name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Lean.Meta.Sym.Simp.DiscrTree.0.Lean.Meta.Sym.pushAllArgs.match_1 | Lean.Meta.Sym.Simp.DiscrTree | (motive : Lean.Expr → Sort u_1) →
(e : Lean.Expr) → ((f a : Lean.Expr) → motive (f.app a)) → ((x : Lean.Expr) → motive x) → motive e | null | false |
IsTopologicalGroup.toHSpace._proof_1 | Mathlib.Topology.Homotopy.HSpaces | ∀ (M : Type u_1) [inst : MulOneClass M] [inst_1 : TopologicalSpace M] [inst_2 : ContinuousMul M],
{ toFun := Function.uncurry Mul.mul, continuous_toFun := ⋯ }.comp
((ContinuousMap.const M 1).prodMk (ContinuousMap.id M)) =
ContinuousMap.id M | null | false |
Std.DHashMap.Internal.Raw₀.toListModel_unionₘ | Std.Data.DHashMap.Internal.WF | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α] [LawfulHashable α]
{m₁ m₂ : Std.DHashMap.Internal.Raw₀ α β},
Std.DHashMap.Internal.Raw.WFImp ↑m₁ →
Std.DHashMap.Internal.Raw.WFImp ↑m₂ →
(Std.DHashMap.Internal.toListModel (↑(m₁.unionₘ m₂)).buckets).Perm
(Std.Interna... | null | true |
Min.mk._flat_ctor | Init.Prelude | {α : Type u} → (α → α → α) → Min α | null | false |
_private.Init.Data.BitVec.Lemmas.0.BitVec.two_mul_toInt_lt._simp_1_4 | Init.Data.BitVec.Lemmas | ∀ (n : ℕ), (0 < n.succ) = True | null | false |
addConj_eq_zero_iff | Mathlib.Algebra.Group.Basic | ∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, a + b + -a = 0 ↔ b = 0 | null | true |
instIsScalarTowerUniformFun | Mathlib.Topology.Algebra.UniformConvergence | ∀ {α : Type u_1} {β : Type u_2} {M : Type u_4} {N : Type u_5} [inst : SMul M N] [inst_1 : SMul M β] [inst_2 : SMul N β]
[IsScalarTower M N β], IsScalarTower M N (UniformFun α β) | null | true |
WittVector.teichmuller_mul_pow_coeff_of_ne | Mathlib.RingTheory.WittVector.TeichmullerSeries | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] {R : Type u_1} [inst : CommRing R] [CharP R p] (x : R) {m n : ℕ},
m ≠ n → ((WittVector.teichmuller p) x * ↑p ^ n).coeff m = 0 | null | true |
SymbolicDynamics.FullShift.Subshift.ofForbidden | Mathlib.Dynamics.SymbolicDynamics.Basic | {A : Type u_1} →
[inst : TopologicalSpace A] →
[inst_1 : Inhabited A] →
{G : Type u_2} →
[inst_2 : AddMonoid G] →
[IsLeftCancelAdd G] →
[DiscreteTopology A] →
Set (SymbolicDynamics.FullShift.Pattern A G) → SymbolicDynamics.FullShift.Subshift A G | The subshift defined by a family of forbidden patterns `F`.
This is a standard way to construct subshifts:
`Subshift.ofForbidden F` consists of all configurations `x : G → A` in which no pattern
`p ∈ F` occurs at any position.
Formally:
* the carrier is `forbidden F` (configurations avoiding `F`),
* it is closed beca... | true |
Qq.synthInstanceQ? | Qq.MetaM | {u : Lean.Level} → (α : Q(Sort u)) → Lean.MetaM (Option Q(«$α»)) | null | true |
_private.Init.Data.Range.Polymorphic.Lemmas.0.Std.Rio.pairwise_toList_upwardEnumerableLt._simp_1_5 | Init.Data.Range.Polymorphic.Lemmas | ∀ {α : Type u} [inst : LT α] [inst_1 : Std.PRange.UpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLT α] {a b : α},
(a < b) = Std.PRange.UpwardEnumerable.LT a b | null | false |
IsPGroup.card_orbit | Mathlib.GroupTheory.PGroup | ∀ {p : ℕ} {G : Type u_1} [inst : Group G],
IsPGroup p G →
∀ [hp : Fact (Nat.Prime p)] {α : Type u_2} [inst_1 : MulAction G α] (a : α) [Finite ↑(MulAction.orbit G a)],
∃ n, Nat.card ↑(MulAction.orbit G a) = p ^ n | null | true |
Finsupp.instEMetricSpaceWithLpOfNNReal | Mathlib.Analysis.Normed.Lp.Finsupp | {ι : Type u_1} →
{X : Type u_2} →
[inst : Zero X] → {p : NNReal} → [Fact (1 ≤ p)] → [EMetricSpace X] → EMetricSpace (WithLp (↑p) (ι →₀ X)) | The L^1 extended metric on `ι`-many copies of a metric space `X` | true |
_private.Std.Data.DTreeMap.Internal.Zipper.0.Std.DTreeMap.Internal.Zipper.size_prependMap | Std.Data.DTreeMap.Internal.Zipper | ∀ {α : Type u} {β : α → Type v} (t : Std.DTreeMap.Internal.Impl α β) (it : Std.DTreeMap.Internal.Zipper α β),
Std.DTreeMap.Internal.Zipper.size✝ (Std.DTreeMap.Internal.Zipper.prependMap t it) =
t.treeSize + Std.DTreeMap.Internal.Zipper.size✝ it | null | true |
UpperHalfPlane.instContinuousGLSMul | Mathlib.Analysis.Complex.UpperHalfPlane.Topology | ContinuousConstSMul (GL (Fin 2) ℝ) UpperHalfPlane | Each element of `GL(2, ℝ)` defines a continuous map `ℍ → ℍ`. | true |
_private.Mathlib.Algebra.Homology.ModelCategory.Injective.0.CochainComplex.Plus.modelCategoryQuillen.instHasFactorizationTrivialCofibrationsFibrations.match_3 | Mathlib.Algebra.Homology.ModelCategory.Injective | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
(K : CochainComplex C ℤ) (n : ℤ) (hn : K.IsStrictlyGE n) (L : CochainComplex C ℤ) (m : ℤ) (hm : L.IsStrictlyGE m)
(motive : ({ obj := K, property := ⋯ } ⟶ { obj := L, property := ⋯ }) → Prop)
(h : { obj := K, proper... | null | false |
Std.ExtDTreeMap.instBEqOfLawfulEqCmpOfTransCmp | Std.Data.ExtDTreeMap.Basic | {α : Type u} →
{β : α → Type v} →
{cmp : α → α → Ordering} →
[Std.LawfulEqCmp cmp] → [Std.TransCmp cmp] → [(k : α) → BEq (β k)] → BEq (Std.ExtDTreeMap α β cmp) | null | true |
Prefunctor.symmetrify_mapReverse | Mathlib.Combinatorics.Quiver.Symmetric | ∀ {U : Type u_1} {V : Type u_2} [inst : Quiver U] [inst_1 : Quiver V] (φ : U ⥤q V), φ.symmetrify.MapReverse | null | true |
VectorFourier.fourierSMulRight._proof_5 | Mathlib.Analysis.Fourier.FourierTransformDeriv | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E], ContinuousConstSMul ℂ E | null | false |
Algebra.Etale.instProd_2 | Mathlib.RingTheory.Etale.Basic | ∀ {R : Type u} [inst : CommRing R] (S : Type u_2) [inst_1 : CommRing S], Algebra.Etale (R × S) S | null | true |
Nat.exists_strictMono | Mathlib.Order.Monotone.Basic | ∀ (α : Type u) [inst : Preorder α] [Nonempty α] [NoMaxOrder α], ∃ f, StrictMono f | If `α` is a nonempty preorder with no maximal elements, then there exists a strictly monotone
function `ℕ → α`. | true |
Metric.infDist_le_dist_of_mem | Mathlib.Topology.MetricSpace.HausdorffDistance | ∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} {x y : α}, y ∈ s → Metric.infDist x s ≤ dist x y | The minimal distance to a set is bounded by the distance to any point in this set. | true |
OrderIso.setCongr_symm_apply | Mathlib.Order.Hom.Set | ∀ {α : Type u_1} [inst : Preorder α] (s t : Set α) (h : s = t) (b : { b // (fun x => x ∈ t) b }),
(RelIso.symm (s.orderIsoOfEq t h)) b = ⟨↑b, ⋯⟩ | **Alias** of `Set.orderIsoOfEq_symm_apply`. | true |
Lean.LevelMVarId.casesOn | Lean.Level | {motive : Lean.LevelMVarId → Sort u} →
(t : Lean.LevelMVarId) → ((name : Lean.Name) → motive { name := name }) → motive t | null | false |
Module.AEval.mapSubmodule._proof_3 | Mathlib.Algebra.Polynomial.Module.AEval | ∀ (R : Type u_2) {A : Type u_3} (M : Type u_1) [inst : CommSemiring R] [inst_1 : Semiring A] (a : A)
[inst_2 : Algebra R A] [inst_3 : AddCommMonoid M] [inst_4 : Module A M] [inst_5 : Module R M]
[inst_6 : IsScalarTower R A M], AddMonoidHomClass (M ≃ₗ[R] Module.AEval R M a) M (Module.AEval R M a) | null | false |
PosNum.succ | Mathlib.Data.Num.Basic | PosNum → PosNum | The successor of a `PosNum`. | true |
AlgebraicGeometry.Scheme.Pullback.Triplet.tensorInl | Mathlib.AlgebraicGeometry.PullbackCarrier | {X Y S : AlgebraicGeometry.Scheme} →
{f : X ⟶ S} → {g : Y ⟶ S} → (T : AlgebraicGeometry.Scheme.Pullback.Triplet f g) → X.residueField T.x ⟶ T.tensor | Given `x : X` and `y : Y` such that `f x = s = g y`, this is the
canonical map `κ(x) ⟶ κ(x) ⊗[κ(s)] κ(y)`. | true |
String.Pos.mk.inj | Init.Data.String.Defs | ∀ {s : String} {offset : String.Pos.Raw} {isValid : String.Pos.Raw.IsValid s offset} {offset_1 : String.Pos.Raw}
{isValid_1 : String.Pos.Raw.IsValid s offset_1},
{ offset := offset, isValid := isValid } = { offset := offset_1, isValid := isValid_1 } → offset = offset_1 | null | true |
Lean.Expr.letE.sizeOf_spec | Lean.Expr | ∀ (declName : Lean.Name) (type value body : Lean.Expr) (nondep : Bool),
sizeOf (Lean.Expr.letE declName type value body nondep) =
1 + sizeOf declName + sizeOf type + sizeOf value + sizeOf body + sizeOf nondep | null | true |
CategoryTheory.CommaMorphism.ctorIdx | Mathlib.CategoryTheory.Comma.Basic | {A : Type u₁} →
{inst : CategoryTheory.Category.{v₁, u₁} A} →
{B : Type u₂} →
{inst_1 : CategoryTheory.Category.{v₂, u₂} B} →
{T : Type u₃} →
{inst_2 : CategoryTheory.Category.{v₃, u₃} T} →
{L : CategoryTheory.Functor A T} →
{R : CategoryTheory.Functor B T} → {X Y... | null | false |
CategoryTheory.Limits.isColimitAux._proof_2 | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{X Y : C} {f : X ⟶ Y} (t : CategoryTheory.Limits.CokernelCofork f)
(desc : (s : CategoryTheory.Limits.CokernelCofork f) → t.pt ⟶ s.pt),
(∀ (s : CategoryTheory.Limits.CokernelCofork f) (m : t.pt ⟶ s.p... | null | false |
_private.Mathlib.Order.Lattice.Nat.0.Nat.sSup_mem.match_1_1 | Mathlib.Order.Lattice.Nat | ∀ {s : Set ℕ} (motive : BddAbove s → Prop) (h₂ : BddAbove s), (∀ (k : ℕ) (hk : k ∈ upperBounds s), motive ⋯) → motive h₂ | null | false |
AbsolutelyContinuousOnInterval.fun_smul | Mathlib.MeasureTheory.Function.AbsolutelyContinuous | ∀ {F : Type u_2} [inst : SeminormedAddCommGroup F] {a b : ℝ} {M : Type u_3} [inst_1 : SeminormedRing M]
[inst_2 : Module M F] [NormSMulClass M F] {f : ℝ → M} {g : ℝ → F},
AbsolutelyContinuousOnInterval f a b →
AbsolutelyContinuousOnInterval g a b → AbsolutelyContinuousOnInterval (fun i => f i • g i) a b | Eta-expanded form of `AbsolutelyContinuousOnInterval.smul`
---
If `f` and `g` are absolutely continuous on `uIcc a b`, then `f • g` is absolutely continuous
on `uIcc a b`. | true |
SubMulAction.rec | Mathlib.GroupTheory.GroupAction.SubMulAction | {R : Type u} →
{M : Type v} →
[inst : SMul R M] →
{motive : SubMulAction R M → Sort u_1} →
((carrier : Set M) →
(smul_mem' : ∀ (c : R) {x : M}, x ∈ carrier → c • x ∈ carrier) →
motive { carrier := carrier, smul_mem' := smul_mem' }) →
(t : SubMulAction R M) → motiv... | null | false |
_private.Mathlib.Geometry.Manifold.IsManifold.Basic.0.IsManifold.instLEInftyOfNatWithTopENat_1._proof_1 | Mathlib.Geometry.Manifold.IsManifold.Basic | ENat.LEInfty 1 | null | false |
Std.ExtDHashMap.getKey?_inter_of_not_mem_right | Std.Data.ExtDHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : α → Type v} {m₁ m₂ : Std.ExtDHashMap α β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] {k : α}, k ∉ m₂ → (m₁ ∩ m₂).getKey? k = none | null | true |
_private.Mathlib.RingTheory.Valuation.ValuativeRel.Basic.0.Valuation.apply_posSubmonoid_ne_zero._simp_1_1 | Mathlib.RingTheory.Valuation.ValuativeRel.Basic | ∀ {R : Type u_2} [inst : Ring R] [inst_1 : ValuativeRel R] (x : ↥(ValuativeRel.posSubmonoid R)),
((ValuativeRel.valuation R) ↑x = 0) = False | null | false |
Lean.Server.ServerTask.waitAny | Lean.Server.ServerTask | {α : Type} →
(tasks : List (Lean.Server.ServerTask α)) →
autoParam (tasks.length > 0) Lean.Server.ServerTask.waitAny._auto_1 → BaseIO α | null | true |
Std.TreeSet.get?_congr | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp] {k k' : α},
cmp k k' = Ordering.eq → t.get? k = t.get? k' | null | true |
Cardinal.add_nat_le_add_nat_iff._simp_1 | Mathlib.SetTheory.Cardinal.Arithmetic | ∀ {α β : Cardinal.{u_1}} (n : ℕ), (α + ↑n ≤ β + ↑n) = (α ≤ β) | null | false |
Bundle.Pretrivialization.continuousAlternatingMap | Mathlib.Topology.VectorBundle.ContinuousAlternatingMap | (𝕜 : Type u_1) →
(ι : Type u_2) →
[inst : NontriviallyNormedField 𝕜] →
{B : Type u_3} →
[inst_1 : TopologicalSpace B] →
{F₁ : Type u_4} →
[inst_2 : NormedAddCommGroup F₁] →
[inst_3 : NormedSpace 𝕜 F₁] →
{E₁ : B → Type u_5} →
[i... | Given trivializations `e₁`, `e₂` for vector bundles `E₁`, `E₂` over a base `B`,
`Pretrivialization.continuousAlternatingMap 𝕜 ι e₁ e₂` is the induced pretrivialization for the
continuous `σ`-semilinear maps from `E₁` to `E₂`. That is, the map which will later become a
trivialization, after the bundle of continuous sem... | true |
_private.Mathlib.Geometry.Manifold.VectorField.LieBracket.0.Filter.EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1 | Mathlib.Geometry.Manifold.VectorField.LieBracket | {𝕜 : Type u_2} →
[inst : NontriviallyNormedField 𝕜] →
{H : Type u_4} →
[inst_1 : TopologicalSpace H] →
{E : Type u_1} →
[inst_2 : NormedAddCommGroup E] →
[inst_3 : NormedSpace 𝕜 E] →
{I : ModelWithCorners 𝕜 E H} →
{M : Type u_3} →
... | null | false |
_private.Mathlib.Algebra.Notation.Indicator.0.Set.mulIndicator_eq_self._simp_1_1 | Mathlib.Algebra.Notation.Indicator | ∀ {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}, (f = g) = ∀ (x : α), f x = g x | null | false |
_private.Batteries.Tactic.Lint.Simp.0.Batteries.Tactic.Lint.simpNF.match_3 | Batteries.Tactic.Lint.Simp | (motive : Lean.Meta.Simp.Result × Lean.Meta.Simp.Stats → Sort u_1) →
(x : Lean.Meta.Simp.Result × Lean.Meta.Simp.Stats) →
((hType' : Lean.Expr) →
(proof? : Option Lean.Expr) →
(cache : Bool) →
(stats : Lean.Meta.Simp.Stats) → motive ({ expr := hType', proof? := proof?, cache := cache... | null | false |
Relation.transGen_eq_self | Mathlib.Logic.Relation | ∀ {α : Type u_1} {r : α → α → Prop} [IsTrans α r], Relation.TransGen r = r | null | true |
CategoryTheory.SimplicialObject.augmentOfIsTerminal._proof_2 | Mathlib.AlgebraicTopology.SimplicialObject.Basic | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] (X : CategoryTheory.SimplicialObject C) {T : C}
(hT : CategoryTheory.Limits.IsTerminal T) ⦃X_1 Y : SimplexCategoryᵒᵖ⦄ (f : X_1 ⟶ Y),
CategoryTheory.CategoryStruct.comp (((CategoryTheory.Functor.id (CategoryTheory.SimplicialObject C)).obj X).map f)
... | null | false |
_private.Init.Data.List.Sublist.0.List.cons_subset_cons._simp_1_1 | Init.Data.List.Sublist | ∀ {α : Type u_1} {b : α} {l : List α} {a : α}, (a ∈ b :: l) = (a = b ∨ a ∈ l) | null | false |
FirstOrder.Language.DefinableSet.coe_compl | Mathlib.ModelTheory.Definability | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {A : Set M} {α : Type u₁} (s : L.DefinableSet A α),
↑sᶜ = (↑s)ᶜ | null | true |
Std.Time.Month.instTransOrdOffset | Std.Time.Date.Unit.Month | Std.TransOrd Std.Time.Month.Offset | null | true |
_private.Batteries.Data.BinaryHeap.Lemmas.0.Batteries.BinaryHeap.heapifyDown_children_of_ge_subtree._proof_1_7 | Batteries.Data.BinaryHeap.Lemmas | ∀ {sz : ℕ} {k : Fin sz}, 2 * ↑k + 1 < sz → 2 * ↑k + 1 < sz | null | false |
not_isLeftRegular_zero | Mathlib.Algebra.GroupWithZero.Regular | ∀ {R : Type u_1} [inst : MulZeroClass R] [nR : Nontrivial R], ¬IsLeftRegular 0 | In a non-trivial ring, the element `0` is not left-regular -- with typeclasses. | true |
DirectSum.instLieRingModuleSubtypeMemSubmodule._proof_3 | Mathlib.Algebra.Lie.Graded | ∀ {ι : Type u_5} {κ : Type u_1} {R : Type u_3} {L : Type u_4} {M : Type u_2} [inst : DecidableEq ι]
[inst_1 : AddCommMonoid ι] [inst_2 : CommRing R] [inst_3 : LieRing L] [inst_4 : LieAlgebra R L]
(ℒ : ι → Submodule R L) [inst_5 : GradedLieAlgebra ℒ] [inst_6 : DecidableEq κ] [inst_7 : AddCommGroup M]
[inst_8 : Mod... | null | false |
MeasureTheory.VectorMeasure.«_aux_Mathlib_MeasureTheory_VectorMeasure_Integral___delab_app_MeasureTheory_VectorMeasure_term∫ᵛ_,_∂[_;_]_1» | Mathlib.MeasureTheory.VectorMeasure.Integral | Lean.PrettyPrinter.Delaborator.Delab | Pretty printer defined by `notation3` command. | false |
FreeAlgebra.cardinalMk_le_max_lift | Mathlib.Algebra.FreeAlgebra.Cardinality | ∀ (R : Type u) [inst : CommSemiring R] (X : Type v),
Cardinal.mk (FreeAlgebra R X) ≤
max (max (Cardinal.lift.{v, u} (Cardinal.mk R)) (Cardinal.lift.{u, v} (Cardinal.mk X))) Cardinal.aleph0 | null | true |
CommMonoid.mul_comm | Mathlib.Algebra.Group.Monoid | ∀ {M : Type u_2} [self : CommMonoid M] (a b : M), a * b = b * a | Multiplication is commutative in a commutative multiplicative magma. | true |
Real.antitone_rpow_of_base_le_one | Mathlib.Analysis.SpecialFunctions.Pow.Real | ∀ {b : ℝ}, 0 < b → b ≤ 1 → Antitone fun x => b ^ x | null | true |
_private.Mathlib.Probability.Process.Stopping.0.MeasureTheory.measurable_stoppedValue._simp_1_4 | Mathlib.Probability.Process.Stopping | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) | null | false |
Float.Model.UnpackedFloat.ExtendedMantissa.accuracy | Init.Data.Float.Model.Unpacked.Round | Float.Model.UnpackedFloat.ExtendedMantissa → Float.Model.UnpackedFloat.Accuracy | Extract the accuracy of the mantissa. | true |
Function.comp_def._simp_2 | Mathlib.Order.OmegaCompletePartialOrder | ∀ {α : Sort u_1} {β : Sort u_2} {δ : Sort u_3} (f : β → δ) (g : α → β), (fun x => f (g x)) = f ∘ g | null | false |
TangentBundle.mem_chart_target_iff._simp_1 | Mathlib.Geometry.Manifold.VectorBundle.Tangent | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_4} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [inst_6 : IsManifold I 1 M] (p : H × E)... | null | false |
CategoryTheory.MorphismProperty.under | Mathlib.CategoryTheory.MorphismProperty.Comma | {T : Type u_3} →
[inst : CategoryTheory.Category.{v_3, u_3} T] →
CategoryTheory.MorphismProperty T → {X : T} → CategoryTheory.MorphismProperty (CategoryTheory.Under X) | The morphism property on `Under X` induced by a morphism property on `C`. | true |
AddGrpCat.hasLimits | Mathlib.Algebra.Category.Grp.Limits | CategoryTheory.Limits.HasLimits AddGrpCat | null | true |
CategoryTheory.ObjectProperty.full_ι | Mathlib.CategoryTheory.ObjectProperty.FullSubcategory | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (P : CategoryTheory.ObjectProperty C), P.ι.Full | null | true |
imageToKernel_arrow_apply | Mathlib.Algebra.Homology.ImageToKernel | ∀ {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {A B C : V}
(f : A ⟶ B) [inst_2 : CategoryTheory.Limits.HasImage f] (g : B ⟶ C) [inst_3 : CategoryTheory.Limits.HasKernel g]
(w : CategoryTheory.CategoryStruct.comp f g = 0) {F : V → V → Type uF} {carrier : ... | null | true |
QuadraticForm.isometryEquivWeightedSumSquaresWeightedSumSquares._proof_4 | Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv | ∀ {ι : Type u_1} {S : Type u_3} {R : Type u_2} [inst : CommSemiring R] [inst_1 : Monoid S]
[inst_2 : DistribMulAction S R] (u : ι → Sˣ) (x : ι → R), (fun x => u • x) ((fun x => u⁻¹ • x) x) = x | null | false |
MulEquiv.instMulEquivClass | Mathlib.Algebra.Group.Equiv.Defs | ∀ {M : Type u_4} {N : Type u_5} [inst : Mul M] [inst_1 : Mul N], MulEquivClass (M ≃* N) M N | null | true |
_private.Init.Data.Vector.Lemmas.0.Array.eraseIdx!.eq_1 | Init.Data.Vector.Lemmas | ∀ {α : Type u} (xs : Array α) (i : ℕ),
xs.eraseIdx! i =
if h : i < xs.size then xs.eraseIdx i h
else panicWithPosWithDecl "Init.Data.Array.Basic" "Array.eraseIdx!" 1867 47 "invalid index" | null | true |
_private.Mathlib.Geometry.Convex.ConvexSpace.Defs.0.Convexity.StdSimplex.support_weights_restrict._simp_1_3 | Mathlib.Geometry.Convex.ConvexSpace.Defs | ∀ {α : Type u_1} {M : Type u_4} [inst : Zero M] {f g : α →₀ M}, (f ≠ g) = ∃ a, f a ≠ g a | null | false |
Finset.mem_filter_univ | Mathlib.Data.Fintype.Defs | ∀ {α : Type u_1} [inst : Fintype α] {p : α → Prop} [inst_1 : DecidablePred p] (x : α),
x ∈ Finset.filter p Finset.univ ↔ p x | null | true |
ContMDiffOn.iterate | Mathlib.Geometry.Manifold.ContMDiff.Basic | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {s : Set M} {n : WithTop ℕ∞} {f : M → M... | The iterates of `C^n` functions on domains are `C^n`. | true |
CategoryTheory.Bicategory.OplaxTrans.ComonadBicat | Mathlib.CategoryTheory.Bicategory.Monad.Basic | (B : Type u) → [CategoryTheory.Bicategory B] → Type (max (max (max (max 0 u) v) 0) w) | The bicategory of comonads in `B`. | true |
CategoryTheory.MorphismProperty.overObj | Mathlib.CategoryTheory.MorphismProperty.Comma | {T : Type u_3} →
[inst : CategoryTheory.Category.{v_3, u_3} T] →
CategoryTheory.MorphismProperty T → {X : T} → CategoryTheory.ObjectProperty (CategoryTheory.Over X) | The object property on `Over X` induced by a morphism property. | true |
PiNat.cylinder_eq_pi | Mathlib.Topology.MetricSpace.PiNat | ∀ {E : ℕ → Type u_1} (x : (n : ℕ) → E n) (n : ℕ), PiNat.cylinder x n = (↑(Finset.range n)).pi fun i => {x i} | null | true |
Pi.nnnorm_def | Mathlib.Analysis.Normed.Group.Constructions | ∀ {ι : Type u_1} {G : ι → Type u_5} [inst : Fintype ι] [inst_1 : (i : ι) → SeminormedAddGroup (G i)]
(f : (i : ι) → G i), ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ | null | true |
_private.Mathlib.Order.Filter.Map.0.Filter.comap_comap._simp_1_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α}, (sᶜ ∈ Filter.comap f l) = ((f '' s)ᶜ ∈ l) | null | false |
SimpleGraph.chromaticNumber_le_sum_right | Mathlib.Combinatorics.SimpleGraph.Sum | ∀ {V : Type u_3} {W : Type u_5} {G : SimpleGraph V} {H : SimpleGraph W}, H.chromaticNumber ≤ (G ⊕g H).chromaticNumber | null | true |
MulAction.fixedBy | Mathlib.GroupTheory.GroupAction.Defs | {M : Type u_1} → (α : Type u_3) → [inst : Monoid M] → [MulAction M α] → M → Set α | `fixedBy m` is the set of elements fixed by `m`. | true |
WithTop.top_ne_natCast._simp_1 | Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | ∀ {α : Type u} [inst : AddMonoidWithOne α] (n : ℕ), (⊤ = ↑n) = False | null | false |
DerivedCategory.triangleOfSESδ_naturality | Mathlib.Algebra.Homology.DerivedCategory.ShortExact | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] {S₁ S₂ : CategoryTheory.ShortComplex (CochainComplex C ℤ)} (hS₁ : S₁.ShortExact)
(hS₂ : S₂.ShortExact) (f : S₁ ⟶ S₂),
CategoryTheory.CategoryStruct.comp (DerivedCategory.triangleOfSESδ hS₁)... | null | true |
NonUnitalStarAlgebra.elemental.isClosed | Mathlib.Topology.Algebra.NonUnitalStarAlgebra | ∀ (R : Type u_1) {A : Type u_2} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
[inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : IsScalarTower R A A] [inst_6 : SMulCommClass R A A]
[inst_7 : StarModule R A] [inst_8 : TopologicalSpace A] [inst_9 : IsSemitopologicalSemiring A]
[ins... | null | true |
Std.TreeMap.compare_maxKey?_modify_eq | 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}
{f : β → β} {km kmm : α} (hkm : t.maxKey? = some km), (t.modify k f).maxKey?.get ⋯ = kmm → cmp kmm km = Ordering.eq | null | true |
MultilinearMap.domCoprod_apply | Mathlib.LinearAlgebra.Multilinear.TensorProduct | ∀ {R : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} [inst : CommSemiring R] {N₁ : Type u_6} [inst_1 : AddCommMonoid N₁]
[inst_2 : Module R N₁] {N₂ : Type u_7} [inst_3 : AddCommMonoid N₂] [inst_4 : Module R N₂] {N : Type u_8}
[inst_5 : AddCommMonoid N] [inst_6 : Module R N] (a : MultilinearMap R (fun x => N) N₁)
(b :... | null | true |
CategoryTheory.Triangulated.TStructure.truncGELTToLTGE_app_pentagon | Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]
[inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C]
(t : CategoryTheory.... | null | true |
_private.Mathlib.CategoryTheory.Monoidal.Mod.0.CategoryTheory.AddMod.Hom.ext.match_1 | Mathlib.CategoryTheory.Monoidal.Mod | ∀ {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}
{inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {A : C} {inst_4 : CategoryTheory.AddMonObj A}
{M N : CategoryTheory.AddMod D ... | null | false |
DerivedCategory.isLE_Q_obj_iff | Mathlib.Algebra.Homology.DerivedCategory.TStructure | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] (K : CochainComplex C ℤ) (n : ℤ), (DerivedCategory.Q.obj K).IsLE n ↔ K.IsLE n | null | true |
Lean.instBEqExtraModUse.beq | Lean.ExtraModUses | Lean.ExtraModUse → Lean.ExtraModUse → Bool | null | true |
Ideal.comap_le_comap_iff_of_surjective | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S] (f : F)
[inst_3 : RingHomClass F R S], Function.Surjective ⇑f → ∀ (I J : Ideal S), Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J | null | true |
Lean.Linter.CodeQuality.Source._sizeOf_inst | Lean.Linter.CodeQuality.Basic | SizeOf Lean.Linter.CodeQuality.Source | null | false |
Std.TreeSet.toList_roo | Std.Data.TreeSet.Slice | ∀ {α : Type u} (cmp : autoParam (α → α → Ordering) Std.TreeSet.toList_roo._auto_1) [Std.TransCmp cmp]
{t : Std.TreeSet α cmp} {lowerBound upperBound : α},
Std.Slice.toList (Std.Roo.Sliceable.mkSlice t (lowerBound<...upperBound)) =
List.filter (fun e => decide ((cmp e lowerBound).isGT = true ∧ (cmp e upperBound)... | null | true |
AEMeasurable.iInf | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α]
{mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α]
{ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α},
(∀ (i : ι), AEMeasura... | null | true |
_private.Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction.0.UpperHalfPlane.denom_cocycle._simp_1_2 | Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False | null | false |
PFunctor.Obj.eta | Mathlib.Data.PFunctor.Univariate.Basic | ∀ {P : PFunctor.{uA, uB}} {α : Type v₁} (x : ↑P α), PFunctor.Obj.mk x.fst x.snd = x | null | true |
_private.Init.Data.Range.Polymorphic.PRange.0.Std.instDecidableEqRoc.decEq._proof_1 | Init.Data.Range.Polymorphic.PRange | ∀ {α : Type u_1} (a a_1 : α), a<...=a_1 = a<...=a_1 | null | false |
Mathlib.Tactic.Ring.Common.evalPow._sunfold | Mathlib.Tactic.Ring.Common | {u : Lean.Level} →
{α : Q(Type u)} →
{bt : Q(«$α») → Type} →
{sα : Q(CommSemiring «$α»)} →
Mathlib.Tactic.Ring.Common.RingCompute bt sα →
Mathlib.Tactic.Ring.Common.RingCompute Mathlib.Tactic.Ring.Common.btℕ Mathlib.Tactic.Ring.Common.sℕ →
{a : Q(«$α»)} →
{b : Q(ℕ... | null | false |
CategoryTheory.Pseudofunctor.StrongTrans.naturality_comp._autoParam | Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo | Lean.Syntax | null | false |
instBooleanAlgebraSubtypeIsIdempotentElem._proof_15 | Mathlib.Algebra.Order.Ring.Idempotent | ∀ {R : Type u_1} [inst : CommRing R] (x x_1 : { a // IsIdempotentElem a }), x ⊓ ⟨1 - ↑x_1, ⋯⟩ = x ⊓ ⟨1 - ↑x_1, ⋯⟩ | null | false |
Geometry.SimplicialComplex.facets | Mathlib.Analysis.Convex.SimplicialComplex.Basic | {𝕜 : Type u_1} →
{E : Type u_2} →
[inst : Ring 𝕜] →
[inst_1 : PartialOrder 𝕜] →
[inst_2 : AddCommGroup E] → [inst_3 : Module 𝕜 E] → Geometry.SimplicialComplex 𝕜 E → Set (Finset E) | A facet of a simplicial complex is a maximal face. | true |
_private.Mathlib.Combinatorics.Matroid.Basic.0.Matroid.singleton_subset_ground | Mathlib.Combinatorics.Matroid.Basic | ∀ {α : Type u_1} {M : Matroid α} {e : α}, e ∈ M.E → {e} ⊆ M.E | null | true |
Lean.Meta.NormCast.Label.ofNat_ctorIdx | Lean.Meta.Tactic.NormCast | ∀ (x : Lean.Meta.NormCast.Label), Lean.Meta.NormCast.Label.ofNat x.ctorIdx = x | null | true |
TopPair.incl._proof_2 | Mathlib.Topology.Category.TopPair | ∀ (X : TopCat),
TopPair.ofHom (CategoryTheory.CategoryStruct.id X) (CategoryTheory.CategoryStruct.id TopPair.snd) ⋯ =
CategoryTheory.CategoryStruct.id (TopPair.ofTopCat X) | null | false |
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