name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
Vector.getElem_zero_flatten | Init.Data.Vector.Find | ∀ {α : Type u_1} {m n : ℕ} {xss : Vector (Vector α m) n} (h : 0 < n * m),
xss.flatten[0] = (Vector.findSome? (fun xs => xs[0]?) xss).get ⋯ |
MonoidHom.coeToMulHom.eq_1 | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} [inst : MulOne M] [inst_1 : MulOne N],
MonoidHom.coeToMulHom = { coe := MonoidHom.toMulHom } |
String.Pos.lt_of_lt_of_le | Init.Data.String.Basic | ∀ {s : String} {p q r : s.Pos}, p < q → q ≤ r → p < r |
HomotopyCategory.instPretriangulatedIntUp | Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | (C : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
[CategoryTheory.Limits.HasBinaryBiproducts C] →
[inst_3 : CategoryTheory.Limits.HasZeroObject C] →
CategoryTheory.Pretriangulated (HomotopyCategory C (ComplexShape.up ℤ)) |
_private.Lean.Meta.Basic.0.Lean.Meta.setInlineAttribute.match_1 | Lean.Meta.Basic | (motive : Except String Lean.Environment → Sort u_1) →
(x : Except String Lean.Environment) →
((env : Lean.Environment) → motive (Except.ok env)) → ((msg : String) → motive (Except.error msg)) → motive x |
_private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.inner_._proof_1 | Mathlib.Analysis.InnerProductSpace.OfNorm | (3 + 1).AtLeastTwo |
summable_sigma_of_nonneg | Mathlib.Topology.Algebra.InfiniteSum.Real | ∀ {α : Type u_4} {β : α → Type u_3} {f : (x : α) × β x → ℝ},
(∀ (x : (x : α) × β x), 0 ≤ f x) →
(Summable f ↔ (∀ (x : α), Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' (y : β x), f ⟨x, y⟩) |
Set.toFinset_empty | Mathlib.Data.Fintype.Sets | ∀ {α : Type u_1} [inst : Fintype ↑∅], ∅.toFinset = ∅ |
Computability.instDecidableEqΓ'.decEq._proof_3 | Mathlib.Computability.Encoding | ∀ (b : Bool), ¬Computability.Γ'.blank = Computability.Γ'.bit b |
CompleteBooleanAlgebra.himp._inherited_default | Mathlib.Order.CompleteBooleanAlgebra | {α : Type u_1} →
(le lt : α → α → Prop) →
(∀ (a : α), le a a) →
(∀ (a b c : α), le a b → le b c → le a c) →
(∀ (a b : α), lt a b ↔ le a b ∧ ¬le b a) →
(∀ (a b : α), le a b → le b a → a = b) →
(sup : α → α → α) →
(∀ (a b : α), le a (sup a b)) →
(∀ (a b : α), le b (sup a b)) → (∀ (a b c : α), le a c → le b c → le (sup a b) c) → (α → α) → α → α → α |
ClosedAddSubgroup | Mathlib.Topology.Algebra.Group.ClosedSubgroup | (G : Type u) → [AddGroup G] → [TopologicalSpace G] → Type u |
Std.Tactic.BVDecide.BVExpr.bitblast.blastExtract.go | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.Extract | {α : Type} →
[inst : Hashable α] →
[inst_1 : DecidableEq α] →
{newWidth : ℕ} →
{aig : Std.Sat.AIG α} →
{w : ℕ} → aig.RefVec w → ℕ → (curr : ℕ) → curr ≤ newWidth → aig.RefVec curr → aig.RefVec newWidth |
_private.Mathlib.RingTheory.TensorProduct.Quotient.0.Algebra.TensorProduct.quotIdealMapEquivTensorQuot._simp_1 | Mathlib.RingTheory.TensorProduct.Quotient | ∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]
[MulHomClass F M N] (f : F) (x y : M), f x * f y = f (x * y) |
Commute.tsum_left | Mathlib.Topology.Algebra.InfiniteSum.Ring | ∀ {ι : Type u_1} {α : Type u_3} {L : SummationFilter ι} [inst : NonUnitalNonAssocSemiring α]
[inst_1 : TopologicalSpace α] [IsTopologicalSemiring α] {f : ι → α} [T2Space α] [L.NeBot] (a : α),
(∀ (i : ι), Commute (f i) a) → Commute (∑'[L] (i : ι), f i) a |
Submodule.dualCoannihilator | Mathlib.LinearAlgebra.Dual.Defs | {R : Type u_1} →
{M : Type u_2} →
[inst : CommSemiring R] →
[inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R (Module.Dual R M) → Submodule R M |
_private.Lean.Elab.BuiltinEvalCommand.0.Lean.Elab.Command.elabEvalCoreUnsafe.match_3 | Lean.Elab.BuiltinEvalCommand | (motive : Option Lean.Elab.Command.EvalAction✝ → Sort u_1) →
(__do_lift : Option Lean.Elab.Command.EvalAction✝¹) →
((act : Lean.Elab.Command.EvalAction✝²) → motive (some act)) →
((x : Option Lean.Elab.Command.EvalAction✝³) → motive x) → motive __do_lift |
MeasurableEmbedding.measurableSet_range | Mathlib.MeasureTheory.MeasurableSpace.Embedding | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β},
MeasurableEmbedding f → MeasurableSet (Set.range f) |
_private.Mathlib.Order.Filter.Map.0.Filter.comap_neBot_iff_frequently._simp_1_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β}, (Filter.comap m f).NeBot = ∀ t ∈ f, ∃ a, m a ∈ t |
_private.Mathlib.CategoryTheory.Category.Pairwise.0.CategoryTheory.instFintypePairwise.match_5.eq_2 | Mathlib.CategoryTheory.Category.Pairwise | ∀ (ι : Type u_1) (motive : CategoryTheory.Pairwise ι → Sort u_2) (a a_1 : ι)
(h_1 : (a : ι) → motive (CategoryTheory.Pairwise.single a))
(h_2 : (a a_2 : ι) → motive (CategoryTheory.Pairwise.pair a a_2)),
(match CategoryTheory.Pairwise.pair a a_1 with
| CategoryTheory.Pairwise.single a => h_1 a
| CategoryTheory.Pairwise.pair a a_2 => h_2 a a_2) =
h_2 a a_1 |
«_aux_ImportGraph_Tools_FindHome___elabRules_command#find_home!__1» | ImportGraph.Tools.FindHome | Lean.Elab.Command.CommandElab |
Option.mem_pmem | Mathlib.Data.Option.Basic | ∀ {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (x : Option α) {a : α} (h : ∀ a ∈ x, p a)
(ha : a ∈ x), f a ⋯ ∈ Option.pmap f x h |
_private.Batteries.Data.List.Lemmas.0.List.getElem_idxOf_eq_idxOfNth_add._proof_1_34 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {x : α} [inst : BEq α] (head : α) (tail : List α) {n s : ℕ}
{h : n < (List.idxsOf x (head :: tail) s).length}, 0 < (List.filter (fun x_1 => x_1 == x) (head :: tail)).length |
_private.Lean.Elab.DocString.0.Lean.Doc.suggestionName.match_1 | Lean.Elab.DocString | (motive : Option Lean.Name → Sort u_1) →
(resolved? : Option Lean.Name) →
((resolved : Lean.Name) → motive (some resolved)) → (Unit → motive none) → motive resolved? |
Topology.IsConstructible.preimage | Mathlib.Topology.Constructible | ∀ {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} {s : Set Y},
Continuous f →
(∀ (s : Set Y), IsOpen s → IsRetrocompact s → IsRetrocompact (f ⁻¹' s)) →
Topology.IsConstructible s → Topology.IsConstructible (f ⁻¹' s) |
Lean.Grind.CommRing.Expr.toPolyC_nc.go | Init.Grind.Ring.CommSolver | ℕ → Lean.Grind.CommRing.Expr → Lean.Grind.CommRing.Poly |
_private.Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple.0.RootPairing.GeckConstruction.instIsIrreducible_aux₂ | Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | ∀ {ι : Type u_1} {K : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Field K] [inst_1 : CharZero K]
[inst_2 : DecidableEq ι] [inst_3 : Fintype ι] [inst_4 : AddCommGroup M] [inst_5 : Module K M]
[inst_6 : AddCommGroup N] [inst_7 : Module K N] {P : RootPairing ι K M N} [inst_8 : P.IsCrystallographic] {b : P.Base}
[P.IsReduced] [P.IsIrreducible]
{U : LieSubmodule K (↥(RootPairing.GeckConstruction.lieAlgebra b)) (↥b.support ⊕ ι → K)} {i : ι},
RootPairing.GeckConstruction.v b i ∈ U → U = ⊤ |
_private.Mathlib.Tactic.TacticAnalysis.Declarations.0.Mathlib.TacticAnalysis.TerminalReplacementOutcome.success.sizeOf_spec | Mathlib.Tactic.TacticAnalysis.Declarations | ∀ (stx : Lean.TSyntax `tactic), sizeOf (Mathlib.TacticAnalysis.TerminalReplacementOutcome.success✝ stx) = 1 + sizeOf stx |
_private.Lean.Widget.TaggedText.0.Lean.Widget.TaggedText.instMonadPrettyFormatStateMTaggedState.match_1 | Lean.Widget.TaggedText | (motive : Lean.Widget.TaggedText.TaggedState✝ → Sort u_1) →
(x : Lean.Widget.TaggedText.TaggedState✝¹) →
((out : Lean.Widget.TaggedText (ℕ × ℕ)) →
(ts : List (ℕ × ℕ × Lean.Widget.TaggedText (ℕ × ℕ))) →
(col : ℕ) → motive { out := out, tagStack := ts, column := col }) →
motive x |
CategoryTheory.PreZeroHypercover.sectionsEquivOfHasPullbacks | Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{S : C} →
(E : CategoryTheory.PreZeroHypercover S) →
[inst_1 : E.HasPullbacks] →
(F : CategoryTheory.Functor Cᵒᵖ (Type u_2)) →
(E.toPreOneHypercover.multicospanIndex F).sections ≃
Subtype (CategoryTheory.Presieve.Arrows.Compatible F E.f) |
LindelofSpace.mk | Mathlib.Topology.Compactness.Lindelof | ∀ {X : Type u_2} [inst : TopologicalSpace X], IsLindelof Set.univ → LindelofSpace X |
Set.Finite.wellFoundedOn | Mathlib.Order.WellFoundedSet | ∀ {α : Type u_2} {r : α → α → Prop} [IsStrictOrder α r] {s : Set α}, s.Finite → s.WellFoundedOn r |
_private.Lean.Elab.StructInst.0.Lean.Elab.Term.StructInst.addParentInstanceFields.match_9 | Lean.Elab.StructInst | (motive : List (Lean.Name × Array Lean.Name) → Sort u_1) →
(worklist : List (Lean.Name × Array Lean.Name)) →
((parentName : Lean.Name) →
(parentFields : Array Lean.Name) →
(worklist' : List (Lean.Name × Array Lean.Name)) → motive ((parentName, parentFields) :: worklist')) →
((x : List (Lean.Name × Array Lean.Name)) → motive x) → motive worklist |
Lean.DeclNameGenerator.noConfusionType | Lean.CoreM | Sort u → Lean.DeclNameGenerator → Lean.DeclNameGenerator → Sort u |
stoneCechEquivalence._proof_5 | Mathlib.Topology.Category.CompHaus.Basic | ∀ (Y : CompHaus), CompactSpace ↑Y.toTop |
maximal_subset_iff | Mathlib.Order.Minimal | ∀ {α : Type u_2} {P : Set α → Prop} {s : Set α}, Maximal P s ↔ P s ∧ ∀ ⦃t : Set α⦄, P t → s ⊆ t → s = t |
Localization.exists_awayMap_bijective_of_localRingHom_bijective | Mathlib.RingTheory.Unramified.LocalRing | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {p : Ideal R}
[inst_3 : p.IsPrime] {q : Ideal S} [inst_4 : q.IsPrime],
p.primesOver S = {q} →
∀ [Module.Finite R S] [inst_6 : q.LiesOver p],
(RingHom.ker (algebraMap R S)).FG →
Function.Bijective ⇑(Localization.localRingHom p q (algebraMap R S) ⋯) →
∃ r ∉ p, ∀ (r' : R), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap R S) r') |
CategoryTheory.CommMon.toMon | Mathlib.CategoryTheory.Monoidal.CommMon_ | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
[inst_2 : CategoryTheory.BraidedCategory C] → CategoryTheory.CommMon C → CategoryTheory.Mon C |
_private.Std.Sync.Channel.0.Std.CloseableChannel.Unbounded.State.mk.noConfusion | Std.Sync.Channel | {α : Type} →
{P : Sort u} →
{values : Std.Queue α} →
{consumers : Std.Queue (Std.CloseableChannel.Consumer✝ α)} →
{closed : Bool} →
{values' : Std.Queue α} →
{consumers' : Std.Queue (Std.CloseableChannel.Consumer✝¹ α)} →
{closed' : Bool} →
{ values := values, consumers := consumers, closed := closed } =
{ values := values', consumers := consumers', closed := closed' } →
(values ≍ values' → consumers ≍ consumers' → closed = closed' → P) → P |
CategoryTheory.MorphismProperty.comp_mem | Mathlib.CategoryTheory.MorphismProperty.Composition | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (W : CategoryTheory.MorphismProperty C)
[W.IsStableUnderComposition] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z),
W f → W g → W (CategoryTheory.CategoryStruct.comp f g) |
linearOrderOfCompares._proof_8 | Mathlib.Order.Compare | ∀ {α : Type u_1} [inst : Preorder α] (cmp : α → α → Ordering),
(∀ (a b : α), (cmp a b).Compares a b) → ∀ (a b : α), a ≤ b ∨ b ≤ a |
_private.Mathlib.GroupTheory.Goursat.0.Subgroup.mk_goursatFst_eq_iff_mk_goursatSnd_eq._simp_1_1 | Mathlib.GroupTheory.Goursat | ∀ {G : Type u_1} [inst : Group G] {N : Subgroup G} [nN : N.Normal] {x y : G}, (↑x = ↑y) = (x / y ∈ N) |
ZMod.χ₈' | Mathlib.NumberTheory.LegendreSymbol.ZModChar | MulChar (ZMod 8) ℤ |
Lean.Lsp.SignatureInformation._sizeOf_1 | Lean.Data.Lsp.LanguageFeatures | Lean.Lsp.SignatureInformation → ℕ |
_private.Mathlib.Topology.Order.0.continuous_sInf_rng._simp_1_1 | Mathlib.Topology.Order | ∀ {α : Type u} {β : Type v} {f : α → β} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace β},
Continuous f = (TopologicalSpace.coinduced f t₁ ≤ t₂) |
ModularForm.mul._proof_2 | Mathlib.NumberTheory.ModularForms.Basic | ∀ {Γ : Subgroup (GL (Fin 2) ℝ)} {k_1 k_2 : ℤ} [inst : Γ.HasDetPlusMinusOne] (f : ModularForm Γ k_1)
(g : ModularForm Γ k_2) {c : OnePoint ℝ},
IsCusp c Γ →
∀ (γ : GL (Fin 2) ℝ),
γ • OnePoint.infty = c →
UpperHalfPlane.IsBoundedAtImInfty (SlashAction.map (k_1 + k_2) γ (f.mul g.toSlashInvariantForm).toFun) |
Real.sin_pi_sub | Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | ∀ (x : ℝ), Real.sin (Real.pi - x) = Real.sin x |
RingQuot.instSemiring | Mathlib.Algebra.RingQuot | {R : Type uR} → [inst : Semiring R] → (r : R → R → Prop) → Semiring (RingQuot r) |
IsLocalization.Away.commutes | Mathlib.RingTheory.Localization.Away.Basic | ∀ {R : Type u_5} [inst : CommSemiring R] (S₁ : Type u_6) (S₂ : Type u_7) (T : Type u_8) [inst_1 : CommSemiring S₁]
[inst_2 : CommSemiring S₂] [inst_3 : CommSemiring T] [inst_4 : Algebra R S₁] [inst_5 : Algebra R S₂]
[inst_6 : Algebra R T] [inst_7 : Algebra S₁ T] [inst_8 : Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T]
(x y : R) [IsLocalization.Away x S₁] [IsLocalization.Away y S₂] [IsLocalization.Away ((algebraMap R S₂) x) T],
IsLocalization.Away ((algebraMap R S₁) y) T |
Subring.list_sum_mem | Mathlib.Algebra.Ring.Subring.Basic | ∀ {R : Type u} [inst : Ring R] (s : Subring R) {l : List R}, (∀ x ∈ l, x ∈ s) → l.sum ∈ s |
CochainComplex.mapBifunctorHomologicalComplexShift₁Iso | Mathlib.Algebra.Homology.BifunctorShift | {C₁ : Type u_1} →
{C₂ : Type u_2} →
{D : Type u_3} →
[inst : CategoryTheory.Category.{v_1, u_1} C₁] →
[inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] →
[inst_2 : CategoryTheory.Category.{v_3, u_3} D] →
[inst_3 : CategoryTheory.Preadditive C₁] →
[inst_4 : CategoryTheory.Limits.HasZeroMorphisms C₂] →
[inst_5 : CategoryTheory.Preadditive D] →
(K₁ : CochainComplex C₁ ℤ) →
(K₂ : CochainComplex C₂ ℤ) →
(F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ D)) →
[inst_6 : F.Additive] →
[inst_7 : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms] →
(x : ℤ) →
((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj
((CategoryTheory.shiftFunctor (HomologicalComplex C₁ (ComplexShape.up ℤ)) x).obj
K₁)).obj
K₂ ≅
(HomologicalComplex₂.shiftFunctor₁ D x).obj
(((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj
K₁).obj
K₂) |
GenContFract.coe_toGenContFract | Mathlib.Algebra.ContinuedFractions.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : Coe α β] {g : GenContFract α},
↑g = { h := Coe.coe g.h, s := Stream'.Seq.map GenContFract.Pair.coeFn g.s } |
CommRingCat.Colimits.Relation.right_distrib | Mathlib.Algebra.Category.Ring.Colimits | ∀ {J : Type v} [inst : CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CommRingCat}
(x y z : CommRingCat.Colimits.Prequotient F),
CommRingCat.Colimits.Relation F ((x.add y).mul z) ((x.mul z).add (y.mul z)) |
UniformConvergenceCLM.neg_apply | Mathlib.Topology.Algebra.Module.StrongTopology | ∀ {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [inst : NormedField 𝕜₁] [inst_1 : NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3}
(F : Type u_4) [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜₁ E] [inst_4 : TopologicalSpace E]
[inst_5 : AddCommGroup F] [inst_6 : Module 𝕜₂ F] [inst_7 : TopologicalSpace F] [inst_8 : IsTopologicalAddGroup F]
(𝔖 : Set (Set E)) (f : UniformConvergenceCLM σ F 𝔖) (x : E), (-f) x = -f x |
Lean.Grind.AC.Seq.sort'_k | Init.Grind.AC | Lean.Grind.AC.Seq → Lean.Grind.AC.Seq → Lean.Grind.AC.Seq |
IsUnifLocDoublingMeasure | Mathlib.MeasureTheory.Measure.Doubling | {α : Type u_1} → [PseudoMetricSpace α] → [inst : MeasurableSpace α] → MeasureTheory.Measure α → Prop |
_private.Mathlib.Topology.Order.0.isClosed_induced._simp_1_1 | Mathlib.Topology.Order | ∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], IsClosed s = IsOpen sᶜ |
Matrix.replicateRow_inj._simp_1 | Mathlib.LinearAlgebra.Matrix.RowCol | ∀ {n : Type u_3} {α : Type v} {ι : Type u_6} [Nonempty ι] {v w : n → α},
(Matrix.replicateRow ι v = Matrix.replicateRow ι w) = (v = w) |
CStarAlgebra.instNegPart | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {A : Type u_1} →
[inst : NonUnitalRing A] →
[inst_1 : Module ℝ A] →
[inst_2 : SMulCommClass ℝ A A] →
[inst_3 : IsScalarTower ℝ A A] →
[inst_4 : StarRing A] →
[inst_5 : TopologicalSpace A] → [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] → NegPart A |
Aesop.ForwardStateStats.mk.noConfusion | Aesop.Stats.Basic | {P : Sort u} →
{ruleStateStats ruleStateStats' : Array Aesop.ForwardRuleStateStats} →
{ ruleStateStats := ruleStateStats } = { ruleStateStats := ruleStateStats' } →
(ruleStateStats = ruleStateStats' → P) → P |
Lean.Server.DirectImports.noConfusionType | Lean.Server.References | Sort u → Lean.Server.DirectImports → Lean.Server.DirectImports → Sort u |
Function.Surjective.addGroup.eq_1 | Mathlib.Algebra.Group.InjSurj | ∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₂] [inst_1 : Zero M₂] [inst_2 : SMul ℕ M₂] [inst_3 : Neg M₂]
[inst_4 : Sub M₂] [inst_5 : SMul ℤ M₂] [inst_6 : AddGroup M₁] (f : M₁ → M₂) (hf : Function.Surjective f)
(one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (inv : ∀ (x : M₁), f (-x) = -f x)
(div : ∀ (x y : M₁), f (x - y) = f x - f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x)
(zpow : ∀ (x : M₁) (n : ℤ), f (n • x) = n • f x),
Function.Surjective.addGroup f hf one mul inv div npow zpow =
{ toSubNegMonoid := Function.Surjective.subNegMonoid f hf one mul inv div npow zpow, neg_add_cancel := ⋯ } |
Aesop.instInhabitedGoalDiff.default | Aesop.RuleTac.GoalDiff | Aesop.GoalDiff |
orderOf_one | Mathlib.GroupTheory.OrderOfElement | ∀ {G : Type u_1} [inst : Monoid G], orderOf 1 = 1 |
_private.Mathlib.Geometry.Euclidean.Inversion.Basic.0.EuclideanGeometry.dist_inversion_center._simp_1_6 | Mathlib.Geometry.Euclidean.Inversion.Basic | ∀ {M₀ : Type u_1} [inst : Mul M₀] [inst_1 : Zero M₀] [NoZeroDivisors M₀] {a b : M₀}, a ≠ 0 → b ≠ 0 → (a * b = 0) = False |
CochainComplex.mappingCocone.triangle_obj₂ | Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
{K L : CochainComplex C ℤ} (φ : K ⟶ L) [inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C],
(CochainComplex.mappingCocone.triangle φ).obj₂ = K |
Asymptotics.isBigO_congr | Mathlib.Analysis.Asymptotics.Defs | ∀ {α : Type u_1} {E : Type u_3} {F : Type u_4} [inst : Norm E] [inst_1 : Norm F] {l : Filter α} {f₁ f₂ : α → E}
{g₁ g₂ : α → F}, f₁ =ᶠ[l] f₂ → g₁ =ᶠ[l] g₂ → (f₁ =O[l] g₁ ↔ f₂ =O[l] g₂) |
Std.Time.Modifier.x.injEq | Std.Time.Format.Basic | ∀ (presentation presentation_1 : Std.Time.OffsetX),
(Std.Time.Modifier.x presentation = Std.Time.Modifier.x presentation_1) = (presentation = presentation_1) |
CategoryTheory.Limits.FormalCoproduct.isoOfComponents._proof_7 | Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] {X Y : CategoryTheory.Limits.FormalCoproduct C}
(e : X.I ≃ Y.I) (h : (i : X.I) → X.obj i ≅ Y.obj (e i)),
CategoryTheory.CategoryStruct.comp
{ f := ⇑e.symm, φ := fun i => CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯) (h (e.symm i)).inv }
{ f := ⇑e, φ := fun i => (h i).hom } =
CategoryTheory.CategoryStruct.id Y |
SeminormedCommRing.mk | Mathlib.Analysis.Normed.Ring.Basic | {α : Type u_5} → [toSeminormedRing : SeminormedRing α] → (∀ (a b : α), a * b = b * a) → SeminormedCommRing α |
CategoryTheory.InducedCategory.hasForget₂._proof_1 | Mathlib.CategoryTheory.ConcreteCategory.Forget | ∀ {C : Type u_1} {D : Type u_4} [inst : CategoryTheory.Category.{u_2, u_4} D] {FD : outParam (D → D → Type u_5)}
{CD : outParam (D → Type u_3)} [inst_1 : outParam ((X Y : D) → FunLike (FD X Y) (CD X) (CD Y))]
[inst_2 : CategoryTheory.ConcreteCategory D FD] (f : C → D),
(CategoryTheory.inducedFunctor f).comp (CategoryTheory.forget D) =
(CategoryTheory.inducedFunctor f).comp (CategoryTheory.forget D) |
CategoryTheory.LaxFunctor.mapComp'.congr_simp | Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
(F : CategoryTheory.LaxFunctor B C) {b₀ b₁ b₂ : B} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (fg : b₀ ⟶ b₂)
(h : CategoryTheory.CategoryStruct.comp f g = fg), F.mapComp' f g fg h = F.mapComp' f g fg h |
CategoryTheory.Limits.IsZero.iso.congr_simp | Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (hX : CategoryTheory.Limits.IsZero X)
(hY : CategoryTheory.Limits.IsZero Y), hX.iso hY = hX.iso hY |
CategoryTheory.Functor.CommShift₂.commShiftObj | 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} →
{inst_4 : CategoryTheory.HasShift C₁ M} →
{inst_5 : CategoryTheory.HasShift C₂ M} →
{inst_6 : CategoryTheory.HasShift D M} →
{G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ D)} →
(h : CategoryTheory.CommShift₂Setup D M) →
[self : G.CommShift₂ h] → (X₁ : C₁) → (G.obj X₁).CommShift M |
Multiset.le_iff_exists_add | Mathlib.Data.Multiset.AddSub | ∀ {α : Type u_1} {s t : Multiset α}, s ≤ t ↔ ∃ u, t = s + u |
Lean.InductiveVal.numNested | Lean.Declaration | Lean.InductiveVal → ℕ |
_private.Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddResult.0.Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper._proof_1_20 | Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddResult | ∀ {n : ℕ}
(acc :
Array Std.Tactic.BVDecide.LRAT.Internal.Assignment ×
Std.Sat.CNF.Clause (Std.Tactic.BVDecide.LRAT.Internal.PosFin n) × Bool × Bool),
acc.1.size = n →
∀ (l : Std.Sat.Literal (Std.Tactic.BVDecide.LRAT.Internal.PosFin n)) (i : Fin n),
↑i < acc.1.size → ↑i < (acc.1.modify (↑l.1) (Std.Tactic.BVDecide.LRAT.Internal.Assignment.addAssignment l.2)).size |
_private.Mathlib.CategoryTheory.Idempotents.Karoubi.0.CategoryTheory.Idempotents.instEssSurjKaroubiToKaroubiOfIsIdempotentComplete._simp_1 | Mathlib.CategoryTheory.Idempotents.Karoubi | ∀ {obj : Type u} [self : CategoryTheory.Category.{v, u} obj] {W X Y Z : obj} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h |
AddLECancellable.tsub_mul | Mathlib.Algebra.Order.Ring.Canonical | ∀ {R : Type u} [inst : NonUnitalNonAssocSemiring R] [inst_1 : PartialOrder R] [CanonicallyOrderedAdd R] [inst_3 : Sub R]
[OrderedSub R] [Std.Total fun x1 x2 => x1 ≤ x2] [MulRightMono R] {a b c : R},
AddLECancellable (b * c) → (a - b) * c = a * c - b * c |
Std.Tactic.BVDecide.Normalize.BitVec.ult_max' | Std.Tactic.BVDecide.Normalize.BitVec | ∀ {w : ℕ} (a : BitVec w), a.ult (-1#w) = !a == -1#w |
BddDistLat.recOn | Mathlib.Order.Category.BddDistLat | {motive : BddDistLat → Sort u} →
(t : BddDistLat) →
((toDistLat : DistLat) →
[isBoundedOrder : BoundedOrder ↑toDistLat] →
motive { toDistLat := toDistLat, isBoundedOrder := isBoundedOrder }) →
motive t |
ContinuousLinearEquiv.arrowCongrEquiv._proof_4 | Mathlib.Topology.Algebra.Module.Equiv | ∀ {R₁ : Type u_7} {R₂ : Type u_1} {R₃ : Type u_2} [inst : Semiring R₁] [inst_1 : Semiring R₂] [inst_2 : Semiring R₃]
{σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [inst_3 : RingHomInvPair σ₁₂ σ₂₁] [inst_4 : RingHomInvPair σ₂₁ σ₁₂]
{σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [inst_5 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {M₁ : Type u_8}
[inst_6 : TopologicalSpace M₁] [inst_7 : AddCommMonoid M₁] {M₂ : Type u_3} [inst_8 : TopologicalSpace M₂]
[inst_9 : AddCommMonoid M₂] {M₃ : Type u_4} [inst_10 : TopologicalSpace M₃] [inst_11 : AddCommMonoid M₃]
{M₄ : Type u_6} [inst_12 : TopologicalSpace M₄] [inst_13 : AddCommMonoid M₄] [inst_14 : Module R₁ M₁]
[inst_15 : Module R₂ M₂] [inst_16 : Module R₃ M₃] {R₄ : Type u_5} [inst_17 : Semiring R₄] [inst_18 : Module R₄ M₄]
{σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [inst_19 : RingHomInvPair σ₃₄ σ₄₃] [inst_20 : RingHomInvPair σ₄₃ σ₃₄]
{σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [inst_21 : RingHomCompTriple σ₂₁ σ₁₄ σ₂₄]
[inst_22 : RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [inst_23 : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] (e₁₂ : M₁ ≃SL[σ₁₂] M₂)
(e₄₃ : M₄ ≃SL[σ₄₃] M₃) (f : M₂ →SL[σ₂₃] M₃), (↑e₄₃).comp (((↑e₄₃.symm).comp (f.comp ↑e₁₂)).comp ↑e₁₂.symm) = f |
Submodule.coe_finsetInf | Mathlib.Algebra.Module.Submodule.Lattice | ∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Type u_4}
(s : Finset ι) (p : ι → Submodule R M), ↑(s.inf p) = ⋂ i ∈ s, ↑(p i) |
Matroid.Indep.mem_closure_iff' | Mathlib.Combinatorics.Matroid.Closure | ∀ {α : Type u_2} {M : Matroid α} {I : Set α} {x : α},
M.Indep I → (x ∈ M.closure I ↔ x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)) |
_private.Mathlib.Combinatorics.Enumerative.IncidenceAlgebra.0.IncidenceAlgebra.moebius_inversion_top._simp_1_7 | Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | ∀ {α : Type u_1} [inst : Preorder α] (a : α), (a ≤ a) = True |
Finset.instLattice._proof_3 | Mathlib.Data.Finset.Lattice.Basic | ∀ {α : Type u_1} [inst : DecidableEq α] (x x_1 x_2 : Finset α), x ≤ x_2 → x_1 ≤ x_2 → ∀ x_3 ∈ x ∪ x_1, x_3 ∈ x_2 |
intervalIntegrable_log_norm_meromorphicOn | Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ},
MeromorphicOn f (Set.uIcc a b) → IntervalIntegrable (fun x => Real.log ‖f x‖) MeasureTheory.volume a b |
_private.Lean.Elab.DeclNameGen.0.Lean.Elab.Command.NameGen.getParentProjArg._sparseCasesOn_8.else_eq | Lean.Elab.DeclNameGen | ∀ {motive : Lean.Name → Sort u} (t : Lean.Name) (str : (pre : Lean.Name) → (str : String) → motive (pre.str str))
(«else» : Nat.hasNotBit 2 t.ctorIdx → motive t) (h : Nat.hasNotBit 2 t.ctorIdx),
Lean.Elab.Command.NameGen.getParentProjArg._sparseCasesOn_8✝ t str «else» = «else» h |
VectorBundleCore.coordChange | Mathlib.Topology.VectorBundle.Basic | {R : Type u_1} →
{B : Type u_2} →
{F : Type u_3} →
[inst : NontriviallyNormedField R] →
[inst_1 : NormedAddCommGroup F] →
[inst_2 : NormedSpace R F] →
[inst_3 : TopologicalSpace B] → {ι : Type u_5} → VectorBundleCore R B F ι → ι → ι → B → F →L[R] F |
Lean.IR.CtorInfo.mk.inj | Lean.Compiler.IR.Basic | ∀ {name : Lean.Name} {cidx size usize ssize : ℕ} {name_1 : Lean.Name} {cidx_1 size_1 usize_1 ssize_1 : ℕ},
{ name := name, cidx := cidx, size := size, usize := usize, ssize := ssize } =
{ name := name_1, cidx := cidx_1, size := size_1, usize := usize_1, ssize := ssize_1 } →
name = name_1 ∧ cidx = cidx_1 ∧ size = size_1 ∧ usize = usize_1 ∧ ssize = ssize_1 |
Matrix.uniqueRingEquiv._proof_2 | Mathlib.LinearAlgebra.Matrix.Unique | ∀ {m : Type u_1} {A : Type u_2} [inst : Unique m] [inst_1 : NonUnitalNonAssocSemiring A] (x y : Matrix m m A),
Matrix.uniqueAddEquiv.toFun (x + y) = Matrix.uniqueAddEquiv.toFun x + Matrix.uniqueAddEquiv.toFun y |
Batteries.RBNode.All.map | Batteries.Data.RBMap.WF | ∀ {α : Type u_1} {β : Type u_2} {p : α → Prop} {q : β → Prop} {f : α → β},
(∀ {x : α}, p x → q (f x)) →
∀ {t : Batteries.RBNode α}, Batteries.RBNode.All p t → Batteries.RBNode.All q (Batteries.RBNode.map f t) |
CochainComplex.homologyδOfTriangle._auto_1 | Mathlib.Algebra.Homology.DerivedCategory.HomologySequence | Lean.Syntax |
_private.Mathlib.Geometry.Manifold.ChartedSpace.0.ChartedSpace.t1Space._simp_1_1 | Mathlib.Geometry.Manifold.ChartedSpace | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) |
Lean.ScopedEnvExtension.ScopedEntries.mk.injEq | Lean.ScopedEnvExtension | ∀ {β : Type} (map map_1 : Lean.SMap Lean.Name (Lean.PArray β)), ({ map := map } = { map := map_1 }) = (map = map_1) |
CategoryTheory.Functor.sheafPullbackConstruction.preservesFiniteLimits | Mathlib.CategoryTheory.Sites.Pullback | ∀ {C : Type u₂} [inst : CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [inst_1 : CategoryTheory.Category.{v₃, u₃} D]
(G : CategoryTheory.Functor C D) (A : Type u₁) [inst_2 : CategoryTheory.Category.{v₁, u₁} A]
(J : CategoryTheory.GrothendieckTopology C) (K : CategoryTheory.GrothendieckTopology D) [inst_3 : G.IsContinuous J K]
[inst_4 : ∀ (F : CategoryTheory.Functor Cᵒᵖ A), G.op.HasLeftKanExtension F] [inst_5 : CategoryTheory.HasSheafify K A]
[CategoryTheory.HasSheafify J A] [CategoryTheory.Limits.PreservesFiniteLimits G.op.lan],
CategoryTheory.Limits.PreservesFiniteLimits (G.sheafPullback A J K) |
OrderIso.arrowCongr | Mathlib.Order.Hom.Basic | {α : Type u_6} →
{β : Type u_7} →
{γ : Type u_8} →
{δ : Type u_9} →
[inst : Preorder α] →
[inst_1 : Preorder β] → [inst_2 : Preorder γ] → [inst_3 : Preorder δ] → α ≃o γ → β ≃o δ → (α →o β) ≃o (γ →o δ) |
_private.Lean.Meta.Sorry.0.Lean.Meta.SorryLabelView.encode.match_1 | Lean.Meta.Sorry | (motive : Option Lean.DeclarationLocation → Sort u_1) →
(x : Option Lean.DeclarationLocation) →
((module : Lean.Name) →
(pos : Lean.Position) →
(charUtf16 : ℕ) →
(endPos : Lean.Position) →
(endCharUtf16 : ℕ) →
motive
(some
{ module := module,
range :=
{ pos := pos, charUtf16 := charUtf16, endPos := endPos, endCharUtf16 := endCharUtf16 } })) →
((x : Option Lean.DeclarationLocation) → motive x) → motive x |
Set.finite_Ico._simp_1 | Mathlib.Order.Interval.Finset.Defs | ∀ {α : Type u_1} [inst : Preorder α] [LocallyFiniteOrder α] (a b : α), (Set.Ico a b).Finite = True |
Lean.Lsp.instHashableInsertReplaceEdit.hash | Lean.Data.Lsp.LanguageFeatures | Lean.Lsp.InsertReplaceEdit → UInt64 |
Aesop.instInhabitedRuleTacDescr.default | Aesop.RuleTac.Descr | Aesop.RuleTacDescr |
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