name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
MeasureTheory.Measure.sum_fintype | Mathlib.MeasureTheory.Measure.Sum | ∀ {α : Type u_1} {ι : Type u_2} {mα : MeasurableSpace α} [inst : Fintype ι] (μ : ι → MeasureTheory.Measure α),
MeasureTheory.Measure.sum μ = ∑ i, μ i | null | true |
LightCondensed.instMonoidalClosedFunctorOppositeLightProfiniteModuleCat._proof_1 | Mathlib.Condensed.Light.Monoidal | ∀ (R : Type u_1) [inst : CommRing R], (CategoryTheory.equivSmallModel LightProfinite).op.congrLeft.functor.IsEquivalence | null | false |
Finset.preimage_add_right_zero' | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : AddGroup α] {b : α}, Finset.preimage 0 (fun x => x + -b) ⋯ = {b} | null | true |
FreeAddMagma.add | Mathlib.Algebra.Free | {α : Type u} → FreeAddMagma α → FreeAddMagma α → FreeAddMagma α | null | true |
Lean.Meta.FindSplitImpl.Context.recOn | Lean.Meta.Tactic.SplitIf | {motive : Lean.Meta.FindSplitImpl.Context → Sort u} →
(t : Lean.Meta.FindSplitImpl.Context) →
((exceptionSet : Lean.ExprSet) →
(kind : Lean.Meta.SplitKind) → motive { exceptionSet := exceptionSet, kind := kind }) →
motive t | null | false |
Mathlib.Tactic.Matrix.DotProductEq.noConfusion | Mathlib.Tactic.Matrix.MulExpand | {P : Sort u} →
{u : Lean.Level} →
{α : Q(Type u)} →
{zα : Q(Zero «$α»)} →
{aα : Q(Add «$α»)} →
{mα : Q(Mul «$α»)} →
{t : Mathlib.Tactic.Matrix.DotProductEq zα aα mα} →
{u' : Lean.Level} →
{α' : Q(Type u')} →
{zα' : Q(Zero «$α'»)} ... | null | false |
Lean.Lsp.FoldingRange.mk.noConfusion | Lean.Data.Lsp.LanguageFeatures | {P : Sort u} →
{startLine endLine : ℕ} →
{kind? : Option Lean.Lsp.FoldingRangeKind} →
{startLine' endLine' : ℕ} →
{kind?' : Option Lean.Lsp.FoldingRangeKind} →
{ startLine := startLine, endLine := endLine, kind? := kind? } =
{ startLine := startLine', endLine := endLine', kin... | null | false |
_private.Mathlib.Algebra.Order.Field.Power.0.Mathlib.Meta.Positivity.evalZPow._sparseCasesOn_20 | Mathlib.Algebra.Order.Field.Power | {motive : Lean.Expr → Sort u} →
(t : Lean.Expr) →
((a : Lean.Literal) → motive (Lean.Expr.lit a)) → (Nat.hasNotBit 512 t.ctorIdx → motive t) → motive t | null | false |
Set.opEquiv_self | Mathlib.Data.Set.Opposite | {α : Type u_1} → (s : Set α) → ↑s.op ≃ ↑s | The members of the opposite of a set are in bijection with the members of the set itself. | true |
Bundle.Pretrivialization.continuousAlternatingMapCoordChange._proof_4 | Mathlib.Topology.VectorBundle.ContinuousAlternatingMap | ∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] {F₂ : Type u_2} [inst_1 : NormedAddCommGroup F₂]
[inst_2 : NormedSpace 𝕜 F₂], ContinuousConstSMul 𝕜 F₂ | null | false |
Complex.sin_surjective | Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | Function.Surjective Complex.sin | null | true |
Module.FinitePresentation.linearEquivMap._proof_3 | Mathlib.Algebra.Module.FinitePresentation | ∀ {R : Type u_1} [inst : CommRing R], IsScalarTower R R R | null | false |
Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits._proof_1 | Mathlib.FieldTheory.IsAlgClosed.Basic | ∀ (A : Type u_1) [Field A], Nontrivial A | null | false |
AddCon.ker.congr_simp | Mathlib.Algebra.Colimit.Module | ∀ {M : Type u_1} {N : Type u_2} {F : Type u_4} [inst : Add M] [inst_1 : Add N] [inst_2 : FunLike F M N]
[inst_3 : AddHomClass F M N] (f f_1 : F), f = f_1 → AddCon.ker f = AddCon.ker f_1 | null | true |
FinBoolAlg.hasForgetToFinBddDistLat._proof_4 | Mathlib.Order.Category.FinBoolAlg | { obj := fun X => ↧↑X.toBoolAlg, map := fun {X Y} f => FinBddDistLat.ofHom (BoolAlg.Hom.hom f.hom),
map_id := FinBoolAlg.hasForgetToFinBddDistLat._proof_1,
map_comp := @FinBoolAlg.hasForgetToFinBddDistLat._proof_2 }.comp
(CategoryTheory.forget FinBddDistLat) =
CategoryTheory.forget FinBoolAlg | null | false |
_private.Mathlib.Order.Interval.Finset.Nat.0.Nat.image_sub_const_Ico._proof_1_3 | Mathlib.Order.Interval.Finset.Nat | ∀ {a b c : ℕ} (x : ℕ), a - c ≤ x ∧ x < b - c → (a ≤ x + c ∧ x + c < b) ∧ x + c - c = x | null | false |
ENNReal.add_halves | Mathlib.Basic.ENNReal.Inv | ∀ (a : ENNReal), a / 2 + a / 2 = a | null | true |
Finset.map_insert | Mathlib.Data.Finset.Image | ∀ {α : Type u_1} {β : Type u_2} [inst : DecidableEq α] [inst_1 : DecidableEq β] (f : α ↪ β) (a : α) (s : Finset α),
Finset.map f (insert a s) = insert (f a) (Finset.map f s) | null | true |
_private.Init.Data.Array.InsertionSort.0.Array.insertionSort.swapLoop.match_1 | Init.Data.Array.InsertionSort | (motive : ℕ → Sort u_1) → (j : ℕ) → (j = 0 → motive 0) → ((j' : ℕ) → j = j'.succ → motive j'.succ) → motive j | null | false |
NNReal.instLinearOrder._aux_6 | Mathlib.Basic.NNReal.Defs | DecidableEq NNReal | null | false |
RingHomId.eq_id | Mathlib.Algebra.Ring.CompTypeclasses | ∀ {R : Type u_4} {inst : Semiring R} {σ : R →+* R} [self : RingHomId σ], σ = RingHom.id R | null | true |
LinearEquiv.piRing | Mathlib.LinearAlgebra.Pi | (R : Type u) →
(M : Type v) →
(ι : Type x) →
[inst : Semiring R] →
(S : Type u_4) →
[Fintype ι] →
[DecidableEq ι] →
[inst_3 : Semiring S] →
[inst_4 : AddCommMonoid M] →
[inst_5 : Module R M] →
[inst_6 : Module ... | Linear equivalence between linear functions `Rⁿ → M` and `Mⁿ`. The spaces `Rⁿ` and `Mⁿ`
are represented as `ι → R` and `ι → M`, respectively, where `ι` is a finite type.
This as an `S`-linear equivalence, under the assumption that `S` acts on `M` commuting with `R`.
When `R` is commutative, we can take this to be the ... | true |
Std.Internal.List.maxKeyD_le_maxKeyD_insertEntry | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
l.isEmpty = false →
∀ {k : α} {v : β k} {fallback : α},
(compare (Std.Internal.List.maxKeyD l fallback)
(Std.Internal.L... | null | true |
Finset.prod_ite_of_false | Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | ∀ {ι : Type u_1} {M : Type u_2} {s : Finset ι} [inst : CommMonoid M] {p : ι → Prop} [inst_1 : DecidablePred p],
(∀ x ∈ s, ¬p x) → ∀ (f g : ι → M), (∏ x ∈ s, if p x then f x else g x) = ∏ x ∈ s, g x | null | true |
Equiv.recOn | Mathlib.Logic.Equiv.Defs | {α : Sort u_1} →
{β : Sort u_2} →
{motive : α ≃ β → Sort u} →
(t : α ≃ β) →
((toFun : α → β) →
(invFun : β → α) →
(left_inv : Function.LeftInverse invFun toFun) →
(right_inv : Function.RightInverse invFun toFun) →
motive { toFun := toFun, i... | null | false |
Continuous.cfcₙ_nnreal'._auto_3 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | Lean.Syntax | null | false |
topologicalAddGroup_inf | Mathlib.Topology.Algebra.Group.Basic | ∀ {G : Type u_1} [inst : AddGroup G] {t₁ t₂ : TopologicalSpace G},
IsTopologicalAddGroup G → IsTopologicalAddGroup G → IsTopologicalAddGroup G | **Alias** of `isTopologicalAddGroup_inf`. | true |
ContinuousLinearMapWOT.continuous_inner_apply | Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E]
[inst_3 : Module 𝕜 E] [inst_4 : NormedAddCommGroup F] [inst_5 : InnerProductSpace 𝕜 F] [CompleteSpace F]
{α : Type u_4} [inst_7 : TopologicalSpace α] {f : α → E →WOT[𝕜] F},
Continuous f → ... | **Alias** of the forward direction of `ContinuousLinearMapWOT.continuous_iff`. | true |
Acc.casesOn | Init.WF | {α : Sort u} →
{r : α → α → Prop} →
{motive : (a : α) → Acc r a → Sort u_1} →
{a : α} → (t : Acc r a) → ((x : α) → (h : ∀ (y : α), r y x → Acc r y) → motive x ⋯) → motive a t | null | false |
Function.Embedding.instAddAction._proof_1 | Mathlib.GroupTheory.GroupAction.Embedding | ∀ {α : Type u_1} {β : Type u_2}, Function.Injective DFunLike.coe | null | false |
Multiset.powerset._proof_1 | Mathlib.Data.Multiset.Powerset | ∀ {α : Type u_1} (x x_1 : List α),
(List.isSetoid α) x x_1 →
Quot.mk (⇑(List.isSetoid (Multiset α))) (Multiset.powersetAux x) =
Quot.mk (⇑(List.isSetoid (Multiset α))) (Multiset.powersetAux x_1) | null | false |
ZFSet.Insert.match_5 | Mathlib.SetTheory.ZFC.Basic | ∀ («Type» : Type u_1) (Func : «Type» → PSet.{u_1}) (Type_1 : Type u_1) (Func_1 : Type_1 → PSet.{u_1})
(b : { «Type» := Type_1, Func := Func_1 }.Type) (motive : (∃ a, (Func a).Equiv (Func_1 b)) → Prop)
(x : ∃ a, (Func a).Equiv (Func_1 b)), (∀ (a : «Type») (ha : (Func a).Equiv (Func_1 b)), motive ⋯) → motive x | null | false |
List.Sublist.flatMap | Mathlib.Data.List.Flatten | ∀ {α : Type u_1} {β : Type u_2} {l₁ l₂ : List α},
l₁.Sublist l₂ → ∀ (f : α → List β), (List.flatMap f l₁).Sublist (List.flatMap f l₂) | null | true |
MeasureTheory.StronglyAdapted.progMeasurable_of_continuous | Mathlib.Probability.Process.Adapted | ∀ {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} [inst : Preorder ι] {f : MeasureTheory.Filtration ι m}
{β : Type u_3} [inst_1 : TopologicalSpace β] {u : ι → Ω → β} [inst_2 : TopologicalSpace ι]
[TopologicalSpace.MetrizableSpace ι] [SecondCountableTopology ι] [inst_5 : MeasurableSpace ι] [OpensMeasurableSpac... | **Alias** of `MeasureTheory.StronglyAdapted.isStronglyProgressive_of_continuous`.
---
A continuous and strongly adapted process is strongly progressive. | true |
Equiv.simpleGraph | Mathlib.Combinatorics.SimpleGraph.Maps | {V : Type u_1} → {W : Type u_2} → V ≃ W → SimpleGraph V ≃ SimpleGraph W | Equivalent types have equivalent simple graphs. | true |
Lean.Compiler.LCNF.getLocalImpureDecls | Lean.Compiler.LCNF.PhaseExt | Lean.CoreM (Array Lean.Name) | null | true |
Std.Internal.List.minKey!_insertEntryIfNew_le_minKey! | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
[inst_4 : Inhabited α] {l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
l.isEmpty = false →
∀ {k : α} {v : β k},
(compare (Std.Internal.List.minKey! (Std.Internal.List.insertEntryIfN... | null | true |
QuadraticAlgebra.instNonUnitalNonAssocSemiring | Mathlib.Algebra.QuadraticAlgebra.Defs | {R : Type u_1} → {a b : R} → [NonUnitalNonAssocSemiring R] → NonUnitalNonAssocSemiring (QuadraticAlgebra R a b) | null | true |
CompactlySupportedContinuousMap.smulc_apply | Mathlib.Topology.ContinuousMap.CompactlySupported | ∀ {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β]
[inst_2 : Zero β] [inst_3 : TopologicalSpace γ] [inst_4 : SMulZeroClass γ β] [inst_5 : ContinuousSMul γ β]
{F : Type u_5} [inst_6 : FunLike F α γ] [inst_7 : ContinuousMapClass F α γ] (f : F)
(g : CompactlySupp... | null | true |
Lean.Meta.Grind.SplitInfo.arg | Lean.Meta.Tactic.Grind.Types | Lean.Expr → Lean.Expr → ℕ → Lean.Expr → Lean.Meta.Grind.SplitSource → Lean.Meta.Grind.SplitInfo | Given applications `a` and `b`, case-split on whether the corresponding
`i`-th arguments are equal or not. The split is only performed if all other
arguments are already known to be equal or are also tagged as split candidates.
| true |
_private.Mathlib.Data.PFun.0.PFun.mem_prodLift._simp_1_6 | Mathlib.Data.PFun | ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, p x ∧ b) = ((∃ x, p x) ∧ b) | null | false |
AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeftIsLimit | Mathlib.Geometry.RingedSpace.OpenImmersion | {X Y Z : AlgebraicGeometry.LocallyRingedSpace} →
(f : X ⟶ Z) →
(g : Y ⟶ Z) →
[H : AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f] →
CategoryTheory.Limits.IsLimit (AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeft f g) | The constructed `pullbackConeOfLeft` is indeed limiting. | true |
List.dropWhile.eq_def | Init.Data.List.TakeDrop | ∀ {α : Type u} (p : α → Bool) (x : List α),
List.dropWhile p x =
match x with
| [] => []
| a :: l =>
match p a with
| true => List.dropWhile p l
| false => a :: l | null | true |
Finsupp.mem_submodule_iff | Mathlib.LinearAlgebra.Finsupp.Pi | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_5}
(S : α → Submodule R M) (x : α →₀ M), x ∈ Finsupp.submodule S ↔ ∀ (i : α), x i ∈ S i | null | true |
Submonoid.val_mem_of_mem_units | Mathlib.Algebra.Group.Submonoid.Units | ∀ {M : Type u_1} [inst : Monoid M] (S : Submonoid M) {x : Mˣ}, x ∈ S.units → ↑x ∈ S | null | true |
Finsupp.mem_neLocus | Mathlib.Data.Finsupp.NeLocus | ∀ {α : Type u_1} {N : Type u_3} [inst : DecidableEq α] [inst_1 : DecidableEq N] [inst_2 : Zero N] {f g : α →₀ N}
{a : α}, a ∈ f.neLocus g ↔ f a ≠ g a | null | true |
Std.DTreeMap.isSome_minKey?_of_mem | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp] {k : α},
k ∈ t → t.minKey?.isSome = true | null | true |
CategoryTheory.Functor.mapTriangleCommShiftIso_inv_app_hom₁ | Mathlib.CategoryTheory.Triangulated.Functor | ∀ {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.HasShift C ℤ]
[inst_3 : CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [inst_4 : F.CommShift ℤ]
[inst_5 : CategoryTheory.Preadditive C] [inst_6 : Ca... | null | true |
Nat.Partrec.Code.ofNatCode.eq_4 | Mathlib.Computability.PartrecCode | Nat.Partrec.Code.ofNatCode 3 = Nat.Partrec.Code.right | null | true |
_private.Init.Data.Int.DivMod.Lemmas.0.Int.fdiv_fmod_unique'._proof_1_1 | Init.Data.Int.DivMod.Lemmas | ∀ {b : ℤ}, b < 0 → ¬0 < -b → False | null | false |
String.Slice.Pattern.Model.IsRevMatch | Init.Data.String.Lemmas.Pattern.Basic | {ρ : Type} → (pat : ρ) → [String.Slice.Pattern.Model.PatternModel pat] → {s : String.Slice} → s.Pos → Prop | Predicate stating that the region between the position `startPos` and the end of the slice
`s` matches the pattern `pat`. Note that there might be a longer match.
| true |
ArchimedeanClass.mk_nonneg_of_le_of_le_of_archimedean | Mathlib.Algebra.Order.Ring.Archimedean | ∀ {R : Type u_1} [inst : LinearOrder R] [inst_1 : CommRing R] [inst_2 : IsStrictOrderedRing R] {S : Type u_3}
[inst_3 : LinearOrder S] [inst_4 : CommRing S] [IsStrictOrderedRing S] [Archimedean S] (f : S →+*o R) {x : R}
{r s : S}, f r ≤ x → x ≤ f s → 0 ≤ ArchimedeanClass.mk x | null | true |
CommRingCat.Colimits.instCommRingColimitType._proof_9 | Mathlib.Algebra.Category.Ring.Colimits | ∀ {J : Type u_1} [inst : CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat)
(x : CommRingCat.Colimits.ColimitType F), x * 1 = x | null | false |
HomologicalComplex.Acyclic.truncGE | Mathlib.Algebra.Homology.Embedding.TruncGEHomology | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
[inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{K : HomologicalComplex C c'} [inst_2 : ∀ (i' : ι'), K.HasHomology i']
[inst_3 : CategoryTheory.Limits.HasZeroObject C],
K... | null | true |
Quaternion.imJ_star | Mathlib.Algebra.Quaternion | ∀ {R : Type u_3} [inst : CommRing R] (a : Quaternion R), (star a).imJ = -a.imJ | null | true |
Lean.Meta.LazyDiscrTree.recOn | Lean.Meta.LazyDiscrTree | {α : Type} →
{motive : Lean.Meta.LazyDiscrTree α → Sort u} →
(t : Lean.Meta.LazyDiscrTree α) →
((tries : Array (Lean.Meta.LazyDiscrTree.Trie α)) →
(roots : Std.HashMap Lean.Meta.LazyDiscrTree.Key Lean.Meta.LazyDiscrTree.TrieIndex) →
motive { tries := tries, roots := roots }) →
... | null | false |
Mathlib.Tactic.Ring.ringCleanupRef | Mathlib.Tactic.Ring.Basic | IO.Ref (Lean.Expr → Lean.MetaM Lean.Expr) | This is a routine which is used to clean up the unsolved subgoal
of a failed `ring1` application. It is overridden in `Mathlib/Tactic/Ring/RingNF.lean`
to apply the `ring_nf` simp set to the goal.
| true |
VitaliFamily.FineSubfamilyOn.index | Mathlib.MeasureTheory.Covering.VitaliFamily | {X : Type u_1} →
[inst : PseudoMetricSpace X] →
{m0 : MeasurableSpace X} →
{μ : MeasureTheory.Measure X} →
{v : VitaliFamily μ} → {f : X → Set (Set X)} → {s : Set X} → v.FineSubfamilyOn f s → Set (X × Set X) | Given `h : v.FineSubfamilyOn f s`, then `h.index` is a set parametrizing a disjoint
covering of almost every `s`. | true |
SimpleGraph.Walk.IsHamiltonian.fintype._proof_1 | Mathlib.Combinatorics.SimpleGraph.Hamiltonian | ∀ {α : Type u_1} [inst : DecidableEq α] {G : SimpleGraph α} {a b : α} {p : G.Walk a b},
p.IsHamiltonian → ∀ (x : α), x ∈ p.support.toFinset | null | false |
Nat.odd_sub._simp_1 | Mathlib.Algebra.Ring.Parity | ∀ {m n : ℕ}, n ≤ m → Odd (m - n) = (Odd m ↔ Even n) | null | false |
CategoryTheory.Functor.elementsFunctor_map | Mathlib.CategoryTheory.Elements | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : CategoryTheory.Functor C (Type w)} (n : X ⟶ Y),
CategoryTheory.Functor.elementsFunctor.map n = (CategoryTheory.NatTrans.mapElements n).toCatHom | null | true |
WithZeroMulInt.toNNReal_le_one_iff | Mathlib.Data.Int.WithZero | ∀ {e : NNReal} {m : WithZero (Multiplicative ℤ)} (he : 1 < e), (WithZeroMulInt.toNNReal ⋯) m ≤ 1 ↔ m ≤ 1 | null | true |
Algebra.transcendental_ringHom_iff_of_comp_eq | Mathlib.RingTheory.Algebraic.Basic | ∀ {R : Type u} {S : Type u_1} {A : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Ring A]
[inst_3 : Algebra R A] {B : Type u_2} [inst_4 : Ring B] [inst_5 : Algebra S B] {FRS : Type u_3} {FAB : Type u_4}
[inst_6 : EquivLike FRS R S] [inst_7 : RingEquivClass FRS R S] [inst_8 : EquivLike FAB A B]
[inst_... | null | true |
padicValRat.of_int | Mathlib.NumberTheory.Padics.PadicVal.Basic | ∀ {p : ℕ} {z : ℤ}, padicValRat p ↑z = ↑(padicValInt p z) | The `p`-adic value of an integer `z ≠ 0` is its `p`-adic value as a rational. | true |
orderBornology_isBounded._simp_1 | Mathlib.Topology.Order.Bornology | ∀ {α : Type u_1} {s : Set α} [inst : Lattice α] [inst_1 : Nonempty α], Bornology.IsBounded s = (BddBelow s ∧ BddAbove s) | null | false |
CategoryTheory.Precoherent.recOn | Mathlib.CategoryTheory.Sites.Coherent.Basic | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{motive : CategoryTheory.Precoherent C → Sort u} →
(t : CategoryTheory.Precoherent C) →
((pullback :
∀ {B₁ B₂ : C} (f : B₂ ⟶ B₁) (α : Type) [Finite α] (X₁ : α → C) (π₁ : (a : α) → X₁ a ⟶ B₁),
CategoryTheor... | null | false |
max_mul_mul_left | Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : Mul α] [MulLeftMono α] (a b c : α), max (a * b) (a * c) = a * max b c | null | true |
ProbabilityTheory.Kernel.ae_compProd_iff | Mathlib.Probability.Kernel.Composition.CompProd | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ]
{η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {p : β × γ → Prop},
MeasurableSe... | null | true |
Equiv.forall_congr' | Mathlib.Logic.Equiv.Defs | ∀ {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (e : α ≃ β),
(∀ (b : β), p (e.symm b) ↔ q b) → ((∀ (a : α), p a) ↔ ∀ (b : β), q b) | null | true |
CategoryTheory.congr_app | Mathlib.CategoryTheory.NatTrans | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C D} {α β : CategoryTheory.NatTrans F G}, α = β → ∀ (X : C), α.app X = β.app X | null | true |
Fintype.linearIndependent_iffₛ | Mathlib.LinearAlgebra.LinearIndependent.Defs | ∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M] [inst_3 : Fintype ι],
LinearIndependent R v ↔ ∀ (f g : ι → R), ∑ i, f i • v i = ∑ i, g i • v i → ∀ (i : ι), f i = g i | null | true |
Group.nilpotencyClass_of_not_nilpotent | Mathlib.GroupTheory.Nilpotent | ∀ {G : Type u_1} [inst : Group G], ¬Group.IsNilpotent G → Group.nilpotencyClass G = 0 | null | true |
CategoryTheory.Functor.PreservesLeftKanExtension.mk._flat_ctor | Mathlib.CategoryTheory.Functor.KanExtension.Preserves | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} A]
[inst_1 : CategoryTheory.Category.{v_2, u_2} B] [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
[inst_3 : CategoryTheory.Category.{v_4, u_4} D] {G : CategoryTheory.Functor B D} {F : CategoryTheory.Functor A B... | null | false |
_private.Mathlib.Data.List.Cycle.0.List.next_eq_getElem._proof_1_7 | Mathlib.Data.List.Cycle | ∀ {α : Type u_1} [inst : DecidableEq α] {l : List α} {a : α},
a ∈ l →
∀ (hl : l ≠ []),
¬(List.idxOf a l + 1) % l.length + 1 ≤ l.dropLast.length →
(List.idxOf a l + 1) % l.length - l.dropLast.length < [l.getLast ⋯].length | null | false |
_private.Mathlib.Algebra.IsPrimePow.0.not_isPrimePow_zero._simp_1_4 | Mathlib.Algebra.IsPrimePow | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap_assoc | Mathlib.AlgebraicGeometry.ColimitsOver | ∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [inst : P.IsStableUnderBaseChange]
[inst_1 : P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1}
[inst_2 : CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover}
[inst_3 : CategoryTheory.Ca... | null | true |
Real.expPartialHomeomorph_target | Mathlib.Analysis.SpecialFunctions.Log.Basic | Real.expPartialHomeomorph.target = Set.Ioi 0 | null | true |
CategoryTheory.Codiscrete.equivFun._proof_2 | Mathlib.AlgebraicTopology.SimplicialSet.NerveCodiscrete | ∀ {X : Type u_1} {n : ℕ},
Function.RightInverse
(fun f =>
{ obj := fun k => { as := f k }, map := fun {X_1 Y} x => ({ as := f X_1 }.iso { as := f Y }).hom, map_id := ⋯,
map_comp := ⋯ })
fun f k => (f.obj k).as | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.0._regBuiltin.BitVec.reduceSignExtend.declare_343._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec.3228954591._hygCtx._hyg.21 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec | IO Unit | null | false |
IsCompl.compl_eq_iff | Mathlib.Order.BooleanAlgebra.Basic | ∀ {α : Type u} {x y z : α} [inst : BooleanAlgebra α], IsCompl x y → (zᶜ = y ↔ z = x) | null | true |
Polynomial.addSubmonoid_closure_setOfPred_eq_monomial | Mathlib.Algebra.Polynomial.Basic | ∀ {R : Type u} [inst : Semiring R], AddSubmonoid.closure {p | ∃ n a, p = (Polynomial.monomial n) a} = ⊤ | Monomials generate the additive monoid of polynomials. | true |
Array.all_iff_forall | Init.Data.Array.Lemmas | ∀ {α : Type u_1} {p : α → Bool} {as : Array α} {start stop : ℕ},
as.all p start stop = true ↔ ∀ (i : ℕ) (x : i < as.size), start ≤ i ∧ i < stop → p as[i] = true | null | true |
AddAction.sigmaFixedByEquivOrbitsProdAddGroup._proof_1 | Mathlib.GroupTheory.GroupAction.Quotient | ∀ (G : Type u_1) (X : Type u_2) [inst : AddGroup G] [inst_1 : AddAction G X] (x : G × X),
x.1 +ᵥ x.2 = x.2 ↔ x.1 +ᵥ x.2 = x.2 | null | false |
_private.Mathlib.Data.Rat.Sqrt.0.Rat.exists_mul_self.match_1_1 | Mathlib.Data.Rat.Sqrt | ∀ (x : ℚ) (motive : (∃ q, q * q = x) → Prop) (x_1 : ∃ q, q * q = x), (∀ (n : ℚ) (hn : n * n = x), motive ⋯) → motive x_1 | null | false |
Mathlib.Tactic._aux_Mathlib_Tactic_Core___macroRules_Mathlib_Tactic_tacticRepeat1__1 | Mathlib.Tactic.Core | Lean.Macro | `repeat1 tac` applies `tac` to main goal at least once. If the application succeeds,
the tactic is applied recursively to the generated subgoals until it eventually fails.
| false |
AlgebraicGeometry.instAddCommGroupObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInType | Mathlib.AlgebraicGeometry.StructureSheaf | {R M : Type u} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
(U : (TopologicalSpace.Opens ↑(AlgebraicGeometry.PrimeSpectrum.Top R))ᵒᵖ) →
AddCommGroup ((AlgebraicGeometry.structureSheafInType R M).obj.obj U) | null | true |
HahnSeries.toOrderTopSubOnePos | Mathlib.RingTheory.HahnSeries.Summable | {Γ : Type u_1} →
{R : Type u_3} →
[inst : AddCommMonoid Γ] →
[inst_1 : LinearOrder Γ] →
[inst_2 : IsOrderedCancelAddMonoid Γ] →
[inst_3 : CommRing R] → {x : HahnSeries Γ R} → 0 < (x - 1).orderTop → ↥(HahnSeries.orderTopSubOnePos Γ R) | Make an element of `orderTopSubOnePos` | true |
infEDist_inv | Mathlib.Analysis.Normed.Group.Pointwise | ∀ {E : Type u_1} [inst : SeminormedCommGroup E] (x : E) (s : Set E), Metric.infEDist x⁻¹ s = Metric.infEDist x s⁻¹ | null | true |
instBornologyPUnit._proof_1 | Mathlib.Topology.Bornology.Basic | ⊥ ≤ Filter.cofinite | null | false |
Lean.SerialMessage.ctorIdx | Lean.Message | Lean.SerialMessage → ℕ | null | false |
Char.lt | Init.Data.Char.Basic | Char → Char → Prop | One character is less than another if its code point is strictly less than the other's.
| true |
Lean.Grind.CommRing.Stepwise.div_cert.eq_1 | Init.Grind.Ring.CommSolver | ∀ (p₁ : Lean.Grind.CommRing.Poly) (k : ℤ) (p : Lean.Grind.CommRing.Poly),
Lean.Grind.CommRing.Stepwise.div_cert p₁ k p = (!k.beq' 0 && (Lean.Grind.CommRing.Poly.mulConst_k k p).beq' p₁) | null | true |
_private.Mathlib.FieldTheory.AlgebraicClosure.0.le_algebraicClosure_iff._simp_1_1 | Mathlib.FieldTheory.AlgebraicClosure | ∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {x : E},
(x ∈ algebraicClosure F E) = IsAlgebraic F x | null | false |
_private.Std.Http.Server.Connection.0.Std.Http.Server.Connection.PollSources.socket | Std.Http.Server.Connection | {α β : Type} → Std.Http.Server.Connection.PollSources✝ α β → Option α | null | true |
_private.Mathlib.Topology.CWComplex.Classical.Basic.0.Topology.RelCWComplex.disjoint_base_iUnion_openCell._simp_1_2 | Mathlib.Topology.CWComplex.Classical.Basic | ∀ {α : Type u_1} {ι : Sort u_4} {s : ι → Set α}, (⋃ i, s i = ∅) = ∀ (i : ι), s i = ∅ | null | false |
CategoryTheory.instIsDenseFunctorOppositeTypeYoneda | Mathlib.CategoryTheory.Functor.KanExtension.Dense | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C], CategoryTheory.yoneda.IsDense | null | true |
Array.forIn' | Init.Data.Array.Basic | {α : Type u} →
{β : Type v} → {m : Type v → Type w} → [Monad m] → (as : Array α) → β → ((a : α) → a ∈ as → β → m (ForInStep β)) → m β | Reference implementation for `forIn'` | true |
Std.ExtDTreeMap.Const.getEntryLT._proof_1 | Std.Data.ExtDTreeMap.Basic | ∀ {α : Type u_1} {cmp : α → α → Ordering} {β : Type u_2} [inst : Std.TransCmp cmp]
(t : Std.ExtDTreeMap α (fun x => β) cmp) (k : α),
(∃ a ∈ t, cmp a k = Ordering.lt) →
∀ (m : Std.DTreeMap α (fun x => β) cmp),
t = Std.ExtDTreeMap.mk m → ∃ a ∈ Std.ExtDTreeMap.mk m, cmp a k = Ordering.lt | null | false |
Real.logb_neg_of_base_lt_one | Mathlib.Analysis.SpecialFunctions.Log.Base | ∀ {b x : ℝ}, 0 < b → b < 1 → 1 < x → Real.logb b x < 0 | null | true |
CategoryTheory.equivEssImageOfReflective_inverse | Mathlib.CategoryTheory.Adjunction.Reflective | ∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{i : CategoryTheory.Functor D C} [inst_2 : CategoryTheory.Reflective i],
CategoryTheory.equivEssImageOfReflective.inverse = i.essImage.ι.comp (CategoryTheory.reflector i) | null | true |
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