name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
AlgebraicGeometry.AffineTargetMorphismProperty.diagonal_of_openCover_source | Mathlib.AlgebraicGeometry.Morphisms.Constructors | ∀ {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [Q.IsLocal] {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
(𝒰 : X.OpenCover) [inst : ∀ (i : 𝒰.I₀), AlgebraicGeometry.IsAffine (𝒰.X i)] [inst_1 : AlgebraicGeometry.IsAffine Y],
(∀ (i j : 𝒰.I₀), Q (CategoryTheory.Limits.pullback.mapDesc (𝒰.f i) (𝒰.f j) f)) → ... | null | true |
String.lt_iff_ltb._simp_1 | Mathlib.Data.String.Basic | ∀ {s₁ s₂ : String}, (s₁ < s₂) = (String.ltb (String.Legacy.iter s₁) (String.Legacy.iter s₂) = true) | null | false |
_private.Mathlib.RingTheory.Polynomial.HilbertPoly.0.Polynomial.preHilbertPoly_eq_choose_add_sub._simp_1_7 | Mathlib.RingTheory.Polynomial.HilbertPoly | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
AddUnits.neg_add_eq_zero._simp_1 | Mathlib.Algebra.Group.Units.Basic | ∀ {α : Type u} [inst : AddMonoid α] {u : AddUnits α} {a : α}, (↑(-u) + a = 0) = (↑u = a) | null | false |
Fin.find?_eq_findRev?_iff | Batteries.Data.Fin.Lemmas | ∀ {n : ℕ} {p : Fin n → Bool}, Fin.find? p = Fin.findRev? p ↔ ∀ (i j : Fin n), p i = true → p j = true → i = j | null | true |
_private.Mathlib.Algebra.Group.Subgroup.Pointwise.0.Subgroup.conjAct_pointwise_smul_iff._simp_1_1 | Mathlib.Algebra.Group.Subgroup.Pointwise | ∀ {G : Type u_1} [inst : Group G] {H K : Subgroup G}, (H = K) = ∀ (x : G), x ∈ H ↔ x ∈ K | null | false |
Std.DTreeMap.Internal.Impl.maxKey?_eq_none_iff | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α β} [Std.TransOrd α],
t.WF → (t.maxKey? = none ↔ t.isEmpty = true) | null | true |
ProbabilityTheory.defaultRatCDF_le_one | Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | ∀ (q : ℚ), ProbabilityTheory.defaultRatCDF q ≤ 1 | null | true |
instAddCommMonoidWeakSpace._proof_6 | Mathlib.Topology.Algebra.Module.Spaces.WeakDual | ∀ (𝕜 : Type u_1) (E : Type u_2) [inst : CommSemiring 𝕜] [inst_1 : TopologicalSpace 𝕜] [inst_2 : ContinuousAdd 𝕜]
[inst_3 : ContinuousConstSMul 𝕜 𝕜] [inst_4 : AddCommMonoid E] [inst_5 : Module 𝕜 E] [inst_6 : TopologicalSpace E]
(a : WeakSpace 𝕜 E), 0 + a = a | null | false |
Lean.KeyedDeclsAttribute.OLeanEntry.mk.injEq | Lean.KeyedDeclsAttribute | ∀ (key : Lean.KeyedDeclsAttribute.Key) (declName : Lean.Name) (key_1 : Lean.KeyedDeclsAttribute.Key)
(declName_1 : Lean.Name),
({ key := key, declName := declName } = { key := key_1, declName := declName_1 }) =
(key = key_1 ∧ declName = declName_1) | null | true |
AddMonoidAlgebra.casesOn | Mathlib.Algebra.MonoidAlgebra.Defs | {R : Type u_8} →
{M : Type u_9} →
[inst : Semiring R] →
{motive : AddMonoidAlgebra R M → Sort u} →
(t : AddMonoidAlgebra R M) → ((coeff : M →₀ R) → motive (AddMonoidAlgebra.ofCoeff coeff)) → motive t | null | false |
Std.ExtHashMap.size_diff_add_size_inter_eq_size_left | Std.Data.ExtHashMap.Lemmas | ∀ {α : Type u} {β : Type v} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.ExtHashMap α β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α], (m₁ \ m₂).size + (m₁ ∩ m₂).size = m₁.size | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesColimit.0.SSet.isColimitCoconeN.desc._proof_1 | Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesColimit | ∀ {X : SSet} (s : CategoryTheory.Limits.Cocone X.functorN) (x : (CategoryTheory.Limits.MultispanShape.prod X.N).L),
CategoryTheory.CategoryStruct.comp
(((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).map SSet.Subcomplex.toSSetFunctor).fst x)
(s.ι.app ((CategoryTheory.Limits.MultispanShape.prod... | null | false |
Std.Rxi.Iterator.noConfusionType | Init.Data.Range.Polymorphic.RangeIterator | Sort u_1 → {α : Type u} → Std.Rxi.Iterator α → {α' : Type u} → Std.Rxi.Iterator α' → Sort u_1 | null | false |
_private.Lean.Meta.LetToHave.0.Lean.Meta.LetToHave.State.noConfusion | Lean.Meta.LetToHave | {P : Sort u} → {t t' : Lean.Meta.LetToHave.State✝} → t = t' → Lean.Meta.LetToHave.State.noConfusionType✝ P t t' | null | false |
groupCohomology.shortComplexH0 | Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {k G : Type u} →
[inst : CommRing k] → [inst_1 : Group G] → Rep.{max u u_1, u, u} k G → CategoryTheory.ShortComplex (ModuleCat k) | The (exact) short complex `A.ρ.invariants ⟶ A ⟶ (G → A)`. | true |
Eq.faithful_of_comp | Mathlib.CategoryTheory.Functor.FullyFaithful | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] {F : CategoryTheory.Functor C D}
{G : CategoryTheory.Functor D E} {H : CategoryTheory.Functor C E} [ℋ : H.Faithful], F.comp G = H → F.Fa... | **Alias** of `CategoryTheory.Functor.Faithful.of_comp_eq`. | true |
_private.Mathlib.RingTheory.Nilpotent.Basic.0.isReduced_iff_pow_one_lt._simp_1_1 | Mathlib.RingTheory.Nilpotent.Basic | ∀ {R : Type u_1} [inst : MonoidWithZero R], IsReduced R = IsRadical 0 | null | false |
Orientation.kahler_map_complex | Mathlib.Analysis.InnerProductSpace.TwoDim | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : Fact (Module.finrank ℝ E = 2)]
(o : Orientation ℝ E (Fin 2)) (f : E ≃ₗᵢ[ℝ] ℂ),
(Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation →
∀ (x y : E), (o.kahler x) y = f y * (starRingEnd ℂ) (f x) | The Kahler form on an oriented real inner product space of dimension 2 can be evaluated in terms
of a complex-number representation of the space. | true |
Ordinal.mul_omega0_dvd | Mathlib.SetTheory.Ordinal.Principal | ∀ {a : Ordinal.{u}}, 0 < a → a < Ordinal.omega0 → ∀ {b : Ordinal.{u}}, Ordinal.omega0 ∣ b → a * b = b | null | true |
Aesop.Options' | Aesop.Options.Internal | Type | null | true |
Algebra.Extension.Hom.id | Mathlib.RingTheory.Extension.Basic | {R : Type u} →
{S : Type v} →
[inst : CommRing R] → [inst_1 : CommRing S] → [inst_2 : Algebra R S] → (P : Algebra.Extension R S) → P.Hom P | The identity hom. | true |
Mathlib.Tactic.FieldSimp.NF.«_aux_Mathlib_Tactic_FieldSimp_Lemmas___macroRules_Mathlib_Tactic_FieldSimp_NF_term_::ᵣ__1» | Mathlib.Tactic.FieldSimp.Lemmas | Lean.Macro | null | false |
CpltSepUniformSpace.t0Space | Mathlib.Topology.Category.UniformSpace | ∀ (X : CpltSepUniformSpace), T0Space X.toUniformSpace.carrier | null | true |
_private.Mathlib.Data.Nat.Fib.Basic.0.Nat.fib_strictMonoOn.match_1_1 | Mathlib.Data.Nat.Fib.Basic | ∀ (motive : (x : ℕ) → x ∈ Set.Ici 2 → (x_2 : ℕ) → x_2 ∈ Set.Ici 2 → x < x_2 → Prop) (x : ℕ) (x_1 : x ∈ Set.Ici 2)
(x_2 : ℕ) (x_3 : x_2 ∈ Set.Ici 2) (x_4 : x < x_2),
(∀ (_m : ℕ) (x : _m + 2 ∈ Set.Ici 2) (_n : ℕ) (x_5 : _n + 2 ∈ Set.Ici 2) (hmn : _m + 2 < _n + 2),
motive _m.succ.succ x _n.succ.succ x_5 hmn) →
... | null | false |
_private.Lean.Compiler.LCNF.Specialize.0.Lean.Compiler.LCNF.Specialize.specializeApp?.match_19 | Lean.Compiler.LCNF.Specialize | (motive : Lean.Compiler.LCNF.LetValue Lean.Compiler.LCNF.Purity.pure → Sort u_1) →
(e : Lean.Compiler.LCNF.LetValue Lean.Compiler.LCNF.Purity.pure) →
((declName : Lean.Name) →
(us : List Lean.Level) →
(args : Array (Lean.Compiler.LCNF.Arg Lean.Compiler.LCNF.Purity.pure)) →
motive (Le... | null | false |
WithBot.toDualTopEquiv._proof_4 | Mathlib.Order.Hom.WithTopBot | ∀ {α : Type u_1} [inst : LE α] {a b : WithBot αᵒᵈ},
{ toFun := fun a => OrderDual.toDual (WithBot.recBotCoe ⊤ (fun a => ↑(OrderDual.ofDual a)) a),
invFun := fun a => WithTop.recTopCoe ⊥ (fun a => ↑(OrderDual.toDual a)) (OrderDual.ofDual a), left_inv := ⋯,
right_inv := ⋯ }
b ≤
{ toFun... | null | false |
_private.Mathlib.NumberTheory.LucasLehmer.0.mersenne_mod_eight._proof_1_1 | Mathlib.NumberTheory.LucasLehmer | ∀ (n : ℕ), mersenne n % 8 = 7 → (2 * mersenne n + 1) % 8 = 7 | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk.0.SzemerediRegularity.edgeDensity_chunk_aux | Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] {P : Finpartition Finset.univ} {G : SimpleGraph α}
[inst_2 : DecidableRel G.Adj] {ε : ℝ} {U V : Finset α} [Nonempty α] (hP : P.IsEquipartition),
P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α →
100 ≤ 4 ^ P.parts.card * ε ^ 5 →
∀ (hU : U ∈ P.p... | null | true |
CategoryTheory.Limits.BinaryFan.isLimitMapConeEquiv | Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryFan | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{D : Type u_1} →
[inst_1 : CategoryTheory.Category.{v_1, u_1} D] →
{F : CategoryTheory.Functor C D} →
{X Y : C} →
{s : CategoryTheory.Limits.BinaryFan X Y} →
CategoryTheory.Limits.IsLimit (F.mapCone s) ≃
... | `F.mapCone s` being limiting is the same as the induced binary fan being limiting. | true |
MonomialOrder.degLex_single_lt_iff | Mathlib.Data.Finsupp.MonomialOrder.DegLex | ∀ {σ : Type u_2} [inst : LinearOrder σ] [inst_1 : WellFoundedGT σ] {a b : σ},
((MonomialOrder.degLex.toSyn fun₀ | a => 1) < MonomialOrder.degLex.toSyn fun₀ | b => 1) ↔ b < a | null | true |
RBTree.RBNode.isOrdered.match_1 | Batteries.Recycling.RBTree.Basic | {α : Type u_1} →
(motive : Option α → Option α → Sort u_2) →
(l r : Option α) → ((l r : α) → motive (some l) (some r)) → ((x x_1 : Option α) → motive x x_1) → motive l r | null | false |
Lean.Meta.Grind.Arith.Linear.RingEqCnstr.mk.sizeOf_spec | Lean.Meta.Tactic.Grind.Arith.Linear.Types | ∀ (p : Lean.Grind.CommRing.Poly) (h : Lean.Meta.Grind.Arith.Linear.RingEqCnstrProof),
sizeOf { p := p, h := h } = 1 + sizeOf p + sizeOf h | null | true |
MeasureTheory.Lp.boundedContinuousFunction._proof_1 | Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions | ∀ (E : Type u_1) [inst : NormedAddCommGroup E], IsTopologicalAddGroup E | null | false |
_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.starRing._proof_1 | Mathlib.Algebra.Star.RingQuot | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)
(hr : ∀ (a b : R), r a b → r (star a) (star b)) (x : RingQuot r), RingQuot.star'✝ r hr (RingQuot.star'✝ r hr x) = x | null | false |
WithBot.unbotD_eq_iff | Mathlib.Order.WithBot | ∀ {α : Type u_1} {d y : α} {x : WithBot α}, WithBot.unbotD d x = y ↔ x = ↑y ∨ x = ⊥ ∧ y = d | null | true |
TopCat.conePtOfConeForget | Mathlib.Topology.Category.TopCat.Limits.Basic | {J : Type v} →
[inst : CategoryTheory.Category.{w, v} J] →
{F : CategoryTheory.Functor J TopCat} → CategoryTheory.Limits.Cone (F.comp (CategoryTheory.forget TopCat)) → Type u | Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)`
of the underlying functor to types, this is the type `c.pt`
with the infimum of the induced topologies by the maps `c.π.app j`. | true |
LinearMap.BilinForm.dualSubmodule._proof_3 | Mathlib.LinearAlgebra.BilinearForm.DualLattice | ∀ {R : Type u_2} {S : Type u_1} {M : Type u_3} [inst : CommRing R] [inst_1 : Field S] [inst_2 : AddCommGroup M]
[inst_3 : Algebra R S] [inst_4 : Module R M] [inst_5 : Module S M] [IsScalarTower R S M] (B : LinearMap.BilinForm S M)
(N : Submodule R M) (r : R), ∀ a ∈ {x | ∀ y ∈ N, (B x) y ∈ 1}, ∀ y ∈ N, (B (r • a)) y... | null | false |
_private.Mathlib.Combinatorics.Matroid.Sum.0.Matroid.sigma._simp_8 | Mathlib.Combinatorics.Matroid.Sum | ∀ {α : Type u_1} {β : α → Type u_4} {p : (a : α) × β a → Prop},
(∀ (x : (a : α) × β a), p x) = ∀ (a : α) (b : β a), p ⟨a, b⟩ | null | false |
RootPairing.EmbeddedG2.twoShortAddLongRoot_eq | Mathlib.LinearAlgebra.RootSystem.Finite.G2 | ∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
[inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (P : RootPairing ι R M N)
[inst_5 : P.EmbeddedG2] [Finite ι] [CharZero R] [IsDomain R],
RootPairing.EmbeddedG2.twoShortAddLongRoot P =
... | null | true |
NonUnitalNormedRing.induced._proof_11 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {F : Type u_3} (R : Type u_1) (S : Type u_2) [inst : FunLike F R S] [inst_1 : NonUnitalNormedRing S] (f : F),
((Filter.comap (fun p => (f p.1, f p.2)) (uniformity S)).lift' fun s => SetRel.comp s s) ≤
Filter.comap (fun p => (f p.1, f p.2)) (uniformity S) | null | false |
_private.Mathlib.Topology.UniformSpace.Basic.0.Filter.HasBasis.uniformSpace_eq_bot._simp_1_1 | Mathlib.Topology.UniformSpace.Basic | ∀ {α : Type ua} {u : UniformSpace α}, (u = ⊥) = (SetRel.id ∈ uniformity α) | null | false |
CategoryTheory.ShortComplex.RightHomologyData.ofHasCokernelOfHasKernel._proof_1 | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : CategoryTheory.Limits.HasCokernel S.f],
CategoryTheory.CategoryStruct.comp S.f (CategoryTheory.Limits.cokernel.π S.f) = 0 | null | false |
Lean.Parser.Term.doAssert.formatter | Lean.Parser.Do | Lean.PrettyPrinter.Formatter | null | true |
CategoryTheory.PreGaloisCategory.natTrans_ext_of_isGalois | Mathlib.CategoryTheory.Galois.Decomposition | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{u₂, u₁} C] [inst_1 : CategoryTheory.GaloisCategory C]
(F : CategoryTheory.Functor C FintypeCat) [CategoryTheory.PreGaloisCategory.FiberFunctor F]
{G : CategoryTheory.Functor C FintypeCat} {t s : F ⟶ G},
(∀ (X : C) [CategoryTheory.PreGaloisCategory.IsGalois X], t.ap... | To check equality of natural transformations `F ⟶ G`, it suffices to check it on
Galois objects. | true |
CategoryTheory.GrothendieckTopology.arrow_trans | Mathlib.CategoryTheory.Sites.Grothendieck | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (J : CategoryTheory.GrothendieckTopology C)
(f : Y ⟶ X) (S R : CategoryTheory.Sieve X),
J.Covers S f → (∀ {Z : C} (g : Z ⟶ X), S.arrows g → J.Covers R g) → J.Covers R f | The transitivity axiom in 'arrow' form: If `S` covers `f` and every arrow in `S` is covered by
`R`, then `R` covers `f`.
| true |
Std.Iter.all_eq_match_step | Init.Data.Iterators.Lemmas.Consumers.Loop | ∀ {α β : Type w} [inst : Std.Iterator α Id β] [Std.Iterators.Finite α Id] [inst_2 : Std.IteratorLoop α Id Id]
[Std.LawfulIteratorLoop α Id Id] {it : Std.Iter β} {p : β → Bool},
Std.Iter.all p it =
match ↑it.step with
| Std.IterStep.yield it' x => if p x = true then Std.Iter.all p it' else false
| Std.It... | null | true |
BitVec.toNat_intMin_of_pos | Init.Data.BitVec.Lemmas | ∀ {w : ℕ}, 0 < w → (BitVec.intMin w).toNat = 2 ^ (w - 1) | null | true |
_private.Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm.0.PiTensorProduct.mapL_add_smul_aux._proof_1_1 | Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | ∀ {ι : Type u_1} (i w : ι), w = i → i = w | null | false |
PMF.apply_ne_top | Mathlib.Probability.ProbabilityMassFunction.Basic | ∀ {α : Type u_1} (p : PMF α) (a : α), p a ≠ ⊤ | null | true |
QuotientGroup.quotientMapSubgroupOfOfLe._proof_1 | Mathlib.GroupTheory.QuotientGroup.Basic | ∀ {G : Type u_1} [inst : Group G] {A' A B' : Subgroup G},
A' ≤ B' → Subgroup.comap A.subtype A' ≤ Subgroup.comap A.subtype B' | null | false |
Algebra.TensorProduct.opAlgEquiv._proof_15 | Mathlib.LinearAlgebra.TensorProduct.Opposite | ∀ (R : Type u_1) (S : Type u_2) (A : Type u_3) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]
[inst_3 : Algebra R S] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [IsScalarTower R S A], SMulCommClass R S Aᵐᵒᵖ | null | false |
Std.TreeSet.Raw.min!_le_of_mem | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp] [inst : Inhabited α],
t.WF → ∀ {k : α}, k ∈ t → (cmp t.min! k).isLE = true | null | true |
SeminormedCommGroup.recOn | Mathlib.Analysis.Normed.Group.Defs | {E : Type u_4} →
{motive : SeminormedCommGroup E → Sort u} →
(t : SeminormedCommGroup E) →
([toNorm : Norm E] →
[toCommGroup : CommGroup E] →
[toPseudoMetricSpace : PseudoMetricSpace E] →
(dist_eq : ∀ (x y : E), dist x y = ‖x⁻¹ * y‖) →
motive
... | null | false |
_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun.0.AbsolutelyContinuousOnInterval.dist_le_of_pairwiseDisjoint_hasSum._simp_1_8 | Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ} {a : Filter α} {c : Filter γ},
Filter.Tendsto f a (Filter.comap g c) = Filter.Tendsto (g ∘ f) a c | null | false |
Std.WP.Spec.forInLoop_invariant | Std.WP.Gadget.ForIn | ∀ {β : Type u} {m : Type u → Type v} {Pred : Type uₚ} {EPred : Type uₑ} [inst : Monad m] [Lean.Order.MonadTail m]
[inst_2 : Std.WP.Assertion Pred] [inst_3 : Std.WP.Assertion EPred] [inst_4 : Std.WP.WPMonad m Pred EPred]
{l : Lean.Loop} {init : β} {f : Unit → β → m (ForInStep β)}
[∀ (P : Pred), Lean.Order.Preserve... | null | true |
MeasurablePow.mk | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : Pow β γ],
(Measurable fun p => p.1 ^ p.2) → MeasurablePow β γ | null | true |
AddCircle.equivIccQuot_comp_mk_eq_toIcoMod | Mathlib.Topology.Instances.AddCircle.Defs | ∀ {𝕜 : Type u_1} [inst : AddCommGroup 𝕜] [inst_1 : LinearOrder 𝕜] [inst_2 : IsOrderedAddMonoid 𝕜] (p a : 𝕜)
[hp : Fact (0 < p)] [inst_3 : Archimedean 𝕜],
⇑(AddCircle.equivIccQuot p a) ∘ Quotient.mk'' = fun x => Quot.mk (AddCircle.EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩ | null | true |
Circle.exp_two_pi_mul_int | Mathlib.Analysis.SpecialFunctions.Complex.Circle | ∀ (n : ℤ), Circle.exp (2 * Real.pi * ↑n) = 1 | null | true |
List.mem_merge | Init.Data.List.Sort.Lemmas | ∀ {α : Type u_1} {le : α → α → Bool} {a : α} {xs ys : List α}, a ∈ xs.merge ys le ↔ a ∈ xs ∨ a ∈ ys | The elements of `merge le xs ys` are exactly the elements of `xs` and `ys`.
| true |
Algebra.SubmersivePresentation.basisKaehlerOfIsCompl.congr_simp | Mathlib.RingTheory.Smooth.StandardSmoothCotangent | ∀ {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : Algebra R S] [inst_3 : Finite σ] (P P_1 : Algebra.SubmersivePresentation R S ι σ) (e_P : P = P_1)
{κ : Type u_5} {f f_1 : κ → ι} (e_f : f = f_1) (hf : Function.Injective f)
(hcompl : IsCompl (Set.rang... | null | true |
_private.Init.Grind.Ordered.Int.0.Lean.Grind.instLawfulOrderLTInt._proof_1 | Init.Grind.Ordered.Int | ∀ (a b : ℤ), ¬(a < b ↔ a ≤ b ∧ ¬b ≤ a) → False | null | false |
_private.Mathlib.RingTheory.Extension.Cotangent.BaseChange.0.Algebra.Extension.tensorCotangentSpace._proof_13 | Mathlib.RingTheory.Extension.Cotangent.BaseChange | ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
(P : Algebra.Extension R S), SMulCommClass R R P.Ring | null | false |
enorm_norm | Mathlib.Analysis.Normed.Group.Real | ∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] (x : E), ‖‖x‖‖ₑ = ‖x‖ₑ | null | true |
apply_ite_left | Mathlib.Basic.Logic.Basic | ∀ {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} (f : α → β → γ) (P : Prop) [inst : Decidable P] (x y : α) (z : β),
f (if P then x else y) z = if P then f x z else f y z | null | true |
Set.inv_mem_centralizer._simp_2 | Mathlib.Algebra.Group.Center | ∀ {M : Type u_1} {S : Set M} [inst : Group M] {a : M}, a ∈ S.centralizer → (a⁻¹ ∈ S.centralizer) = True | null | false |
ENNReal.zpow_sub | Mathlib.Basic.ENNReal.Inv | ∀ {x : ENNReal}, x ≠ 0 → x ≠ ⊤ → ∀ (m n : ℤ), x ^ (m - n) = x ^ m * (x ^ n)⁻¹ | null | true |
ProfiniteAddGrp.mk.inj | Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic | ∀ {toProfinite : Profinite} {addGroup : AddGroup ↑toProfinite.toTop}
{topologicalAddGroup : IsTopologicalAddGroup ↑toProfinite.toTop} {toProfinite_1 : Profinite}
{addGroup_1 : AddGroup ↑toProfinite_1.toTop} {topologicalAddGroup_1 : IsTopologicalAddGroup ↑toProfinite_1.toTop},
{ toProfinite := toProfinite, addGrou... | null | true |
Nat.lcm_mul_right_right_eq_mul_of_lcm_eq_mul | Init.Data.Nat.Lcm | ∀ {n m k : ℕ}, n.lcm m = n * m → n.lcm (m * k) = n.lcm k * m | null | true |
_private.Init.Data.Range.Polymorphic.Lemmas.0.Std.Rxc.Iterator.toList_eq_match.match_1.eq_2 | Init.Data.Range.Polymorphic.Lemmas | ∀ {α : Type u_1} (motive : Option α → Sort u_2) (a : α) (h_1 : Unit → motive none) (h_2 : (a : α) → motive (some a)),
(match some a with
| none => h_1 ()
| some a => h_2 a) =
h_2 a | null | true |
LinearPMap.add_domain | Mathlib.LinearAlgebra.LinearPMap | ∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] {σ : R →+* S} {E : Type u_4} [inst_2 : AddCommGroup E]
[inst_3 : Module R E] {F : Type u_5} [inst_4 : AddCommGroup F] [inst_5 : Module S F] (f g : E →ₛₗ.[σ] F),
(f + g).domain = f.domain ⊓ g.domain | null | true |
_private.Init.Data.Range.Polymorphic.NatLemmas.0.Nat.toList_rcc_eq_cons_iff._proof_1_1 | Init.Data.Range.Polymorphic.NatLemmas | ∀ {m n a : ℕ}, m = a → ¬(m < n + 1 ↔ m ≤ n) → False | null | false |
MeasureTheory.Integrable.enorm_toL1 | Mathlib.MeasureTheory.Function.L1Space.AEEqFun | ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
{f : α → β} (hf : MeasureTheory.Integrable f μ), ‖MeasureTheory.Integrable.toL1 f hf‖ₑ = ∫⁻ (a : α), ‖f a‖ₑ ∂μ | null | true |
Polynomial.wfDvdMonoid | Mathlib.RingTheory.Polynomial.UniqueFactorization | ∀ {R : Type u_1} [inst : CommSemiring R] [NoZeroDivisors R] [WfDvdMonoid R], WfDvdMonoid (Polynomial R) | null | true |
Field.inv_zero | Mathlib.Algebra.Field.Defs | ∀ {K : Type u} [self : Field K], 0⁻¹ = 0 | The inverse of `0` is `0` by convention. | true |
LinearPMap.instVAdd | Mathlib.LinearAlgebra.LinearPMap | {R : Type u_1} →
{S : Type u_2} →
[inst : Ring R] →
[inst_1 : Ring S] →
{σ : R →+* S} →
{E : Type u_4} →
[inst_2 : AddCommGroup E] →
[inst_3 : Module R E] →
{F : Type u_5} → [inst_4 : AddCommGroup F] → [inst_5 : Module S F] → VAdd (E →ₛₗ[σ] F) (E →... | null | true |
_private.Mathlib.MeasureTheory.Measure.Decomposition.Hahn.0.MeasureTheory.hahn_decomposition._simp_1_5 | Mathlib.MeasureTheory.Measure.Decomposition.Hahn | ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α] [AddRightMono α] {a b c : α}, (a - c ≤ b) = (a ≤ b + c) | null | false |
UpperSemicontinuousWithinAt.sup | Mathlib.Topology.Semicontinuity.Basic | ∀ {α : Type u_4} {β : Type u_5} [inst : TopologicalSpace α] [inst_1 : LinearOrder β] {f g : α → β} {s : Set α} {a : α},
UpperSemicontinuousWithinAt f s a →
UpperSemicontinuousWithinAt g s a → UpperSemicontinuousWithinAt (fun x => max (f x) (g x)) s a | null | true |
CategoryTheory.uliftCoyonedaEquiv_symm_map | Mathlib.CategoryTheory.Yoneda | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y)
{F : CategoryTheory.Functor C (Type (max w v₁))} (t : F.obj X),
CategoryTheory.uliftCoyonedaEquiv.symm ((CategoryTheory.ConcreteCategory.hom (F.map f)) t) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.uliftCoyoneda.{w, v₁, ... | null | true |
_private.Init.Data.List.Sublist.0.List.infix_cons.match_1_1 | Init.Data.List.Sublist | ∀ {α : Type u_1} {l₁ l₂ : List α} (motive : l₁ <:+: l₂ → Prop) (x : l₁ <:+: l₂),
(∀ (l₁' l₂' : List α) (h : l₁' ++ l₁ ++ l₂' = l₂), motive ⋯) → motive x | null | false |
_private.Mathlib.Algebra.Homology.QuasiIso.0.instIsIsoHomologyMapOfQuasiIsoAt._simp_1 | Mathlib.Algebra.Homology.QuasiIso | ∀ {ι : Type u_1} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L)
(i : ι) [inst_2 : K.HasHomology i] [inst_3 : L.HasHomology i],
QuasiIsoAt f i = CategoryTheory.ShortComplex.QuasiIso ((Homolog... | null | false |
_private.Mathlib.Tactic.Linter.Lint.0.Mathlib.Linter.DupNamespaceLinter.dupNamespace._sparseCasesOn_3 | Mathlib.Tactic.Linter.Lint | {α : Type u} →
{motive : List α → Sort u_1} → (t : List α) → motive [] → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t | null | false |
Quiver.reverse_inj._simp_1 | Mathlib.Combinatorics.Quiver.Symmetric | ∀ {V : Type u_2} [inst : Quiver V] [h : Quiver.HasInvolutiveReverse V] {a b : V} (f g : a ⟶ b),
(Quiver.reverse f = Quiver.reverse g) = (f = g) | null | false |
connectedComponentsEquivZerothHomotopy._proof_1 | Mathlib.Topology.Connected.LocallyPathConnected | ∀ {X : Type u_1} [inst : TopologicalSpace X] [LocallyPathConnectedSpace X] (x1 x2 : X), x1 ≈ x2 → Joined x1 x2 | null | false |
groupHomology.cyclesMap_comp_isoCycles₁_hom_assoc | Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | ∀ {k G H : Type u} [inst : CommRing k] [inst_1 : Group G] [inst_2 : Group H] {A : Rep.{u, u, u} k G}
{B : Rep.{u, u, u} k H} (f : G →* H) (φ : A ⟶ Rep.res f B) {Z : ModuleCat k} (h : ↧↥(groupHomology.cycles₁ B) ⟶ Z),
CategoryTheory.CategoryStruct.comp (groupHomology.cyclesMap f φ 1)
(CategoryTheory.CategorySt... | null | true |
_private.Mathlib.MeasureTheory.Function.SimpleFunc.0.MeasureTheory.SimpleFunc.map_preimage._simp_1_1 | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {α : Type u} {β : Type v} {f : α → β} {s : Set β} {a : α}, (f a ∈ s) = (a ∈ f ⁻¹' s) | null | false |
Aesop.StatsFileRecord.position | Aesop.Stats.File | Aesop.StatsFileRecord → Option Lean.Position | null | true |
_private.Lean.Elab.ConfigEval.DeriveEvalConfigItem.0.Lean.Elab.ConfigEval.hasToOmit.match_1 | Lean.Elab.ConfigEval.DeriveEvalConfigItem | (motive : Lean.Ident × Lean.Name → Sort u_1) →
(x : Lean.Ident × Lean.Name) → ((fst : Lean.Ident) → (name : Lean.Name) → motive (fst, name)) → motive x | null | false |
_private.Mathlib.Algebra.Ring.Parity.0.Nat.odd_add._proof_1_1 | Mathlib.Algebra.Ring.Parity | ∀ {m n : ℕ}, Odd (m + n) ↔ (Odd m ↔ Even n) | null | false |
_private.Mathlib.Analysis.Complex.Hadamard.0.Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁'._simp_1_7 | Mathlib.Analysis.Complex.Hadamard | ∀ {α : Sort u_1} {p : α → Prop} {q : (∃ x, p x) → Prop}, (∀ (h : ∃ x, p x), q h) = ∀ (x : α) (h : p x), q ⋯ | null | false |
CategoryTheory.Pseudofunctor.whiskerLeft_mapId_inv | Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
(F : CategoryTheory.Pseudofunctor B C) {a b : B} (f : a ⟶ b),
CategoryTheory.Bicategory.whiskerLeft (F.map f) (F.mapId b).inv =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor (F.m... | null | true |
_private.Mathlib.GroupTheory.Perm.Sign.0.Equiv.Perm.signBijAux_injOn._simp_1_2 | Mathlib.GroupTheory.Perm.Sign | ∀ {α : Type u_1} {β : α → Type u_4} {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂}, (⟨a₁, b₁⟩ = ⟨a₂, b₂⟩) = (a₁ = a₂ ∧ b₁ ≍ b₂) | null | false |
Nat.shiftLeft_shiftRight | Init.Data.Nat.Lemmas | ∀ (x n : ℕ), x <<< n >>> n = x | null | true |
LinearOrder.mkOfGroupCone._proof_3 | Mathlib.Algebra.Order.Group.Cone | ∀ {S : Type u_2} {G : Type u_1} [inst : CommGroup G] [inst_1 : SetLike S G] (C : S) [inst_2 : GroupConeClass S G]
[inst_3 : DecidablePred fun x => x ∈ C] (a b : G), (if a ≤ b then a else b) = if a ≤ b then a else b | null | false |
Real.arccos_le_pi_div_two | Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | ∀ {x : ℝ}, Real.arccos x ≤ Real.pi / 2 ↔ 0 ≤ x | null | true |
List.Nat.antidiagonalTuple.match_1 | Mathlib.Data.Fin.Tuple.NatAntidiagonal | (motive : ℕ → ℕ → Sort u_1) →
(x x_1 : ℕ) → (Unit → motive 0 0) → ((n : ℕ) → motive 0 n.succ) → ((k n : ℕ) → motive k.succ n) → motive x x_1 | null | false |
_private.Mathlib.CategoryTheory.MorphismProperty.Basic.0.CategoryTheory.MorphismProperty.map_inverseImage_le.match_1_1 | Mathlib.CategoryTheory.MorphismProperty.Basic | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (P : CategoryTheory.MorphismProperty D)
(F : CategoryTheory.Functor C D) (x x_1 : D) (x_2 : x ⟶ x_1) (motive : (P.inverseImage F).map F x_2 → Prop)
(x_3 : (P.inverseImage F).map F x_2),
... | null | false |
Differential.differentialFiniteDimensional._proof_2 | Mathlib.FieldTheory.Differential.Basic | ∀ (F : Type u_2) [inst : Field F] [inst_1 : CharZero F] (K : Type u_1) [inst_2 : Field K] [inst_3 : Algebra F K]
[inst_4 : FiniteDimensional F K], F⟮⋯.choose⟯ = ⊤ | null | false |
Finset.prod_Ioo_mul_eq_prod_Ico | Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite | ∀ {α : Type u_1} {M : Type u_2} [inst : CommMonoid M] {f : α → M} {a b : α} [inst_1 : PartialOrder α]
[inst_2 : LocallyFiniteOrder α], a < b → (∏ x ∈ Finset.Ioo a b, f x) * f a = ∏ x ∈ Finset.Ico a b, f x | null | true |
Derivation.compAlgebraMap | Mathlib.RingTheory.Derivation.Basic | {R : Type u_1} →
(A : Type u_2) →
{B : Type u_3} →
{M : Type u_4} →
[inst : CommSemiring R] →
[inst_1 : CommSemiring A] →
[inst_2 : CommSemiring B] →
[inst_3 : AddCommMonoid M] →
[inst_4 : Algebra R A] →
[inst_5 : Algebra R B] →
... | For a tower `R → A → B` and an `R`-derivation `B → M`, we may compose with `A → B` to obtain an
`R`-derivation `A → M`. | true |
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