name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.Bicategory.Adjunction.homEquiv₁_apply | Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] {b c d : B} {l : b ⟶ c} {r : c ⟶ b}
(adj : CategoryTheory.Bicategory.Adjunction l r) {g : b ⟶ d} {h : c ⟶ d}
(γ : g ⟶ CategoryTheory.CategoryStruct.comp l h),
adj.homEquiv₁ γ =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft r γ)
... | null | true |
Std.Time.Year.instDecidableLeOffset | Std.Time.Date.Unit.Year | {x y : Std.Time.Year.Offset} → Decidable (x ≤ y) | null | true |
_private.Init.Data.Iterators.Lemmas.Consumers.Loop.0.Std.Iter.toArray_eq_match_step.match_1.eq_1 | Init.Data.Iterators.Lemmas.Consumers.Loop | ∀ {α β : Type u_1} (motive : Std.IterStep (Std.Iter β) β → Sort u_2) (it' : Std.Iter β) (out : β)
(h_1 : (it' : Std.Iter β) → (out : β) → motive (Std.IterStep.yield it' out))
(h_2 : (it' : Std.Iter β) → motive (Std.IterStep.skip it')) (h_3 : Unit → motive Std.IterStep.done),
(match Std.IterStep.yield it' out with... | null | true |
Lean.ExternEntry.ctorIdx | Lean.Compiler.ExternAttr | Lean.ExternEntry → ℕ | null | false |
Std.IterM.toArray.go.match_1 | Init.Data.Iterators.Consumers.Monadic.Collect | {α β : Type u_1} →
{m : Type u_1 → Type u_2} →
[inst : Std.Iterator α m β] →
(it : Std.IterM m β) →
(motive : it.Step → Sort u_3) →
(x : it.Step) →
((it' : Std.IterM m β) →
(out : β) →
(h : it.IsPlausibleStep (Std.IterStep.yield it' out)) → mot... | null | false |
AbsoluteValue.IsEquiv.rfl | Mathlib.Analysis.AbsoluteValue.Equivalence | ∀ {R : Type u_1} [inst : Semiring R] {S : Type u_2} [inst_1 : Semiring S] [inst_2 : PartialOrder S]
{v : AbsoluteValue R S}, v.IsEquiv v | null | true |
Filter.Germ.instInvolutiveInv._proof_1 | Mathlib.Order.Filter.Germ.Basic | ∀ {α : Type u_1} {l : Filter α} {G : Type u_2} [inst : InvolutiveInv G] (q : Quotient (l.germSetoid G)), q⁻¹⁻¹ = q | null | false |
CategoryTheory.MorphismProperty.bijective_respectsIso | Mathlib.CategoryTheory.MorphismProperty.Concrete | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type u_2}
[inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC],
(CategoryTheory.MorphismProperty.bijective C).RespectsIso | null | true |
_private.Mathlib.Algebra.Algebra.Subalgebra.Lattice.0.Algebra.adjoin_insert_adjoin._simp_1_2 | Mathlib.Algebra.Algebra.Subalgebra.Lattice | ∀ {α : Type u_1} {s t : Set α} {a : α}, (insert a s ⊆ t) = (a ∈ t ∧ s ⊆ t) | null | false |
Algebra.Generators.Hom.casesOn | Mathlib.RingTheory.Extension.Generators | {R : Type u} →
{S : Type v} →
{ι : Type w} →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
{P : Algebra.Generators R S ι} →
{R' : Type u_1} →
{S' : Type u_2} →
{ι' : Type u_3} →
[inst_3... | null | false |
Std.Sat.AIG.Decl.atom.noConfusion | Std.Sat.AIG.Basic | {α : Type} →
{P : Sort u} → {idx idx' : α} → Std.Sat.AIG.Decl.atom idx = Std.Sat.AIG.Decl.atom idx' → (idx ≍ idx' → P) → P | null | false |
OrderDual.instAddCommGroupWithOne._proof_1 | Mathlib.Algebra.Order.Ring.Synonym | ∀ {R : Type u_1} [inst : AddCommGroupWithOne R], autoParam (↑0 = 0) AddMonoidWithOne.natCast_zero._autoParam | null | false |
Finset.sum_partition | Mathlib.Algebra.BigOperators.Group.Finset.Basic | ∀ {ι : Type u_1} {M : Type u_4} {s : Finset ι} [inst : AddCommMonoid M] {f : ι → M} (R : Setoid ι)
[inst_1 : DecidableRel ⇑R], ∑ x ∈ s, f x = ∑ xbar ∈ Finset.image (Quotient.mk R) s, ∑ y ∈ s with ⟦y⟧ = xbar, f y | A sum can be partitioned into a sum of sums, each equivalent under a setoid. | true |
_private.Std.Data.DHashMap.Internal.AssocList.Lemmas.0.Std.DHashMap.Internal.AssocList.getKey!.eq_1 | Std.Data.DHashMap.Internal.AssocList.Lemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Inhabited α] (a : α),
Std.DHashMap.Internal.AssocList.getKey! a Std.DHashMap.Internal.AssocList.nil =
panicWithPosWithDecl "Std.Data.DHashMap.Internal.AssocList.Basic" "Std.DHashMap.Internal.AssocList.getKey!" 168 11
"key is not present in hash table" | null | true |
_private.Mathlib.SetTheory.Ordinal.Basic.0.Ordinal.type_eq_zero_iff_isEmpty.match_1_1 | Mathlib.SetTheory.Ordinal.Basic | ∀ {α : Type u_1} {r : α → α → Prop} (motive : Nonempty (r ≃r emptyRelation) → Prop) (x : Nonempty (r ≃r emptyRelation)),
(∀ (s : r ≃r emptyRelation), motive ⋯) → motive x | null | false |
NumberField.mixedEmbedding.norm_eq_zero_iff | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] {x : NumberField.mixedEmbedding.mixedSpace K},
NumberField.mixedEmbedding.norm x = 0 ↔ ∃ w, (NumberField.mixedEmbedding.normAtPlace w) x = 0 | null | true |
AlgebraicIndependent.extendScalars | Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure | ∀ {ι : Type u_1} {R : Type u_2} {A : Type u_4} {x : ι → A} (S : Type u_5) [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : CommRing A] [inst_3 : Algebra R S] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [IsScalarTower R S A]
[NoZeroDivisors S], AlgebraicIndependent R x → ∀ [alg : Algebra.IsAlgebraic R S], Algeb... | null | true |
Lean.Lsp.instFileSourcePlainTermGoalParams | Lean.Server.FileSource | Lean.Lsp.FileSource Lean.Lsp.PlainTermGoalParams | null | true |
AddCommGroup.neg_modEq_neg._simp_1 | Mathlib.Algebra.Group.ModEq | ∀ {G : Type u_1} [inst : AddCommGroup G] {p a b : G}, (-a ≡ -b [PMOD p]) = (a ≡ b [PMOD p]) | null | false |
ValuationSubring.algebra._proof_4 | Mathlib.RingTheory.Valuation.AlgebraInstances | ∀ {K : Type u_2} [inst : Field K] (v : Valuation K (WithZero (Multiplicative ℤ))) (L : Type u_3) [inst_1 : Field L]
[inst_2 : Algebra K L] (E : Type u_1) [inst_3 : Field E] [inst_4 : Algebra K E] [inst_5 : Algebra L E]
[inst_6 : IsScalarTower K L E] (x y : ↥(integralClosure (↥v.valuationSubring) L)),
↑⟨(algebraMa... | null | false |
Lean.Meta.Simp.Simprocs.post | Lean.Meta.Tactic.Simp.Types | Lean.Meta.Simprocs → Lean.Meta.Simp.SimprocTree | null | true |
DoubleQuot.coe_quotQuotEquivQuotOfLEₐ | Mathlib.RingTheory.Ideal.Quotient.Operations | ∀ (R : Type u) {A : Type u_1} [inst : CommSemiring R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {I J : Ideal A}
(h : I ≤ J), ⇑(DoubleQuot.quotQuotEquivQuotOfLEₐ R h) = ⇑(DoubleQuot.quotQuotEquivQuotOfLE h) | null | true |
Lean.Elab.Term.MutualClosure.LetRecClosure._sizeOf_1 | Lean.Elab.MutualDef | Lean.Elab.Term.MutualClosure.LetRecClosure → ℕ | null | false |
_private.Lean.Meta.Tactic.Grind.Arith.Linear.Util.0.Lean.Grind.Linarith.Poly.updateOccs.go | Lean.Meta.Tactic.Grind.Arith.Linear.Util | Lean.Grind.Linarith.Var → Lean.Grind.Linarith.Poly → Lean.Meta.Grind.Arith.Linear.LinearM Unit | null | true |
ImportGraph.Lake.Glob.Modules.mk.injEq | ImportGraph.Util.Modules | ∀ (glob : Lake.Glob) (dir : System.FilePath) (glob_1 : Lake.Glob) (dir_1 : System.FilePath),
({ glob := glob, dir := dir } = { glob := glob_1, dir := dir_1 }) = (glob = glob_1 ∧ dir = dir_1) | null | true |
CategoryTheory.HasLiftingProperty.transfiniteComposition.hasLiftingPropertyFixedBot_ι_app_bot | Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} [inst_1 : LinearOrder J] [inst_2 : OrderBot J]
{F : CategoryTheory.Functor J C} {c : CategoryTheory.Limits.Cocone F} (hc : CategoryTheory.Limits.IsColimit c)
{X Y : C} {p : X ⟶ Y} {g : c.pt ⟶ Y} [F.IsWellOrderContinuous] [inst_4 : SuccOrder J] [W... | null | true |
WithTopology | Mathlib.Topology.Defs.Basic | (X : Type u_1) → TopologicalSpace X → Type u_1 | A one-field structure wrapper for `X` with the topology coinduced from `t`. | true |
_private.Mathlib.GroupTheory.Perm.Cycle.Concrete.0.Cycle.support_formPerm._simp_1_1 | Mathlib.GroupTheory.Perm.Cycle.Concrete | ∀ {a b : ℕ}, (a.succ ≤ b.succ) = (a ≤ b) | null | false |
Finset.pow_right_strictMono | Mathlib.Geometry.Group.Growth.LinearLowerBound | ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G},
1 ∈ X → (↑(Subgroup.closure ↑X)).Infinite → StrictMono fun n => X ^ n | null | true |
_private.Mathlib.Tactic.Determinant.Bird.Cert.0.Mathlib.Tactic.Determinant.isZeroVal.match_1 | Mathlib.Tactic.Determinant.Bird.Cert | {u : Lean.Level} →
{α : Q(Type u)} →
{rα : Q(CommRing «$α»)} →
(motive : (e : Q(«$α»)) → Mathlib.Tactic.Determinant.CertVal rα e → Sort u_1) →
(e : Q(«$α»)) →
(val : Mathlib.Tactic.Determinant.CertVal rα e) →
(Unit → motive q(0) Mathlib.Tactic.Ring.Common.ExSum.zero) →
... | null | false |
DistLat.instCategory._proof_2 | Mathlib.Order.Category.DistLat | ∀ {X Y : DistLat} (f : X.Hom Y), { hom' := { hom' := LatticeHom.id ↑Y }.hom'.comp f.hom' } = f | null | false |
Mathlib.Tactic.BicategoryLike.WhiskerLeft.tgtM._unsafe_rec | Mathlib.Tactic.CategoryTheory.Coherence.Normalize | {m : Type → Type} →
[Monad m] →
[Mathlib.Tactic.BicategoryLike.MonadMor₁ m] →
Mathlib.Tactic.BicategoryLike.WhiskerLeft → m Mathlib.Tactic.BicategoryLike.Mor₁ | null | false |
ImportGraph.Shake.Bitset.foldr | ImportGraph.Shake.Basic | {α : Type} → ImportGraph.Shake.Bitset → α → (α → ℕ → α) → α | Fold over the set bit indices in a `Bitset` from high to low. This is more efficient for large
bitsets than `Bitset.foldl`. | true |
SimplexCategory.Truncated.incl._proof_1 | Mathlib.AlgebraicTopology.SimplexCategory.Defs | ∀ (n m : ℕ), n ≤ m → ∀ (x : SimplexCategory), x.len ≤ n → x.len ≤ m | null | false |
zpow_right_inj₀ | Mathlib.Algebra.Order.GroupWithZero.Basic | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : LinearOrder G₀] {a : G₀} {m n : ℤ} [PosMulStrictMono G₀]
[ZeroLEOneClass G₀], 0 < a → a ≠ 1 → (a ^ m = a ^ n ↔ m = n) | null | true |
_private.Mathlib.NumberTheory.RatFunc.Ostrowski.0.RatFunc.uniformizingPolynomial_isUniformizer.match_1_4 | Mathlib.NumberTheory.RatFunc.Ostrowski | ∀ {K : Type u_2} {Γ : Type u_1} [inst : Field K] [inst_1 : LinearOrderedCommGroupWithZero Γ]
{v : Valuation (RatFunc K) Γ} [inst_2 : v.IsNontrivial] [inst_3 : Valuation.IsTrivialOn K v] (hle : v RatFunc.X ≤ 1)
(h0 : v ↑(RatFunc.uniformizingPolynomial hle) ≠ 0) (γ : Γˣ) (motive : (∃ k, ⋯.unit ^ k = γ) → Prop)
(x :... | null | false |
Aesop.BuiltinRules.destructProductsCore | Aesop.BuiltinRules.DestructProducts | Lean.MVarId → Lean.Meta.TransparencyMode → Aesop.BaseM (Lean.MVarId × Array Aesop.Script.LazyStep) | null | true |
Fin.encodeFun_decodeFun | Batteries.Data.Fin.Coding | ∀ {n m : ℕ} (x : Fin (n ^ m)), Fin.encodeFun (Fin.decodeFun x) = x | null | true |
Lean.Meta.Sym.Internal.mkFVarS | Lean.Meta.Sym.AlphaShareBuilder | {m : Type → Type} → [Lean.Meta.Sym.Internal.MonadShareCommon m] → Lean.FVarId → m Lean.Expr | null | true |
Set.decidableInsert | Mathlib.Data.Set.Basic | {α : Type u} → (s : Set α) → (a b : α) → [Decidable (a = b)] → [Decidable (a ∈ s)] → Decidable (a ∈ insert b s) | null | true |
_private.Mathlib.RingTheory.LocalRing.ResidueField.Basic.0.IsLocalRing.ResidueField.instIsScalarTower_1._simp_1 | Mathlib.RingTheory.LocalRing.ResidueField.Basic | ∀ (R : Type u) (S : Type v) (A : Type w) [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Semiring A]
[inst_3 : Algebra R S] [inst_4 : Algebra S A] [inst_5 : Algebra R A] [IsScalarTower R S A] (x : R),
(algebraMap S A) ((algebraMap R S) x) = (algebraMap R A) x | null | false |
_private.Init.Data.BitVec.Bitblast.0.BitVec.fastUmulOverflow._proof_1_20 | Init.Data.BitVec.Bitblast | ∀ (w : ℕ) (x y : BitVec (w + 1 + 1)),
¬x.clz.toNat + y.clz.toNat ≤ w →
¬w + 1 < y.clz.toNat → ¬w + 1 + 1 - x.clz.toNat + (w + 1 + 1 - y.clz.toNat) ≤ w + 1 + 1 + 1 → False | null | false |
_private.Mathlib.MeasureTheory.Integral.Average.0.MeasureTheory.setLAverage_const._simp_1_1 | Mathlib.MeasureTheory.Integral.Average | ∀ {α : Type u_1} {m : MeasurableSpace α}, MeasurableSet Set.univ = True | null | false |
Ordinal.ofNat_lt_card._simp_1 | Mathlib.SetTheory.Ordinal.Basic | ∀ {o : Ordinal.{u_1}} {n : ℕ} [inst : n.AtLeastTwo], (OfNat.ofNat n < o.card) = (OfNat.ofNat n < o) | null | false |
Matroid.IsBasis'.contract_isBasis'_sdiff_sdiff_of_subset | Mathlib.Combinatorics.Matroid.Minor.Contract | ∀ {α : Type u_1} {M : Matroid α} {I J X : Set α}, M.IsBasis' I X → J ⊆ I → (M.contract J).IsBasis' (I \ J) (X \ J) | null | true |
AddSubgroup.map_iInf | Mathlib.Algebra.Group.Subgroup.Map | ∀ {G : Type u_1} [inst : AddGroup G] {N : Type u_4} [inst_1 : AddGroup N] {ι : Sort u_6} [Nonempty ι] (f : G →+ N),
Function.Injective ⇑f → ∀ (s : ι → AddSubgroup G), AddSubgroup.map f (iInf s) = ⨅ i, AddSubgroup.map f (s i) | null | true |
MeasureTheory.Integrable.enorm | Mathlib.MeasureTheory.Function.L1Space.Integrable | ∀ {α : Type u_1} {ε : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace ε]
[inst_1 : ContinuousENorm ε] {f : α → ε}, MeasureTheory.Integrable f μ → MeasureTheory.Integrable (fun x => ‖f x‖ₑ) μ | null | true |
MvPolynomial.ACounit_surjective | Mathlib.Algebra.MvPolynomial.Counit | ∀ (A : Type u_1) (B : Type u_2) [inst : CommSemiring A] [inst_1 : CommSemiring B] [inst_2 : Algebra A B],
Function.Surjective ⇑(MvPolynomial.ACounit A B) | null | true |
Pi.evalNonUnitalStarAlgHom._proof_5 | Mathlib.Algebra.Star.StarAlgHom | ∀ {ι : Type u_2} (A : ι → Type u_1) (j : ι) [inst : (i : ι) → NonUnitalNonAssocSemiring (A i)]
[inst_1 : (i : ι) → Star (A i)] (x : (i : ι) → A i),
(Pi.evalMulHom A j).toFun (star x) = (Pi.evalMulHom A j).toFun (star x) | null | false |
_private.Lean.Server.Rpc.RequestHandling.0.Lean.Server.handleRpcCall.match_3 | Lean.Server.Rpc.RequestHandling | (motive : Except String Lean.Server.RpcProcedure → Sort u_1) →
(x : Except String Lean.Server.RpcProcedure) →
((x : Lean.Server.RpcProcedure) → motive (Except.ok x)) → ((e : String) → motive (Except.error e)) → motive x | null | false |
Turing.PartrecToTM2.contStack.eq_def | Mathlib.Computability.TuringMachine.ToPartrec | ∀ (x : Turing.ToPartrec.Cont),
Turing.PartrecToTM2.contStack x =
match x with
| Turing.ToPartrec.Cont.halt => []
| Turing.ToPartrec.Cont.cons₁ a ns k => ns :: Turing.PartrecToTM2.contStack k
| Turing.ToPartrec.Cont.cons₂ ns k => ns :: Turing.PartrecToTM2.contStack k
| Turing.ToPartrec.Cont.comp a ... | null | true |
AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app | Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | ∀ {X : AlgebraicGeometry.Scheme} (I : X.IdealSheafData) (U : ↑X.affineOpens),
AlgebraicGeometry.Scheme.Hom.app I.subschemeι ↑U =
CategoryTheory.CategoryStruct.comp (CommRingCat.ofHom (Ideal.Quotient.mk (I.ideal U))) (I.subschemeObjIso U).inv | null | true |
cfc_imaginaryPart._auto_3 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | Lean.Syntax | null | false |
Algebra.Extension.toBaseChange | 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} →
(T : Type u_1) → [inst_3 : CommRing T] → [inst_4 : Algebra R T] → P.Hom P.baseChange | The canonical hom from `P` to its base change `P.baseChange`. | true |
IsDiffImmersionAt.mfderiv_injective | Mathlib.Geometry.Manifold.ImmersionDiff | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {E' : Type u} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup E'] [inst_4 : NormedSpace 𝕜 E'] {H : Type u_6}
[inst_5 : TopologicalSpace H] {H' : Type u_7} [inst_6 : TopologicalSpace H'] {I : ModelWithCorne... | null | true |
FourierInvPair.rec | Mathlib.Analysis.Fourier.Notation | {E : Type u_5} →
{F : Type u_6} →
[inst : FourierTransform F E] →
[inst_1 : FourierTransformInv E F] →
{motive : FourierInvPair E F → Sort u} →
((fourier_fourierInv_eq : ∀ (f : E), FourierTransform.fourier (FourierTransformInv.fourierInv f) = f) →
motive ⋯) →
(t :... | null | false |
isGLB_inv | Mathlib.Algebra.Order.Group.Pointwise.Bounds | ∀ {G : Type u_2} [inst : Group G] [inst_1 : Preorder G] [MulLeftMono G] [MulRightMono G] {s : Set G} {a : G},
IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ | null | true |
HomologicalComplex.dTo | Mathlib.Algebra.Homology.HomologicalComplex | {ι : Type u_1} →
{V : Type u} →
[inst : CategoryTheory.Category.{v, u} V] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] →
{c : ComplexShape ι} → (C : HomologicalComplex V c) → (j : ι) → C.xPrev j ⟶ C.X j | The differential mapping into `C.X j`, or zero if there isn't one.
| true |
IsLprojection.Subtype.zero._proof_3 | Mathlib.Analysis.Normed.Module.MStructure | ∀ {X : Type u_1} [inst : NormedAddCommGroup X] {M : Type u_2} [inst_1 : Ring M] [inst_2 : Module M X], IsLprojection X 0 | null | false |
_private.Mathlib.SetTheory.Ordinal.Notation.0.ONote.opowAux.match_1.eq_1 | Mathlib.SetTheory.Ordinal.Notation | ∀ (motive : ℕ → ℕ → Sort u_1) (x : ℕ) (h_1 : (x : ℕ) → motive x 0) (h_2 : (m : ℕ) → motive 0 m.succ)
(h_3 : (k m : ℕ) → motive k.succ m),
(match x, 0 with
| x, 0 => h_1 x
| 0, m.succ => h_2 m
| k.succ, m => h_3 k m) =
h_1 x | null | true |
Bracket.casesOn | Mathlib.Algebra.Notation.Bracket | {L : Type u_1} →
{M : Type u_2} →
{motive : Bracket L M → Sort u} →
(t : Bracket L M) → ((bracket : L → M → M) → motive { bracket := bracket }) → motive t | null | false |
MeasureTheory.hasFiniteIntegral_const_iff_isFiniteMeasure | Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
{c : β}, c ≠ 0 → (MeasureTheory.HasFiniteIntegral (fun x => c) μ ↔ MeasureTheory.IsFiniteMeasure μ) | null | true |
TopologicalSpace.Compacts.isClosedEmbedding_singleton | Mathlib.Topology.Sets.VietorisTopology | ∀ {α : Type u_1} [inst : TopologicalSpace α] [T2Space α], Topology.IsClosedEmbedding fun x => {x} | null | true |
ModuleCat.ctorIdx | Mathlib.Algebra.Category.ModuleCat.Basic | {R : Type u} → {inst : Ring R} → ModuleCat R → ℕ | null | false |
Asymptotics.isLittleO_top._simp_1 | Mathlib.Analysis.Asymptotics.Lemmas | ∀ {α : Type u_1} {F' : Type u_7} {E'' : Type u_9} [inst : SeminormedAddCommGroup F'] [inst_1 : NormedAddCommGroup E'']
{g' : α → F'} {f'' : α → E''}, f'' =o[⊤] g' = ∀ (x : α), f'' x = 0 | null | false |
SeparatelyContinuousAdd.mk | Mathlib.Topology.Algebra.Monoid.Defs | ∀ {M : Type u_1} [inst : TopologicalSpace M] [inst_1 : Add M],
(∀ {a : M}, Continuous fun x => a + x) → (∀ {a : M}, Continuous fun x => x + a) → SeparatelyContinuousAdd M | null | true |
localizedModuleIsLocalizedModule | Mathlib.Algebra.Module.LocalizedModule.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M : Type u_2} [inst_1 : AddCommMonoid M]
[inst_2 : Module R M], IsLocalizedModule S (LocalizedModule.mkLinearMap S M) | null | true |
Commute.span_left | Mathlib.LinearAlgebra.Span.Defs | ∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonUnitalNonAssocSemiring A] [inst_2 : Module R A]
[IsScalarTower R A A] [SMulCommClass R A A] {s : Set A} {x : A},
(∀ y ∈ s, Commute y x) → ∀ y ∈ Submodule.span R s, Commute y x | In a non-unital algebra, if every element of a set `s` commutes with `x`, then every element of
`Submodule.span R s` commutes with `x`. | true |
Lean.Meta.ACLt.ReduceMode.noConfusionType | Lean.Meta.ACLt | Sort v✝ → Lean.Meta.ACLt.ReduceMode → Lean.Meta.ACLt.ReduceMode → Sort v✝ | null | true |
Fin.encodeProd | Batteries.Data.Fin.Coding | {m n : ℕ} → Fin m × Fin n → Fin (m * n) | Encode `Fin m × Fin n` into `Fin (m * n)`. | true |
CategoryTheory.ParametrizedAdjunction._sizeOf_1 | Mathlib.CategoryTheory.Adjunction.Parametrized | {C₁ : Type u₁} →
{C₂ : Type u₂} →
{C₃ : Type u₃} →
{inst : CategoryTheory.Category.{v₁, u₁} C₁} →
{inst_1 : CategoryTheory.Category.{v₂, u₂} C₂} →
{inst_2 : CategoryTheory.Category.{v₃, u₃} C₃} →
{F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₃)} →
{G ... | null | false |
Std.Internal.List.maxKey?_of_perm | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{l l' : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l → l.Perm l' → Std.Internal.List.maxKey? l = Std.Internal.List.maxKey? l' | null | true |
CategoryTheory.Limits.CoproductsFromFiniteFiltered.isColimitFiniteSubproductsCocone._proof_1 | Mathlib.CategoryTheory.Limits.Constructions.Filtered | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {α : Type u_1}
[inst_1 : CategoryTheory.Limits.HasFiniteCoproducts C] (f : α → C)
[inst_2 : CategoryTheory.Limits.HasColimitsOfShape (Finset (CategoryTheory.Discrete α)) C]
[inst_3 : CategoryTheory.Limits.HasCoproduct f] (j : Finset (CategoryTheory.Di... | null | false |
NNReal.nhds_zero | Mathlib.Topology.Instances.NNReal.Lemmas | nhds 0 = ⨅ a, ⨅ (_ : a ≠ 0), Filter.principal (Set.Iio a) | null | true |
Relation.fun_eq_comp | Mathlib.Logic.Relation | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → β → Prop} (f : γ → α),
Relation.Comp (fun x1 x2 => f x1 = x2) r = fun x => r (f x) | null | true |
CategoryTheory.IsPreconnected | Mathlib.CategoryTheory.IsConnected | (J : Type u₁) → [CategoryTheory.Category.{v₁, u₁} J] → Prop | A possibly empty category for which every functor to a discrete category is constant.
| true |
_private.Mathlib.Combinatorics.SimpleGraph.CycleGraph.0.SimpleGraph.cycleGraph.getVert_cycleCons._proof_1_11 | Mathlib.Combinatorics.SimpleGraph.CycleGraph | ∀ {n : ℕ} (n_1 n_2 : ℕ) (hm : n_2 + 1 < n + 3),
↑((SimpleGraph.cycleGraph.cycleCons✝ n ⟨n_2 + 1, ⋯⟩).getVert (n_1 + 1)) < n + 3 | null | false |
_private.Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable.0.Multiset.aestronglyMeasurable_fun_prod._simp_1_1 | Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | ∀ {α : Type u_7} {M : α → Type u_8} [inst : (a : α) → CommMonoid (M a)] (a : α) (s : Multiset ((a : α) → M a)),
(Multiset.map (fun f => f a) s).prod = s.prod a | null | false |
IsCyclotomicExtension.norm_zeta_sub_one_of_isPrimePow | Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | ∀ {n : ℕ} [inst : NeZero n] {K : Type u} (L : Type v) [inst_1 : Field K] [inst_2 : Field L] [inst_3 : Algebra K L],
IsPrimePow n →
∀ [inst_4 : IsCyclotomicExtension {n} K L],
Irreducible (Polynomial.cyclotomic n K) →
n ≠ 2 → (Algebra.norm K) (IsCyclotomicExtension.zeta n K L - 1) = ↑n.minFac | If `IsPrimePow n`, `n ≠ 2` and `Irreducible (cyclotomic n K)` (in particular for `K = ℚ`),
then the norm of `zeta n K L - 1` is `n.minFac`. | true |
AddSubmonoid.pi_mem_of_single_mem_aux | Mathlib.Algebra.Group.Submonoid.Finite | ∀ {η : Type u_1} {f : η → Type u_2} [inst : (i : η) → AddZeroClass (f i)] {S : Type u_3}
[inst_1 : SetLike S ((i : η) → f i)] [AddSubmonoidClass S ((i : η) → f i)] [inst_3 : DecidableEq η] (I : Finset η)
{H : S} (x : (i : η) → f i), (∀ i ∉ I, x i = 0) → (∀ i ∈ I, Pi.single i (x i) ∈ H) → x ∈ H | null | true |
Std.ExtDTreeMap.Const.get?_modify_self | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} [inst : Std.TransCmp cmp] {β : Type v} {t : Std.ExtDTreeMap α (fun x => β) cmp}
{k : α} {f : β → β},
Std.ExtDTreeMap.Const.get? (Std.ExtDTreeMap.Const.modify t k f) k = Option.map f (Std.ExtDTreeMap.Const.get? t k) | null | true |
_private.Init.Data.String.Lemmas.Pattern.String.Basic.0.String.Slice.Pattern.Model.ForwardSliceSearcher.revMatchesAt_iff_splits.match_1_3 | Init.Data.String.Lemmas.Pattern.String.Basic | ∀ {pat s : String.Slice} {pos : s.Pos}
(motive : (∃ startPos t₁ t₂, startPos.Splits t₁ (pat.copy ++ t₂) ∧ pos.Splits (t₁ ++ pat.copy) t₂) → Prop)
(x : ∃ startPos t₁ t₂, startPos.Splits t₁ (pat.copy ++ t₂) ∧ pos.Splits (t₁ ++ pat.copy) t₂),
(∀ (e : s.Pos) (t₁ t₂ : String) (ht₁ : e.Splits t₁ (pat.copy ++ t₂)) (ht₂ ... | null | false |
CategoryTheory.Over.starPullbackIsoStar._proof_7 | Mathlib.CategoryTheory.Comma.Over.Pullback | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasBinaryProducts C]
[CategoryTheory.Limits.HasPullbacks C] {X Y : C} (f : X ⟶ Y) (Z : C),
CategoryTheory.Limits.HasPullback ((CategoryTheory.Over.star Y).obj Z).hom f | null | false |
OreLocalization.instFieldNonZeroDivisors._proof_7 | Mathlib.RingTheory.OreLocalization.Ring | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : Nontrivial R] [inst_2 : NoZeroDivisors R]
[inst_3 : OreLocalization.OreSet (nonZeroDivisors R)] (a : OreLocalization (nonZeroDivisors R) R), a ≠ 0 → a * a⁻¹ = 1 | null | false |
EuclideanDomain.lcm_eq_zero_iff | Mathlib.Algebra.EuclideanDomain.Basic | ∀ {R : Type u} [inst : EuclideanDomain R] [inst_1 : DecidableEq R] {x y : R},
EuclideanDomain.lcm x y = 0 ↔ x = 0 ∨ y = 0 | null | true |
Std.DTreeMap.getKey?_union_of_not_mem_left | 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},
k ∉ t₁ → (t₁ ∪ t₂).getKey? k = t₂.getKey? k | null | true |
CategoryTheory.Limits.ColimitPresentation.Total.Hom.comp_base | Mathlib.CategoryTheory.Presentable.ColimitPresentation | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type u_1} {I : J → Type u_2}
[inst_1 : CategoryTheory.Category.{v_1, u_1} J] [inst_2 : (j : J) → CategoryTheory.Category.{u_3, u_2} (I j)]
{D : CategoryTheory.Functor J C} {P : (j : J) → CategoryTheory.Limits.ColimitPresentation (I j) (D.obj j)}
{k l m... | null | true |
Polynomial.isRoot_cyclotomic_iff | Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | ∀ {R : Type u_1} [inst : CommRing R] {n : ℕ} [IsDomain R] [NeZero ↑n] {μ : R},
(Polynomial.cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n | null | true |
_private.Init.Data.Range.Polymorphic.PRange.0.Std.instDecidableEqRoo.decEq._proof_3 | Init.Data.Range.Polymorphic.PRange | ∀ {α : Type u_1} (a a_1 b b_1 : α), ¬a = b → a<...a_1 = b<...b_1 → False | null | false |
Prod.instBornology | Mathlib.Topology.Bornology.Constructions | {α : Type u_1} → {β : Type u_2} → [Bornology α] → [Bornology β] → Bornology (α × β) | null | true |
DirectSum.instLieAlgebraSubtypeMemSubmodule | Mathlib.Algebra.Lie.Graded | {ι : Type u_1} →
{R : Type u_5} →
{L : Type u_6} →
[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] → LieAlgebra... | null | true |
CategoryTheory.Comonad.comparison_obj_A | Mathlib.CategoryTheory.Monad.Adjunction | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{L : CategoryTheory.Functor C D} {R : CategoryTheory.Functor D C} (h : L ⊣ R) (X : C),
((CategoryTheory.Comonad.comparison h).obj X).A = L.obj X | null | true |
_private.Mathlib.Analysis.Calculus.FDeriv.OfCompLeft.0.HasFDerivAtFilter.of_comp_aux | Mathlib.Analysis.Calculus.FDeriv.OfCompLeft | ∀ {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] {g : E → F} {f : F → G} {h : E → G} {... | null | true |
_private.Mathlib.RingTheory.Polynomial.Basic.0.Polynomial.Monic.geom_sum._simp_1_2 | Mathlib.RingTheory.Polynomial.Basic | ∀ {α : Type u_1} [inst : AddMonoidWithOne α] [inst_1 : PartialOrder α] [AddLeftMono α] [ZeroLEOneClass α] [CharZero α]
{m n : ℕ}, (↑m < ↑n) = (m < n) | null | false |
CategoryTheory.isIso_left_of_isIso_biprod_map | Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z)
[CategoryTheory.IsIso (CategoryTheory.Limits.biprod.map f g)], CategoryTheory.IsIso f | If
```
(f 0)
(0 g)
```
is invertible, then `f` is invertible.
| true |
FormalMultilinearSeries.const_smul_sum | Mathlib.Analysis.Analytic.ConvergenceRadius | ∀ {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [inst : Semiring 𝕜] [inst_1 : AddCommMonoid E] [inst_2 : AddCommMonoid F]
[inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 F] [inst_5 : TopologicalSpace E] [inst_6 : TopologicalSpace F]
[inst_7 : ContinuousAdd E] [inst_8 : ContinuousAdd F] [inst_9 : ContinuousConstSMul 𝕜 ... | null | true |
Filter.eventually_forall_ge_atTop | Mathlib.Order.Filter.AtTopBot.Finite | ∀ {α : Type u_2} [inst : Preorder α] {p : α → Prop},
(∀ᶠ (x : α) in Filter.atTop, ∀ (y : α), x ≤ y → p y) ↔ ∀ᶠ (x : α) in Filter.atTop, p x | null | true |
Array.lt_toList | Init.Data.Array.Lex.Lemmas | ∀ {α : Type u_1} [inst : LT α] {xs ys : Array α}, xs.toList < ys.toList ↔ xs < ys | null | true |
CategoryTheory.MonoidalCategory.«_aux_Mathlib_CategoryTheory_Monoidal_Category___macroRules_CategoryTheory_MonoidalCategory_term_◁__1» | Mathlib.CategoryTheory.Monoidal.Category | Lean.Macro | null | false |
AddCommGroup.mem_primaryComponent_iff_addOrderOf | Mathlib.GroupTheory.Torsion | ∀ {G : Type u_1} [inst : AddCommGroup G] {p : ℕ} [Fact (Nat.Prime p)] {g : G},
g ∈ AddCommGroup.primaryComponent G p ↔ ∃ n, addOrderOf g = p ^ n | For prime `p`, `g` lies in the additive `p`-primary component iff its additive
order is a power of `p`. | true |
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