name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.67M | allowCompletion bool 2
classes |
|---|---|---|---|
LieSubmodule.lowerCentralSeries_eq_lcs_comap | Mathlib.Algebra.Lie.Nilpotent | ∀ {R : Type u} {L : Type v} {M : Type w} [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] (k : ℕ) (N : LieSubmodule R L M)
[LieModule R L M], LieModule.lowerCentralSeries R L (↥N) k = LieSubmodule.comap N.incl (LieSubmodule.lcs k N) | true |
HopfAlgCat.noConfusionType | Mathlib.Algebra.Category.HopfAlgCat.Basic | Sort u_1 →
{R : Type u} → [inst : CommRing R] → HopfAlgCat R → {R' : Type u} → [inst' : CommRing R'] → HopfAlgCat R' → Sort u_1 | false |
CategoryTheory.ShortComplex.instPreservesLimitsOfShapeπ₂ | Mathlib.Algebra.Homology.ShortComplex.Limits | ∀ {J : Type u_1} {C : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} J]
[inst_1 : CategoryTheory.Category.{v_2, u_2} C] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[CategoryTheory.Limits.HasLimitsOfShape J C],
CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.ShortComplex.π₂ | true |
CategoryTheory.Idempotents.functorExtension₁._proof_1 | Mathlib.CategoryTheory.Idempotents.FunctorExtension | ∀ (C : Type u_1) (D : Type u_4) [inst : CategoryTheory.Category.{u_2, u_1} C]
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] (F : CategoryTheory.Functor C (CategoryTheory.Idempotents.Karoubi D)),
CategoryTheory.Idempotents.FunctorExtension₁.map (CategoryTheory.CategoryStruct.id F) =
CategoryTheory.CategoryStruct.id (CategoryTheory.Idempotents.FunctorExtension₁.obj F) | false |
ContinuousLinearMap.instSMul._proof_1 | Mathlib.Topology.Algebra.Module.LinearMap | ∀ {R₁ : Type u_4} {R₂ : Type u_5} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_1}
[inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_2} [inst_4 : TopologicalSpace M₂]
[inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂] {S₂ : Type u_3}
[inst_8 : DistribSMul S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂),
Continuous fun x => c • (↑f).toFun x | false |
_private.Mathlib.Combinatorics.Additive.VerySmallDoubling.0.Finset.doubling_lt_golden_ratio._simp_1_23 | Mathlib.Combinatorics.Additive.VerySmallDoubling | ∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p) | false |
NormedField.instRankLeOneNNRealValuation._proof_3 | Mathlib.Topology.Algebra.Valued.NormedValued | ∀ {K : Type u_1} [hK : NormedField K] [inst : IsUltrametricDist K], StrictMono ⇑MonoidWithZeroHom.ValueGroup₀.embedding | false |
Matrix.diagonal_mulVec_single | Mathlib.Data.Matrix.Mul | ∀ {n : Type u_3} {R : Type u_7} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : NonUnitalNonAssocSemiring R]
(v : n → R) (j : n) (x : R), (Matrix.diagonal v).mulVec (Pi.single j x) = Pi.single j (v j * x) | true |
_private.Mathlib.Analysis.Asymptotics.Theta.0.Asymptotics.isTheta_const_const_iff._simp_1_2 | Mathlib.Analysis.Asymptotics.Theta | ∀ {α : Type u_1} {E'' : Type u_9} {F'' : Type u_10} [inst : NormedAddCommGroup E''] [inst_1 : NormedAddCommGroup F'']
{c : E''} {c' : F''} (l : Filter α) [l.NeBot], ((fun _x => c) =O[l] fun _x => c') = (c' = 0 → c = 0) | false |
Lean.Doc.instBEqListItem.beq | Lean.DocString.Types | {α : Type u_1} → [BEq α] → Lean.Doc.ListItem α → Lean.Doc.ListItem α → Bool | true |
Std.DTreeMap.isSome_maxKey?_iff_isEmpty_eq_false | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp],
t.maxKey?.isSome = true ↔ t.isEmpty = false | true |
Mathlib.Tactic.Monoidal.instMonadNormalizeNaturalityMonoidalM.match_1 | Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | (ctx : Mathlib.Tactic.Monoidal.Context) →
(motive : Option Q(CategoryTheory.MonoidalCategory unknown_1) → Sort u_1) →
(x : Option Q(CategoryTheory.MonoidalCategory unknown_1)) →
((_monoidal : Q(CategoryTheory.MonoidalCategory unknown_1)) → motive (some _monoidal)) →
((x : Option Q(CategoryTheory.MonoidalCategory unknown_1)) → motive x) → motive x | false |
ProbabilityTheory.preCDF_le_one | Mathlib.Probability.Kernel.Disintegration.CondCDF | ∀ {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ],
∀ᵐ (a : α) ∂ρ.fst, ∀ (r : ℚ), ProbabilityTheory.preCDF ρ r a ≤ 1 | true |
Metric.Snowflaking.preimage_toSnowflaking_emetricBall | Mathlib.Topology.MetricSpace.Snowflaking | ∀ {X : Type u_1} {α : ℝ} {hα₀ : 0 < α} {hα₁ : α ≤ 1} [inst : PseudoEMetricSpace X] (x : Metric.Snowflaking X α hα₀ hα₁)
(d : ENNReal),
⇑Metric.Snowflaking.toSnowflaking ⁻¹' Metric.eball x d = Metric.eball (Metric.Snowflaking.ofSnowflaking x) (d ^ α⁻¹) | true |
Finset.min_union | Mathlib.Data.Finset.Max | ∀ {α : Type u_2} [inst : LinearOrder α] {s t : Finset α}, (s ∪ t).min = min s.min t.min | true |
Lean.Compiler.LCNF.Code.collectUsed | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} → Lean.Compiler.LCNF.Code pu → optParam Lean.FVarIdHashSet ∅ → Lean.FVarIdHashSet | true |
CategoryTheory.Subgroupoid.instSetLikeSigmaHom | Mathlib.CategoryTheory.Groupoid.Subgroupoid | {C : Type u} → [inst : CategoryTheory.Groupoid C] → SetLike (CategoryTheory.Subgroupoid C) ((c : C) × (d : C) × (c ⟶ d)) | true |
ENat.one_lt_card._simp_1 | Mathlib.SetTheory.Cardinal.Finite | ∀ {α : Type u_1} [Nontrivial α], (1 < ENat.card α) = True | false |
Lean.Expr.isDIte | Lean.Util.Recognizers | Lean.Expr → Bool | true |
edist_lt_top | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u_3} [inst : PseudoMetricSpace α] (x y : α), edist x y < ⊤ | true |
CategoryTheory.Functor.PullbackObjObj.mk.inj | Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj | ∀ {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {inst : CategoryTheory.Category.{v₁, u₁} C₁}
{inst_1 : CategoryTheory.Category.{v₂, u₂} C₂} {inst_2 : CategoryTheory.Category.{v₃, u₃} C₃}
{G : CategoryTheory.Functor C₁ᵒᵖ (CategoryTheory.Functor C₃ C₂)} {X₁ Y₁ : C₁} {f₁ : X₁ ⟶ Y₁} {X₃ Y₃ : C₃}
{f₃ : X₃ ⟶ Y₃} {pt : C₂} {fst : pt ⟶ (G.obj (Opposite.op X₁)).obj X₃} {snd : pt ⟶ (G.obj (Opposite.op Y₁)).obj Y₃}
{isPullback : CategoryTheory.IsPullback fst snd ((G.obj (Opposite.op X₁)).map f₃) ((G.map f₁.op).app Y₃)} {pt_1 : C₂}
{fst_1 : pt_1 ⟶ (G.obj (Opposite.op X₁)).obj X₃} {snd_1 : pt_1 ⟶ (G.obj (Opposite.op Y₁)).obj Y₃}
{isPullback_1 : CategoryTheory.IsPullback fst_1 snd_1 ((G.obj (Opposite.op X₁)).map f₃) ((G.map f₁.op).app Y₃)},
{ pt := pt, fst := fst, snd := snd, isPullback := isPullback } =
{ pt := pt_1, fst := fst_1, snd := snd_1, isPullback := isPullback_1 } →
pt = pt_1 ∧ fst ≍ fst_1 ∧ snd ≍ snd_1 | true |
FirstOrder.Language.DefinableSet.coe_bot | Mathlib.ModelTheory.Definability | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {A : Set M} {α : Type u₁}, ↑⊥ = ∅ | true |
ArchimedeanClass.orderHom | Mathlib.Algebra.Order.Archimedean.Class | {M : Type u_1} →
[inst : AddCommGroup M] →
[inst_1 : LinearOrder M] →
[inst_2 : IsOrderedAddMonoid M] →
{N : Type u_2} →
[inst_3 : AddCommGroup N] →
[inst_4 : LinearOrder N] →
[inst_5 : IsOrderedAddMonoid N] → (M →+o N) → ArchimedeanClass M →o ArchimedeanClass N | true |
_private.Mathlib.CategoryTheory.Shift.Localization.0.CategoryTheory.Functor.commShiftOfLocalization._simp_1 | Mathlib.CategoryTheory.Shift.Localization | ∀ {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} (self : CategoryTheory.NatTrans F G) ⦃X Y : C⦄ (f : X ⟶ Y) {Z : D}
(h : G.obj Y ⟶ Z),
CategoryTheory.CategoryStruct.comp (self.app X) (CategoryTheory.CategoryStruct.comp (G.map f) h) =
CategoryTheory.CategoryStruct.comp (F.map f) (CategoryTheory.CategoryStruct.comp (self.app Y) h) | false |
ExteriorAlgebra.ιInv | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | {R : Type u1} →
[inst : CommRing R] → {M : Type u2} → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → ExteriorAlgebra R M →ₗ[R] M | true |
_private.Lean.Meta.InferType.0.Lean.Meta.inferMVarType | Lean.Meta.InferType | Lean.MVarId → Lean.MetaM Lean.Expr | true |
AddOpposite.instCommMonoid._proof_2 | Mathlib.Algebra.Group.Opposite | ∀ {α : Type u_1} [inst : CommMonoid α] (x x_1 : αᵃᵒᵖ), AddOpposite.unop (x * x_1) = AddOpposite.unop (x * x_1) | false |
ENormedCommMonoid.toESeminormedCommMonoid | Mathlib.Analysis.Normed.Group.Defs | {E : Type u_8} → {inst : TopologicalSpace E} → [self : ENormedCommMonoid E] → ESeminormedCommMonoid E | true |
_private.Init.Data.Nat.Lcm.0.Nat.lcm_pos._simp_1_1 | Init.Data.Nat.Lcm | ∀ {n : ℕ}, (n ≠ 0) = (0 < n) | false |
Equiv.algebra | Mathlib.Algebra.Algebra.TransferInstance | (R : Type u_1) →
{α : Type u_2} →
{β : Type u_3} →
[inst : CommSemiring R] →
(e : α ≃ β) →
[inst_1 : Semiring β] →
have x := e.semiring;
[Algebra R β] → Algebra R α | true |
ZMod.intCast_cast_mul | Mathlib.Data.ZMod.Basic | ∀ {n : ℕ} (x y : ZMod n), (x * y).cast = x.cast * y.cast % ↑n | true |
Lean.Elab.InlayHintLinkLocation._sizeOf_inst | Lean.Elab.InfoTree.InlayHints | SizeOf Lean.Elab.InlayHintLinkLocation | false |
Lean.Meta.Grind.EMatch.State.recOn | Lean.Meta.Tactic.Grind.Types | {motive : Lean.Meta.Grind.EMatch.State → Sort u} →
(t : Lean.Meta.Grind.EMatch.State) →
((thmMap : Lean.Meta.Grind.EMatchTheoremsArray) →
(gmt : ℕ) →
(thms newThms : Lean.PArray Lean.Meta.Grind.EMatchTheorem) →
(numInstances numDelayedInstances num : ℕ) →
(preInstances : Lean.Meta.Grind.PreInstanceSet) →
(nextThmIdx : ℕ) →
(matchEqNames : Lean.PHashSet Lean.Name) →
(delayedThmInsts :
Lean.PHashMap Lean.Meta.Sym.ExprPtr (List Lean.Meta.Grind.DelayedTheoremInstance)) →
motive
{ thmMap := thmMap, gmt := gmt, thms := thms, newThms := newThms, numInstances := numInstances,
numDelayedInstances := numDelayedInstances, num := num, preInstances := preInstances,
nextThmIdx := nextThmIdx, matchEqNames := matchEqNames,
delayedThmInsts := delayedThmInsts }) →
motive t | false |
Vector.append_assoc_symm | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {n m k : ℕ} {xs : Vector α n} {ys : Vector α m} {zs : Vector α k},
xs ++ (ys ++ zs) = Vector.cast ⋯ (xs ++ ys ++ zs) | true |
_private.Mathlib.Data.Seq.Parallel.0.Computation.BisimO.match_1.splitter._sparseCasesOn_3 | Mathlib.Data.Seq.Parallel | {α : Type u} →
{β : Type v} →
{motive : α ⊕ β → Sort u_1} →
(t : α ⊕ β) → ((val : α) → motive (Sum.inl val)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t | false |
List.anyM_pure | Init.Data.List.Monadic | ∀ {m : Type → Type u_1} {α : Type u_2} [inst : Monad m] [LawfulMonad m] {p : α → Bool} {as : List α},
List.anyM (fun x => pure (p x)) as = pure (as.any p) | true |
Option.forIn_toList | Init.Data.Option.List | ∀ {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [inst : Monad m] (o : Option α) (b : β)
(f : α → β → m (ForInStep β)), forIn o.toList b f = forIn o b f | true |
Filter.le_limsup_of_frequently_le' | Mathlib.Order.LiminfLimsup | ∀ {α : Type u_6} {β : Type u_7} [inst : CompleteLattice β] {f : Filter α} {u : α → β} {x : β},
(∃ᶠ (a : α) in f, x ≤ u a) → x ≤ Filter.limsup u f | true |
MeasureTheory.posConvolution._proof_1 | Mathlib.Analysis.Convolution | ∀ {F : Type u_1} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F], SMulCommClass ℝ ℝ F | false |
Shrink.instNonUnitalCommRing | Mathlib.Algebra.Ring.Shrink | {α : Type u_1} → [inst : Small.{v, u_1} α] → [NonUnitalCommRing α] → NonUnitalCommRing (Shrink.{v, u_1} α) | true |
injective_frobenius._simp_1 | Mathlib.FieldTheory.Perfect | ∀ (R : Type u_1) (p : ℕ) [inst : CommSemiring R] [inst_1 : ExpChar R p] [PerfectRing R p],
Function.Injective ⇑(frobenius R p) = True | false |
ULift.distribMulAction'._proof_2 | Mathlib.Algebra.Module.ULift | ∀ {R : Type u_3} {M : Type u_2} [inst : Monoid R] [inst_1 : AddMonoid M] [inst_2 : DistribMulAction R M] (a : R)
(x y : ULift.{u_1, u_2} M), a • (x + y) = a • x + a • y | false |
CategoryTheory.Limits.colimitLimitToLimitColimitCone._proof_4 | Mathlib.CategoryTheory.Limits.ColimitLimit | ∀ {J : Type u_2} {K : Type u_5} [inst : CategoryTheory.Category.{u_1, u_2} J]
[inst_1 : CategoryTheory.Category.{u_6, u_5} K] {C : Type u_4} [inst_2 : CategoryTheory.Category.{u_3, u_4} C]
[CategoryTheory.Limits.HasLimitsOfShape J C] [inst_4 : CategoryTheory.Limits.HasColimitsOfShape K C]
(G : CategoryTheory.Functor J (CategoryTheory.Functor K C)),
CategoryTheory.Limits.HasLimit
((CategoryTheory.Functor.curry.obj (CategoryTheory.Functor.uncurry.obj G)).comp CategoryTheory.Limits.colim) | false |
RelHom.instFintype | Mathlib.Data.Fintype.Pi | {α : Type u_3} →
{β : Type u_4} →
[Fintype α] →
[Fintype β] →
[DecidableEq α] →
{r : α → α → Prop} → {s : β → β → Prop} → [DecidableRel r] → [DecidableRel s] → Fintype (r →r s) | true |
CategoryTheory.Limits.widePushoutShapeOp._proof_3 | Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | ∀ (J : Type u_1) {X Y Z : CategoryTheory.Limits.WidePushoutShape J} (f : X ⟶ Y) (g : Y ⟶ Z),
CategoryTheory.Limits.widePushoutShapeOpMap J X Z (CategoryTheory.CategoryStruct.comp f g) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.widePushoutShapeOpMap J X Y f)
(CategoryTheory.Limits.widePushoutShapeOpMap J Y Z g) | false |
Lean.Elab.Command.InductiveElabStep2.prefinalize | Lean.Elab.MutualInductive | Lean.Elab.Command.InductiveElabStep2 →
List Lean.Name →
Array Lean.Expr → (Lean.Expr → Lean.MetaM Lean.Expr) → Lean.Elab.TermElabM Lean.Elab.Command.InductiveElabStep3 | true |
Std.DTreeMap.containsThenInsert_snd | 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}
{v : β k}, (t.containsThenInsert k v).2 = t.insert k v | true |
UInt32.toNat_ofNat_of_lt | Init.Data.UInt.Lemmas | ∀ {n : ℕ}, n < UInt32.size → UInt32.toNat (OfNat.ofNat n) = n | true |
Subgroup.map_symm_eq_iff_map_eq | Mathlib.Algebra.Group.Subgroup.Map | ∀ {G : Type u_1} [inst : Group G] (K : Subgroup G) {N : Type u_5} [inst_1 : Group N] {H : Subgroup N} {e : G ≃* N},
Subgroup.map (↑e.symm) H = K ↔ Subgroup.map (↑e) K = H | true |
Std.Iterators.Types.Flatten.IsPlausibleStep.rec | Init.Data.Iterators.Combinators.Monadic.FlatMap | ∀ {α α₂ β : Type w} {m : Type w → Type w'} [inst : Std.Iterator α m (Std.IterM m β)] [inst_1 : Std.Iterator α₂ m β]
{motive :
(it : Std.IterM m β) →
(step : Std.IterStep (Std.IterM m β) β) → Std.Iterators.Types.Flatten.IsPlausibleStep it step → Prop},
(∀ {it₁ it₁' : Std.IterM m (Std.IterM m β)} {it₂' : Std.IterM m β}
(a : it₁.IsPlausibleStep (Std.IterStep.yield it₁' it₂')),
motive { internalState := { it₁ := it₁, it₂ := none } }
(Std.IterStep.skip { internalState := { it₁ := it₁', it₂ := some it₂' } }) ⋯) →
(∀ {it₁ it₁' : Std.IterM m (Std.IterM m β)} (a : it₁.IsPlausibleStep (Std.IterStep.skip it₁')),
motive { internalState := { it₁ := it₁, it₂ := none } }
(Std.IterStep.skip { internalState := { it₁ := it₁', it₂ := none } }) ⋯) →
(∀ {it₁ : Std.IterM m (Std.IterM m β)} (a : it₁.IsPlausibleStep Std.IterStep.done),
motive { internalState := { it₁ := it₁, it₂ := none } } Std.IterStep.done ⋯) →
(∀ {it₁ : Std.IterM m (Std.IterM m β)} {it₂ it₂' : Std.IterM m β} {b : β}
(a : it₂.IsPlausibleStep (Std.IterStep.yield it₂' b)),
motive { internalState := { it₁ := it₁, it₂ := some it₂ } }
(Std.IterStep.yield { internalState := { it₁ := it₁, it₂ := some it₂' } } b) ⋯) →
(∀ {it₁ : Std.IterM m (Std.IterM m β)} {it₂ it₂' : Std.IterM m β}
(a : it₂.IsPlausibleStep (Std.IterStep.skip it₂')),
motive { internalState := { it₁ := it₁, it₂ := some it₂ } }
(Std.IterStep.skip { internalState := { it₁ := it₁, it₂ := some it₂' } }) ⋯) →
(∀ {it₁ : Std.IterM m (Std.IterM m β)} {it₂ : Std.IterM m β} (a : it₂.IsPlausibleStep Std.IterStep.done),
motive { internalState := { it₁ := it₁, it₂ := some it₂ } }
(Std.IterStep.skip { internalState := { it₁ := it₁, it₂ := none } }) ⋯) →
∀ {it : Std.IterM m β} {step : Std.IterStep (Std.IterM m β) β}
(t : Std.Iterators.Types.Flatten.IsPlausibleStep it step), motive it step t | false |
Lean.Meta.Simp.instInhabitedContext | Lean.Meta.Tactic.Simp.Types | Inhabited Lean.Meta.Simp.Context | true |
_private.Mathlib.MeasureTheory.Integral.IntegrableOn.0.MeasureTheory.integrableAtFilter_atBot_iff.match_1_1 | Mathlib.MeasureTheory.Integral.IntegrableOn | ∀ {α : Type u_1} {ε : Type u_2} {mα : MeasurableSpace α} {f : α → ε} {μ : MeasureTheory.Measure α}
[inst : TopologicalSpace ε] [inst_1 : ContinuousENorm ε] [inst_2 : Preorder α]
(motive : MeasureTheory.IntegrableAtFilter f Filter.atBot μ → Prop)
(x : MeasureTheory.IntegrableAtFilter f Filter.atBot μ),
(∀ (s : Set α) (hs : s ∈ Filter.atBot) (hi : MeasureTheory.IntegrableOn f s μ), motive ⋯) → motive x | false |
Fintype.one_lt_card_iff_nontrivial | Mathlib.Data.Fintype.EquivFin | ∀ {α : Type u_1} [inst : Fintype α], 1 < Fintype.card α ↔ Nontrivial α | true |
Order.isSuccPrelimit_iff_of_noMax | Mathlib.Order.SuccPred.Limit | ∀ {α : Type u_1} {a : α} [inst : Preorder α] [inst_1 : SuccOrder α] [IsSuccArchimedean α] [NoMaxOrder α],
Order.IsSuccPrelimit a ↔ IsMin a | true |
Std.Roo.noConfusionType | Init.Data.Range.Polymorphic.PRange | Sort u_1 → {α : Type u} → Std.Roo α → {α' : Type u} → Std.Roo α' → Sort u_1 | false |
Set.instCompleteAtomicBooleanAlgebra._proof_5 | Mathlib.Data.Set.BooleanAlgebra | ∀ {α : Type u_1} (a : Set α), ⊥ ≤ a | false |
CategoryTheory.Limits.WalkingMulticospan.Hom.casesOn | Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {J : CategoryTheory.Limits.MulticospanShape} →
{motive : (x x_1 : CategoryTheory.Limits.WalkingMulticospan J) → x.Hom x_1 → Sort u} →
{x x_1 : CategoryTheory.Limits.WalkingMulticospan J} →
(t : x.Hom x_1) →
((A : CategoryTheory.Limits.WalkingMulticospan J) →
motive A A (CategoryTheory.Limits.WalkingMulticospan.Hom.id A)) →
((b : J.R) →
motive (CategoryTheory.Limits.WalkingMulticospan.left (J.fst b))
(CategoryTheory.Limits.WalkingMulticospan.right b)
(CategoryTheory.Limits.WalkingMulticospan.Hom.fst b)) →
((b : J.R) →
motive (CategoryTheory.Limits.WalkingMulticospan.left (J.snd b))
(CategoryTheory.Limits.WalkingMulticospan.right b)
(CategoryTheory.Limits.WalkingMulticospan.Hom.snd b)) →
motive x x_1 t | false |
Lean.Grind.CommRing.Mon.mult.injEq | Init.Grind.Ring.CommSolver | ∀ (p : Lean.Grind.CommRing.Power) (m : Lean.Grind.CommRing.Mon) (p_1 : Lean.Grind.CommRing.Power)
(m_1 : Lean.Grind.CommRing.Mon),
(Lean.Grind.CommRing.Mon.mult p m = Lean.Grind.CommRing.Mon.mult p_1 m_1) = (p = p_1 ∧ m = m_1) | true |
Equiv.Perm.two_le_card_support_cycleOf_iff._simp_1 | Mathlib.GroupTheory.Perm.Cycle.Factors | ∀ {α : Type u_2} {f : Equiv.Perm α} {x : α} [inst : DecidableEq α] [inst_1 : Fintype α],
(2 ≤ (f.cycleOf x).support.card) = (f x ≠ x) | false |
LinearIndependent.repr | 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] → LinearIndependent R v → ↥(Submodule.span R (Set.range v)) →ₗ[R] ι →₀ R | true |
Lean.Meta.LiftLetsConfig.noConfusion | Init.MetaTypes | {P : Sort u} → {t t' : Lean.Meta.LiftLetsConfig} → t = t' → Lean.Meta.LiftLetsConfig.noConfusionType P t t' | false |
_private.Batteries.Data.DList.Lemmas.0.Batteries.DList.push.match_1.eq_1 | Batteries.Data.DList.Lemmas | ∀ {α : Type u_1} (motive : Batteries.DList α → α → Sort u_2) (f : List α → List α) (h : ∀ (l : List α), f l = f [] ++ l)
(a : α)
(h_1 :
(f : List α → List α) → (h : ∀ (l : List α), f l = f [] ++ l) → (a : α) → motive { apply := f, invariant := h } a),
(match { apply := f, invariant := h }, a with
| { apply := f, invariant := h }, a => h_1 f h a) =
h_1 f h a | true |
Functor._aux_Mathlib_Control_Functor___unexpand_Functor_mapConstRev_1 | Mathlib.Control.Functor | Lean.PrettyPrinter.Unexpander | false |
AlgebraicGeometry.StructureSheaf.sectionsSubalgebra | Mathlib.AlgebraicGeometry.StructureSheaf | {R : Type u} →
(A : Type u) →
[inst : CommRing R] →
[inst_1 : CommRing A] →
[inst_2 : Algebra R A] →
(U : TopologicalSpace.Opens ↑(AlgebraicGeometry.PrimeSpectrum.Top R)) →
Subalgebra R ((x : ↥U) → AlgebraicGeometry.StructureSheaf.Localizations A ↑x) | true |
Ideal.cotangentToQuotientSquare | Mathlib.RingTheory.Ideal.Cotangent | {R : Type u} → [inst : CommRing R] → (I : Ideal R) → I.Cotangent →ₗ[R] R ⧸ I ^ 2 | true |
Std.TreeMap.Raw.Equiv.getEntryLE!_eq | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap.Raw α β cmp} [Std.TransCmp cmp]
[inst : Inhabited (α × β)] {k : α}, t₁.WF → t₂.WF → t₁.Equiv t₂ → t₁.getEntryLE! k = t₂.getEntryLE! k | true |
iSup_psigma' | Mathlib.Order.CompleteLattice.Basic | ∀ {α : Type u_1} [inst : CompleteLattice α] {ι : Sort u_8} {κ : ι → Sort u_9} (f : (i : ι) → κ i → α),
⨆ i, ⨆ j, f i j = ⨆ ij, f ij.fst ij.snd | true |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.getD_erase._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {x : Ord α} {x_1 : BEq α} [Std.LawfulBEqOrd α] {a b : α}, (compare a b = Ordering.eq) = ((a == b) = true) | false |
String.rawEndPos.eq_1 | Init.Data.String.Iterator | ∀ (s : String), s.rawEndPos = { byteIdx := s.utf8ByteSize } | true |
CategoryTheory.Limits.colimitIsoFlipCompColim_hom_app | Mathlib.CategoryTheory.Limits.FunctorCategory.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type u₁} [inst_1 : CategoryTheory.Category.{v₁, u₁} J]
{K : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} K] [inst_3 : CategoryTheory.Limits.HasColimitsOfShape J C]
(F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K),
(CategoryTheory.Limits.colimitIsoFlipCompColim F).hom.app X =
(CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation F X).hom | true |
CategoryTheory.Cat.Hom.ext | Mathlib.CategoryTheory.Category.Cat | ∀ {C D : CategoryTheory.Cat} {x y : C.Hom D}, x.toFunctor = y.toFunctor → x = y | true |
Frm.carrier | Mathlib.Order.Category.Frm | Frm → Type u_1 | true |
_private.Mathlib.Analysis.Convex.StrictCombination.0.StrictConvex.centerMass_mem_interior._simp_1_4 | Mathlib.Analysis.Convex.StrictCombination | ∀ {α : Type u_1} [inst : AddZeroClass α] [inst_1 : LT α] [AddLeftStrictMono α] [AddLeftReflectLT α] (a : α) {b : α},
(a < a + b) = (0 < b) | false |
instAddCommGroupFreeAbelianGroup._aux_17 | Mathlib.GroupTheory.FreeAbelianGroup | (α : Type u_1) → ℤ → FreeAbelianGroup α → FreeAbelianGroup α | false |
Std.DHashMap.Internal.Raw₀.Const.get?ₘ | Std.Data.DHashMap.Internal.Model | {α : Type u} → {β : Type v} → [BEq α] → [Hashable α] → (Std.DHashMap.Internal.Raw₀ α fun x => β) → α → Option β | true |
IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletionIntegers_apply | Mathlib.RingTheory.DedekindDomain.AdicValuation | ∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsDedekindDomain R] (K : Type u_2) [inst_2 : Field K]
[inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R),
↑((algebraMap R ↥(IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v)) r) =
↑((WithVal.equiv (IsDedekindDomain.HeightOneSpectrum.valuation K v)).symm ((algebraMap R K) r)) | true |
Subfield.relrank.eq_1 | Mathlib.FieldTheory.Relrank | ∀ {E : Type v} [inst : Field E] (A B : Subfield E), A.relrank B = Module.rank ↥(A ⊓ B) ↥(Subfield.extendScalars ⋯) | true |
Lean.Server.Completion.ContextualizedCompletionInfo.mk._flat_ctor | Lean.Server.Completion.CompletionUtils | Lean.Server.Completion.HoverInfo →
Lean.Elab.ContextInfo → Lean.Elab.CompletionInfo → Lean.Server.Completion.ContextualizedCompletionInfo | false |
List.toAssocList'._sunfold | Lean.Data.AssocList | {α : Type u} → {β : Type v} → List (α × β) → Lean.AssocList α β | false |
SymmetricAlgebra.algHom._proof_1 | Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | ∀ (R : Type u_1) (M : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
IsScalarTower R R M | false |
Lean.Compiler.LCNF.JoinPointCommonArgs.AnalysisCtx._sizeOf_inst | Lean.Compiler.LCNF.JoinPoints | SizeOf Lean.Compiler.LCNF.JoinPointCommonArgs.AnalysisCtx | false |
IsLocalization.Away.exists_isIntegral_mul_of_isIntegral_algebraMap | Mathlib.RingTheory.Localization.Integral | ∀ {R : Type u_5} {S : Type u_6} {Sₘ : Type u_7} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing Sₘ]
[inst_3 : Algebra R S] [inst_4 : Algebra S Sₘ] [inst_5 : Algebra R Sₘ] [IsScalarTower R S Sₘ] {r : S},
IsIntegral R r →
∀ [IsLocalization.Away r Sₘ] {x : S}, IsIntegral R ((algebraMap S Sₘ) x) → ∃ n, IsIntegral R (r ^ n * x) | true |
CompositionSeries.Equivalent.trans | Mathlib.Order.JordanHolder | ∀ {X : Type u} [inst : Lattice X] [inst_1 : JordanHolderLattice X] {s₁ s₂ s₃ : CompositionSeries X},
s₁.Equivalent s₂ → s₂.Equivalent s₃ → s₁.Equivalent s₃ | true |
Filter.EventuallyLE.rfl | Mathlib.Order.Filter.Basic | ∀ {α : Type u} {β : Type v} [inst : Preorder β] {l : Filter α} {f : α → β}, f ≤ᶠ[l] f | true |
_private.Lean.Compiler.Old.0.Lean.Compiler.getDeclNamesForCodeGen.match_1 | Lean.Compiler.Old | (motive : Lean.Declaration → Sort u_1) →
(x : Lean.Declaration) →
((name : Lean.Name) →
(levelParams : List Lean.Name) →
(type value : Lean.Expr) →
(hints : Lean.ReducibilityHints) →
(safety : Lean.DefinitionSafety) →
(all : List Lean.Name) →
motive
(Lean.Declaration.defnDecl
{ name := name, levelParams := levelParams, type := type, value := value, hints := hints,
safety := safety, all := all })) →
((name : Lean.Name) →
(levelParams : List Lean.Name) →
(type value : Lean.Expr) →
(isUnsafe : Bool) →
(all : List Lean.Name) →
motive
(Lean.Declaration.opaqueDecl
{ name := name, levelParams := levelParams, type := type, value := value, isUnsafe := isUnsafe,
all := all })) →
((name : Lean.Name) →
(levelParams : List Lean.Name) →
(type : Lean.Expr) →
(isUnsafe : Bool) →
motive
(Lean.Declaration.axiomDecl
{ name := name, levelParams := levelParams, type := type, isUnsafe := isUnsafe })) →
((defs : List Lean.DefinitionVal) → motive (Lean.Declaration.mutualDefnDecl defs)) →
((x : Lean.Declaration) → motive x) → motive x | false |
IsCoprime.mono | Mathlib.RingTheory.Coprime.Basic | ∀ {R : Type u} [inst : CommSemiring R] {x y z w : R}, x ∣ y → z ∣ w → IsCoprime y w → IsCoprime x z | true |
_private.Mathlib.Algebra.Module.Presentation.Tensor.0.Module.Relations.tensor.match_1.eq_1 | Mathlib.Algebra.Module.Presentation.Tensor | ∀ {A : Type u_5} [inst : CommRing A] (relations₁ : Module.Relations A) (relations₂ : Module.Relations A)
(motive : relations₁.R × relations₂.G ⊕ relations₁.G × relations₂.R → Sort u_6) (r₁ : relations₁.R)
(g₂ : relations₂.G) (h_1 : (r₁ : relations₁.R) → (g₂ : relations₂.G) → motive (Sum.inl (r₁, g₂)))
(h_2 : (g₁ : relations₁.G) → (r₂ : relations₂.R) → motive (Sum.inr (g₁, r₂))),
(match Sum.inl (r₁, g₂) with
| Sum.inl (r₁, g₂) => h_1 r₁ g₂
| Sum.inr (g₁, r₂) => h_2 g₁ r₂) =
h_1 r₁ g₂ | true |
LinearMap.baseChange_comp | Mathlib.LinearAlgebra.TensorProduct.Tower | ∀ {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} {P : Type u_6} [inst : CommSemiring R]
[inst_1 : Semiring A] [inst_2 : Algebra R A] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid N]
[inst_5 : AddCommMonoid P] [inst_6 : Module R M] [inst_7 : Module R N] [inst_8 : Module R P] (f : M →ₗ[R] N)
(g : N →ₗ[R] P), LinearMap.baseChange A (g ∘ₗ f) = LinearMap.baseChange A g ∘ₗ LinearMap.baseChange A f | true |
IsSl2Triple | Mathlib.Algebra.Lie.Sl2 | {L : Type u_2} → [LieRing L] → L → L → L → Prop | true |
SSet.PtSimplex.MulStruct.ctorIdx | Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | {X : SSet} →
{n : ℕ} →
{x : X.obj (Opposite.op (SimplexCategory.mk 0))} → {f g fg : X.PtSimplex n x} → {i : Fin n} → f.MulStruct g fg i → ℕ | false |
UInt32.ofBitVec_add | Init.Data.UInt.Lemmas | ∀ (a b : BitVec 32), { toBitVec := a + b } = { toBitVec := a } + { toBitVec := b } | true |
bddAbove_range_mul | Mathlib.Algebra.Order.GroupWithZero.Bounds | ∀ {α : Type u_1} {β : Type u_2} [Nonempty α] {u v : α → β} [inst : Preorder β] [inst_1 : Zero β] [inst_2 : Mul β]
[PosMulMono β] [MulPosMono β],
BddAbove (Set.range u) → 0 ≤ u → BddAbove (Set.range v) → 0 ≤ v → BddAbove (Set.range (u * v)) | true |
Complex.tendsto_norm_tan_of_cos_eq_zero | Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv | ∀ {x : ℂ}, Complex.cos x = 0 → Filter.Tendsto (fun x => ‖Complex.tan x‖) (nhdsWithin x {x}ᶜ) Filter.atTop | true |
Subtype.t0Space | Mathlib.Topology.Separation.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {p : X → Prop}, T0Space (Subtype p) | true |
groupHomology.cycles₁IsoOfIsTrivial.eq_1 | Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | ∀ {k G : Type u} [inst : CommRing k] [inst_1 : Group G] (A : Rep.{u, u, u} k G) [inst_2 : A.IsTrivial],
groupHomology.cycles₁IsoOfIsTrivial A = (LinearEquiv.ofTop (groupHomology.cycles₁ A) ⋯).toModuleIso | true |
Matroid.subsingleton_indep._auto_1 | Mathlib.Combinatorics.Matroid.Loop | Lean.Syntax | false |
InfHom.withBot_toFun | Mathlib.Order.Hom.WithTopBot | ∀ {α : Type u_1} {β : Type u_2} [inst : SemilatticeInf α] [inst_1 : SemilatticeInf β] (f : InfHom α β) (a : WithBot α),
f.withBot a = WithBot.map (⇑f) a | true |
CategoryTheory.normalOfIsPushoutSndOfNormal._proof_4 | Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S} [gn : CategoryTheory.NormalEpi g]
(comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k)
(t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)),
0 = CategoryTheory.CategoryStruct.comp 0 f →
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ h ⋯) =
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Cofork.ofπ h ⋯) | false |
Std.TreeSet.Raw.max?_eq_none_iff._simp_1 | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp],
t.WF → (t.max? = none) = (t.isEmpty = true) | false |
two_mul_le_add_mul_sq | Mathlib.Algebra.Order.Field.Basic | ∀ {α : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {a b ε : α},
0 < ε → 2 * a * b ≤ ε * a ^ 2 + ε⁻¹ * b ^ 2 | true |
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