name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
IsDedekindDomain.HeightOneSpectrum.under._proof_1 | Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | ∀ (A : Type u_1) [inst : CommRing A] {B : Type u_2} [inst_1 : CommRing B] [inst_2 : Algebra A B]
(w : IsDedekindDomain.HeightOneSpectrum B), (Ideal.under A w.asIdeal).IsPrime | null | false |
_private.Init.Data.String.Lemmas.Pattern.Char.0.String.Slice.Pattern.Model.Char.isValidSearchFrom_iff_isValidSearchFrom_beq._simp_1_2 | Init.Data.String.Lemmas.Pattern.Char | ∀ {c : Char} {s : String.Slice} {pos pos' : s.Pos},
String.Slice.Pattern.Model.IsLongestMatchAt c pos pos' =
String.Slice.Pattern.Model.IsLongestMatchAt (fun x => x == c) pos pos' | null | false |
_private.Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order.0.isStrictlyPositive_add._proof_1_1 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | ∀ {A : Type u_1} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {a b : A},
IsStrictlyPositive a ∧ 0 ≤ b ∨ 0 ≤ a ∧ IsStrictlyPositive b → IsStrictlyPositive (a + b) | null | false |
_private.Mathlib.Algebra.Star.Module.0.selfAdjointPart_comp_subtype_skewAdjoint.match_1_1 | Mathlib.Algebra.Star.Module | ∀ (R : Type u_2) {A : Type u_1} [inst : Semiring R] [inst_1 : StarMul R] [inst_2 : TrivialStar R]
[inst_3 : AddCommGroup A] [inst_4 : Module R A] [inst_5 : StarAddMonoid A] [inst_6 : StarModule R A]
(motive : ↥(skewAdjoint.submodule R A) → Prop) (x : ↥(skewAdjoint.submodule R A)),
(∀ (x : A) (hx : star x = -x), m... | null | false |
Std.ExtDTreeMap.Const.alter_eq_empty_iff | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : Std.ExtDTreeMap α (fun x => β) cmp} [inst : Std.TransCmp cmp]
{k : α} {f : Option β → Option β},
Std.ExtDTreeMap.Const.alter t k f = ∅ ↔ (t = ∅ ∨ t.size = 1 ∧ k ∈ t) ∧ f (Std.ExtDTreeMap.Const.get? t k) = none | null | true |
Set.image_list_prod._f | Mathlib.Algebra.Group.Pointwise.Set.BigOperators | ∀ {α : Type u_2} {β : Type u_3} {F : Type u_4} [inst : FunLike F α β] [inst_1 : Monoid α] [inst_2 : Monoid β]
[MonoidHomClass F α β] (f : F) (x : List (Set α)) (f_1 : List.below x),
⇑f '' x.prod = (List.map (fun s => ⇑f '' s) x).prod | null | false |
CategoryTheory.Sheaf.instPreservesFiniteLimitsFunctorOppositeSheafToPresheafOfHasFiniteLimits | Mathlib.CategoryTheory.Sites.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {D : Type w}
[inst_1 : CategoryTheory.Category.{w', w} D] [CategoryTheory.Limits.HasFiniteLimits D],
CategoryTheory.Limits.PreservesFiniteLimits (CategoryTheory.sheafToPresheaf J D) | null | true |
AddCon.addSubgroup_quotientAddGroupCon | Mathlib.GroupTheory.QuotientGroup.Defs | ∀ {G : Type u_1} [inst : AddGroup G] (H : AddSubgroup G) [inst_1 : H.Normal], (QuotientAddGroup.con H).addSubgroup = H | null | true |
ContinuousMultilinearMap.uniformContinuous_restrictScalars | Mathlib.Topology.Algebra.Module.Multilinear.Topology | ∀ {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [inst : NormedField 𝕜]
[inst_1 : (i : ι) → TopologicalSpace (E i)] [inst_2 : (i : ι) → AddCommGroup (E i)]
[inst_3 : (i : ι) → Module 𝕜 (E i)] [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F] [inst_6 : UniformSpace F]
[inst_7 : IsUniformAddGroup... | null | true |
BoxIntegral.unitPartition.prepartition_isHenstock | Mathlib.Analysis.BoxIntegral.UnitPartition | ∀ {ι : Type u_1} (n : ℕ) [inst : NeZero n] [inst_1 : Fintype ι] (B : BoxIntegral.Box ι),
(BoxIntegral.unitPartition.prepartition n B).IsHenstock | null | true |
Finset.gcd_congr | Mathlib.Algebra.GCDMonoid.Finset | ∀ {α : Type u_2} {β : Type u_3} [inst : CommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] {s₁ s₂ : Finset β}
{f g : β → α}, s₁ = s₂ → (∀ a ∈ s₂, f a = g a) → s₁.gcd f = s₂.gcd g | null | true |
LowerSet.notMem_bot._simp_1 | Mathlib.Order.UpperLower.CompleteLattice | ∀ {α : Type u_1} [inst : LE α] {a : α}, (a ∈ ⊥) = False | null | false |
Std.DHashMap.Equiv.constGet_eq | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m₁ m₂ : Std.DHashMap α fun x => β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] {k : α} (hk : k ∈ m₁) (h : m₁.Equiv m₂),
Std.DHashMap.Const.get m₁ k hk = Std.DHashMap.Const.get m₂ k ⋯ | null | true |
CategoryTheory.NatTrans.removeOp._proof_2 | Mathlib.CategoryTheory.Opposites | ∀ {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] {F G : CategoryTheory.Functor C D} (α : F.op ⟶ G.op) (X Y : C)
(f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp (G.map f) (α.app (Opposite.op Y)).unop =
CategoryTheory.CategoryStruct.co... | null | false |
Valuation.mem_nhds_iff | Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology | ∀ {R : Type u_1} [inst : Ring R] [inst_1 : ValuativeRel R] {Γ₀ : Type u_3} [inst_2 : LinearOrderedCommGroupWithZero Γ₀]
[inst_3 : TopologicalSpace R] (v : Valuation R Γ₀) [v.Compatible] [IsValuativeTopology R] {s : Set R} {x : R},
s ∈ nhds x ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s | null | true |
_private.Std.Http.Protocol.H1.Message.0.Std.Http.Protocol.H1.Message.Head.getSize.match_3 | Std.Http.Protocol.H1.Message | (motive : Option Bool → Option (Array Std.Http.Header.Value) → Sort u_1) →
(x : Option Bool) →
(contentLength : Option (Array Std.Http.Header.Value)) →
(Unit → motive (some true) none) →
((x : Option Bool) → (x_1 : Option (Array Std.Http.Header.Value)) → motive x x_1) → motive x contentLength | null | false |
Std.TreeMap.getKeyD_insert_self | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {a fallback : α}
{b : β}, (t.insert a b).getKeyD a fallback = a | null | true |
_private.Init.Data.String.Basic.0.String.Slice.utf8ByteSize_slice._proof_1_2 | Init.Data.String.Basic | ∀ {s : String.Slice} {newStart newEnd : s.Pos},
¬s.startInclusive.offset.byteIdx + newEnd.offset.byteIdx -
(s.startInclusive.offset.byteIdx + newStart.offset.byteIdx) =
newEnd.offset.byteIdx - newStart.offset.byteIdx →
False | null | false |
Representation.instAddCommGroupAsModule._proof_8 | Mathlib.RepresentationTheory.Basic | ∀ {k : Type u_1} {G : Type u_2} {V : Type u_3} [inst : Ring k] [inst_1 : Monoid G] [inst_2 : AddCommGroup V]
[inst_3 : Module k V] (ρ : Representation k G V),
autoParam (∀ (a : ρ.asModule), 0 • a = 0) SubNegMonoid.zsmul_zero'._autoParam | null | false |
PositiveLinearMap.toOrderHom_comp | Mathlib.Algebra.Order.Module.PositiveLinearMap | ∀ {R : Type u_1} {E₁ : Type u_2} {E₂ : Type u_3} {E₃ : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid E₁]
[inst_2 : PartialOrder E₁] [inst_3 : AddCommMonoid E₂] [inst_4 : PartialOrder E₂] [inst_5 : AddCommMonoid E₃]
[inst_6 : PartialOrder E₃] [inst_7 : Module R E₁] [inst_8 : Module R E₂] [inst_9 : Module R E... | null | true |
_private.Lean.Elab.DeclNameGen.0.Lean.Elab.Command.NameGen.mkBaseNameCore.visit'.eq_def | Lean.Elab.DeclNameGen | ∀ (e : Lean.Expr) (omitTopForall : Bool),
Lean.Elab.Command.NameGen.mkBaseNameCore.visit'✝ e omitTopForall =
match e with
| Lean.Expr.const name us => do
modify fun st =>
{ seen := Lean.Elab.Command.NameGen.MkNameState.seen✝ st,
consts := (Lean.Elab.Command.NameGen.MkNameState.cons... | null | true |
TopCat.ι₁_fst_assoc | Mathlib.Topology.Category.TopCat.Monoidal | ∀ (X : TopCat) {Z : TopCat} (h : X ⟶ Z),
CategoryTheory.CategoryStruct.comp TopCat.ι₁
(CategoryTheory.CategoryStruct.comp (CategoryTheory.SemiCartesianMonoidalCategory.fst X TopCat.I) h) =
h | null | true |
CategoryTheory.rightExactFunctor | Mathlib.CategoryTheory.Limits.ExactFunctor | (C : Type u₁) →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
(D : Type u₂) →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] → CategoryTheory.ObjectProperty (CategoryTheory.Functor C D) | Right-exactness, as a property of objects in `C ⥤ D`. | true |
Mathlib.Tactic.ClickSuggestions.RwKind.hasBVars.elim | Mathlib.Tactic.ClickSuggestions.Util | {motive : Mathlib.Tactic.ClickSuggestions.RwKind → Sort u} →
(t : Mathlib.Tactic.ClickSuggestions.RwKind) →
t.ctorIdx = 0 → motive Mathlib.Tactic.ClickSuggestions.RwKind.hasBVars → motive t | null | false |
DilationEquiv.coe_one | Mathlib.Topology.MetricSpace.DilationEquiv | ∀ {X : Type u_1} [inst : PseudoEMetricSpace X], ⇑1 = id | null | true |
OrderType.instOfNat | Mathlib.Order.Types.Arithmetic | (n : ℕ) → OfNat OrderType.{0} n | null | true |
Std.TreeSet.getD_diff_of_mem_right | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeSet α cmp} [Std.TransCmp cmp] {k fallback : α},
k ∈ t₂ → (t₁ \ t₂).getD k fallback = fallback | null | true |
List.length_mapFinIdx | Init.Data.List.MapIdx | ∀ {α : Type u_1} {β : Type u_2} {as : List α} {f : (i : ℕ) → α → i < as.length → β}, (as.mapFinIdx f).length = as.length | null | true |
Subalgebra.normedCommRing._proof_1 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {𝕜 : Type u_2} [inst : CommRing 𝕜] {E : Type u_1} [inst_1 : NormedCommRing E] [inst_2 : Algebra 𝕜 E]
(s : Subalgebra 𝕜 E) {x y : ↥s}, dist x y = 0 → x = y | null | false |
RelSeries.head_fromListIsChain | Mathlib.Order.RelSeries | ∀ {α : Type u_1} {r : SetRel α α} (l : List α) (l_ne_nil : l ≠ []) (hl : List.IsChain (fun x1 x2 => (x1, x2) ∈ r) l),
(RelSeries.fromListIsChain l l_ne_nil hl).head = l.head l_ne_nil | null | true |
_private.Mathlib.Tactic.Hint.0.Mathlib.Tactic.Hint._aux_Mathlib_Tactic_Hint___elabRules_Mathlib_Tactic_Hint_registerHintStx_1._sparseCasesOn_1 | Mathlib.Tactic.Hint | {α : Type u} →
{motive : Option α → Sort u_1} →
(t : Option α) → ((val : α) → motive (some val)) → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
Graph.IsLoopAt.mono | Mathlib.Combinatorics.Graph.Subgraph | ∀ {α : Type u_1} {β : Type u_2} {x : α} {e : β} {G H : Graph α β}, H ≤ G → H.IsLoopAt e x → G.IsLoopAt e x | null | true |
_private.Mathlib.Data.Seq.Basic.0.Stream'.Seq.take.match_1.splitter | Mathlib.Data.Seq.Basic | {α : Type u_1} →
(motive : ℕ → Stream'.Seq α → Sort u_2) →
(x : ℕ) →
(x_1 : Stream'.Seq α) →
((x : Stream'.Seq α) → motive 0 x) → ((n : ℕ) → (s : Stream'.Seq α) → motive n.succ s) → motive x x_1 | null | true |
IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots | Mathlib.NumberTheory.Cyclotomic.Basic | ∀ {A : Type u} {B : Type v} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
{ζ : B} {n : ℕ} [NeZero n],
IsPrimitiveRoot ζ n →
Algebra.adjoin A ((Polynomial.cyclotomic n A).rootSet B) = Algebra.adjoin A {b | ∃ a ∈ {n}, a ≠ 0 ∧ b ^ a = 1} | null | true |
ExteriorAlgebra.ι_inj._simp_1 | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | ∀ (R : Type u1) [inst : CommRing R] {M : Type u2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (x y : M),
((ExteriorAlgebra.ι R) x = (ExteriorAlgebra.ι R) y) = (x = y) | null | false |
CategoryTheory.Sheaf.H.longSequenceHom._auto_1 | Mathlib.CategoryTheory.Sites.SheafCohomology.ExactSequences | Lean.Syntax | null | false |
CategoryTheory.NatTrans.leftOpWhiskerRight_assoc | Mathlib.CategoryTheory.Opposites | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C Dᵒᵖ} {E : Type u_1} [inst_2 : CategoryTheory.Category.{v_1, u_1} E]
{H : CategoryTheory.Functor E C} (α : F ⟶ G) {Z : CategoryTheory.Functor Eᵒᵖ D} (h : (H.comp F).... | null | true |
NumberField.IsCMField.complexConj_eq_self_iff | Mathlib.NumberTheory.NumberField.CMField | ∀ (K : Type u_1) [inst : Field K] [inst_1 : CharZero K] [inst_2 : NumberField.IsCMField K]
[inst_3 : Algebra.IsIntegral ℚ K] (x : K),
(NumberField.IsCMField.complexConj K) x = x ↔ x ∈ NumberField.maximalRealSubfield K | An element of `K` is fixed by the complex conjugation iff it lies in `K⁺`.
| true |
_private.Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape.0.CategoryTheory.ObjectProperty.limitsOfShape_isEmpty_iff.match_1_1 | Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] (P : CategoryTheory.ObjectProperty C) (J : Type u_2)
[inst_1 : CategoryTheory.Category.{u_1, u_2} J] (X : C) (motive : P.limitsOfShape J X → Prop)
(x : P.limitsOfShape J X),
(∀ (f : CategoryTheory.Functor J C) (p : (CategoryTheory.Functor.const J).obj... | null | false |
IsSemitopologicalSemiring | Mathlib.Topology.Algebra.Ring.Basic | (R : Type u_2) → [TopologicalSpace R] → [NonUnitalNonAssocSemiring R] → Prop | A semitopological semiring is a semiring `R` where addition is jointly continuous and
multiplication is continuous in each variable separately.
We allow for non-unital and non-associative semirings as well.
The `IsSemitopologicalSemiring` class should *only* be instantiated in the presence of a
`NonUnitalNonAssocSemir... | true |
LieAlgebra.IsKilling.disjoint_ker_weight_corootSpace | Mathlib.Algebra.Lie.Weights.Killing | ∀ {K : Type u_2} {L : Type u_3} [inst : LieRing L] [inst_1 : Field K] [inst_2 : LieAlgebra K L]
[inst_3 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_4 : H.IsCartanSubalgebra] [LieAlgebra.IsKilling K L]
[LieModule.IsTriangularizable K (↥H) L] [inst_7 : CharZero K] (α : LieModule.Weight K (↥H) L),
Disjoin... | null | true |
edist_nndist | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u} [inst : PseudoMetricSpace α] (x y : α), edist x y = ↑(nndist x y) | Express `edist` in terms of `nndist` | true |
List.forM_nil | Init.Data.List.Control | ∀ {m : Type u_1 → Type u_2} {α : Type u_3} [inst : Monad m] {f : α → m PUnit.{u_1 + 1}}, forM [] f = pure PUnit.unit | null | true |
CategoryTheory.CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse._proof_2 | Mathlib.CategoryTheory.Comma.Over.Basic | ∀ {T : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} T] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] (F : CategoryTheory.Functor T D) (Y : D) (X : T)
{Y_1 Z : CategoryTheory.CostructuredArrow ((CategoryTheory.Over.forget X).comp F) Y} (m : Y_1 ⟶ Z),
CategoryTheory.CategoryStruct.comp
... | null | false |
PNat.XgcdType.mk.sizeOf_spec | Mathlib.Data.PNat.Xgcd | ∀ (wp x y zp ap bp : ℕ),
sizeOf { wp := wp, x := x, y := y, zp := zp, ap := ap, bp := bp } =
1 + sizeOf wp + sizeOf x + sizeOf y + sizeOf zp + sizeOf ap + sizeOf bp | null | true |
linearOrderOfSTO._proof_1 | Mathlib.Order.RelClasses | ∀ {α : Type u_1} (r : α → α → Prop) [IsStrictTotalOrder α r] (x y : α), ¬r y x ↔ x = y ∨ r x y | null | false |
CategoryTheory.ProjectiveResolution.quasiIso._autoParam | Mathlib.CategoryTheory.Preadditive.Projective.Resolution | Lean.Syntax | null | false |
bihimp_comm | Mathlib.Order.SymmDiff | ∀ {α : Type u_2} [inst : GeneralizedHeytingAlgebra α] (a b : α), bihimp a b = bihimp b a | null | true |
Lean.Expr.proj.elim | Lean.Expr | {motive : Lean.Expr → Sort u} →
(t : Lean.Expr) →
t.ctorIdx = 11 →
((typeName : Lean.Name) → (idx : ℕ) → (struct : Lean.Expr) → motive (Lean.Expr.proj typeName idx struct)) →
motive t | null | false |
CategoryTheory.PreZeroHypercoverFamily._sizeOf_inst | Mathlib.CategoryTheory.Sites.Hypercover.ZeroFamily | (C : Type u) →
{inst : CategoryTheory.Category.{v, u} C} → [SizeOf C] → SizeOf (CategoryTheory.PreZeroHypercoverFamily C) | null | false |
Lean.Lsp.InitializationOptions.ctorIdx | Lean.Data.Lsp.InitShutdown | Lean.Lsp.InitializationOptions → ℕ | null | false |
instDecidableIrrationalSqrtOfNatReal | Mathlib.NumberTheory.Real.Irrational | {n : ℕ} → [inst : n.AtLeastTwo] → Decidable (Irrational √(OfNat.ofNat n)) | This can be used as
```lean
unseal Nat.sqrt.iter in
example : Irrational √24 := by decide
```
| true |
mul_le_mul_of_nonpos_of_nonneg' | Mathlib.Algebra.Order.Ring.Unbundled.Basic | ∀ {R : Type u} [inst : Semiring R] [inst_1 : Preorder R] {a b c d : R} [ExistsAddOfLE R] [PosMulMono R] [MulPosMono R]
[AddRightMono R] [AddRightReflectLE R], c ≤ a → b ≤ d → 0 ≤ a → d ≤ 0 → a * b ≤ c * d | null | true |
ZeroAtInftyContinuousMap.rec | Mathlib.Topology.ContinuousMap.ZeroAtInfty | {α : Type u} →
{β : Type v} →
[inst : TopologicalSpace α] →
[inst_1 : Zero β] →
[inst_2 : TopologicalSpace β] →
{motive : ZeroAtInftyContinuousMap α β → Sort u_1} →
((toContinuousMap : C(α, β)) →
(zero_at_infty' : Filter.Tendsto toContinuousMap.toFun (Filter.coc... | null | false |
CategoryTheory.ShortComplex.Exact.isIso_imageToKernel | Mathlib.CategoryTheory.Abelian.Exact | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Abelian C]
(S : CategoryTheory.ShortComplex C), S.Exact → CategoryTheory.IsIso (imageToKernel S.f S.g ⋯) | null | true |
Lean.KeyedDeclsAttribute.ExtensionState.declNames | Lean.KeyedDeclsAttribute | {γ : Type} → Lean.KeyedDeclsAttribute.ExtensionState γ → Lean.PHashSet Lean.Name | null | true |
LinearMap.mem_submoduleImage._simp_1 | Mathlib.Algebra.Module.Submodule.Range | ∀ {R : Type u_1} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_10}
[inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] {O : Submodule R M} {ϕ : ↥O →ₗ[R] M'} {N : Submodule R M} {x : M'},
(x ∈ ϕ.submoduleImage N) = ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x | null | false |
Submonoid.closure_insert_one | Mathlib.Algebra.Group.Submonoid.Basic | ∀ {M : Type u_1} [inst : MulOneClass M] (s : Set M), Submonoid.closure (insert 1 s) = Submonoid.closure s | null | true |
Lean.Meta.Grind.Arith.Cutsat.SymbolicIntInterval.isFinite | Lean.Meta.Tactic.Grind.Arith.Cutsat.ToIntInfo | Lean.Meta.Grind.Arith.Cutsat.SymbolicIntInterval → Bool | null | true |
Std.DHashMap.Const.size_le_size_insertMany_list | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m : Std.DHashMap α fun x => β} [EquivBEq α]
[LawfulHashable α] {l : List (α × β)}, m.size ≤ (Std.DHashMap.Const.insertMany m l).size | null | true |
AdjoinRoot.instCommRing | Mathlib.RingTheory.AdjoinRoot | {R : Type u_1} → [inst : CommRing R] → (f : Polynomial R) → CommRing (AdjoinRoot f) | null | true |
_private.Batteries.Data.Float.Lemmas.0.instPartialEquivBEqFloat_batteries._proof_2 | Batteries.Data.Float.Lemmas | ∀ {a b c : Float}, (a == b) = true → (b == c) = true → (a == c) = true | null | false |
SimpleGraph.Walk.support_toPath_subset | Mathlib.Combinatorics.SimpleGraph.Paths | ∀ {V : Type u} {G : SimpleGraph V} {u v : V} [inst : DecidableEq V] (p : G.Walk u v), (↑p.toPath).support ⊆ p.support | **Alias** of `SimpleGraph.Walk.support_toPath_subset_support`. | true |
MeasurableSet.image_of_measurable_injOn | Mathlib.MeasureTheory.Constructions.Polish.Basic | ∀ {γ : Type u_3} {α : Type u_4} [inst : MeasurableSpace α] {s : Set γ} {f : γ → α}
[MeasurableSpace.CountablySeparated α] [inst_2 : MeasurableSpace γ] [StandardBorelSpace γ],
MeasurableSet s → Measurable f → Set.InjOn f s → MeasurableSet (f '' s) | The Lusin-Souslin theorem: if `s` is Borel-measurable in a standard Borel space,
then its image under a measurable injective map taking values in a
countably separate measurable space is also Borel-measurable. | true |
_private.Mathlib.LinearAlgebra.Pi.0.LinearMap.pi_eq_zero._simp_1_2 | Mathlib.LinearAlgebra.Pi | ∀ {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}, (f = g) = ∀ (x : α), f x = g x | null | false |
RelEmbedding.recOn | Mathlib.Order.RelIso.Basic | {α : Type u_5} →
{β : Type u_6} →
{r : α → α → Prop} →
{s : β → β → Prop} →
{motive : r ↪r s → Sort u} →
(t : r ↪r s) →
((toEmbedding : α ↪ β) →
(map_rel_iff' : ∀ {a b : α}, s (toEmbedding a) (toEmbedding b) ↔ r a b) →
motive { toEmbedding := t... | null | false |
Std.Internal.Do.CompleteLattice.le_iff_forall_le_3 | Std.Internal.Do.Order.Lemmas | ∀ {l : Type u} [inst : Lean.Order.CompleteLattice l] {σ₁ σ₂ σ₃ : Type v} {P Q : σ₁ → σ₂ → σ₃ → l},
Lean.Order.PartialOrder.rel P Q ↔
∀ (s₁ : σ₁) (s₂ : σ₂) (s₃ : σ₃), Lean.Order.PartialOrder.rel (P s₁ s₂ s₃) (Q s₁ s₂ s₃) | null | true |
IdealFilter.isTorsion_def | Mathlib.RingTheory.IdealFilter.Basic | ∀ {A : Type u_1} [inst : Ring A] (F : IdealFilter A) (M : Type u_2) [inst_1 : AddCommMonoid M] [inst_2 : Module A M],
F.IsTorsion M ↔ ∀ (m : M), F.IsTorsionElem m | null | true |
_private.Mathlib.Algebra.Homology.HomotopyCategory.HomComplex.0.CochainComplex.HomComplex.Cocycle.homOf._simp_1 | Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | ∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a - b = 0) = (a = b) | null | false |
_private.Init.Data.String.Decode.0.ByteArray.utf8DecodeChar?.val_assemble₁_le._proof_1_1 | Init.Data.String.Decode | ∀ {w : UInt8}, w.toNat < 128 → ¬w.toNat ≤ 127 → False | null | false |
Concept.instPartialOrder | Mathlib.Order.Concept | {α : Type u_2} → {β : Type u_3} → {r : α → β → Prop} → PartialOrder (Concept α β r) | null | true |
_private.Init.Data.List.Lemmas.0.List.length_pos_iff_exists_mem.match_1_1 | Init.Data.List.Lemmas | ∀ {α : Type u_1} {l : List α} (motive : (∃ a, a ∈ l) → Prop) (x : ∃ a, a ∈ l),
(∀ (w : α) (h : w ∈ l), motive ⋯) → motive x | null | false |
_private.Mathlib.Algebra.Torsor.Basic.0.Equiv.right_vsub_pointReflection._simp_1_1 | Mathlib.Algebra.Torsor.Basic | ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a : α) (n : ℕ), -(n • a) = n • -a | null | false |
CategoryTheory.Limits.WidePullbackShape.Hom.id.elim | Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {J : Type w} →
{motive : (a a_1 : CategoryTheory.Limits.WidePullbackShape J) → a.Hom a_1 → Sort u} →
{a a_1 : CategoryTheory.Limits.WidePullbackShape J} →
(t : a.Hom a_1) →
t.ctorIdx = 0 →
((X : CategoryTheory.Limits.WidePullbackShape J) →
motive X X (CategoryTheory.Limits.Wi... | null | false |
_private.Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup.0.SubMulAction.fixingSubgroup_map_conj_eq._simp_1_2 | Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | ∀ {G : Type u_1} [inst : DivInvMonoid G] (x : G), x⁻¹ = x ^ (-1) | null | false |
Fin.dfoldrM.loop._unsafe_rec | Batteries.Data.Fin.Basic | {m : Type u_1 → Type u_2} →
[Monad m] →
(n : ℕ) →
(α : Fin (n + 1) → Type u_1) →
((i : Fin n) → α i.succ → m (α i.castSucc)) → (i : ℕ) → (h : i < n + 1) → α ⟨i, h⟩ → m (α 0) | null | false |
MeasureTheory.aecover_closedBall | Mathlib.MeasureTheory.Integral.IntegralEqImproper | ∀ {α : Type u_1} {ι : Type u_2} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι}
[inst_1 : PseudoMetricSpace α] [OpensMeasurableSpace α] {x : α} {r : ι → ℝ},
Filter.Tendsto r l Filter.atTop → MeasureTheory.AECover μ l fun i => Metric.closedBall x (r i) | null | true |
CartanMatrix.isSimplyLaced_D | Mathlib.LinearAlgebra.Matrix.Cartan.Basic | ∀ (n : ℕ), (CartanMatrix.D n).IsSimplyLaced | null | true |
BitVec.twoPow | Init.Data.BitVec.Basic | (w : ℕ) → ℕ → BitVec w | `twoPow w i` is the bitvector `2^i` if `i < w`, and `0` otherwise. In other words, it is 2 to the
power `i`.
From the bitwise point of view, it has the `i`th bit as `1` and all other bits as `0`.
| true |
_private.Init.Data.Array.Monadic.0.Array.foldlM_filterMap.match_1.eq_1 | Init.Data.Array.Monadic | ∀ {β : Type u_1} (motive : Option β → Sort u_2) (b : β) (h_1 : (b : β) → motive (some b)) (h_2 : Unit → motive none),
(match some b with
| some b => h_1 b
| none => h_2 ()) =
h_1 b | null | true |
AddSubgroup.normalCore_le | Mathlib.Algebra.Group.Subgroup.Basic | ∀ {G : Type u_1} [inst : AddGroup G] (H : AddSubgroup G), H.normalCore ≤ H | null | true |
LocalSubring.noConfusion | Mathlib.RingTheory.LocalRing.LocalSubring | {P : Sort u} →
{R : Type u_1} →
{inst : CommRing R} →
{t : LocalSubring R} →
{R' : Type u_1} →
{inst' : CommRing R'} →
{t' : LocalSubring R'} → R = R' → inst ≍ inst' → t ≍ t' → LocalSubring.noConfusionType P t t' | null | false |
_private.Mathlib.Analysis.Complex.CanonicalDecomposition.0.Complex.canonicalFactor_ne_zero._simp_1_3 | Mathlib.Analysis.Complex.CanonicalDecomposition | ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.closedBall x ε) = (dist y x ≤ ε) | null | false |
_private.Batteries.Data.String.Legacy.0.String.Legacy.posOfAux._proof_1 | Batteries.Data.String.Legacy | ∀ (s : String) (stopPos pos : String.Pos.Raw),
pos < stopPos → stopPos.byteIdx - (String.Pos.Raw.next s pos).byteIdx < stopPos.byteIdx - pos.byteIdx | null | false |
Set.smul_set_univ | Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | ∀ {α : Type u_2} {β : Type u_3} [inst : Group α] [inst_1 : MulAction α β] {a : α}, a • Set.univ = Set.univ | null | true |
_private.Mathlib.MeasureTheory.Function.SimpleFunc.0.MeasureTheory.SimpleFunc.support_eq._simp_1_1 | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {ι : Type u_1} {M : Type u_3} [inst : Zero M] {f : ι → M} {x : ι}, (x ∈ Function.support f) = (f x ≠ 0) | null | false |
_private.Mathlib.Algebra.GroupWithZero.Range.0.MonoidWithZeroHom.valueGroup_eq_range._simp_1_8 | Mathlib.Algebra.GroupWithZero.Range | ∀ {M₀ : Type u_2} [inst : MonoidWithZero M₀] [Nontrivial M₀] (u : M₀ˣ), (↑u = 0) = False | null | false |
UInt16.reduceLT | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | Lean.Meta.Simp.Simproc | null | true |
List.pop_toArray | Init.Data.List.ToArray | ∀ {α : Type u_1} (l : List α), l.toArray.pop = l.dropLast.toArray | null | true |
DilationEquiv.symm_bijective | Mathlib.Topology.MetricSpace.DilationEquiv | ∀ {X : Type u_1} {Y : Type u_2} [inst : PseudoEMetricSpace X] [inst_1 : PseudoEMetricSpace Y],
Function.Bijective DilationEquiv.symm | null | true |
Valuation.HasExtension.instIsLocalHomValuationInteger | Mathlib.RingTheory.Valuation.Extension | ∀ {R : Type u_1} [inst : CommRing R] {ΓR : Type u_6} [inst_1 : LinearOrderedCommGroupWithZero ΓR] {vR : Valuation R ΓR}
{S : Type u_9} {ΓS : Type u_10} [inst_2 : CommRing S] [inst_3 : LinearOrderedCommGroupWithZero ΓS]
[inst_4 : Algebra R S] [IsLocalHom (algebraMap R S)] {vS : Valuation S ΓS} [inst_6 : vR.HasExtens... | null | true |
Vector.instOrientedCmpCompareLex | Init.Data.Ord.Vector | ∀ {α : Type u_1} {cmp : α → α → Ordering} [Std.OrientedCmp cmp] {n : ℕ}, Std.OrientedCmp (Vector.compareLex cmp) | null | true |
_private.Mathlib.Tactic.Ring.Basic.0.Mathlib.Tactic.Ring.ExSum.evalNatCast.match_3 | Mathlib.Tactic.Ring.Basic | {v : Lean.Level} →
{β : Q(Type v)} →
(sβ : Q(CommSemiring «$β»)) →
(motive : (a : Q(ℕ)) → Mathlib.Tactic.Ring.ExSum sβ a → Sort u_1) →
(a : Q(ℕ)) →
(va : Mathlib.Tactic.Ring.ExSum sβ a) →
(Unit → motive q(0) Mathlib.Tactic.Ring.Common.ExSum.zero) →
((a b : Q(«$β»)... | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.0.BitVec.reduceUMod._regBuiltin.BitVec.reduceUMod.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.2858813359._hygCtx._hyg.27 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec | IO Unit | null | false |
DerivedCategory.descShortComplex_triangleOfSESδ | Mathlib.Algebra.Homology.DerivedCategory.ShortExact | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] {S : CategoryTheory.ShortComplex (CochainComplex C ℤ)} (hS : S.ShortExact),
CategoryTheory.CategoryStruct.comp (DerivedCategory.Q.map (CochainComplex.mappingCone.descShortComplex S))
(D... | null | true |
DFinsupp.instDFunLike | Mathlib.Data.DFinsupp.Defs | {ι : Type u} → {β : ι → Type v} → [inst : (i : ι) → Zero (β i)] → DFunLike (Π₀ (i : ι), β i) ι β | null | true |
Mathlib.Meta.FunProp.Mor.Arg.noConfusionType | Mathlib.Tactic.FunProp.Mor | Sort u → Mathlib.Meta.FunProp.Mor.Arg → Mathlib.Meta.FunProp.Mor.Arg → Sort u | null | false |
SimpleGraph.isMaximalIndepSet_compl._simp_1 | Mathlib.Combinatorics.SimpleGraph.Clique | ∀ {α : Type u_3} {G : SimpleGraph α} (s : Finset α), Maximal Gᶜ.IsIndepSet ↑s = Maximal G.IsClique ↑s | null | false |
Std.Format.group.injEq | Init.Data.Format.Basic | ∀ (a : Std.Format) (behavior : Std.Format.FlattenBehavior) (a_1 : Std.Format) (behavior_1 : Std.Format.FlattenBehavior),
(a.group behavior = a_1.group behavior_1) = (a = a_1 ∧ behavior = behavior_1) | null | true |
RingQuot.instRing._proof_8 | Mathlib.Algebra.RingQuot | ∀ {R : Type u_1} [inst : Ring R] (r : R → R → Prop) (n : ℕ),
{ toQuot := Quot.mk (RingQuot.Rel r) ↑↑n } = { toQuot := Quot.mk (RingQuot.Rel r) ↑n } | null | false |
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