name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
ModuleCat.linearOverField._proof_2 | Mathlib.Algebra.Category.ModuleCat.Algebra | ∀ {k : Type u_3} [inst : Field k] {A : Type u_2} [inst_1 : Ring A] [inst_2 : Algebra k A] (X Y Z : ModuleCat A) (r : k)
(f : X ⟶ Y) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp (r • f) g = r • CategoryTheory.CategoryStruct.comp f g | null | false |
HahnSeries.SummableFamily.embDomain_of_notMem_range | Mathlib.RingTheory.HahnSeries.Summable | ∀ {Γ : Type u_1} {R : Type u_3} {α : Type u_5} {β : Type u_6} [inst : PartialOrder Γ] [inst_1 : AddCommMonoid R]
(s : HahnSeries.SummableFamily Γ R α) (f : α ↪ β) {b : β}, b ∉ Set.range ⇑f → (s.embDomain f) b = 0 | null | true |
ContDiffWithinAt.sinh | Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E}
{n : WithTop ℕ∞}, ContDiffWithinAt ℝ n f s x → ContDiffWithinAt ℝ n (fun x => Real.sinh (f x)) s x | null | true |
_private.Init.Data.Vector.Count.0.Vector.count_le_count_map._simp_1_1 | Init.Data.Vector.Count | ∀ {α : Type u_2} [inst : BEq α] [LawfulBEq α] {β : Type u_1} [inst_2 : BEq β] [LawfulBEq β] {xs : Array α} {f : α → β}
{x : α}, (Array.count x xs ≤ Array.count (f x) (Array.map f xs)) = True | null | false |
_private.Lean.Meta.Tactic.Grind.Theorems.0.Lean.Meta.Grind.Theorems.mk.noConfusion | Lean.Meta.Tactic.Grind.Theorems | {α : Type} →
{P : Sort u} →
{smap : Lean.PHashMap Lean.Name (List α)} →
{origins erased : Lean.PHashSet Lean.Meta.Grind.Origin} →
{omap : Lean.PHashMap Lean.Meta.Grind.Origin (List α)} →
{smap' : Lean.PHashMap Lean.Name (List α)} →
{origins' erased' : Lean.PHashSet Lean.Meta.Gr... | null | false |
CategoryTheory.RigidCategory.ctorIdx | Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {C : Type u} →
{inst : CategoryTheory.Category.{v, u} C} →
{inst_1 : CategoryTheory.MonoidalCategory C} → CategoryTheory.RigidCategory C → ℕ | null | false |
Array.all_subtype | Init.Data.Array.Attach | ∀ {α : Type u_1} {stop : ℕ} {p : α → Prop} {xs : Array { x // p x }} {f : { x // p x } → Bool} {g : α → Bool},
(∀ (x : α) (h : p x), f ⟨x, h⟩ = g x) → stop = xs.size → xs.all f 0 stop = xs.unattach.all g | null | true |
gcd_comm | Mathlib.Algebra.GCDMonoid.Basic | ∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] (a b : α), gcd a b = gcd b a | null | true |
_private.Lean.Meta.LitValues.0.Lean.Meta.getFloat32Lit? | Lean.Meta.LitValues | Lean.Expr → Lean.MetaM (Option Float32) | Recognize a non-negated `Float32` literal: an `OfScientific` or `OfNat` application. | true |
Lean.Elab.Tactic.TacticParsedSnapshotInner.mk.injEq | Lean.Elab.Term.TermElabM | ∀ {α : Type} (toSnapshot : Lean.Language.Snapshot) (stx : Lean.Syntax) (inner? : Option (Lean.Language.SnapshotTask α))
(finished : Lean.Language.SnapshotTask Lean.Elab.Tactic.TacticFinishedSnapshot)
(toSnapshot_1 : Lean.Language.Snapshot) (stx_1 : Lean.Syntax) (inner?_1 : Option (Lean.Language.SnapshotTask α))
(... | null | true |
BoxIntegral.Box.instPartialOrder._proof_3 | Mathlib.Analysis.BoxIntegral.Box.Basic | ∀ {ι : Type u_1} (a b : BoxIntegral.Box ι), a < b ↔ a ≤ b ∧ ¬b ≤ a | null | false |
_private.Std.Data.ExtDHashMap.Lemmas.0.Std.ExtDHashMap.map_eq_empty_iff._simp_1_2 | Std.Data.ExtDHashMap.Lemmas | ∀ {a b : Bool}, (a = true ↔ b = true) = (a = b) | null | false |
groupHomology.H0π_comp_H0Iso_hom_assoc | Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | ∀ {k G : Type u} [inst : CommRing k] [inst_1 : Group G] (A : Rep.{u, u, u} k G) {Z : ModuleCat k}
(h : (Rep.coinvariantsFunctor k G).obj A ⟶ Z),
CategoryTheory.CategoryStruct.comp (groupHomology.H0π A)
(CategoryTheory.CategoryStruct.comp (groupHomology.H0Iso A).hom h) =
CategoryTheory.CategoryStruct.comp ... | null | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.String.0._regBuiltin.String.reduceOfList.declare_18._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.String.3166094940._hygCtx._hyg.20 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.String | IO Unit | null | false |
Finsupp.support_indicator_subset | Mathlib.Data.Finsupp.Indicator | ∀ {ι : Type u_1} {α : Type u_2} [inst : Zero α] (s : Finset ι) (f : (i : ι) → i ∈ s → α),
(Finsupp.indicator s f).support ⊆ s | null | true |
AddSubsemigroup.centerToAddOpposite._proof_4 | Mathlib.GroupTheory.Submonoid.Center | ∀ {M : Type u_1} [inst : Add M] (r : ↥(AddSubsemigroup.center Mᵃᵒᵖ)), AddOpposite.unop ↑r ∈ Set.addCenter M | null | false |
Substring.Raw.Valid.data_drop | Batteries.Data.String.Lemmas | ∀ {s : Substring.Raw}, s.Valid → ∀ (n : ℕ), (s.drop n).toString.toList = List.drop n s.toString.toList | null | true |
_private.Mathlib.Algebra.Exact.Sequence.0.Module.sum_neg_one_pow_finrank_eq_zero_of_exact._proof_1_3 | Mathlib.Algebra.Exact.Sequence | ∀ {k : Type u_1} [inst : DivisionRing k] {n : ℕ} (V : Fin (n + 2) → Type u_2)
[inst_1 : (i : Fin (n + 2)) → AddCommGroup (V i)] [inst_2 : (i : Fin (n + 2)) → Module k (V i)]
(f : (i : Fin (n + 1)) → V i.castSucc →ₗ[k] V i.succ) (i : Fin n),
Module.finrank k ↥(f i.succ).range + Module.finrank k ↥(f i.succ).ker = M... | null | false |
QuasiconcaveOn.dual | Mathlib.Analysis.Convex.Quasiconvex | ∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : LE β] [inst_4 : SMul 𝕜 E] {s : Set E} {f : E → β},
QuasiconcaveOn 𝕜 s f → QuasiconvexOn 𝕜 s (⇑OrderDual.toDual ∘ f) | null | true |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
_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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
_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.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 |
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 |
UInt16.reduceLT | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | Lean.Meta.Simp.Simproc | 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 |
MeasureTheory.VectorMeasure.etvDist._proof_1 | Mathlib.MeasureTheory.VectorMeasure.TotalVariation | ∀ {M : Type u_1} [inst : NormedAddCommGroup M], T2Space M | null | false |
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 |
Sym2.rec.match_1 | Mathlib.Data.Sym.Sym2 | {α : Type u_1} → (motive : α × α → Sort u_2) → (x : α × α) → ((a b : α) → motive (a, b)) → motive x | null | false |
MvPolynomial.weightedHomogeneousComponent_eq_self | Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | ∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] {σ : Type u_3} [inst_1 : AddCommMonoid M] {w : σ → M} {n : M}
{p : MvPolynomial σ R},
MvPolynomial.IsWeightedHomogeneous w p n → (MvPolynomial.weightedHomogeneousComponent w n) p = p | null | true |
_private.Mathlib.Algebra.Homology.Localization.0.HomotopyCategory.quotient_map_mem_quasiIso_iff._simp_1_1 | Mathlib.Algebra.Homology.Localization | ∀ {ι : Type u_1} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c}
[inst_2 : CategoryTheory.CategoryWithHomology C] (f : K ⟶ L), HomologicalComplex.quasiIso C c f = QuasiIso f | null | false |
ByteArray.forIn.loop._proof_4 | Init.Data.ByteArray.Basic | ∀ (as : ByteArray), ∀ i < as.size, i ≤ as.size | null | false |
AddMonoidAlgebra.opRingEquiv_symm_single | Mathlib.Algebra.MonoidAlgebra.Opposite | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : Add M] (r : Rᵐᵒᵖ) (x : Mᵃᵒᵖ),
AddMonoidAlgebra.opRingEquiv.symm (AddMonoidAlgebra.single x r) =
MulOpposite.op (AddMonoidAlgebra.single (AddOpposite.unop x) (MulOpposite.unop r)) | null | true |
CategoryTheory.Functor.mapHomologicalComplexIdIso._proof_2 | Mathlib.Algebra.Homology.Additive | ∀ {ι : Type u_1} (W₁ : Type u_3) [inst : CategoryTheory.Category.{u_2, u_3} W₁]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms W₁] (c : ComplexShape ι) (K : HomologicalComplex W₁ c) (i j : ι),
c.Rel i j →
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Iso.refl ((((CategoryTheory.Functor.id W₁).map... | null | false |
cfcₙ.congr_simp | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | ∀ {R : Type u_3} {A : Type u_4} {p p_1 : A → Prop} (e_p : p = p_1) [inst : CommSemiring R] [inst_1 : Nontrivial R]
[inst_2 : StarRing R] [inst_3 : MetricSpace R] [inst_4 : IsTopologicalSemiring R] [inst_5 : ContinuousStar R]
[inst_6 : NonUnitalRing A] [inst_7 : StarRing A] [inst_8 : TopologicalSpace A] [inst_9 : Mo... | null | true |
Matrix.specialOrthogonalGroup | Mathlib.LinearAlgebra.UnitaryGroup | (n : Type u) →
[inst : DecidableEq n] → [inst_1 : Fintype n] → (R : Type v) → [inst_2 : CommRing R] → Submonoid (Matrix n n R) | `Matrix.specialOrthogonalGroup n` is the group of orthogonal `n` by `n` where the determinant
is one. (This definition is only correct if 2 is invertible.) | true |
Equiv.mulOneClass | Mathlib.Algebra.Group.TransferInstance | {α : Type u_2} → {β : Type u_3} → α ≃ β → [MulOneClass β] → MulOneClass α | Transfer `MulOneClass` across an `Equiv` | true |
_private.Mathlib.Combinatorics.Matroid.Map.0.Matroid.mapSetEmbedding._simp_1 | Mathlib.Combinatorics.Matroid.Map | ∀ {α : Type u} {s : Set α}, (s ⊆ ∅) = (s = ∅) | null | false |
IsCyclotomicExtension.Rat.galEquivZMod.eq_1 | Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | ∀ (n : ℕ) [inst : NeZero n] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField K]
[hK : IsCyclotomicExtension {n} ℚ K],
IsCyclotomicExtension.Rat.galEquivZMod n K = IsCyclotomicExtension.autEquivPow K ⋯ | null | true |
_private.Mathlib.Data.List.Sort.0.List.erase_orderedInsert_of_notMem._proof_1_1 | Mathlib.Data.List.Sort | ∀ {α : Type u_1} {r : α → α → Prop} [inst : DecidableRel r] [inst_1 : DecidableEq α] {x : α},
(List.orderedInsert r x []).erase x = [] | null | false |
_private.Mathlib.RingTheory.Ideal.MinimalPrime.Basic.0.Ideal.map_sup_mem_minimalPrimes_of_map_quotientMk_mem_minimalPrimes.match_1_1 | Mathlib.RingTheory.Ideal.MinimalPrime.Basic | ∀ {R : Type u_2} [inst : CommSemiring R] {S : Type u_1} [inst_1 : CommRing S] [inst_2 : Algebra R S] {I : Ideal R}
{J : Ideal S} (q : Ideal S) (motive : q.IsPrime ∧ Ideal.map (algebraMap R S) I ⊔ J ≤ q → Prop)
(x : q.IsPrime ∧ Ideal.map (algebraMap R S) I ⊔ J ≤ q),
(∀ (left : q.IsPrime) (hleq : Ideal.map (algebra... | null | false |
Module.Basis.toMatrix_smul | Mathlib.LinearAlgebra.Matrix.Basis | ∀ {ι : Type u_1} {R₁ : Type u_9} {S : Type u_10} [inst : CommSemiring R₁] [inst_1 : Semiring S] [inst_2 : Algebra R₁ S]
[inst_3 : Fintype ι] [inst_4 : DecidableEq ι] (x : S) (b : Module.Basis ι R₁ S) (w : ι → S),
b.toMatrix (x • w) = (Algebra.leftMulMatrix b) x * b.toMatrix w | null | true |
ENNReal.ofReal_essSup | Mathlib.MeasureTheory.Function.EssSup | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
Filter.IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae μ) f →
Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae μ) f →
ENNReal.ofReal (essSup f μ) = essSup (fun a => ENNReal.ofReal (f a)) μ | null | true |
_private.Mathlib.RingTheory.LocalRing.ResidueField.Ideal.0.instIsFractionRingQuotientIdealResidueField._simp_3 | Mathlib.RingTheory.LocalRing.ResidueField.Ideal | ∀ {R : Type u} [inst : Ring R] {I : Ideal R} [inst_1 : I.IsTwoSided] (x y : R),
(x - y ∈ I) = ((Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y) | null | false |
HomotopyEquiv.toHomologyIso._proof_1 | Mathlib.Algebra.Homology.Homotopy | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C] {ι : Type u_3}
{c : ComplexShape ι} {K L : HomologicalComplex C c} (h : HomotopyEquiv K L) (i : ι) [inst_2 : K.HasHomology i]
[inst_3 : L.HasHomology i],
CategoryTheory.CategoryStruct.comp (HomologicalComplex.ho... | null | false |
IsOpen.is_const_of_deriv_eq_zero | Mathlib.Analysis.Calculus.MeanValue | ∀ {𝕜 : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup G] [inst_2 : NormedSpace 𝕜 G] {f : 𝕜 → G}
{s : Set 𝕜},
IsOpen s → IsPreconnected s → DifferentiableOn 𝕜 f s → Set.EqOn (deriv f) 0 s → ∀ {x y : 𝕜}, x ∈ s → y ∈ s → f x = f y | null | true |
Std.IterM.fold_filterMapM | Init.Data.Iterators.Lemmas.Combinators.Monadic.FilterMap | ∀ {α β γ δ : Type w} {m : Type w → Type w'} {n : Type w → Type w''} [inst : Std.Iterator α m β]
[Std.Iterators.Finite α m] [inst_2 : Monad m] [LawfulMonad m] [inst_4 : Monad n] [inst_5 : MonadAttach n]
[LawfulMonad n] [WeaklyLawfulMonadAttach n] [inst_8 : Std.IteratorLoop α m n] [Std.LawfulIteratorLoop α m n]
[in... | null | true |
CategoryTheory.PreGaloisCategory.PointedGaloisObject.Hom.mk | Mathlib.CategoryTheory.Galois.Prorepresentability | {C : Type u₁} →
[inst : CategoryTheory.Category.{u₂, u₁} C] →
[inst_1 : CategoryTheory.GaloisCategory C] →
{F : CategoryTheory.Functor C FintypeCat} →
{A B : CategoryTheory.PreGaloisCategory.PointedGaloisObject F} →
(val : A.obj ⟶ B.obj) →
autoParam ((CategoryTheory.ConcreteCat... | null | true |
Std.DTreeMap.Internal.Impl.getKeyLE?.eq_1 | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] (k : α),
Std.DTreeMap.Internal.Impl.getKeyLE? k = Std.DTreeMap.Internal.Impl.getKeyLE?.go k none | null | true |
Std.WP.Spec.repeatM.match_1 | Std.WP.Triple.SpecLemmas | {α β : Type u_1} →
(motive : α ⊕ β → Sort u_2) →
(r : α ⊕ β) → ((a' : α) → motive (Sum.inl a')) → ((b : β) → motive (Sum.inr b)) → motive r | null | false |
MulAction.stabilizer_smul_eq_right | Mathlib.GroupTheory.GroupAction.Defs | ∀ {G : Type u_1} {β : Type u_3} [inst : Group G] [inst_1 : MulAction G β] {α : Type u_4} [inst_2 : Group α]
[inst_3 : MulAction α β] [SMulCommClass G α β] (a : α) (b : β),
MulAction.stabilizer G (a • b) = MulAction.stabilizer G b | null | true |
_private.Init.Data.BitVec.Bitblast.0.BitVec.DivModState.lawful_init._simp_1_3 | Init.Data.BitVec.Bitblast | ∀ {n : ℕ} {x y : BitVec n}, (x < y) = (x.toNat < y.toNat) | null | false |
map_eq_zero._simp_1 | Mathlib.Algebra.GroupWithZero.Units.Lemmas | ∀ {G₀ : Type u_3} {M₀' : Type u_4} {F : Type u_6} [inst : GroupWithZero G₀] [inst_1 : MulZeroOneClass M₀']
[Nontrivial M₀'] [inst_3 : FunLike F G₀ M₀'] [MonoidWithZeroHomClass F G₀ M₀'] (f : F) {a : G₀}, (f a = 0) = (a = 0) | null | false |
MeasureTheory.VectorMeasure.exists_variation_le_add | Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | ∀ {X : Type u_1} {V : Type u_2} {mX : MeasurableSpace X} [inst : TopologicalSpace V] [inst_1 : ENormedAddCommMonoid V]
[inst_2 : T2Space V] (μ : MeasureTheory.VectorMeasure X V) {s : Set X},
MeasurableSet s →
∀ {ε : NNReal},
0 < ε →
μ.variation s ≠ ⊤ →
∃ P,
(∀ t ∈ P, t ⊆ s) ∧... | Measure version of `preVariation.exists_Finpartition_sum_ge`. | true |
_private.Batteries.Data.BinaryHeap.WF.0.Batteries.BinaryHeap.WF.Children.set_of_ge_child._proof_1_5 | Batteries.Data.BinaryHeap.WF | ∀ {sz : ℕ} {k : Fin sz}, ↑k + 1 ≤ sz → ↑k < sz | null | false |
Lean.Expr.NumApps.State.counters | Lean.Util.NumApps | Lean.Expr.NumApps.State → Lean.NameMap ℕ | null | true |
_private.Mathlib.RingTheory.Coalgebra.CoassocSimps.0.CoassocSimps._aux_Mathlib_RingTheory_Coalgebra_CoassocSimps___unexpand_TensorProduct_map_1 | Mathlib.RingTheory.Coalgebra.CoassocSimps | Lean.PrettyPrinter.Unexpander | null | false |
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