤ ε → Metric.diam (Metric.cthickening ε s) ≤ Metric.diam s + 2 * ε | null | true |
CategoryTheory.Monoidal.instMonoidalTransportedInverseEquivalenceTransported._proof_4 | Mathlib.CategoryTheory.Monoidal.Transport | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] [inst_1 : CategoryTheory.MonoidalCategory C]
{D : Type u_4} [inst_2 : CategoryTheory.Category.{u_2, u_4} D] (e : C ≌ D)
{X₁ Y₁ X₂ Y₂ : CategoryTheory.Monoidal.Transported e} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂),
e.inverse.map (CategoryTheory.MonoidalCategorySt... | null | false |
Module.Grassmannian._sizeOf_inst | Mathlib.RingTheory.Grassmannian | (R : Type u) →
{inst : CommRing R} →
(M : Type v) →
{inst_1 : AddCommGroup M} →
{inst_2 : Module R M} → (k : ℕ) → [SizeOf R] → [SizeOf M] → SizeOf (Module.Grassmannian R M k) | null | false |
CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_π_app | Mathlib.CategoryTheory.Limits.Constructions.Filtered | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {α : Type w}
[inst_1 : CategoryTheory.Limits.HasFiniteProducts C]
[inst_2 : CategoryTheory.Limits.HasLimitsOfShape (Finset (CategoryTheory.Discrete α))ᵒᵖ C]
(F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) (j : CategoryTheory.Discrete α),
(Categ... | null | true |
instCommGroupAbelianization._proof_23 | Mathlib.GroupTheory.Abelianization.Defs | ∀ (G : Type u_1) [inst : Group G] (a b : Abelianization G), a * b = b * a | null | false |
Function.IsFixedPt.image_iterate | Mathlib.Dynamics.FixedPoints.Basic | ∀ {α : Type u_1} {f : α → α} {s : Set α},
Function.IsFixedPt (Set.image f) s → ∀ (n : ℕ), Function.IsFixedPt (Set.image f^[n]) s | null | true |
_private.Mathlib.SetTheory.Cardinal.Cofinality.Club.0.not_isStationary_iff._simp_1_2 | Mathlib.SetTheory.Cardinal.Cofinality.Club | ∀ {α : Type u_1} [inst : SemilatticeInf α] [inst_1 : OrderBot α] {a b : α}, Disjoint a b = (a ⊓ b = ⊥) | null | false |
WittVector.instCommRing._proof_9 | Mathlib.RingTheory.WittVector.Basic | ∀ (p : ℕ) (R : Type u_1) [inst : CommRing R] [inst_1 : Fact (Nat.Prime p)] (z : ℤ)
(x : WittVector p (MvPolynomial R ℤ)),
WittVector.mapFun (⇑(MvPolynomial.counit R)) (z • x) = z • WittVector.mapFun (⇑(MvPolynomial.counit R)) x | null | false |
Lean.Compiler.LCNF.Simp.ConstantFold.folderExt | Lean.Compiler.LCNF.Simp.ConstantFold | Lean.PersistentEnvExtension Lean.Compiler.LCNF.Simp.ConstantFold.FolderOleanEntry
Lean.Compiler.LCNF.Simp.ConstantFold.FolderEntry
(List Lean.Compiler.LCNF.Simp.ConstantFold.FolderEntry ×
Lean.SMap Lean.Name Lean.Compiler.LCNF.Simp.ConstantFold.Folder) | null | true |
ULift.seminormedRing._proof_17 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {α : Type u_2} [inst : SeminormedRing α] (a : ULift.{u_1, u_2} α), -a + a = 0 | null | false |
Nat.IsAtMostAlmostPrime.mul | Mathlib.NumberTheory.AlmostPrime | ∀ {k l m n : ℕ}, k.IsAtMostAlmostPrime m → l.IsAtMostAlmostPrime n → (k + l).IsAtMostAlmostPrime (m * n) | null | true |
_private.Mathlib.LinearAlgebra.Goursat.0.Submodule.goursat._proof_1_10 | Mathlib.LinearAlgebra.Goursat | ∀ {R : Type u_1} [inst : Ring R], RingHomInvPair (RingHom.id R) (RingHom.id R) | null | false |
Aesop.Nanos.nanos | Aesop.Nanos | Aesop.Nanos → ℕ | null | true |
Set.Ioc_add_bij | Mathlib.Algebra.Order.Interval.Set.Monoid | ∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M] [ExistsAddOfLE M]
(a b d : M), Set.BijOn (fun x => x + d) (Set.Ioc a b) (Set.Ioc (a + d) (b + d)) | null | true |
_private.Batteries.Data.UnionFind.Basic.0.Batteries.UnionFind.setParentBump_rankD_lt._proof_1 | Batteries.Data.UnionFind.Basic | ∀ {arr' : Array Batteries.UFNode} {arr : Array Batteries.UFNode} {x : Fin arr.size} {y : Fin arr.size} {i : ℕ},
¬↑x < arr.size → False | null | false |
Lake.VerComparator.mk.inj | Lake.Util.Version | ∀ {ver : Lake.StdVer} {op : Lake.ComparatorOp} {includeSuffixes : Bool} {ver_1 : Lake.StdVer} {op_1 : Lake.ComparatorOp}
{includeSuffixes_1 : Bool},
{ ver := ver, op := op, includeSuffixes := includeSuffixes } =
{ ver := ver_1, op := op_1, includeSuffixes := includeSuffixes_1 } →
ver = ver_1 ∧ op = op_1 ∧... | null | true |
Polynomial.Monic.add_of_left | Mathlib.Algebra.Polynomial.Monic | ∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R}, p.Monic → q.degree < p.degree → (p + q).Monic | null | true |
setOf_riemannianEDist_lt_subset_nhds' | Mathlib.Geometry.Manifold.Riemannian.Basic | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
(I : ModelWithCorners ℝ E H) {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
[inst_5 : Bundle.RiemannianBundle fun x => TangentSpace I x] [inst_6 : IsManifold I 1 M]
[IsC... | **Alias** of `setOfPred_riemannianEDist_lt_subset_nhds'`.
---
Any neighborhood of `x` contains all the points which are close enough to `x` for the
Riemannian distance, `ℝ≥0∞` version. | true |
MaxTropical.instMonoidOfAddMonoid.eq_1 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u} [inst : AddMonoid R],
MaxTropical.instMonoidOfAddMonoid =
{ toOne := MaxTropical.instOneOfZero, toMul := MaxTropical.instMulOfAdd, mul_assoc := ⋯, one_mul := ⋯, mul_one := ⋯,
npow := fun n x => x ^ n, npow_zero := ⋯, npow_succ := ⋯ } | null | true |
Bundle.Pullback.lift | Mathlib.Data.Bundle | {B : Type u_1} →
{F : Type u_2} →
{E : B → Type u_3} → {B' : Type u_4} → (f : B' → B) → Bundle.TotalSpace F (f *ᵖ E) → Bundle.TotalSpace F E | The base map `f : B' → B` lifts to a canonical map on the total spaces. | true |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic.0.WeierstrassCurve.Projective.Y_eq_of_equiv._simp_1_2 | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | ∀ {R : Type r} [inst : CommRing R] (P : Fin 3 → R) (u : R), (u • P) 1 = u * P 1 | null | false |
Matrix.diagonalLinearMap._proof_1 | Mathlib.Data.Matrix.Basic | ∀ (n : Type u_2) (α : Type u_1) [inst : DecidableEq n] [inst_1 : AddCommMonoid α] (x y : n → α),
(↑(Matrix.diagonalAddMonoidHom n α)).toFun (x + y) =
(↑(Matrix.diagonalAddMonoidHom n α)).toFun x + (↑(Matrix.diagonalAddMonoidHom n α)).toFun y | null | false |
Turing.TM1.SupportsStmt.eq_2 | Mathlib.Computability.TuringMachine.PostTuringMachine | ∀ {Γ : Type u_1} {Λ : Type u_2} {σ : Type u_3} (S : Finset Λ) (a : Γ → σ → Γ) (q : Turing.TM1.Stmt Γ Λ σ),
Turing.TM1.SupportsStmt S (Turing.TM1.Stmt.write a q) = Turing.TM1.SupportsStmt S q | null | true |
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