name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Computability.«term_≡ᵀ_» | Mathlib.Computability.TuringDegree | Lean.TrailingParserDescr | `f` is Turing equivalent to `g` if `f` is reducible to `g` and `g` is reducible to `f`.
| true |
Vector.push_inj_left | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n : ℕ} {a : α} {xs ys : Vector α n}, xs.push a = ys.push a ↔ xs = ys | null | true |
Lean.mkPtrSet | Lean.Util.PtrSet | {α : Type} → optParam ℕ 64 → Lean.PtrSet α | null | true |
LinearMap.FiniteRangeSetoid.setoid | Mathlib.Algebra.Module.LinearMap.FiniteRange | {K : Type u_1} →
{V : Type u_2} →
{V₂ : Type u_3} →
[inst : CommRing K] →
[inst_1 : AddCommGroup V] →
[inst_2 : Module K V] → [inst_3 : AddCommGroup V₂] → [inst_4 : Module K V₂] → Setoid (V →ₗ[K] V₂) | This is the equivalence relation on linear maps such that `u ≈ v` precisely
when `u - v` is a linear map with noetherian range. We allow ourself this slightly abusive name
because the more natural definition (`u - v` has finitely generated range) only yields a
well-behaved relation (more precisely, an additive congruen... | true |
WittVector.nsmul_coeff | Mathlib.RingTheory.WittVector.Defs | ∀ {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [inst : CommRing R] (m : ℕ) (x : WittVector p R) (n : ℕ),
(m • x).coeff n = WittVector.peval (WittVector.wittNSMul p m n) ![x.coeff] | null | true |
MeasureTheory.instMetrizableSpaceProbabilityMeasure | Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | ∀ (X : Type u_2) [inst : TopologicalSpace X] [TopologicalSpace.PseudoMetrizableSpace X]
[TopologicalSpace.SeparableSpace X] [inst_3 : MeasurableSpace X] [inst_4 : BorelSpace X],
TopologicalSpace.MetrizableSpace (MeasureTheory.ProbabilityMeasure X) | The topology of convergence in distribution on a separable Borel space is metrizable. | true |
Subgroup.IsSubnormal.recOn | Mathlib.GroupTheory.IsSubnormal | ∀ {G : Type u_1} [inst : Group G] {motive : (a : Subgroup G) → a.IsSubnormal → Prop} {a : Subgroup G}
(t : a.IsSubnormal),
motive ⊤ ⋯ →
(∀ (H K : Subgroup G) (h_le : H ≤ K) (hSubn : K.IsSubnormal) (hN : (H.subgroupOf K).Normal),
motive K hSubn → motive H ⋯) →
motive a t | null | false |
CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHom._default | Mathlib.CategoryTheory.Monoidal.Action.Basic | {C : Type u_1} →
{D : Type u_2} →
{inst : CategoryTheory.Category.{v_1, u_1} C} →
{inst_1 : CategoryTheory.Category.{v_2, u_2} D} →
{inst_2 : CategoryTheory.MonoidalCategoryStruct C} →
(actionObj : C → D → D) →
({c c' : C} → (c ⟶ c') → (d : D) → actionObj c d ⟶ actionObj c' d) ... | null | false |
Subtype.forall_set_subtype | Mathlib.Data.Set.Image | ∀ {α : Type u_1} {t : Set α} (p : Set α → Prop), (∀ (s : Set ↑t), p (Subtype.val '' s)) ↔ ∀ s ⊆ t, p s | null | true |
Lean.Lsp.instFileSourceSignatureHelpParams | Lean.Server.FileSource | Lean.Lsp.FileSource Lean.Lsp.SignatureHelpParams | null | true |
_private.Mathlib.Computability.TuringDegree.0.instPreorderPFunNat | Mathlib.Computability.TuringDegree | Preorder (ℕ →. ℕ) | null | true |
SimpleGraph.Subgraph.botIso._proof_2 | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u_1} {G : SimpleGraph V} (x : ↑⊥.verts), (False.elim ⋯).elim = x | null | false |
CategoryTheory.Grothendieck.map._proof_2 | Mathlib.CategoryTheory.Grothendieck | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] {F G : CategoryTheory.Functor C CategoryTheory.Cat}
(α : F ⟶ G) (X : CategoryTheory.Grothendieck F),
{ base := (CategoryTheory.CategoryStruct.id X).base,
fiber :=
CategoryTheory.CategoryStruct.comp ((CategoryTheory.eqToHom ⋯).toNatTrans.ap... | null | false |
Option.filter_some | Init.Data.Option.Lemmas | ∀ {α : Type u_1} {p : α → Bool} {a : α}, Option.filter p (some a) = if p a = true then some a else none | null | true |
_private.Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper.0.AlgebraicGeometry.Proj.isSeparated._simp_5 | Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | ∀ {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T),
CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.map g) (AlgebraicGeometry.Spec.map f) =
AlgebraicGeometry.Spec.map (CategoryTheory.CategoryStruct.comp f g) | null | false |
_private.Mathlib.Algebra.Ring.Int.Parity.0.Int.sq_emod_four._proof_1_1 | Mathlib.Algebra.Ring.Int.Parity | ∀ (k : ℤ), (2 * k) ^ 2 % 4 = 2 * k % 2 | null | false |
String.Slice.contains_char_eq_contains_beq | Init.Data.String.Lemmas.Pattern.Char | ∀ {c : Char} {s : String.Slice}, s.contains c = s.contains fun x => x == c | null | true |
MvPowerSeries.instInv | Mathlib.RingTheory.MvPowerSeries.Inverse | {σ : Type u_1} → {k : Type u_3} → [Field k] → Inv (MvPowerSeries σ k) | null | true |
CategoryTheory.SimplicialObject.Splitting.IndexSet.id | Mathlib.AlgebraicTopology.SimplicialObject.Split | (Δ : SimplexCategoryᵒᵖ) → CategoryTheory.SimplicialObject.Splitting.IndexSet Δ | The distinguished element in `Splitting.IndexSet Δ` which corresponds to the
identity of `Δ`. | true |
Lean.Elab.Tactic.closeMainGoal | Lean.Elab.Tactic.Basic | Lean.Name → Lean.Expr → optParam Bool true → Lean.Elab.Tactic.TacticM Unit | Closes main goal using the given expression.
If `checkUnassigned == true`, then `val` must not contain unassigned metavariables.
Returns `true` if `val` was successfully used to close the goal.
| true |
Turing.TM1to0.trAux._sunfold | Mathlib.Computability.TuringMachine.PostTuringMachine | {Γ : Type u_1} →
{Λ : Type u_2} →
{σ : Type u_3} →
(M : Λ → Turing.TM1.Stmt Γ Λ σ) → Γ → Turing.TM1.Stmt Γ Λ σ → σ → Turing.TM1to0.Λ' M × Turing.TM0.Stmt Γ | null | false |
Lean.Elab.GoalsAtResult | Lean.Server.InfoUtils | Type | null | true |
_private.Mathlib.Algebra.Group.Pointwise.Set.ListOfFn.0.Set.mem_list_prod._simp_1_2 | Mathlib.Algebra.Group.Pointwise.Set.ListOfFn | ∀ {α : Type u} {P : List α → Prop}, (∃ l, P l) = ∃ n f, P (List.ofFn f) | null | false |
ContinuousAlternatingMap.piLIE._proof_6 | Mathlib.Analysis.Normed.Module.Alternating.Basic | ∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] {ι' : Type u_2} {F : ι' → Type u_3}
[inst_1 : (i' : ι') → SeminormedAddCommGroup (F i')] [inst_2 : (i' : ι') → NormedSpace 𝕜 (F i')],
SMulCommClass 𝕜 𝕜 ((i : ι') → F i) | null | false |
Function.IsFixedPt.eq_1 | Mathlib.Order.OmegaCompletePartialOrder | ∀ {α : Type u₁} (f : α → α) (x : α), Function.IsFixedPt f x = (f x = x) | null | true |
Dvd.noConfusion | Init.Prelude | {P : Sort u} →
{α : Type u_1} → {t : Dvd α} → {α' : Type u_1} → {t' : Dvd α'} → α = α' → t ≍ t' → Dvd.noConfusionType P t t' | null | false |
FirstOrder.Language.BoundedFormula.all_iff_not_ex_not | Mathlib.ModelTheory.Equivalence | ∀ {L : FirstOrder.Language} {T : L.Theory} {α : Type w} {n : ℕ} (φ : L.BoundedFormula α (n + 1)),
T.Iff φ.all φ.not.ex.not | null | true |
Lean.Meta.Grind.Arith.Cutsat.DvdCnstrProof.cooper₁.elim | Lean.Meta.Tactic.Grind.Arith.Cutsat.Types | {motive_7 : Lean.Meta.Grind.Arith.Cutsat.DvdCnstrProof → Sort u} →
(t : Lean.Meta.Grind.Arith.Cutsat.DvdCnstrProof) →
t.ctorIdx = 9 →
((c : Lean.Meta.Grind.Arith.Cutsat.CooperSplit) →
motive_7 (Lean.Meta.Grind.Arith.Cutsat.DvdCnstrProof.cooper₁ c)) →
motive_7 t | null | false |
Lean.Lsp.instToJsonChangeAnnotation.toJson | Lean.Data.Lsp.Basic | Lean.Lsp.ChangeAnnotation → Lean.Json | null | true |
Lean.Lsp.instFromJsonCallHierarchyPrepareParams.fromJson | Lean.Data.Lsp.LanguageFeatures | Lean.Json → Except String Lean.Lsp.CallHierarchyPrepareParams | null | true |
HomologicalComplex.homotopyCofiber.XIsoBiprod.congr_simp | Mathlib.Algebra.Homology.HomotopyCofiber | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {ι : Type u_2}
{c : ComplexShape ι} {F G : HomologicalComplex C c} (φ : F ⟶ G) [inst_2 : HomologicalComplex.HasHomotopyCofiber φ]
[inst_3 : DecidableRel c.Rel] (i j : ι) (hij : c.Rel i j)
[inst_4 : CategoryTheor... | null | true |
Algebra.WeaklyQuasiFiniteAt.of_quasiFiniteAt_residueField | Mathlib.RingTheory.QuasiFinite.Weakly | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
(q : Ideal S) [inst_3 : q.IsPrime] [inst_4 : p.IsPrime] [q.LiesOver p] (Q : Ideal (p.Fiber S)) [inst_6 : Q.IsPrime],
Ideal.comap Algebra.TensorProduct.includeRight.toRingHom Q = q →
∀ [Algebra.QuasiFin... | Use `Algebra.QuasiFinite.of_quasiFiniteAt_residueField` instead
for `Algebra.QuasiFiniteAt R q`. | true |
Std.TreeSet.Raw.maxD_eq_iff_mem_and_forall | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp] [Std.LawfulEqCmp cmp],
t.WF → t.isEmpty = false → ∀ {km fallback : α}, t.maxD fallback = km ↔ km ∈ t ∧ ∀ k ∈ t, (cmp k km).isLE = true | null | true |
AntitoneOn.Ico | Mathlib.Order.Interval.Set.Monotone | ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] {f g : α → β} {s : Set α},
AntitoneOn f s → MonotoneOn g s → MonotoneOn (fun x => Set.Ico (f x) (g x)) s | null | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter.0.SimplexCategory.σ₀Iter_succ._proof_1_4 | Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | ∀ (i : ℕ) {n m : ℕ} (h : n + (i + 1) = m) (k : Fin ({ len := m }.len + 1)),
(CategoryTheory.ConcreteCategory.hom (SimplexCategory.σ₀Iter i ⋯)) k ≤ Fin.castSucc 0 →
(CategoryTheory.ConcreteCategory.hom (SimplexCategory.σ₀Iter i ⋯)) k ≠ Fin.last (n + 1) | null | false |
List.reduceOption_cons_of_some | Mathlib.Data.List.ReduceOption | ∀ {α : Type u_1} (x : α) (l : List (Option α)), (some x :: l).reduceOption = x :: l.reduceOption | null | true |
Ordinal.invVeblen₂_gamma | Mathlib.SetTheory.Ordinal.Veblen | ∀ (o : Ordinal.{u_1}), o.gamma.invVeblen₂ = 0 | null | true |
IsJordan.mk._flat_ctor | Mathlib.Algebra.Jordan.Basic | ∀ {A : Type u_1} [inst : Mul A],
(∀ (a b : A), a * b * a = a * (b * a)) →
(∀ (a b : A), a * a * (a * b) = a * (a * a * b)) →
(∀ (a b : A), a * a * (b * a) = a * a * b * a) →
(∀ (a b : A), a * b * (a * a) = a * (b * (a * a))) →
(∀ (a b : A), b * a * (a * a) = b * (a * a) * a) → IsJordan A | null | false |
CompactlySupportedContinuousMap.instInf._proof_2 | Mathlib.Topology.ContinuousMap.CompactlySupported | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : SemilatticeInf β] [inst_2 : Zero β]
[inst_3 : TopologicalSpace β] (f g : CompactlySupportedContinuousMap α β), HasCompactSupport (⇑f ⊓ ⇑g) | null | false |
_private.Mathlib.Algebra.Algebra.Bilinear.0.LinearMap.pow_mulLeft.match_1_1 | Mathlib.Algebra.Algebra.Bilinear | ∀ (motive : ℕ → Prop) (n : ℕ), (∀ (_ : Unit), motive 0) → (∀ (n : ℕ), motive n.succ) → motive n | null | false |
_private.Mathlib.Data.Nat.ChineseRemainder.0.Nat.modEq_list_map_prod_iff._simp_1_2 | Mathlib.Data.Nat.ChineseRemainder | ∀ {k : ℕ} {l : List ℕ}, k.Coprime l.prod = ∀ n ∈ l, k.Coprime n | null | false |
_private.Mathlib.RingTheory.Coalgebra.GroupLike.0.isGroupLikeElem_self._simp_1_1 | Mathlib.RingTheory.Coalgebra.GroupLike | ∀ (R : Type u_2) {A : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A]
[inst_3 : Coalgebra R A] (a : A),
IsGroupLikeElem R a = (CoalgebraStruct.counit a = 1 ∧ CoalgebraStruct.comul a = a ⊗ₜ[R] a) | null | false |
CategoryTheory.ComposableArrows.fourδ₁Toδ₀_app_zero | Mathlib.CategoryTheory.ComposableArrows.Four | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {i₀ i₁ i₂ i₃ i₄ : C} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂)
(f₃ : i₂ ⟶ i₃) (f₄ : i₃ ⟶ i₄) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂),
(CategoryTheory.ComposableArrows.fourδ₁Toδ₀ f₁ f₂ f₃ f₄ f₁₂ h₁₂).app 0 = f₁ | null | true |
MeasureTheory.average_smul_const | Mathlib.MeasureTheory.Integral.Average | ∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
(μ : MeasureTheory.Measure α) {𝕜 : Type u_4} [inst_2 : RCLike 𝕜] [inst_3 : NormedSpace 𝕜 E] [CompleteSpace E]
(f : α → 𝕜) (c : E), ⨍ (a : α), f a • c ∂μ = (⨍ (a : α), f a ∂μ) • c | null | true |
Combinatorics.Line._sizeOf_inst | Mathlib.Combinatorics.HalesJewett | (α : Type u_5) → (ι : Type u_6) → [SizeOf α] → [SizeOf ι] → SizeOf (Combinatorics.Line α ι) | null | false |
FiniteArchimedeanClass.lift_mk | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {α : Type u_2}
(f : { a // a ≠ 0 } → α)
(h : ∀ (a b : { a // a ≠ 0 }), FiniteArchimedeanClass.mk ↑a ⋯ = FiniteArchimedeanClass.mk ↑b ⋯ → f a = f b) {a : M}
(ha : a ≠ 0), FiniteArchimedeanClass.lift f h (FiniteArchime... | null | true |
Mathlib.Meta.NormNum.isInt_lcm | Mathlib.Tactic.NormNum.GCD | ∀ {x y nx ny : ℤ} {z : ℕ},
Mathlib.Meta.NormNum.IsInt x nx →
Mathlib.Meta.NormNum.IsInt y ny → nx.lcm ny = z → Mathlib.Meta.NormNum.IsNat (x.lcm y) z | null | true |
FirstOrder.Language.Sentence.cardGe.eq_1 | Mathlib.ModelTheory.Semantics | ∀ (L : FirstOrder.Language) (n : ℕ),
FirstOrder.Language.Sentence.cardGe L n =
(List.foldr (fun x1 x2 => x1 ⊓ x2) ⊤
(List.map
(fun ij =>
(((FirstOrder.Language.var ∘ Sum.inr) ij.1).bdEqual ((FirstOrder.Language.var ∘ Sum.inr) ij.2)).not)
(List.filter (fun ij => decide (ij.1... | null | true |
ZeroAtInftyContinuousMap.unitizationEquiv._proof_1 | Mathlib.Topology.ContinuousMap.ZeroAtInftyUnitization | ∀ (X : Type u_2) (R : Type u_1) [inst : TopologicalSpace X] [inst_1 : R1Space X] [inst_2 : TopologicalSpace R]
[inst_3 : AddCommGroup R] [inst_4 : IsTopologicalAddGroup R] (f : Unitization R (ZeroAtInftyContinuousMap X R)),
Unitization.mk
((ContinuousMap.const (OnePoint X) f.toProd.1 + f.toProd.2.toOnePoint) ... | null | false |
DirectLimit.NonUnitalStarRing.of._proof_4 | Mathlib.Algebra.Colimit.DirectLimit | ∀ {ι : Type u_2} [inst : Preorder ι] (G : ι → Type u_1) {T : ⦃i j : ι⦄ → i ≤ j → Type u_3}
(f : (x x_1 : ι) → (h : x ≤ x_1) → T h) [inst_1 : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)]
[inst_2 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] [inst_3 : IsDirectedOrder ι]
[inst_4 : (i : ι) → NonUnitalNonA... | null | false |
Subsemigroup.instCompleteLattice._proof_14 | Mathlib.Algebra.Group.Subsemigroup.Basic | ∀ {M : Type u_1} [inst : Mul M] (x : Subsemigroup M), ∀ x_1 ∈ ⊥, x_1 ∈ x | null | false |
_private.Mathlib.Tactic.DeriveEncodable.0.Mathlib.Deriving.Encodable.instEncodableS | Mathlib.Tactic.DeriveEncodable | Encodable Mathlib.Deriving.Encodable.S✝ | null | true |
iSup_subtype' | Mathlib.Order.CompleteLattice.Basic | ∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α},
⨆ i, ⨆ (h : p i), f i h = ⨆ x, f ↑x ⋯ | null | true |
BitVec.msb_twoPow | Init.Data.BitVec.Lemmas | ∀ {i w : ℕ}, (BitVec.twoPow w i).msb = (decide (i < w) && decide (i = w - 1)) | null | true |
GrpCat.SurjectiveOfEpiAuxs.tau._proof_1 | Mathlib.Algebra.Category.Grp.EpiMono | ∀ {A B : GrpCat} (f : A ⟶ B), ∃ y, y • ↑(GrpCat.Hom.hom f).range = ↑(GrpCat.Hom.hom f).range | null | false |
MulAction.isPreprimitive_stabilizer_of_surjective | Mathlib.GroupTheory.Perm.MaximalSubgroups | ∀ {M : Type u_1} {α : Type u_2} [inst : Group M] [inst_1 : MulAction M α] (s : Set α),
Function.Surjective MulAction.toPerm → MulAction.IsPreprimitive ↥(MulAction.stabilizer M s) ↑s | In the permutation group, the stabilizer of any set
acts primitively on that set. | true |
HomologicalComplex.dFrom_eq | 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) {i j : ι}
(r : c.Rel i j), C.dFrom i = CategoryTheory.CategoryStruct.comp (C.d i j) (C.xNextIso r).inv | null | true |
_private.Std.Data.DTreeMap.Internal.Model.0.Std.DTreeMap.Internal.Impl.entryAtIdx?.match_1.eq_1 | Std.Data.DTreeMap.Internal.Model | ∀ (motive : Ordering → Sort u_1) (h_1 : Unit → motive Ordering.lt) (h_2 : Unit → motive Ordering.eq)
(h_3 : Unit → motive Ordering.gt),
(match Ordering.lt with
| Ordering.lt => h_1 ()
| Ordering.eq => h_2 ()
| Ordering.gt => h_3 ()) =
h_1 () | null | true |
Turing.PartrecToTM2.move₂ | Mathlib.Computability.TuringMachine.ToPartrec | (Turing.PartrecToTM2.Γ' → Bool) →
Turing.PartrecToTM2.K' → Turing.PartrecToTM2.K' → Turing.PartrecToTM2.Λ' → Turing.PartrecToTM2.Λ' | Move elements from `k₁` to `k₂` without reversion, by performing a double move via the `rev`
stack. | true |
Lean.Widget.RpcEncodablePacket.«_@».Lean.Widget.UserWidget.577854155._hygCtx._hyg.1.recOn | Lean.Widget.UserWidget | {motive : Lean.Widget.RpcEncodablePacket✝ → Sort u} →
(t : Lean.Widget.RpcEncodablePacket✝) → ((widgets : Lean.Json) → motive { widgets := widgets }) → motive t | null | false |
Mathlib.Notation3.mkScopedMatcher | Mathlib.Util.Notation3 | Lean.Name →
Lean.Name → Lean.Term → Array Lean.Name → OptionT Lean.Elab.TermElabM (List Mathlib.Notation3.DelabKey × Lean.Term) | Create a `Term` that represents a matcher for `scoped` notation.
Fails in the `OptionT` sense if a matcher couldn't be constructed.
Also returns a delaborator key like in `mkExprMatcher`.
Reminder: `$lit:ident : (scoped $scopedId:ident => $scopedTerm:Term)` | true |
_private.Mathlib.Algebra.Lie.Weights.Cartan.0.LieAlgebra.mem_zeroRootSubalgebra._simp_1_1 | Mathlib.Algebra.Lie.Weights.Cartan | ∀ {R : Type u_2} {L : Type u_3} (M : Type u_4) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
[inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [inst_6 : LieModule R L M]
[inst_7 : LieRing.IsNilpotent L] (χ : L → R) (m : M),
(m ∈ LieModule.genWeightSpace M χ) = ∀ (x ... | null | false |
_private.Mathlib.ModelTheory.Semantics.0.FirstOrder.Language.model_distinctConstantsTheory._simp_1_1 | Mathlib.ModelTheory.Semantics | ∀ {α : Type u} {β : Type v} (f : α → β) (s : Set α) (y : β), (y ∈ f '' s) = ∃ x ∈ s, f x = y | null | false |
Ordinal.omega | Mathlib.SetTheory.Cardinal.Aleph | Ordinal.{u_1} ↪o Ordinal.{u_1} | The `omega` function gives the infinite initial ordinals listed by their ordinal index.
`omega 0 = ω`, `omega 1 = ω₁` is the first uncountable ordinal, and so on.
This is not to be confused with the first infinite ordinal `Ordinal.omega0`.
For a version including finite ordinals, see `Ordinal.preOmega`.
Conventions... | true |
AddChar.circleEquivComplex._proof_5 | Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | ∀ {α : Type u_1} [inst : AddCommGroup α] [inst_1 : Finite α] (ψ : AddChar α ℂ),
(fun ψ => AddChar.toMonoidHomEquiv.symm (Circle.coeHom.comp ψ.toMonoidHom))
((fun ψ => { toFun := fun a => ⟨ψ a, ⋯⟩, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ }) ψ) =
ψ | null | false |
Lean.Parser.Term.subst.parenthesizer | Lean.Parser.Term | Lean.PrettyPrinter.Parenthesizer | null | true |
Ideal.map_sup_comap_of_surjective | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S] (f : F)
[inst_3 : RingHomClass F R S],
Function.Surjective ⇑f → ∀ (I J : Ideal S), Ideal.map f (Ideal.comap f I ⊔ Ideal.comap f J) = I ⊔ J | null | true |
Homeomorph.mulRight | Mathlib.Topology.Algebra.Group.Basic | {G : Type u_1} → [inst : TopologicalSpace G] → [inst_1 : Group G] → [SeparatelyContinuousMul G] → G → G ≃ₜ G | Multiplication from the right in a topological group as a homeomorphism. | true |
CategoryTheory.SimplicialObject.Splitting.toKaroubiNondegComplexIsoN₁_hom_f_PInfty_assoc | Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X : CategoryTheory.SimplicialObject C} (s : X.Splitting)
[inst_1 : CategoryTheory.Preadditive C] {Z : ChainComplex C ℕ}
(h : AlgebraicTopology.AlternatingFaceMapComplex.obj X ⟶ Z),
CategoryTheory.CategoryStruct.comp s.toKaroubiNondegComplexIsoN₁.hom.... | null | true |
Batteries.Tactic.Lint.isAutoDecl | Batteries.Tactic.Lint.Basic | {m : Type → Type} → [Monad m] → [Lean.MonadEnv m] → Lean.Name → m Bool | Returns true if `decl` is an automatically generated declaration.
Also returns true if `decl` is an internal name or created during macro
expansion.
See `Lean.Environment.isAutoDecl` for an identical pure version of this function on the environment.
| true |
CategoryTheory.Mon.tensorUnit_mul | Mathlib.CategoryTheory.Monoidal.Mon | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C],
CategoryTheory.MonObj.mul =
(CategoryTheory.MonoidalCategoryStruct.leftUnitor (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).hom | null | true |
Turing.TM2to1.trStmts₁.eq_3 | Mathlib.Computability.TuringMachine.StackTuringMachine | ∀ {K : Type u_1} {Γ : K → Type u_2} {Λ : Type u_3} {σ : Type u_4} (k : K) (f : σ → Option (Γ k) → σ)
(q : Turing.TM2.Stmt Γ Λ σ),
Turing.TM2to1.trStmts₁ (Turing.TM2.Stmt.pop k f q) =
{Turing.TM2to1.Λ'.go k (Turing.TM2to1.StAct.pop f) q, Turing.TM2to1.Λ'.ret q} ∪ Turing.TM2to1.trStmts₁ q | null | true |
_private.Lean.Parser.Term.0.Lean.Parser.Term.proj._regBuiltin.Lean.Parser.Term.proj_1 | Lean.Parser.Term | IO Unit | null | false |
_private.Mathlib.MeasureTheory.Integral.SetToL1.ChangeMeasure.0.MeasureTheory.setToFun_top_smul_measure._simp_1_3 | Mathlib.MeasureTheory.Integral.SetToL1.ChangeMeasure | (¬False) = True | null | false |
Module.DirectLimit.of._proof_3 | Mathlib.Algebra.Colimit.Module | ∀ (R : Type u_3) [inst : Semiring R] (ι : Type u_1) [inst_1 : Preorder ι] (G : ι → Type u_2)
[inst_2 : (i : ι) → AddCommMonoid (G i)] [inst_3 : (i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j)
[inst_4 : DecidableEq ι] (x : R) (x_1 : DirectSum ι G),
(↑(Module.DirectLimit.moduleCon f).mk').toFun (x... | null | false |
_private.Mathlib.Tactic.ClickSuggestions.Unfold.0.Mathlib.Tactic.ClickSuggestions.unfoldProjDefaultInst?.match_10 | Mathlib.Tactic.ClickSuggestions.Unfold | (motive : Option Lean.ConstantInfo → Sort u_1) →
(x : Option Lean.ConstantInfo) →
((ci : Lean.ConstructorVal) → motive (some (Lean.ConstantInfo.ctorInfo ci))) →
((x : Option Lean.ConstantInfo) → motive x) → motive x | null | false |
Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave'_1 | Init.Tactics | Lean.Macro | Similar to `have`, but using `refine'` | false |
Real.analyticOn_cos | Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | ∀ {s : Set ℝ}, AnalyticOn ℝ Real.cos s | The function `Real.cos` is real analytic. | true |
DirectLimit.instMulDistribMulActionOfMulActionHomClass._proof_4 | Mathlib.Algebra.Colimit.DirectLimit | ∀ {R : Type u_4} {ι : Type u_1} [inst : Preorder ι] {G : ι → Type u_2} {T : ⦃i j : ι⦄ → i ≤ j → Type u_3}
{f : (x x_1 : ι) → (h : x ≤ x_1) → T h} [inst_1 : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)]
[inst_2 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] [inst_3 : IsDirectedOrder ι] [inst_4 : Nonempty ι... | null | false |
HasMFDerivWithinAt._proof_2 | Mathlib.Geometry.Manifold.MFDeriv.Defs | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜], RingHomInvPair (RingHom.id 𝕜) (RingHom.id 𝕜) | null | false |
_private.Mathlib.CategoryTheory.Triangulated.Subcategory.0.CategoryTheory.ObjectProperty.extensionProduct_retractClosure_retractClosure_le._proof_1_3 | Mathlib.CategoryTheory.Triangulated.Subcategory | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.HasShift C ℤ] (A B : C)
(f₃ : B ⟶ (CategoryTheory.shiftFunctor C 1).obj A) (A' : C) (a₁ : A ⟶ A') (B' : C) (a₃ : B ⟶ B') (b₃ : B' ⟶ B),
CategoryTheory.CategoryStruct.comp a₃ b₃ = CategoryTheory.CategoryStruct.id B →
Category... | null | false |
Subgroup.rightCosetEquivSubgroup | Mathlib.GroupTheory.Coset.Basic | {α : Type u_1} → [inst : Group α] → {s : Subgroup α} → (g : α) → ↑(MulOpposite.op g • ↑s) ≃ ↥s | The natural bijection between a right coset `s * g` and `s`. | true |
Lean.CollectFVars.State.fvarIds._default | Lean.Util.CollectFVars | Array Lean.FVarId | null | false |
Lean.Lsp.instDecidableEqCompletionItemKind._proof_2 | Lean.Data.Lsp.LanguageFeatures | ∀ (x y : Lean.Lsp.CompletionItemKind), ¬x.ctorIdx = y.ctorIdx → x = y → False | null | false |
_private.Init.Data.String.Pattern.String.0.String.Slice.Pattern.ForwardSliceSearcher.buildTable.go.induct_unfolding | Init.Data.String.Pattern.String | ∀ (pat : String.Slice)
(motive :
(table : Array ℕ) →
0 < table.size →
table.size ≤ pat.utf8ByteSize →
(∀ (i : ℕ) (hi : i < table.size), table[i] ≤ i) → Vector ℕ pat.utf8ByteSize → Prop),
(∀ (table : Array ℕ) (ht₀ : 0 < table.size) (ht : table.size ≤ pat.utf8ByteSize)
(h : ∀ (i : ℕ)... | null | true |
RatFunc.instCommRing._proof_5 | Mathlib.FieldTheory.RatFunc.Basic | ∀ (K : Type u_1) [inst : CommRing K], Nat.unaryCast 0 = 0 | null | false |
HahnSeries.instIsScalarTower | Mathlib.RingTheory.HahnSeries.Addition | ∀ {Γ : Type u_1} {R : Type u_3} [inst : PartialOrder Γ] {V : Type u_8} [inst_1 : Monoid R] [inst_2 : AddMonoid V]
[inst_3 : DistribMulAction R V] {S : Type u_9} [inst_4 : Monoid S] [inst_5 : DistribMulAction S V] [inst_6 : SMul R S]
[IsScalarTower R S V], IsScalarTower R S (HahnSeries Γ V) | null | true |
Array.toListLitAux._f | Init.Data.Array.GetLit | {α : Type u_1} →
(xs : Array α) →
(n : ℕ) →
xs.size = n →
(x : ℕ) → Nat.below (motive := fun x => x ≤ xs.size → List α → List α) x → x ≤ xs.size → List α → List α | null | false |
Computation.liftRel_pure_right._simp_1 | Mathlib.Data.Seq.Computation | ∀ {α : Type u} {β : Type v} (R : α → β → Prop) (ca : Computation α) (b : β),
Computation.LiftRel R ca (Computation.pure b) = ∃ a ∈ ca, R a b | null | false |
Float.Model.Format.mk | Init.Data.Float.Model.Format.Basic | (mantissaBitsWithoutImplicit : ℕ) →
autoParam (0 < mantissaBitsWithoutImplicit) Float.Model.Format.hm._autoParam →
(exponentBits : ℕ) → autoParam (2 ≤ exponentBits) Float.Model.Format.he._autoParam → Float.Model.Format | null | true |
Lean.Parser.testParseFile | Lean.Parser.Module | Lean.Environment → System.FilePath → IO Lean.Syntax | null | true |
Set.Finite.eq_insert_of_subset_of_encard_eq_succ | Mathlib.Data.Set.Card | ∀ {α : Type u_1} {s t : Set α}, s.Finite → s ⊆ t → t.encard = s.encard + 1 → ∃ a, t = insert a s | null | true |
Lean.Expr.getRevArg!._sunfold | Lean.Expr | Lean.Expr → ℕ → Lean.Expr | null | false |
Set.pi.eq_1 | Mathlib.Data.Set.Prod | ∀ {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) (t : (i : ι) → Set (α i)), s.pi t = {f | ∀ i ∈ s, f i ∈ t i} | null | true |
HomologicalComplex.homologyι_singleObjOpcyclesSelfIso_inv | Mathlib.Algebra.Homology.SingleHomology | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [inst_3 : DecidableEq ι] (c : ComplexShape ι) (j : ι)
(A : C),
CategoryTheory.CategoryStruct.comp (((HomologicalComplex.single C c j).obj A).... | null | true |
_private.Init.Data.List.Lemmas.0.List.map_eq_nil_iff.match_1_1 | Init.Data.List.Lemmas | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} (motive : (l : List α) → List.map f l = [] → Prop) (l : List α)
(x : List.map f l = []), (∀ (x : List.map f [] = []), motive [] x) → motive l x | null | false |
CategoryTheory.Abelian.SpectralObject.d_d._auto_5 | Mathlib.Algebra.Homology.SpectralObject.Differentials | Lean.Syntax | null | false |
Array.countP_push_of_neg | Init.Data.Array.Count | ∀ {α : Type u_1} {p : α → Bool} {a : α} {xs : Array α}, ¬p a = true → Array.countP p (xs.push a) = Array.countP p xs | null | true |
Complex.HadamardThreeLines.norm_invInterpStrip | Mathlib.Analysis.Complex.Hadamard | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) {ε : ℝ},
ε > 0 →
‖Complex.HadamardThreeLines.invInterpStrip f z ε‖ =
(ε + Complex.HadamardThreeLines.sSupNormIm f 0) ^ (z.re - 1) *
(ε + Complex.HadamardThreeLines.sSupNormIm f 1) ^ (-z.re) | Useful rewrite for the absolute value of `invInterpStrip` | true |
Submodule.moduleSubmodule._proof_1 | Mathlib.RingTheory.Ideal.Operations | ∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(b : Submodule R M), 1 • b = b | null | false |
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