name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.Lax.OplaxTrans.naturality_id | Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
{F G : CategoryTheory.LaxFunctor B C} (self : CategoryTheory.Lax.OplaxTrans F G) (a : B),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (self.app a))
(self.natural... | null | true |
_private.Mathlib.Order.Filter.Basic.0.Filter.principal_eq_iff_eq._simp_1_2 | Mathlib.Order.Filter.Basic | ∀ {α : Type u_1} {s t : Set α}, (s ∈ Filter.principal t) = (t ⊆ s) | null | false |
LinearEquiv.piRing_apply | Mathlib.LinearAlgebra.Pi | ∀ {R : Type u} {M : Type v} {ι : Type x} [inst : Semiring R] (S : Type u_4) [inst_1 : Fintype ι]
[inst_2 : DecidableEq ι] [inst_3 : Semiring S] [inst_4 : AddCommMonoid M] [inst_5 : Module R M] [inst_6 : Module S M]
[inst_7 : SMulCommClass R S M] (f : (ι → R) →ₗ[R] M) (i : ι), (LinearEquiv.piRing R M ι S) f i = f (P... | null | true |
ModularForm.IsCuspForm.congr_simp | Mathlib.NumberTheory.ModularForms.CuspFormSubmodule | ∀ {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ} [inst : Γ.HasDetOne] (f f_1 : ModularForm Γ k),
f = f_1 → f.IsCuspForm = f_1.IsCuspForm | null | true |
continuous_barycentric_coord | Mathlib.Analysis.Normed.Affine.AddTorsorBases | ∀ {ι : Type u_1} {𝕜 : Type u_2} {E : Type u_3} {P : Type u_4} [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] [inst_4 : MetricSpace P] [inst_5 : NormedAddTorsor E P]
[FiniteDimensional 𝕜 E] (b : AffineBasis ι 𝕜 P) (i : ι), Continuous ⇑(b.coord i) | null | true |
ConvexCone.gc_hull_coe | Mathlib.Geometry.Convex.Cone.Basic | ∀ {R : Type u_2} {M : Type u_4} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : AddCommMonoid M]
[inst_3 : SMul R M], GaloisConnection (ConvexCone.hull R) SetLike.coe | null | true |
Std.DTreeMap.Internal.Impl.getEntryGE?.eq_1 | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] (k : α),
Std.DTreeMap.Internal.Impl.getEntryGE? k = Std.DTreeMap.Internal.Impl.getEntryGE?.go k none | null | true |
ConvexCone.coe_strictlyPositive | Mathlib.Geometry.Convex.Cone.Basic | ∀ (R : Type u_2) (M : Type u_4) [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : AddCommGroup M]
[inst_3 : PartialOrder M] [inst_4 : IsOrderedAddMonoid M] [inst_5 : Module R M] [inst_6 : PosSMulStrictMono R M],
↑(ConvexCone.strictlyPositive R M) = Set.Ioi 0 | null | true |
Std.TreeMap.Raw.Equiv.inter_left | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap.Raw α β cmp} [Std.TransCmp cmp]
{t₃ : Std.TreeMap.Raw α β cmp}, t₁.WF → t₂.WF → t₃.WF → t₁.Equiv t₂ → (t₁ ∩ t₃).Equiv (t₂ ∩ t₃) | null | true |
CategoryTheory.Limits.ι_colimitCompWhiskeringLeftIsoCompColimit_hom | Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
{J : Type u₁} [inst_2 : CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂}
[inst_3 : CategoryTheory.Category.{v₂, u₂} K] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C))
(G : CategoryTheo... | null | true |
Ring.DimensionLEOne.not_lt_lt | Mathlib.RingTheory.DedekindDomain.Basic | ∀ {R : Type u_1} [inst : CommRing R] [Ring.DimensionLEOne R] (p₀ p₁ p₂ : Ideal R) [hp₁ : p₁.IsPrime] [hp₂ : p₂.IsPrime],
¬(p₀ < p₁ ∧ p₁ < p₂) | null | true |
ProbabilityTheory.IsMarkovKernel.casesOn | Mathlib.Probability.Kernel.Defs | {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} →
{mβ : MeasurableSpace β} →
{κ : ProbabilityTheory.Kernel α β} →
{motive : ProbabilityTheory.IsMarkovKernel κ → Sort u} →
(t : ProbabilityTheory.IsMarkovKernel κ) →
((isProbabilityMeasure : ∀ (a : α), Mea... | null | false |
ModularForm.norm_eq_zero_iff | Mathlib.NumberTheory.ModularForms.NormTrace | ∀ {𝒢 : Subgroup (GL (Fin 2) ℝ)} (ℋ : Subgroup (GL (Fin 2) ℝ)) {F : Type u_1} [inst : FunLike F UpperHalfPlane ℂ] {k : ℤ}
(f : F) [inst_1 : 𝒢.IsFiniteRelIndex ℋ] [inst_2 : ℋ.HasDetPlusMinusOne] [inst_3 : ModularFormClass F 𝒢 k],
ModularForm.norm ℋ f = 0 ↔ ⇑f = 0 | null | true |
MonoidHom.submonoidMap | Mathlib.Algebra.Group.Submonoid.Operations | {M : Type u_1} →
{N : Type u_2} →
[inst : MulOneClass M] → [inst_1 : MulOneClass N] → (f : M →* N) → (M' : Submonoid M) → ↥M' →* ↥(Submonoid.map f M') | The `MonoidHom` from a `Submonoid` to its image.
See `MulEquiv.SubmonoidMap` for a variant for `MulEquiv`s. | true |
SkewPolynomial.coeff_ofNat_succ | Mathlib.Algebra.SkewPolynomial.Basic | ∀ {R : Type u_1} [inst : Semiring R] [MulSemiringAction (Multiplicative ℕ) R] (a n : ℕ) [h : a.AtLeastTwo],
(OfNat.ofNat a).coeff (n + 1) = 0 | null | true |
Subsemiring.closure_le._simp_1 | Mathlib.Algebra.Ring.Subsemiring.Basic | ∀ {R : Type u} [inst : NonAssocSemiring R] {s : Set R} {t : Subsemiring R}, (Subsemiring.closure s ≤ t) = (s ⊆ ↑t) | null | false |
_private.Mathlib.Combinatorics.Additive.RuzsaCovering.0.Finset.ruzsa_covering_mul._simp_1_4 | Mathlib.Combinatorics.Additive.RuzsaCovering | ∀ {α : Type u_2} {s t : Finset α}, (¬Disjoint s t) = ∃ a ∈ s, a ∈ t | null | false |
CategoryTheory.Functor.prod'CompSnd_inv_app | Mathlib.CategoryTheory.Products.Basic | ∀ {A : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} A] {B : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} B]
{C : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} C] (F : CategoryTheory.Functor A B)
(G : CategoryTheory.Functor A C) (X : A), (F.prod'CompSnd G).inv.app X = CategoryTheory.CategoryStruct.... | null | true |
definition._proof_1._@.Mathlib.Analysis.Normed.Module.FiniteDimension.3513382408._hygCtx._hyg.2 | Mathlib.Analysis.Normed.Module.FiniteDimension | RingHomInvPair (RingHom.id ℝ) (RingHom.id ℝ) | null | false |
CategoryTheory.Limits.Types.isColimitOfMulticoequalizerDiagram'._proof_2 | Mathlib.CategoryTheory.Limits.Types.Multicoequalizer | ∀ {X : Type u_1} {ι : Type u_2} {A : Set X} {U : ι → Set X} {V : ι → ι → Set X}
(c : CompleteLattice.MulticoequalizerDiagram A U V) (x x_1 : ι),
CategoryTheory.CategoryStruct.comp (Set.functorToTypes.mapIso (CategoryTheory.eqToIso ⋯)).hom
((c.multispanIndex.map Set.functorToTypes).fst (x_1, x)) =
Category... | null | false |
zero_pow_eq_zero | Mathlib.Algebra.GroupWithZero.Basic | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {n : ℕ} [Nontrivial M₀], 0 ^ n = 0 ↔ n ≠ 0 | null | true |
MeasureTheory.IsAddFundamentalDomain.vadd_of_comm | Mathlib.MeasureTheory.Group.FundamentalDomain | ∀ {G : Type u_1} {α : Type u_3} [inst : AddGroup G] [inst_1 : AddAction G α] [inst_2 : MeasurableSpace α] {s : Set α}
{μ : MeasureTheory.Measure α} {G' : Type u_6} [inst_3 : AddGroup G'] [inst_4 : AddAction G' α]
[MeasurableConstVAdd G' α] [MeasureTheory.VAddInvariantMeasure G' α μ] [VAddCommClass G' G α],
Measur... | null | true |
CategoryTheory.CartesianMonoidalCategory.fullSubcategory_tensorProductIsBinaryProduct_lift_hom | Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{P : CategoryTheory.ObjectProperty C} [inst_2 : P.IsClosedUnderLimitsOfShape (CategoryTheory.Discrete PEmpty.{1})]
[inst_3 : P.IsClosedUnderLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.Walk... | null | true |
Lean.Meta.ExtractLetsConfig | Init.MetaTypes | Type | Configuration for the `extract_lets` tactic.
| true |
_private.Mathlib.Analysis.SpecialFunctions.Complex.Arg.0.Complex.arg_lt_arg_add_two_pi._proof_1_2 | Mathlib.Analysis.SpecialFunctions.Complex.Arg | ∀ (x y : ℂ), x.arg < y.arg + 2 * Real.pi | null | false |
Mathlib.CrossRef.stacksTagParser.parenthesizer | Mathlib.Tactic.CrossRefAttribute | Lean.PrettyPrinter.Parenthesizer | null | true |
AddMonoidAlgebra.commAlgEquiv_single_zero_single | Mathlib.Algebra.MonoidAlgebra.Basic | ∀ {R : Type u_1} {A : Type u_4} {M : Type u_7} {N : Type u_8} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : Algebra R A] [inst_3 : AddMonoid M] [inst_4 : AddMonoid N] (m : M),
(AddMonoidAlgebra.commAlgEquiv R) (AddMonoidAlgebra.single 0 (AddMonoidAlgebra.single m 1)) =
AddMonoidAlgebra.single m 1 | null | true |
Decidable.imp_or' | Init.PropLemmas | ∀ {b : Prop} {a : Sort u_1} {c : Prop} [Decidable b], (∀ (a : a), b ∨ c) ↔ (∀ (a : a), b) ∨ ∀ (a : a), c | null | true |
CategoryTheory.Abelian.SpectralObject.pOpcycles_δFromOpcycles_assoc | Mathlib.Algebra.Homology.SpectralObject.Cycles | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ : ℤ)
(hn₁ : n₀ + 1 = n₁) {Z : C} (h : (X.H n... | null | true |
_private.Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots.0.IsPrimitiveRoot.coe_submonoidClass_iff._simp_1_1 | Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | ∀ {M : Type u_1} [inst : CommMonoid M] (ζ : M) (k : ℕ), IsPrimitiveRoot ζ k = (ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l) | null | false |
instLargeCategoryLocale._aux_3 | Mathlib.Topology.Category.Locale | (X : Locale) → X ⟶ X | null | false |
reflTransGen_of_succ | Mathlib.Order.SuccPred.Relation | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : SuccOrder α] [IsSuccArchimedean α] (r : α → α → Prop) {n m : α},
(∀ i ∈ Set.Ico n m, r i (Order.succ i)) → (∀ i ∈ Set.Ico m n, r (Order.succ i) i) → Relation.ReflTransGen r n m | `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ succ i` and `succ i ~ i`
for all `i` between `n` and `m`. | true |
Lean.Meta.CanUnfoldPredicateConfig._sizeOf_inst | Lean.Meta.Basic | SizeOf Lean.Meta.CanUnfoldPredicateConfig | null | false |
Plausible.NoShrink.get | Plausible.Sampleable | {α : Type u_1} → Plausible.NoShrink α → α | null | true |
ContDiffOn.smulRight | Mathlib.Analysis.Calculus.ContDiff.Comp | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
[inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {s : Set E} {n : WithTop ℕ∞} {f : E →... | null | true |
Coheyting.boundary_top | Mathlib.Order.Heyting.Boundary | ∀ {α : Type u_1} [inst : CoheytingAlgebra α], Coheyting.boundary ⊤ = ⊥ | null | true |
_private.Mathlib.Analysis.Calculus.Deriv.Inverse.0.HasDerivWithinAt.eventually_notMem._simp_1_7 | Mathlib.Analysis.Calculus.Deriv.Inverse | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
_private.Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass.0.PeriodPair.weierstrassPExceptSummand_of_notMem._proof_1_3 | Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | ∀ (L : PeriodPair), ∀ l₀ ∉ L.lattice, L.weierstrassPExceptSummand l₀ = L.weierstrassPSummand | null | false |
Finset.union_sdiff_of_subset | Mathlib.Data.Finset.SDiff | ∀ {α : Type u_1} [inst : DecidableEq α] {s t : Finset α}, s ⊆ t → s ∪ t \ s = t | null | true |
ISize.toNatClampNeg_ofInt_of_two_pow_numBits | Init.Data.SInt.Lemmas | ∀ {n : ℤ},
-2 ^ (System.Platform.numBits - 1) ≤ n →
n < 2 ^ (System.Platform.numBits - 1) → (ISize.ofInt n).toNatClampNeg = n.toNat | null | true |
_private.Lean.Parser.Term.0.Lean.Parser.Term.let._regBuiltin.Lean.Parser.Term.letOptUsedOnly.formatter_17 | Lean.Parser.Term | IO Unit | null | false |
ImportGraph.Lean.Import.FormatBehavior | ImportGraph.Imports.Pretty | Type | A configuration option guiding the behavior of import block pretty-printing functions (e.g
`ImportGraph.Lean.Import.pretty`) which determines the order and grouping of imports. | true |
ISize.neg_zero | Init.Data.SInt.Lemmas | -0 = 0 | null | true |
WithBot.addMonoid._aux_4 | Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {α : Type u_1} → [AddMonoid α] → ℕ → WithBot α → WithBot α | null | false |
fontaineThetaInvertP._proof_1 | Mathlib.RingTheory.Perfectoid.BDeRham | ∀ (R : Type u_1) [inst : CommRing R] (p : ℕ) [inst_1 : Fact (Nat.Prime p)] [inst_2 : Fact ¬IsUnit ↑p]
[inst_3 : IsAdicComplete (Ideal.span {↑p}) R],
IsUnit (((algebraMap R (Localization.Away ↑p)).comp (WittVector.fontaineTheta R p)) ↑p) | null | false |
CategoryTheory.GrothendieckTopology.mem_sieves_iff_coe._simp_1 | Mathlib.CategoryTheory.Sites.Grothendieck | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : C} {S : CategoryTheory.Sieve X}
(J : CategoryTheory.GrothendieckTopology C), (S ∈ J.sieves X) = (S ∈ J X) | null | false |
_private.Lean.Elab.Term.TermElabM.0.Lean.Elab.Term.mkCoe.match_5 | Lean.Elab.Term.TermElabM | (motive : Lean.Exception → Sort u_1) →
(x : Lean.Exception) →
((ref : Lean.Syntax) → (msg : Lean.MessageData) → motive (Lean.Exception.error ref msg)) →
((x : Lean.Exception) → motive x) → motive x | null | false |
SimpleGraph.IsFiveWheelLike.notMem_right | Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | ∀ {α : Type u_1} [inst : DecidableEq α] {G : SimpleGraph α} {r k : ℕ} {v w₁ w₂ : α} {s t : Finset α},
G.IsFiveWheelLike r k v w₁ w₂ s t → v ∉ t | null | true |
PrimeSpectrum.BasicConstructibleSetData.mk._flat_ctor | Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {R : Type u_1} → R → (n : ℕ) → (Fin n → R) → PrimeSpectrum.BasicConstructibleSetData R | null | false |
Measurable.measurableEmbedding | Mathlib.MeasureTheory.Constructions.Polish.Basic | ∀ {γ : Type u_3} {α : Type u_4} [inst : MeasurableSpace α] {f : γ → α} [MeasurableSpace.CountablySeparated α]
[inst_2 : MeasurableSpace γ] [StandardBorelSpace γ], Measurable f → Function.Injective f → MeasurableEmbedding f | An injective measurable function from a standard Borel space to a
countably separated measurable space is a measurable embedding. | true |
NormedField.noConfusion | Mathlib.Analysis.Normed.Field.Basic | {P : Sort u} →
{α : Type u_3} →
{t : NormedField α} → {α' : Type u_3} → {t' : NormedField α'} → α = α' → t ≍ t' → NormedField.noConfusionType P t t' | null | false |
Std.Internal.UV.System.instReprRUsage | Std.Internal.UV.System | Repr Std.Internal.UV.System.RUsage | null | true |
Finset.card_eq_of_bijective | Mathlib.Data.Finset.Card | ∀ {α : Type u_1} {s : Finset α} {n : ℕ} (f : (i : ℕ) → i < n → α),
(∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a) →
(∀ (i : ℕ) (h : i < n), f i h ∈ s) → (∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j) → s.card = n | See also `card_bij`.
TODO: consider deprecating, since this has been unused in mathlib for a long time and is just a
special case of `card_bij`.
| true |
DenseRange.zsmul_of_ergodic_add_left | Mathlib.Dynamics.Ergodic.Action.OfMinimal | ∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : TopologicalSpace G] [IsTopologicalAddGroup G]
[inst_3 : MeasurableSpace G] [OpensMeasurableSpace G] {μ : MeasureTheory.Measure G} [μ.IsOpenPosMeasure] {g : G},
Ergodic (fun x => g + x) μ → DenseRange fun x => x • g | If the left addition of `g` is ergodic
with respect to a measure which is positive on nonempty open sets,
then the integer multiples of `g` are dense in `G`. | true |
minimalPrimes_eq_minimals | Mathlib.RingTheory.Ideal.MinimalPrime.Basic | ∀ {R : Type u_1} [inst : CommSemiring R], minimalPrimes R = {x | Minimal Ideal.IsPrime x} | null | true |
Set.pow_subset_pow_mul_of_sq_subset_mul | Mathlib.Algebra.Group.Pointwise.Set.Basic | ∀ {α : Type u_2} [inst : Monoid α] {s t : Set α} {n : ℕ}, s ^ 2 ⊆ t * s → n ≠ 0 → s ^ n ⊆ t ^ (n - 1) * s | null | true |
derivedSeries_zero | Mathlib.GroupTheory.Solvable | ∀ (G : Type u_1) [inst : Group G], derivedSeries G 0 = ⊤ | null | true |
Real.summable_one_div_int_add_rpow | Mathlib.Analysis.PSeries | ∀ (a s : ℝ), (Summable fun n => 1 / |↑n + a| ^ s) ↔ 1 < s | null | true |
CategoryTheory.ShortComplex.ShortExact.extClass_comp_assoc | Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : CategoryTheory.HasExt C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact) {Y : C} {n : ℕ}
(γ : CategoryTheory.Abelian.Ext S.X₂ Y n) {n' : ℕ} {h : 1 + n = n'},
hS.extClass.comp ((CategoryTheory.Abelian.Ext... | null | true |
Valuation.Uniformizer.recOn | Mathlib.RingTheory.Valuation.Discrete.Basic | {Γ : Type u_1} →
[inst : LinearOrderedCommGroupWithZero Γ] →
{A : Type u_2} →
[inst_1 : Ring A] →
{v : Valuation A Γ} →
[hv : v.IsRankOneDiscrete] →
{motive : v.Uniformizer → Sort u} →
(t : v.Uniformizer) →
((val : ↥v.integer) →
... | null | false |
SkewMonoidAlgebra.single.eq_1 | Mathlib.Algebra.SkewMonoidAlgebra.Lift | ∀ {k : Type u_1} {G : Type u_2} [inst : AddMonoid k] (a : G) (b : k),
SkewMonoidAlgebra.single a b = { coeff := fun₀ | a => b } | null | true |
CategoryTheory.Limits.prod.map.eq_1 | Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W X Y Z : C}
[inst_1 : CategoryTheory.Limits.HasBinaryProduct W X] [inst_2 : CategoryTheory.Limits.HasBinaryProduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z),
CategoryTheory.Limits.prod.map f g = CategoryTheory.Limits.limMap (CategoryTheory.Limits.mapPair f g) | null | true |
BoxIntegral.IntegrationParams.toFilterDistortion_neBot | Mathlib.Analysis.BoxIntegral.Partition.Filter | ∀ {ι : Type u_1} [inst : Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι),
(l.toFilterDistortion I I.distortion).NeBot | null | true |
_private.Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks.0.SimpleGraph.Walk.isSubwalk_iff_darts_isInfix._proof_1_17 | Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | ∀ {V : Type u_1} {G : SimpleGraph V} {u v u' v' : V} {p₁ : G.Walk u v} {p₂ : G.Walk u' v'} (k : ℕ),
p₁.support.length + k ≤ p₂.support.length → ∀ i < p₁.darts.length, k + i + 1 < p₂.support.length | null | false |
PredOrder.ofCore._proof_2 | Mathlib.Order.SuccPred.Basic | ∀ {α : Type u_1} [inst : LinearOrder α] (pred : α → α),
(∀ {a : α}, ¬IsMin a → ∀ (b : α), b < a ↔ b ≤ pred a) → ∀ {a : α}, ¬IsMin a → ¬a ≤ pred a | null | false |
ProbabilityTheory.Kernel.compProd_assoc | Mathlib.Probability.Kernel.Composition.CompProd | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} {δ : Type u_4} {mδ : MeasurableSpace δ} {κ : ProbabilityTheory.Kernel α β}
{η : ProbabilityTheory.Kernel (α × β) γ} {ξ : ProbabilityTheory.Kernel (α × β × γ) δ},
(κ.compProd (η.compProd (ξ.com... | `Kernel.compProd` is associative. We have to insert `MeasurableEquiv.prodAssoc` in two places
because the products of types `α × β × γ` and `(α × β) × γ` are different. | true |
SimplexCategory.σ₀Iter_succ'_assoc | Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | ∀ (i : ℕ) {n m : ℕ} (h : autoParam (n + i = m) SimplexCategory.σ₀Iter_succ'._auto_1) {Z : SimplexCategory}
(h_1 : { len := n } ⟶ Z),
CategoryTheory.CategoryStruct.comp (SimplexCategory.σ₀Iter (i + 1) ⋯) h_1 =
CategoryTheory.CategoryStruct.comp (SimplexCategory.σ 0)
(CategoryTheory.CategoryStruct.comp (Sim... | null | true |
CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'.eq_1 | Mathlib.CategoryTheory.Limits.IndYoneda | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{u₂, u₁} C] {I : Type v₁} [inst_1 : CategoryTheory.Category.{v₂, v₁} I]
(F : CategoryTheory.Functor Iᵒᵖ C) [inst_2 : CategoryTheory.Limits.HasColimit F],
CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda' F =
CategoryTheory.coyoneda.mapIso (CategoryTheory.Li... | null | true |
_private.Mathlib.Topology.CWComplex.Classical.Basic.0.Topology.RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell.match_1_3 | Mathlib.Topology.CWComplex.Classical.Basic | ∀ {X : Type u_1} [t : TopologicalSpace X] {C D : Set X} [inst : Topology.RelCWComplex C D]
(I : (m : ℕ) → Finset (Topology.RelCWComplex.cell C m)) (m : ℕ) (motive : ↥(I m) → Prop) (x : ↥(I m)),
(∀ (j : Topology.RelCWComplex.cell C m) (property : j ∈ I m), motive ⟨j, property⟩) → motive x | null | false |
InfTopHom.instSemilatticeInf._proof_1 | Mathlib.Order.Hom.BoundedLattice | ∀ {α : Type u_1} {β : Type u_2} [inst : Min α] [inst_1 : Top α] [inst_2 : SemilatticeInf β] [inst_3 : OrderTop β],
Function.Injective DFunLike.coe | null | false |
_private.Lean.Meta.Tactic.Grind.AC.PP.0.Lean.Meta.Grind.AC.ppStruct?.match_1 | Lean.Meta.Tactic.Grind.AC.PP | (motive : Option Lean.Expr → Sort u_1) →
(x : Option Lean.Expr) → ((u : Lean.Expr) → motive (some u)) → ((x : Option Lean.Expr) → motive x) → motive x | null | false |
AddCommMonoid.toNatModule._proof_3 | Mathlib.Algebra.Module.NatInt | ∀ {M : Type u_1} [inst : AddCommMonoid M] (a : M), 0 • a = 0 | null | false |
instCommRingBDeRhamPlus._proof_31 | Mathlib.RingTheory.Perfectoid.BDeRham | ∀ (R : Type u_1) [inst : CommRing R] (p : ℕ) [inst_1 : Fact (Nat.Prime p)] [inst_2 : Fact ¬IsUnit ↑p]
[inst_3 : IsAdicComplete (Ideal.span {↑p}) R],
autoParam (∀ (n : ℕ), ↑(n + 1) = ↑n + 1) AddMonoidWithOne.natCast_succ._autoParam | null | false |
Seminorm.preimage_metric_ball | Mathlib.Analysis.Normed.Module.Seminorm.Basic | ∀ {𝕜 : Type u_3} {E : Type u_7} [inst : SeminormedRing 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
(p : Seminorm 𝕜 E) {r : ℝ}, ⇑p ⁻¹' Metric.ball 0 r = {x | p x < r} | null | true |
_private.Mathlib.GroupTheory.Index.0.Subgroup.index_eq_two_iff_exists_notMem_and._simp_1_1 | Mathlib.GroupTheory.Index | ∀ {G : Type u_1} [inst : Group G] {H : Subgroup G}, (H.index = 2) = ∃ a, ∀ (b : G), Xor (b * a ∈ H) (b ∈ H) | null | false |
ConditionallyCompleteLinearOrderBot.lt._inherited_default | Mathlib.Order.ConditionallyCompleteLattice.Defs | {α : Type u_2} → (α → α → Prop) → α → α → Prop | null | false |
_private.Mathlib.Algebra.Module.Presentation.Tautological.0.Module.Presentation.tautologicalRelations.match_1.eq_2 | Mathlib.Algebra.Module.Presentation.Tautological | ∀ (A : Type u_2) (M : Type u_1) (motive : Module.Presentation.tautological.R A M → Sort u_3) (a : A) (m : M)
(h_1 : (m₁ m₂ : M) → motive (Module.Presentation.tautological.R.add m₁ m₂))
(h_2 : (a : A) → (m : M) → motive (Module.Presentation.tautological.R.smul a m)),
(match Module.Presentation.tautological.R.smul ... | null | true |
MonoidHom.FixedPointFree.commutatorMap_injective | Mathlib.GroupTheory.FixedPointFree | ∀ {F : Type u_1} {G : Type u_2} [inst : Group G] [inst_1 : FunLike F G G] [MonoidHomClass F G G] {φ : F},
MonoidHom.FixedPointFree ⇑φ → Function.Injective (MonoidHom.commutatorMap ⇑φ) | null | true |
Subgroup.instMulActionSubtypeMem._proof_1 | Mathlib.Algebra.Group.Subgroup.Actions | ∀ {G : Type u_1} [inst : Group G], SubmonoidClass (Subgroup G) G | null | false |
Std.DTreeMap.Internal.Impl.Const.get!_empty | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {β : Type v} [Std.TransOrd α] [inst : Inhabited β] {a : α},
Std.DTreeMap.Internal.Impl.Const.get! Std.DTreeMap.Internal.Impl.empty a = default | null | true |
CategoryTheory.Limits.hasCokernels_of_hasCoequalizers | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[CategoryTheory.Limits.HasCoequalizers C], CategoryTheory.Limits.HasCokernels C | null | true |
LightDiagram'.mk.inj | Mathlib.Topology.Category.LightProfinite.Basic | ∀ {diagram diagram_1 : CategoryTheory.Functor ℕᵒᵖ FintypeCat.Skeleton},
{ diagram := diagram } = { diagram := diagram_1 } → diagram = diagram_1 | null | true |
Std.DTreeMap.Equiv.getEntryLE?_eq | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.DTreeMap α β cmp} [Std.TransCmp cmp] {k : α},
t₁.Equiv t₂ → t₁.getEntryLE? k = t₂.getEntryLE? k | null | true |
FiniteField.pow_finrank_eq_card | Mathlib.FieldTheory.Finite.GaloisField | ∀ (p : ℕ) [inst : Fact (Nat.Prime p)] (k : Type u_3) [inst_1 : AddCommGroup k] [inst_2 : Fintype k]
[inst_3 : Module (ZMod p) k], p ^ Module.finrank (ZMod p) k = Fintype.card k | null | true |
signedDist_le_dist | Mathlib.Geometry.Euclidean.SignedDist | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] (v : V) (p q : P), ((signedDist v) p) q ≤ dist p q | null | true |
Lean.Meta.LazyDiscrTree.pushRootCase | Lean.Meta.LazyDiscrTree | Std.HashMap Lean.Meta.LazyDiscrTree.Key Lean.Meta.LazyDiscrTree.TrieIndex →
Lean.Meta.LazyDiscrTree.Key →
Array Lean.Expr → Array Lean.Meta.LazyDiscrTree.PartialMatch → Array Lean.Meta.LazyDiscrTree.PartialMatch | Add partial match to cases if discriminator tree root map has potential matches.
| true |
CategoryTheory.AddMod.Hom.mk._flat_ctor | Mathlib.CategoryTheory.Monoidal.Mod | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{D : Type u₂} →
[inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] →
{A : C} →
[inst_4 : Cat... | null | false |
Subgroup.commutator_commutator_eq_bot_of_rotate | Mathlib.GroupTheory.Commutator.Basic | ∀ {G : Type u_1} [inst : Group G] {H₁ H₂ H₃ : Subgroup G}, ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥ → ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥ → ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ | **The Three Subgroups Lemma** (via the Hall-Witt identity) | true |
ComplexShape.ctorIdx | Mathlib.Algebra.Homology.ComplexShape | {ι : Type u_1} → ComplexShape ι → ℕ | null | false |
ProofWidgets.Jsx.delabHtmlOfComponent | ProofWidgets.Data.Html | Lean.PrettyPrinter.Delaborator.Delab | null | true |
ModuleCat.homAddEquiv._proof_1 | Mathlib.Algebra.Category.ModuleCat.Basic | ∀ {R : Type u_2} [inst : Ring R] {M N : ModuleCat R} (x x_1 : M ⟶ N),
ModuleCat.homEquiv.toFun (x + x_1) = ModuleCat.homEquiv.toFun (x + x_1) | null | false |
Nat.Partition.ext_iff | Mathlib.Combinatorics.Enumerative.Partition.Basic | ∀ {n : ℕ} {x y : n.Partition}, x = y ↔ x.parts = y.parts | null | true |
IsDedekindDomain.HeightOneSpectrum.adicCompletion.toCompletion_ofCompletion | Mathlib.RingTheory.DedekindDomain.AdicValuation | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] (K : Type u_2) [inst_2 : Field K]
[inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
(x : (IsDedekindDomain.HeightOneSpectrum.valuation K v).Completion),
(IsDedekindDomain.HeightOneSpectrum.adicCompleti... | null | true |
DifferentiableWithinAt.csin | Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {f : E → ℂ} {x : E} {s : Set E},
DifferentiableWithinAt ℂ f s x → DifferentiableWithinAt ℂ (fun x => Complex.sin (f x)) s x | null | true |
Lean.Grind.CommRing.Mon.degreeOf.eq_1 | Init.Grind.Ring.CommSolver | ∀ (x : Lean.Grind.CommRing.Var), Lean.Grind.CommRing.Mon.unit.degreeOf x = 0 | null | true |
ENormedMonoid.recOn | Mathlib.Analysis.Normed.Group.Defs | {E : Type u_4} →
[inst : TopologicalSpace E] →
{motive : ENormedMonoid E → Sort u} →
(t : ENormedMonoid E) →
([toESeminormedMonoid : ESeminormedMonoid E] →
(enorm_eq_zero : ∀ (x : E), ‖x‖ₑ = 0 ↔ x = 1) →
motive { toESeminormedMonoid := toESeminormedMonoid, enorm_eq_zero := ... | null | false |
Nat.cast_add | Mathlib.Data.Nat.Cast.Defs | ∀ {R : Type u_1} [inst : AddMonoidWithOne R] (m n : ℕ), ↑(m + n) = ↑m + ↑n | null | true |
MeasCat.Integral._proof_1 | Mathlib.MeasureTheory.Category.MeasCat | ∀ (μ : MeasureTheory.Measure (MeasureTheory.Measure ENNReal)),
∫⁻ (x : ENNReal), x ∂μ.join = ∫⁻ (x : ENNReal), x ∂MeasureTheory.Measure.map (fun m => ∫⁻ (x : ENNReal), x ∂m) μ | null | false |
CategoryTheory.GrothendieckTopology.trivial | Mathlib.CategoryTheory.Sites.Grothendieck | (C : Type u) → [inst : CategoryTheory.Category.{v, u} C] → CategoryTheory.GrothendieckTopology C | The trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is
also known as the indiscrete, coarse, or chaotic topology.
See [MM92] Chapter III, Section 2, example (a), or
https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies
| true |
CliffordAlgebra.toBaseChange_ofBaseChange | Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | ∀ {R : Type u_1} (A : Type u_2) {V : Type u_3} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : AddCommGroup V]
[inst_3 : Algebra R A] [inst_4 : Module R V] [inst_5 : Invertible 2] (Q : QuadraticForm R V)
(x : TensorProduct R A (CliffordAlgebra Q)),
(CliffordAlgebra.toBaseChange A Q) ((CliffordAlgebra.ofBaseCh... | null | true |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.