name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CochainComplex.mappingConeHomOfDegreewiseSplitIso | Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {C : Type u_1} →
[inst : CategoryTheory.Category.{v, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
(S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) →
(σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) →
[inst_2 : CategoryTheory.Limits.HasBinaryB... | The canonical isomorphism `mappingCone (homOfDegreewiseSplit S σ) ≅ S.X₂⟦(1 : ℤ)⟧`. | true |
AddSubmonoid.instSetLike.eq_1 | Mathlib.Algebra.Group.Submonoid.Defs | ∀ {M : Type u_1} [inst : AddZeroClass M], AddSubmonoid.instSetLike = { coe := fun s => s.carrier, coe_injective := ⋯ } | null | true |
Lean.Meta.InductionSubgoal.fields | Lean.Meta.Tactic.Induction | Lean.Meta.InductionSubgoal → Array Lean.Expr | null | true |
Prefunctor.symmetrify_map | Mathlib.Combinatorics.Quiver.Symmetric | ∀ {U : Type u_1} {V : Type u_2} [inst : Quiver U] [inst_1 : Quiver V] (φ : U ⥤q V) {X Y : Quiver.Symmetrify U}
(a : (X ⟶ Y) ⊕ (Y ⟶ X)), φ.symmetrify.map a = Sum.map φ.map φ.map a | null | true |
Tuple.lt_card_le_iff_apply_le_of_monotone | Mathlib.Data.Fin.Tuple.Sort | ∀ {n : ℕ} {α : Type u_1} {j : Fin n} {f : Fin n → α} [inst : Preorder α] {a : α} [inst_1 : DecidableLE α],
Monotone f → (↑j < {i | f i ≤ a}.card ↔ f j ≤ a) | If `f₀ ≤ f₁ ≤ f₂ ≤ ⋯` is a sorted `n`-tuple of elements of `α`, then for any `j : Fin n` and
`a : α` we have `j < #{i | fᵢ ≤ a}` iff `fⱼ ≤ a`. | true |
TopologicalSpace.denseRange_denseSeq | Mathlib.Topology.Bases | ∀ (α : Type u) [t : TopologicalSpace α] [inst : TopologicalSpace.SeparableSpace α] [inst_1 : Nonempty α],
DenseRange (TopologicalSpace.denseSeq α) | The sequence `TopologicalSpace.denseSeq α` has dense range. | true |
ProbabilityTheory.gaussianReal_sub_const | Mathlib.Probability.Distributions.Gaussian.Real | ∀ {μ : ℝ} {v : NNReal} {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} {X : Ω → ℝ},
ProbabilityTheory.HasLaw X (ProbabilityTheory.gaussianReal μ v) P →
∀ (y : ℝ), ProbabilityTheory.HasLaw (fun ω => X ω - y) (ProbabilityTheory.gaussianReal (μ - y) v) P | If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X - y`
has Gaussian law with mean `μ - y` and variance `v`. | true |
GrpCat.sectionsSubgroup._proof_2 | Mathlib.Algebra.Category.Grp.Limits | ∀ {J : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} J] (F : CategoryTheory.Functor J GrpCat)
{a b : (j : J) → ↑((F.comp (CategoryTheory.forget₂ GrpCat MonCat)).obj j)},
a ∈ (MonCat.sectionsSubmonoid (F.comp (CategoryTheory.forget₂ GrpCat MonCat))).carrier →
b ∈ (MonCat.sectionsSubmonoid (F.comp (Categor... | null | false |
_private.Init.Data.List.Monadic.0.List.mapM'.match_1.splitter | Init.Data.List.Monadic | {α : Type u_1} →
(motive : List α → Sort u_2) →
(x : List α) → (Unit → motive []) → ((a : α) → (l : List α) → motive (a :: l)) → motive x | null | true |
CategoryTheory.Limits.WalkingMultispan.proxyType | Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | CategoryTheory.Limits.MultispanShape → Type (max w w') | A "proxy type" equivalent to `CategoryTheory.Limits.WalkingMultispan` that is constructed from `Unit`, `PLift`, `Sigma`, `Empty`, and `Sum`. See `CategoryTheory.Limits.WalkingMultispan.proxyTypeEquiv` for the equivalence. (Generated by the `proxy_equiv%` elaborator.) | true |
CategoryTheory.EnrichedCategory.id_comp._autoParam | Mathlib.CategoryTheory.Enriched.Basic | Lean.Syntax | null | false |
CategoryTheory.Functor.hasStrongEpiMonoFactorisations_imp_of_isEquivalence | Mathlib.CategoryTheory.Limits.Shapes.Images | ∀ {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] (F : CategoryTheory.Functor C D) [F.IsEquivalence]
[h : CategoryTheory.Limits.HasStrongEpiMonoFactorisations C], CategoryTheory.Limits.HasStrongEpiMonoFactorisations D | null | true |
Set.countable_iff_exists_surjective | Mathlib.Data.Set.Countable | ∀ {α : Type u} {s : Set α}, s.Nonempty → (s.Countable ↔ ∃ f, Function.Surjective f) | A non-empty set is countable iff there exists a surjection from the
natural numbers onto the subtype induced by the set.
| true |
LocallyConstant.instAddZeroClass.eq_1 | Mathlib.Topology.LocallyConstant.Algebra | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : AddZeroClass Y],
LocallyConstant.instAddZeroClass = Function.Injective.addZeroClass DFunLike.coe ⋯ ⋯ ⋯ | null | true |
OrderIso.sumLexIioIci._proof_1 | Mathlib.Order.Hom.Lex | ∀ {α : Type u_1} [inst : LinearOrder α] (x : α), Set.Ici x = {y | ¬y < x} | null | false |
ContDiffWithinAt._proof_3 | Mathlib.Analysis.Calculus.ContDiff.Defs | ∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F], SMulCommClass 𝕜 𝕜 F | null | false |
Batteries.Tactic.GeneralizeProofs.withGeneralizedProofs | Batteries.Tactic.GeneralizeProofs | {α : Type} →
[Nonempty α] →
Lean.Expr →
Option Lean.Expr →
(Array Lean.Expr → Array Lean.Expr → Lean.Expr → Batteries.Tactic.GeneralizeProofs.MGen α) →
Batteries.Tactic.GeneralizeProofs.MGen α | Generalizes the proofs in the type `e` and runs `k` in a local context with these propositions.
This continuation `k` is passed
1. an array of fvars for the propositions
2. an array of proof terms (extracted from `e`) that prove these propositions
3. the generalized `e`, which refers to these fvars
The `propToFVar` ma... | true |
Submonoid.toCommMonoid.eq_1 | Mathlib.Algebra.Group.Submonoid.Defs | ∀ {M : Type u_5} [inst : CommMonoid M] (S : Submonoid M), S.toCommMonoid = SubmonoidClass.toCommMonoid S | null | true |
Fin.instUpwardEnumerable | Init.Data.Range.Polymorphic.Fin | {n : ℕ} → Std.PRange.UpwardEnumerable (Fin n) | null | true |
SimpleGraph.isVertexCover_empty._simp_1 | Mathlib.Combinatorics.SimpleGraph.VertexCover | ∀ {V : Type u_1} {G : SimpleGraph V}, G.IsVertexCover ∅ = (G = ⊥) | null | false |
_private.Mathlib.Probability.Distributions.Fernique.0.ProbabilityTheory.Fernique.lintegral_exp_mul_sq_norm_le_mul._simp_1_13 | Mathlib.Probability.Distributions.Fernique | ∀ {a : ENNReal}, (0 < a.toReal) = (0 < a ∧ a < ⊤) | null | false |
Mathlib.Tactic.ClickSuggestions.GrwKey.numGoals | Mathlib.Tactic.ClickSuggestions.GRewrite | Mathlib.Tactic.ClickSuggestions.GrwKey → ℕ | The number of side goals created. | true |
Metric.infEDist_biUnion | Mathlib.Topology.MetricSpace.HausdorffDistance | ∀ {α : Type u} [inst : PseudoEMetricSpace α] {ι : Type u_2} (f : ι → Set α) (I : Set ι) (x : α),
Metric.infEDist x (⋃ i ∈ I, f i) = ⨅ i ∈ I, Metric.infEDist x (f i) | null | true |
Aesop.instInhabitedRuleBuilderOptions | Aesop.Builder.Basic | Inhabited Aesop.RuleBuilderOptions | null | true |
LieModuleEquiv.mk._flat_ctor | Mathlib.Algebra.Lie.Basic | {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 : AddCommGroup N] →
[inst_4 : Module R M] →
[inst_5 : Module R N] →
... | null | false |
DFinsupp.equivFunOnFintype._proof_3 | Mathlib.Data.DFinsupp.Defs | ∀ {ι : Type u_1} {β : ι → Type u_2} [inst : (i : ι) → Zero (β i)] [inst_1 : Fintype ι] (x : Π₀ (i : ι), β i) (x_1 : ι),
x_1 ∈ Finset.univ.val ∨ x x_1 = 0 | null | false |
CategoryTheory.ShrinkHoms.comp_def | Mathlib.CategoryTheory.EssentiallySmall | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C]
{X Y Z : CategoryTheory.ShrinkHoms.{u} C} (f : Shrink.{w, v} (X.fromShrinkHoms ⟶ Y.fromShrinkHoms))
(g : Shrink.{w, v} (Y.fromShrinkHoms ⟶ Z.fromShrinkHoms)),
CategoryTheory.CategoryStruct.comp f g =
(e... | null | true |
ProbabilityTheory.IndepSets.union | Mathlib.Probability.Independence.Basic | ∀ {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s₁ s₂ s' : Set (Set Ω)},
ProbabilityTheory.IndepSets s₁ s' μ → ProbabilityTheory.IndepSets s₂ s' μ → ProbabilityTheory.IndepSets (s₁ ∪ s₂) s' μ | null | true |
CategoryTheory.Limits.FormalCoproduct.eval_obj_obj | Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] (A : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} A]
[inst_2 : CategoryTheory.Limits.HasCoproducts A] (F : CategoryTheory.Functor C A)
(X : CategoryTheory.Limits.FormalCoproduct C),
((CategoryTheory.Limits.FormalCoproduct.eval C A).obj F).obj X = ∐ f... | null | true |
RingHom.Smooth.holdsForLocalizationAway | Mathlib.RingTheory.RingHom.Smooth | RingHom.HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => RingHom.Smooth | null | true |
List.tail_append_of_ne_nil | Init.Data.List.Lemmas | ∀ {α : Type u_1} {xs ys : List α}, xs ≠ [] → (xs ++ ys).tail = xs.tail ++ ys | null | true |
ShrinkingLemma.PartialRefinement.instCoeFunForallSet | Mathlib.Topology.ShrinkingLemma | {ι : Type u_1} →
{X : Type u_2} →
[inst : TopologicalSpace X] →
{u : ι → Set X} →
{s : Set X} → {p : Set X → Prop} → CoeFun (ShrinkingLemma.PartialRefinement u s p) fun x => ι → Set X | null | true |
Lean.Meta.Grind.Extension.addEMatchTheorem | Lean.Meta.Tactic.Grind.EMatchTheorem | Lean.Meta.Grind.Extension →
Lean.Name →
ℕ →
List Lean.Expr →
Lean.Meta.Grind.EMatchTheoremKind →
Bool →
optParam Lean.AttributeKind Lean.AttributeKind.global →
List Lean.Meta.Grind.EMatchTheoremConstraint → Lean.MetaM Unit | Adds an E-matching theorem to the environment.
See `mkEMatchTheorem`.
| true |
MeasureTheory.SignedMeasure.haveLebesgueDecomposition_neg | Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | ∀ {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α)
[s.HaveLebesgueDecomposition μ], (-s).HaveLebesgueDecomposition μ | null | true |
Equiv.swap_apply_eq_iff | Mathlib.Logic.Equiv.Basic | ∀ {α : Sort u_1} [inst : DecidableEq α] {x y z w : α}, (Equiv.swap x y) z = w ↔ z = (Equiv.swap x y) w | null | true |
TopologicalSpace.Compacts.lipschitz_prod | Mathlib.Topology.MetricSpace.Closeds | ∀ {α : Type u_1} {β : Type u_2} [inst : EMetricSpace α] [inst_1 : EMetricSpace β], LipschitzWith 1 fun p => p.1 ×ˢ p.2 | null | true |
Mathlib.Meta.Positivity.Strictness.nonnegative.injEq | Mathlib.Tactic.Positivity.Core | ∀ {u : Lean.Level} {α : Q(Type u)} {zα : Q(Zero «$α»)} {pα : Q(PartialOrder «$α»)} {e : Q(«$α»)}
(pf pf_1 : Q(0 ≤ «$e»)),
(Mathlib.Meta.Positivity.Strictness.nonnegative pf = Mathlib.Meta.Positivity.Strictness.nonnegative pf_1) =
(pf = pf_1) | null | true |
DirectLimit.map₀RingHom._proof_2 | Mathlib.Algebra.Colimit.DirectLimit | ∀ {ι : Type u_2} [inst : Preorder ι] {G : ι → Type u_1} {T : ⦃i j : ι⦄ → i ≤ j → Type u_3}
(f : (x x_1 : ι) → (h : x ≤ x_1) → T h) [inst_1 : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)]
[inst_2 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] [inst_3 : IsDirectedOrder ι] [inst_4 : Nonempty ι]
[inst_5 : (... | null | false |
_private.Mathlib.Analysis.Convex.DoublyStochasticMatrix.0.convex_doublyStochastic._simp_1_1 | Mathlib.Analysis.Convex.DoublyStochasticMatrix | ∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p) | null | false |
Mathlib.Meta.FunProp.TheoremForm.comp | Mathlib.Tactic.FunProp.Theorems | Mathlib.Meta.FunProp.TheoremForm | null | true |
IsAntichain.preimage_relIso | Mathlib.Order.Antichain | ∀ {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {t : Set β},
IsAntichain r' t → ∀ (φ : r ≃r r'), IsAntichain r (⇑φ ⁻¹' t) | null | true |
NonemptyInterval.fst_sub | Mathlib.Algebra.Order.Interval.Basic | ∀ {α : Type u_2} [inst : Preorder α] [inst_1 : AddCommSemigroup α] [inst_2 : Sub α] [inst_3 : OrderedSub α]
[inst_4 : AddLeftMono α] (s t : NonemptyInterval α), (s - t).toProd.1 = s.toProd.1 - t.toProd.2 | null | true |
Set.extremePoints_eq_self | Mathlib.Analysis.Convex.Extreme | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : SMul 𝕜 E] [Subsingleton E] (A : Set E), Set.extremePoints 𝕜 A = A | null | true |
Bipointed.hasForget._proof_7 | Mathlib.CategoryTheory.Category.Bipointed | ∀ {X Y : Bipointed} (f : X.HomSubtype Y), ↑f X.toProd.2 = Y.toProd.2 | null | false |
SimpleGraph.Walk.Nil.rec | Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {V : Type u} →
{G : SimpleGraph V} →
{v : V} →
{motive : {w : V} → (a : G.Walk v w) → a.Nil → Sort u_1} →
motive SimpleGraph.Walk.nil ⋯ → {w : V} → {a : G.Walk v w} → (t : a.Nil) → motive a t | null | false |
IsCyclotomicExtension.Rat.Three.eq_one_or_neg_one_of_unit_of_congruent | Mathlib.NumberTheory.NumberField.Cyclotomic.Three | ∀ {K : Type u_1} [inst : Field K] {ζ : K} (hζ : IsPrimitiveRoot ζ 3) (u : (NumberField.RingOfIntegers K)ˣ)
[inst_1 : NumberField K] [IsCyclotomicExtension {3} ℚ K], (∃ n, (hζ.toInteger - 1) ^ 2 ∣ ↑u - ↑n) → u = 1 ∨ u = -1 | If a unit `u` is congruent to an integer modulo `λ ^ 2`, then `u = 1` or `u = -1`.
This is a special case of the so-called *Kummer's lemma*. | true |
Lean.Linter.LinterOptions.ctorIdx | Lean.Linter.Init | Lean.Linter.LinterOptions → ℕ | null | false |
SkewPolynomial.sum_monomial | Mathlib.Algebra.SkewPolynomial.Basic | ∀ {R : Type u_1} [inst : Semiring R] (f : SkewPolynomial R), (f.sum fun a => ⇑(SkewPolynomial.monomial a)) = f | null | true |
SummationFilter.hasSum_symmetricIcc_iff | Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | ∀ {α : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f : ℤ → α} {a : α},
HasSum f a (SummationFilter.symmetricIcc ℤ) ↔
Filter.Tendsto (fun N => ∑ n ∈ Finset.Icc (-↑N) ↑N, f n) Filter.atTop (nhds a) | null | true |
Lean.Server.FileWorker.InlayHintState.mk | Lean.Server.FileWorker.InlayHints | Array Lean.Elab.InlayHintInfo → ℕ → Option ℕ → Bool → Lean.Server.FileWorker.InlayHintState | null | true |
Matrix.updateRow_reindex | Mathlib.LinearAlgebra.Matrix.RowCol | ∀ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [inst : DecidableEq l]
[inst_1 : DecidableEq m] (A : Matrix m n α) (i : l) (r : o → α) (e : m ≃ l) (f : n ≃ o),
((Matrix.reindex e f) A).updateRow i r = (Matrix.reindex e f) (A.updateRow (e.symm i) fun j => r (f j)) | null | true |
_private.Mathlib.LinearAlgebra.RootSystem.Chain.0.RootPairing.root_add_nsmul_mem_range_iff_le_chainTopCoeff._proof_1_5 | Mathlib.LinearAlgebra.RootSystem.Chain | ∀ {ι : Type u_2} {R : Type u_3} {M : Type u_1} {N : Type u_4} [inst : Finite ι] [inst_1 : CommRing R]
[inst_2 : CharZero R] [inst_3 : IsDomain R] [inst_4 : AddCommGroup M] [inst_5 : Module R M] [inst_6 : AddCommGroup N]
[inst_7 : Module R N] {P : RootPairing ι R M N} [inst_8 : P.IsCrystallographic] {i j : ι}
(h :... | null | false |
Polynomial.monic_zero_iff_subsingleton' | Mathlib.Algebra.Polynomial.Monic | ∀ {R : Type u} [inst : Semiring R], Polynomial.Monic 0 ↔ (∀ (f g : Polynomial R), f = g) ∧ ∀ (a b : R), a = b | null | true |
Submodule.dualCoannihilator_bot | Mathlib.LinearAlgebra.Dual.Defs | ∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
⊥.dualCoannihilator = ⊤ | null | true |
Filter.add_bot | Mathlib.Order.Filter.Pointwise | ∀ {α : Type u_2} [inst : Add α] {f : Filter α}, f + ⊥ = ⊥ | null | true |
Filter.ofCardinalUnion._proof_1 | Mathlib.Order.Filter.CardinalInter | ∀ {α : Type u_1} {c : Cardinal.{u_1}} (l : Set (Set α)),
(∀ (S : Set (Set α)), Cardinal.mk ↑S < c → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l) →
∀ (S : Set (Set α)), Cardinal.mk ↑S < c → S ⊆ {s | sᶜ ∈ l} → ⋂₀ S ∈ {s | sᶜ ∈ l} | null | false |
MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_semiconj | Mathlib.Dynamics.Ergodic.Ergodic | ∀ {α : Type u_1} {m : MeasurableSpace α} {f : α → α} {μ : MeasureTheory.Measure α} {β : Type u_2}
{m' : MeasurableSpace β} {μ' : MeasureTheory.Measure β} {g : α → β},
MeasureTheory.MeasurePreserving g μ μ' → PreErgodic f μ → ∀ {f' : β → β}, Function.Semiconj g f f' → PreErgodic f' μ' | null | true |
Interval.subtractionCommMonoid._proof_2 | Mathlib.Algebra.Order.Interval.Basic | ∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : PartialOrder α] [inst_2 : IsOrderedAddMonoid α] (a : Interval α),
zsmulRec nsmulRec 0 a = 0 | null | false |
SSet.RelativeMorphism.Homotopy.postcomp_h | Mathlib.AlgebraicTopology.SimplicialSet.RelativeMorphism | ∀ {X Y Z : SSet} {A : X.Subcomplex} {B : Y.Subcomplex} {φ : A.toSSet ⟶ B.toSSet} {f g : SSet.RelativeMorphism A B φ}
{C : Z.Subcomplex} {ψ : B.toSSet ⟶ C.toSSet} (h : f.Homotopy g) (f' : SSet.RelativeMorphism B C ψ)
{φψ : A.toSSet ⟶ C.toSSet} (fac : CategoryTheory.CategoryStruct.comp φ ψ = φψ),
(h.postcomp f' fac... | null | true |
Ring.ofMinimalAxioms._proof_13 | Mathlib.Algebra.Ring.MinimalAxioms | ∀ {R : Type u_1} [inst : Add R] [inst_1 : Neg R] [inst_2 : Zero R] [inst_3 : One R]
(add_assoc : ∀ (a b c : R), a + b + c = a + (b + c)) (zero_add : ∀ (a : R), 0 + a = a)
(neg_add_cancel : ∀ (a : R), -a + a = 0) (n : ℕ), (Int.negSucc n).castDef = -↑(n + 1) | null | false |
IsLocallyConstant.of_constant | Mathlib.Topology.LocallyConstant.Basic | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] (f : X → Y), (∀ (x y : X), f x = f y) → IsLocallyConstant f | null | true |
_private.Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer.0.alternatingGroup.commutator_perm_le._simp_1_7 | Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | ∀ {S : Type u_3} [inst : CommMagma S] (a b : S), Commute a b = True | null | false |
String.decEq._proof_1 | Init.Prelude | ∀ (a : List UInt8) (isValidUTF8 : { data := { toList := a } }.IsValidUTF8),
{ toByteArray := { data := { toList := a } }, isValidUTF8 := isValidUTF8 } =
{ toByteArray := { data := { toList := a } }, isValidUTF8 := isValidUTF8 } | null | false |
Set.eq_restrict_iff | Mathlib.Data.Set.Restrict | ∀ {α : Type u_1} {π : α → Type u_6} {s : Set α} {f : (a : ↑s) → π ↑a} {g : (a : α) → π a},
f = s.restrict g ↔ ∀ (a : α) (ha : a ∈ s), f ⟨a, ha⟩ = g a | null | true |
CategoryTheory.Functor.hasColimit_map_comp_ι_comp_grothendieckProj | Mathlib.CategoryTheory.Functor.KanExtension.Adjunction | ∀ {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] (L : CategoryTheory.Functor C D) {H : Type u_3}
[inst_2 : CategoryTheory.Category.{v_3, u_3} H] (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F]
{X Y : D} (f : X ⟶ Y),
... | null | true |
IsPrimitiveRoot.casesOn | Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {M : Type u_1} →
[inst : CommMonoid M] →
{ζ : M} →
{k : ℕ} →
{motive : IsPrimitiveRoot ζ k → Sort u} →
(t : IsPrimitiveRoot ζ k) →
((pow_eq_one : ζ ^ k = 1) → (dvd_of_pow_eq_one : ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l) → motive ⋯) → motive t | null | false |
_private.Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity.0.chevalley_mvPolynomial_mvPolynomial._simp_1_9 | Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | ∀ {R : Type u_2} {S : Type u_3} {σ : Type u_4} [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Module R S]
{M : Submodule R S} {p : MvPolynomial σ S},
(p ∈ MvPolynomial.coeffsIn σ M) = ∀ (i : σ →₀ ℕ), MvPolynomial.coeff i p ∈ M | null | false |
differentiableAt_iff_comp_const_sub | Mathlib.Analysis.Calculus.Deriv.Add | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {a b : 𝕜},
DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x => f (b - x)) (b - a) | null | true |
AddMonCat.id_apply | Mathlib.Algebra.Category.MonCat.Basic | ∀ (M : AddMonCat) (x : ↑M), (CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id M)) x = x | null | true |
AlgEquiv.aut | Mathlib.Algebra.Algebra.Equiv | {R : Type uR} →
{A₁ : Type uA₁} → [inst : CommSemiring R] → [inst_1 : Semiring A₁] → [inst_2 : Algebra R A₁] → Group (A₁ ≃ₐ[R] A₁) | [Stacks Tag 09HR](https://stacks.math.columbia.edu/tag/09HR) | true |
IO.Process.runCmdWithInput'.match_1 | Batteries.Lean.IO.Process | (cmd : String) →
(args : Array String) →
(motive :
{ stdin := IO.Process.Stdio.piped, stdout := IO.Process.Stdio.piped, stderr := IO.Process.Stdio.piped,
cmd := cmd, args := args }.stdin.toHandleType ×
IO.Process.Child
{ stdin := IO.Process.Stdio.null,
... | null | false |
CategoryTheory.SmallCategoryOfSet.mk.sizeOf_spec | Mathlib.CategoryTheory.SmallRepresentatives | ∀ {Ω : Type w} [inst : SizeOf Ω] (obj : Set Ω) (hom : ↑obj → ↑obj → Set Ω) (id : (X : ↑obj) → ↑(hom X X))
(comp : {X Y Z : ↑obj} → ↑(hom X Y) → ↑(hom Y Z) → ↑(hom X Z))
(id_comp :
autoParam (∀ {X Y : ↑obj} (f : ↑(hom X Y)), comp (id X) f = f) CategoryTheory.SmallCategoryOfSet.id_comp._autoParam)
(comp_id :
... | null | true |
Set.Nonempty.of_vsub_right | Mathlib.Algebra.Group.Pointwise.Set.Scalar | ∀ {α : Type u_2} {β : Type u_3} [inst : VSub α β] {s t : Set β}, (s -ᵥ t).Nonempty → t.Nonempty | null | true |
OmegaCompletePartialOrder.ContinuousHom.ωScottContinuous.map | Mathlib.Order.OmegaCompletePartialOrder | ∀ {α : Type u_2} [inst : OmegaCompletePartialOrder α] {β γ : Type u_6} {f : β → γ} {g : α → Part β},
OmegaCompletePartialOrder.ωScottContinuous g → OmegaCompletePartialOrder.ωScottContinuous fun x => f <$> g x | null | true |
Int.ediv_nonneg | Init.Data.Int.DivMod.Lemmas | ∀ {a b : ℤ}, 0 ≤ a → 0 ≤ b → 0 ≤ a / b | null | true |
_private.Mathlib.AlgebraicTopology.ExtraDegeneracy.0.CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_16 | Mathlib.AlgebraicTopology.ExtraDegeneracy | ∀ {n : ℕ} (i : Fin (n + 1)) (k : Fin (n + 1 + 1)), i.succ.rev = k → ↑k + 1 + ↑i = n + 1 | null | false |
ShrinkingLemma.PartialRefinement.mk.noConfusion | Mathlib.Topology.ShrinkingLemma | {ι : Type u_1} →
{X : Type u_2} →
{inst : TopologicalSpace X} →
{u : ι → Set X} →
{s : Set X} →
{p : Set X → Prop} →
{P : Sort u} →
{toFun : ι → Set X} →
{carrier : Set ι} →
{isOpen : ∀ (i : ι), IsOpen (toFun i)} →
... | null | false |
Lean.Meta.Canonicalizer.State | Lean.Meta.Canonicalizer | Type | State for the `CanonM` monad.
| true |
_private.Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory.0.CategoryTheory.Limits.Concrete.Pi.map_ext.match_1_1 | Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | ∀ {J : Type u_1} (motive : CategoryTheory.Discrete J → Prop) (h : CategoryTheory.Discrete J),
(∀ (j : J), motive { as := j }) → motive h | null | false |
IsOpen.measure_eq_zero_iff | Mathlib.MeasureTheory.Measure.OpenPos | ∀ {X : Type u_1} [inst : TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure]
{U : Set X}, IsOpen U → (μ U = 0 ↔ U = ∅) | null | true |
_private.Lean.Elab.PreDefinition.Structural.BRecOn.0.Lean.Elab.Structural.replaceRecApps.loop.match_8 | Lean.Elab.PreDefinition.Structural.BRecOn | (recArgInfos : Array Lean.Elab.Structural.RecArgInfo) →
(motive : Option (Fin recArgInfos.size) → Sort u_1) →
(x : Option (Fin recArgInfos.size)) →
((fnIdx : Fin recArgInfos.size) → motive (some fnIdx)) →
((x : Option (Fin recArgInfos.size)) → motive x) → motive x | null | false |
CategoryTheory.FunctorToTypes.binaryCoproductIso._proof_1 | Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] (F G : CategoryTheory.Functor C (Type u_2)),
CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.pair F G) | null | false |
LieHom.coe_mk | Mathlib.Algebra.Lie.Basic | ∀ {R : Type u} {L₁ : Type v} {L₂ : Type w} [inst : CommRing R] [inst_1 : LieRing L₁] [inst_2 : LieAlgebra R L₁]
[inst_3 : LieRing L₂] [inst_4 : LieAlgebra R L₂] (f : L₁ → L₂) (h₁ : ∀ (x y : L₁), f (x + y) = f x + f y)
(h₂ :
∀ (m : R) (x : L₁),
{ toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) ... | null | true |
Valuation.instLinearOrderedCommGroupWithZeroMrange._proof_8 | Mathlib.RingTheory.Valuation.Archimedean | ∀ {F : Type u_2} {Γ₀ : Type u_1} [inst : Field F] [inst_1 : LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀}
(a : ↥(MonoidHom.mrange v)), ⊥ ≤ a | null | false |
Lean.Meta.Sym.Simp.MethodsRef.toMethods | Lean.Meta.Sym.Simp.SimpM | Lean.Meta.Sym.Simp.MethodsRef → Lean.Meta.Sym.Simp.Methods | null | true |
continuousFunctionalCalculus._proof_2 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | ContinuousMul ℂ | null | false |
Sigma.Lex.preorder.match_1 | Mathlib.Data.Sigma.Order | ∀ {ι : Type u_2} {α : ι → Type u_1} (motive : (Σₗ (i : ι), α i) → Prop) (x : Σₗ (i : ι), α i),
(∀ (fst : ι) (a : α fst), motive ⟨fst, a⟩) → motive x | null | false |
_private.Mathlib.Lean.Meta.RefinedDiscrTree.Encode.0.Lean.Meta.RefinedDiscrTree.etaPossibilities._sunfold | Mathlib.Lean.Meta.RefinedDiscrTree.Encode | Lean.Expr →
List Lean.FVarId →
Bool →
Lean.Meta.RefinedDiscrTree.LazyEntry →
ReaderT Lean.Meta.RefinedDiscrTree.Context✝ Lean.MetaM
(List (Lean.Meta.RefinedDiscrTree.Key × Lean.Meta.RefinedDiscrTree.LazyEntry)) | null | false |
_private.Init.Data.String.Slice.0.String.Slice.Pos.skipWhile.match_1.eq_1 | Init.Data.String.Slice | ∀ {s : String.Slice} (motive : Option s.Pos → Sort u_1) (nextCurr : s.Pos)
(h_1 : (nextCurr : s.Pos) → motive (some nextCurr)) (h_2 : Unit → motive none),
(match some nextCurr with
| some nextCurr => h_1 nextCurr
| none => h_2 ()) =
h_1 nextCurr | null | true |
Std.Channel.Sync.recv | Std.Sync.Channel | {α : Type} → [Inhabited α] → Std.Channel.Sync α → BaseIO α | Receive a value from the channel, blocking until the transmission could be completed.
| true |
MeasureTheory.Integrable.uniformIntegrable_condExp | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_2} [MeasureTheory.IsFiniteMeasure μ]
{g : α → ℝ},
MeasureTheory.Integrable g μ →
∀ {ℱ : ι → MeasurableSpace α}, (∀ (i : ι), ℱ i ≤ m0) → MeasureTheory.UniformIntegrable (fun i => μ[g | ℱ i]) 1 μ | Given an integrable function `g`, the conditional expectations of `g` with respect to
a sequence of sub-σ-algebras is uniformly integrable. | true |
_private.Mathlib.RingTheory.Congruence.Basic.0.RingCon.instIsCentralScalarQuotient._proof_1 | Mathlib.RingTheory.Congruence.Basic | ∀ {α : Type u_1} {R : Type u_2} [inst : Add R] [inst_1 : MulOneClass R] [inst_2 : SMul α R]
[inst_3 : IsScalarTower α R R] (c : RingCon R) [inst_4 : SMul αᵐᵒᵖ R] [inst_5 : IsCentralScalar α R],
IsCentralScalar α c.Quotient | null | false |
CategoryTheory.Functor.CommShift₂.mk._flat_ctor | Mathlib.CategoryTheory.Shift.CommShiftTwo | {C₁ : Type u_1} →
{C₂ : Type u_3} →
{D : Type u_5} →
[inst : CategoryTheory.Category.{v_1, u_1} C₁] →
[inst_1 : CategoryTheory.Category.{v_3, u_3} C₂] →
[inst_2 : CategoryTheory.Category.{v_5, u_5} D] →
{M : Type u_6} →
[inst_3 : AddCommMonoid M] →
... | null | false |
AddCommMagma | Mathlib.Algebra.Group.Defs | Type u → Type u | A commutative additive magma is a type with an addition which commutes. | true |
Nat.Partrec'.below.comp | Mathlib.Computability.PartrecBasis | ∀ {motive : {n : ℕ} → (a : List.Vector ℕ n →. ℕ) → Nat.Partrec' a → Prop} {m n : ℕ} {f : List.Vector ℕ n →. ℕ}
(g : Fin n → List.Vector ℕ m →. ℕ) (a : Nat.Partrec' f) (a_1 : ∀ (i : Fin n), Nat.Partrec' (g i)),
Nat.Partrec'.below a →
motive f a → (∀ (i : Fin n), Nat.Partrec'.below ⋯) → (∀ (i : Fin n), motive (g ... | null | true |
CategoryTheory.ShortComplex.LeftHomologyData.map._proof_2 | Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | ∀ {C : Type u_4} {D : Type u_2} [inst : CategoryTheory.Category.{u_3, u_4} C]
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] {S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData)
(F : CategoryTheory.Fun... | null | false |
FirstOrder.Language.BoundedFormula.realize_liftAt_one | Mathlib.ModelTheory.Semantics | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {α : Type u'} {n m : ℕ} {φ : L.BoundedFormula α n}
{v : α → M} {xs : Fin (n + 1) → M},
m ≤ n →
((FirstOrder.Language.BoundedFormula.liftAt 1 m φ).Realize v xs ↔
φ.Realize v (xs ∘ fun i => if ↑i < m then i.castSucc else i.succ)) | null | true |
Lean.Meta.FunIndParamKind.dropped.elim | Lean.Meta.Tactic.FunIndInfo | {motive : Lean.Meta.FunIndParamKind → Sort u} →
(t : Lean.Meta.FunIndParamKind) → t.ctorIdx = 0 → motive Lean.Meta.FunIndParamKind.dropped → motive t | null | false |
Shrink.continuousLinearEquiv_symm_apply | Mathlib.Topology.Algebra.Module.TransferInstance | ∀ (R : Type u_1) (α : Type u_2) [inst : Small.{v, u_2} α] [inst_1 : AddCommMonoid α] [inst_2 : TopologicalSpace α]
[inst_3 : Semiring R] [inst_4 : Module R α] (a : α), (Shrink.continuousLinearEquiv R α).symm a = (equivShrink α) a | null | true |
Std.HashSet.get!_inter_of_not_mem_right | Std.Data.HashSet.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.HashSet α} [EquivBEq α] [LawfulHashable α]
[inst : Inhabited α] {k : α}, k ∉ m₂ → (m₁ ∩ m₂).get! k = default | null | true |
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