name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
Lean.ScopedEnvExtension.State.mk.injEq | Lean.ScopedEnvExtension | ∀ {σ : Type} (state : σ) (activeScopes : Lean.NameSet) (delimitsLocal : Bool) (state_1 : σ)
(activeScopes_1 : Lean.NameSet) (delimitsLocal_1 : Bool),
({ state := state, activeScopes := activeScopes, delimitsLocal := delimitsLocal } =
{ state := state_1, activeScopes := activeScopes_1, delimitsLocal := delimitsLocal_1 }) =
(state = state_1 ∧ activeScopes = activeScopes_1 ∧ delimitsLocal = delimitsLocal_1) |
_private.Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts.0.CategoryTheory.hasCoproduct_fin | Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C]
[CategoryTheory.Limits.HasInitial C] (n : ℕ) (f : Fin n → C), CategoryTheory.Limits.HasCoproduct f |
Lean.Parser.Command.structExplicitBinder | Lean.Parser.Command | Lean.Parser.Parser |
Lean.Meta.Contradiction.Config.emptyType | Lean.Meta.Tactic.Contradiction | Lean.Meta.Contradiction.Config → Bool |
CoxeterSystem.length_wordProd_le | Mathlib.GroupTheory.Coxeter.Length | ∀ {B : Type u_1} {W : Type u_2} [inst : Group W] {M : CoxeterMatrix B} (cs : CoxeterSystem M W) (ω : List B),
cs.length (cs.wordProd ω) ≤ ω.length |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point.0._aux_Mathlib_AlgebraicGeometry_EllipticCurve_Projective_Point___macroRules__private_Mathlib_AlgebraicGeometry_EllipticCurve_Projective_Point_0_termZ_1 | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | Lean.Macro |
Pi.Colex.instCompleteLinearOrderColexForall._proof_10 | Mathlib.Order.CompleteLattice.PiLex | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : LinearOrder ι] [inst_1 : (i : ι) → CompleteLinearOrder (α i)]
[inst_2 : WellFoundedGT ι] (s : Set (Colex ((i : ι) → α i))) (a : Colex ((i : ι) → α i)),
(∀ b ∈ s, a ≤ b) → a ≤ sInf s |
MoritaEquivalence.mk.injEq | Mathlib.RingTheory.Morita.Basic | ∀ {R : Type u₀} [inst : CommSemiring R] {A : Type u₁} [inst_1 : Ring A] [inst_2 : Algebra R A] {B : Type u₂}
[inst_3 : Ring B] [inst_4 : Algebra R B] (eqv : ModuleCat A ≌ ModuleCat B)
(linear : autoParam (CategoryTheory.Functor.Linear R eqv.functor) MoritaEquivalence.linear._autoParam)
(eqv_1 : ModuleCat A ≌ ModuleCat B)
(linear_1 : autoParam (CategoryTheory.Functor.Linear R eqv_1.functor) MoritaEquivalence.linear._autoParam),
({ eqv := eqv, linear := linear } = { eqv := eqv_1, linear := linear_1 }) = (eqv = eqv_1) |
HomotopicalAlgebra.FibrantObject.homMk_id | Mathlib.AlgebraicTopology.ModelCategory.Bifibrant | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : HomotopicalAlgebra.CategoryWithFibrations C]
[inst_2 : CategoryTheory.Limits.HasTerminal C] (X : C) [inst_3 : HomotopicalAlgebra.IsFibrant X],
HomotopicalAlgebra.FibrantObject.homMk (CategoryTheory.CategoryStruct.id X) =
CategoryTheory.CategoryStruct.id (HomotopicalAlgebra.FibrantObject.mk X) |
Equiv.Perm.sign_inv | Mathlib.GroupTheory.Perm.Sign | ∀ {α : Type u} [inst : DecidableEq α] [inst_1 : Fintype α] (f : Equiv.Perm α), Equiv.Perm.sign f⁻¹ = Equiv.Perm.sign f |
RootableBy.mk._flat_ctor | Mathlib.GroupTheory.Divisible | {A : Type u_1} →
{α : Type u_2} →
[inst : Monoid A] →
[inst_1 : Pow A α] →
[inst_2 : Zero α] →
(root : A → α → A) →
(∀ (a : A), root a 0 = 1) → (∀ {n : α} (a : A), n ≠ 0 → root a n ^ n = a) → RootableBy A α |
HahnEmbedding.Seed.hahnCoeff_apply | Mathlib.Algebra.Order.Module.HahnEmbedding | ∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : IsOrderedRing K] [inst_3 : Archimedean K]
{M : Type u_2} [inst_4 : AddCommGroup M] [inst_5 : LinearOrder M] [inst_6 : IsOrderedAddMonoid M]
[inst_7 : Module K M] [inst_8 : IsOrderedModule K M] {R : Type u_3} [inst_9 : AddCommGroup R]
[inst_10 : LinearOrder R] [inst_11 : Module K R] (seed : HahnEmbedding.Seed K M R) {x : ↥seed.baseDomain}
{f : Π₀ (c : FiniteArchimedeanClass M), ↥(seed.stratum c)},
(↑x = f.sum fun c => ⇑(seed.stratum c).subtype) →
∀ (c : FiniteArchimedeanClass M), (seed.hahnCoeff x) c = (seed.coeff c) (f c) |
Std.TreeMap.getKeyLT | Std.Data.TreeMap.AdditionalOperations | {α : Type u} →
{β : Type v} →
{cmp : α → α → Ordering} →
[Std.TransCmp cmp] → (t : Std.TreeMap α β cmp) → (k : α) → (∃ a ∈ t, cmp a k = Ordering.lt) → α |
TensorProduct.induction_on | Mathlib.LinearAlgebra.TensorProduct.Defs | ∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_7} {N : Type u_8} [inst_1 : AddCommMonoid M]
[inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] {motive : TensorProduct R M N → Prop}
(z : TensorProduct R M N),
motive 0 →
(∀ (x : M) (y : N), motive (x ⊗ₜ[R] y)) →
(∀ (x y : TensorProduct R M N), motive x → motive y → motive (x + y)) → motive z |
Finset.isPWO_support_addAntidiagonal | Mathlib.Data.Finset.MulAntidiagonal | ∀ {α : Type u_1} [inst : AddCommMonoid α] [inst_1 : PartialOrder α] [inst_2 : IsOrderedCancelAddMonoid α] {s t : Set α}
{hs : s.IsPWO} {ht : t.IsPWO}, {a | (Finset.addAntidiagonal hs ht a).Nonempty}.IsPWO |
_private.Mathlib.Data.Analysis.Filter.0.Filter.Realizer.ne_bot_iff._simp_1_1 | Mathlib.Data.Analysis.Filter | ∀ {α : Type u} {s : Set α}, (¬s.Nonempty) = (s = ∅) |
CategoryTheory.Limits.hasFiniteLimits_of_hasLimitsLimits_of_createsFiniteLimits | Mathlib.CategoryTheory.Limits.Preserves.Creates.Finite | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(F : CategoryTheory.Functor C D) [CategoryTheory.Limits.HasFiniteLimits D]
[CategoryTheory.Limits.CreatesFiniteLimits F], CategoryTheory.Limits.HasFiniteLimits C |
_private.Init.Data.SInt.Lemmas.0.Int32.toInt64_lt._simp_1_2 | Init.Data.SInt.Lemmas | ∀ {x y : Int64}, (x < y) = (x.toInt < y.toInt) |
SSet.finite_of_hasDimensionLT | Mathlib.AlgebraicTopology.SimplicialSet.Finite | ∀ (X : SSet) (d : ℕ) [X.HasDimensionLT d], (∀ i < d, Finite ↑(X.nonDegenerate i)) → X.Finite |
Language.one_add_self_mul_kstar_eq_kstar | Mathlib.Computability.Language | ∀ {α : Type u_1} (l : Language α), 1 + l * KStar.kstar l = KStar.kstar l |
Int16.add_eq_left._simp_1 | Init.Data.SInt.Lemmas | ∀ {a b : Int16}, (a + b = a) = (b = 0) |
Std.ExtDHashMap.getKey_eq_getKey! | Std.Data.ExtDHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : α → Type v} {m : Std.ExtDHashMap α β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] [inst_2 : Inhabited α] {a : α} {h : a ∈ m}, m.getKey a h = m.getKey! a |
_private.Lean.Meta.LitValues.0.Lean.Meta.getListLitOf?.match_3 | Lean.Meta.LitValues | {α : Type} →
(motive : Option (Option (Array α)) → Sort u_1) →
(x : Option (Option (Array α))) → (Unit → motive none) → ((a : Option (Array α)) → motive (some a)) → motive x |
_private.Mathlib.Order.Filter.Map.0.Filter.compl_mem_kernMap._simp_1_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {m : α → β} {f : Filter α} {s : Set β},
(s ∈ Filter.kernMap m f) = ∃ t, tᶜ ∈ f ∧ m '' t = sᶜ |
divp_mul_eq_mul_divp | Mathlib.Algebra.Group.Units.Basic | ∀ {α : Type u} [inst : CommMonoid α] (x y : α) (u : αˣ), x /ₚ u * y = x * y /ₚ u |
Std.Do.SPred.Tactic.instIsPure | Std.Do.SPred.DerivedLaws | ∀ {φ : Prop} {σ : Type u_1} {s : σ} (σs : List (Type u_1)) (P : Std.Do.SPred (σ :: σs))
[inst : Std.Do.SPred.Tactic.IsPure P φ], Std.Do.SPred.Tactic.IsPure (P s) φ |
String.Slice.RevByteIterator.ctorIdx | Init.Data.String.Iterate | String.Slice.RevByteIterator → ℕ |
NonUnitalSubsemiring.corner._proof_4 | Mathlib.RingTheory.Idempotents | ∀ {R : Type u_1} (e : R) [inst : NonUnitalSemiring R] {a b : R},
a ∈ (Subsemigroup.corner e).carrier → b ∈ (Subsemigroup.corner e).carrier → a * b ∈ (Subsemigroup.corner e).carrier |
Std.DTreeMap.Internal.Impl.get_insertIfNew! | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α β} [inst : Std.TransOrd α]
[inst_1 : Std.LawfulEqOrd α] (h : t.WF) {k a : α} {v : β k} {h₁ : a ∈ Std.DTreeMap.Internal.Impl.insertIfNew! k v t},
(Std.DTreeMap.Internal.Impl.insertIfNew! k v t).get a h₁ =
if h₂ : compare k a = Ordering.eq ∧ k ∉ t then cast ⋯ v else t.get a ⋯ |
Std.Tactic.BVDecide.LRAT.Internal.DefaultClause.delete_iff | Std.Tactic.BVDecide.LRAT.Internal.Clause | ∀ {n : ℕ} (c : Std.Tactic.BVDecide.LRAT.Internal.DefaultClause n)
(l l' : Std.Sat.Literal (Std.Tactic.BVDecide.LRAT.Internal.PosFin n)),
l' ∈ (c.delete l).toList ↔ l' ≠ l ∧ l' ∈ c.toList |
_private.Mathlib.FieldTheory.IsAlgClosed.Basic.0.IsAlgClosed.liftAux | Mathlib.FieldTheory.IsAlgClosed.Basic | (K : Type u) →
[inst : Field K] →
(L : Type v) →
(M : Type w) →
[inst_1 : Field L] →
[inst_2 : Algebra K L] →
[inst_3 : Field M] → [inst_4 : Algebra K M] → [IsAlgClosed M] → [Algebra.IsAlgebraic K L] → L →ₐ[K] M |
_private.Mathlib.GroupTheory.Subgroup.Centralizer.0.Subgroup.normalizerMonoidHom_ker._simp_1_6 | Mathlib.GroupTheory.Subgroup.Centralizer | ∀ {G : Type u_3} [inst : Group G] {a b c : G}, (a = b * c⁻¹) = (a * c = b) |
BestFirstNode.ctorIdx | Mathlib.Deprecated.MLList.BestFirst | {α : Sort u_1} → {ω : Type u_2} → {prio : α → Thunk ω} → {ε : α → Type} → BestFirstNode prio ε → ℕ |
_private.Lean.Meta.Basic.0.Lean.Meta.isSyntheticMVar.match_1 | Lean.Meta.Basic | (motive : Lean.MetavarKind → Sort u_1) →
(__do_lift : Lean.MetavarKind) →
(Unit → motive Lean.MetavarKind.synthetic) →
(Unit → motive Lean.MetavarKind.syntheticOpaque) → ((x : Lean.MetavarKind) → motive x) → motive __do_lift |
CategoryTheory.Pseudofunctor.DescentData.toDescentDataCompPullFunctorIso | Mathlib.CategoryTheory.Sites.Descent.DescentData | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat) →
{ι : Type t} →
{S : C} →
{X : ι → C} →
{f : (i : ι) → X i ⟶ S} →
{S' : C} →
{p : S' ⟶ S} →
{ι' : Type t'} →
{X' : ι' → C} →
{f' : (j : ι') → X' j ⟶ S'} →
{α : ι' → ι} →
{p' : (j : ι') → X' j ⟶ X (α j)} →
(w :
∀ (j : ι'),
CategoryTheory.CategoryStruct.comp (p' j) (f (α j)) =
CategoryTheory.CategoryStruct.comp (f' j) p) →
(F.toDescentData f).comp (CategoryTheory.Pseudofunctor.DescentData.pullFunctor F w) ≅
(F.map p.op.toLoc).toFunctor.comp (F.toDescentData f') |
MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuousMap_continuous_integral | Mathlib.MeasureTheory.Measure.ProbabilityMeasure | ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : TopologicalSpace Ω] [inst_2 : OpensMeasurableSpace Ω]
{X : Type u_2} [inst_3 : TopologicalSpace X] {μs : X → MeasureTheory.ProbabilityMeasure Ω} [CompactSpace Ω],
Continuous μs ↔ ∀ (f : C(Ω, ℝ)), Continuous fun x => ∫ (ω : Ω), f ω ∂↑(μs x) |
Metric.frontier_thickening_disjoint | Mathlib.Topology.MetricSpace.Thickening | ∀ {α : Type u} [inst : PseudoEMetricSpace α] (A : Set α),
Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r A)) |
Lean.Elab.Term.MatchExpr.ElseAlt.mk.sizeOf_spec | Lean.Elab.MatchExpr | ∀ (rhs : Lean.Syntax), sizeOf { rhs := rhs } = 1 + sizeOf rhs |
_private.Mathlib.Tactic.NormNum.NatFactorial.0.Mathlib.Meta.NormNum.evalNatDescFactorial._proof_2 | Mathlib.Tactic.NormNum.NatFactorial | ∀ (x y z : Q(ℕ)), «$x» =Q «$z» + «$y» |
denselyOrdered_multiplicative_iff | Mathlib.GroupTheory.ArchimedeanDensely | ∀ {X : Type u_2} [inst : LT X], DenselyOrdered (Multiplicative X) ↔ DenselyOrdered X |
Lean.Lsp.SemanticTokenType.method.sizeOf_spec | Lean.Data.Lsp.LanguageFeatures | sizeOf Lean.Lsp.SemanticTokenType.method = 1 |
GrpCat.uliftFunctor_preservesLimitsOfSize | Mathlib.Algebra.Category.Grp.Ulift | CategoryTheory.Limits.PreservesLimitsOfSize.{w', w, u, max u v, u + 1, max (u + 1) (v + 1)} GrpCat.uliftFunctor |
MeasureTheory.regular_inv_iff | Mathlib.MeasureTheory.Group.Measure | ∀ {G : Type u_1} [inst : MeasurableSpace G] [inst_1 : TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G}
[inst_3 : Group G] [IsTopologicalGroup G], μ.inv.Regular ↔ μ.Regular |
_private.Std.Data.DTreeMap.Internal.Operations.0.Std.DTreeMap.Internal.Impl.alter.match_3.eq_3 | Std.Data.DTreeMap.Internal.Operations | ∀ (motive : Ordering → Sort u_1) (h_1 : Ordering.eq = Ordering.lt → motive Ordering.lt)
(h_2 : Ordering.eq = Ordering.gt → motive Ordering.gt) (h_3 : Ordering.eq = Ordering.eq → motive Ordering.eq),
(match h : Ordering.eq with
| Ordering.lt => h_1 h
| Ordering.gt => h_2 h
| Ordering.eq => h_3 h) =
h_3 ⋯ |
ULift.semiring._proof_5 | Mathlib.Algebra.Ring.ULift | ∀ {R : Type u_2} [inst : Semiring R] (x : ULift.{u_1, u_2} R), Monoid.npow 0 x = 1 |
_private.Mathlib.FieldTheory.Extension.0.IntermediateField.nonempty_algHom_adjoin_of_splits.match_1_1 | Mathlib.FieldTheory.Extension | ∀ {F : Type u_3} {E : Type u_1} {K : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Field K]
[inst_3 : Algebra F E] [inst_4 : Algebra F K] {S : Set E}
(motive : (∃ φ, φ.comp (IntermediateField.inclusion ⋯) = ⊥.emb) → Prop)
(x : ∃ φ, φ.comp (IntermediateField.inclusion ⋯) = ⊥.emb),
(∀ (φ : ↥(IntermediateField.adjoin F S) →ₐ[F] K) (h : φ.comp (IntermediateField.inclusion ⋯) = ⊥.emb), motive ⋯) →
motive x |
CategoryTheory.Limits.CatCospanTransform.inv_whiskerRight | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | ∀ {A : Type u₁} {B : Type u₂} {C : Type u₃} {A' : Type u₄} {B' : Type u₅} {C' : Type u₆} {A'' : Type u₇} {B'' : Type u₈}
{C'' : Type u₉} [inst : CategoryTheory.Category.{v₁, u₁} A] [inst_1 : CategoryTheory.Category.{v₂, u₂} B]
[inst_2 : CategoryTheory.Category.{v₃, u₃} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B}
[inst_3 : CategoryTheory.Category.{v₄, u₄} A'] [inst_4 : CategoryTheory.Category.{v₅, u₅} B']
[inst_5 : CategoryTheory.Category.{v₆, u₆} C'] {F' : CategoryTheory.Functor A' B'} {G' : CategoryTheory.Functor C' B'}
[inst_6 : CategoryTheory.Category.{v₇, u₇} A''] [inst_7 : CategoryTheory.Category.{v₈, u₈} B'']
[inst_8 : CategoryTheory.Category.{v₉, u₉} C''] {F'' : CategoryTheory.Functor A'' B''}
{G'' : CategoryTheory.Functor C'' B''} {ψ ψ' : CategoryTheory.Limits.CatCospanTransform F G F' G'} (η : ψ ⟶ ψ')
[inst_9 : CategoryTheory.IsIso η] {φ : CategoryTheory.Limits.CatCospanTransform F' G' F'' G''},
CategoryTheory.inv (CategoryTheory.Limits.CatCospanTransformMorphism.whiskerRight η φ) =
CategoryTheory.Limits.CatCospanTransformMorphism.whiskerRight (CategoryTheory.inv η) φ |
EReal.abs_neg | Mathlib.Data.EReal.Inv | ∀ (x : EReal), (-x).abs = x.abs |
Mathlib.Tactic.Ring.ExProd.cast._unsafe_rec | Mathlib.Tactic.Ring.Common | {u : Lean.Level} →
{α : Q(Type u)} →
{sα : Q(CommSemiring «$α»)} →
{v : Lean.Level} →
{β : Q(Type v)} →
{sβ : Q(CommSemiring «$β»)} →
{a : Q(«$α»)} → Mathlib.Tactic.Ring.ExProd sα a → (a : Q(«$β»)) × Mathlib.Tactic.Ring.ExProd sβ a |
Std.DHashMap.Raw.get!_inter_of_mem_right | Std.Data.DHashMap.RawLemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] {m₁ m₂ : Std.DHashMap.Raw α β}
[inst_2 : LawfulBEq α], m₁.WF → m₂.WF → ∀ {k : α} [inst_3 : Inhabited (β k)], k ∈ m₂ → (m₁ ∩ m₂).get! k = m₁.get! k |
NumberField.mixedEmbedding.convexBodySum.congr_simp | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | ∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] (B B_1 : ℝ),
B = B_1 →
∀ (a a_1 : NumberField.mixedEmbedding.mixedSpace K),
a = a_1 → NumberField.mixedEmbedding.convexBodySum K B a = NumberField.mixedEmbedding.convexBodySum K B_1 a_1 |
AntivaryOn.neg_left | Mathlib.Algebra.Order.Monovary | ∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : AddCommGroup α] [inst_1 : Preorder α] [IsOrderedAddMonoid α]
[inst_3 : PartialOrder β] {s : Set ι} {f : ι → α} {g : ι → β}, AntivaryOn f g s → MonovaryOn (-f) g s |
_private.Lean.Elab.Tactic.Change.0.Lean.Elab.Tactic.evalChange.match_1 | Lean.Elab.Tactic.Change | (motive : Lean.Expr × List Lean.MVarId → Sort u_1) →
(__discr : Lean.Expr × List Lean.MVarId) →
((hTy' : Lean.Expr) → (mvars : List Lean.MVarId) → motive (hTy', mvars)) → motive __discr |
Lean.Parser.Term.debugAssert._regBuiltin.Lean.Parser.Term.debugAssert.parenthesizer_11 | Lean.Parser.Term | IO Unit |
Ideal.IsHomogeneous.iSup₂ | Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | ∀ {ι : Type u_1} {σ : Type u_2} {A : Type u_3} [inst : Semiring A] [inst_1 : DecidableEq ι] [inst_2 : AddMonoid ι]
[inst_3 : SetLike σ A] [inst_4 : AddSubmonoidClass σ A] {𝒜 : ι → σ} [inst_5 : GradedRing 𝒜] {κ : Sort u_4}
{κ' : κ → Sort u_5} {f : (i : κ) → κ' i → Ideal A},
(∀ (i : κ) (j : κ' i), Ideal.IsHomogeneous 𝒜 (f i j)) → Ideal.IsHomogeneous 𝒜 (⨆ i, ⨆ j, f i j) |
List.max?.eq_1 | Init.Data.List.MinMax | ∀ {α : Type u} [inst : Max α], [].max? = none |
Equiv.Perm.toList_ne_singleton | Mathlib.GroupTheory.Perm.Cycle.Concrete | ∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] (p : Equiv.Perm α) (x y : α), p.toList x ≠ [y] |
CategoryTheory.Limits.IsZero.iso._proof_1 | Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {X Y : C} (hX : CategoryTheory.Limits.IsZero X),
CategoryTheory.CategoryStruct.comp (hX.to_ Y) (hX.from_ Y) = CategoryTheory.CategoryStruct.id X |
LowerSet.instDiv._proof_1 | Mathlib.Algebra.Order.UpperLower | ∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : Preorder α] [IsOrderedMonoid α] (s t : LowerSet α),
IsLowerSet (s.carrier / ↑t) |
_private.Mathlib.Algebra.Free.0.FreeMagma.length_pos.match_1_1 | Mathlib.Algebra.Free | ∀ {α : Type u_1} (motive : FreeMagma α → Prop) (x : FreeMagma α),
(∀ (a : α), motive (FreeMagma.of a)) → (∀ (y z : FreeMagma α), motive (y.mul z)) → motive x |
Membership.mem.out | Mathlib.Data.Set.Operations | ∀ {α : Type u} {a : α} {p : α → Prop}, a ∈ {x | p x} → p a |
CategoryTheory.Limits.pullbackConeOfLeftIso_fst | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Iso | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[inst_1 : CategoryTheory.IsIso f],
(CategoryTheory.Limits.pullbackConeOfLeftIso f g).fst = CategoryTheory.CategoryStruct.comp g (CategoryTheory.inv f) |
QuotientAddGroup.lift._proof_2 | Mathlib.GroupTheory.QuotientGroup.Defs | ∀ {G : Type u_1} {M : Type u_2} [inst : AddGroup G] [inst_1 : AddMonoid M] (N : AddSubgroup G) [nN : N.Normal]
(φ : G →+ M), N ≤ φ.ker → QuotientAddGroup.con N ≤ AddCon.ker φ |
SignType.ofNat | Mathlib.Data.Sign.Defs | ℕ → SignType |
Filter.Germ.instSemigroup._proof_1 | Mathlib.Order.Filter.Germ.Basic | ∀ {α : Type u_1} {l : Filter α} {M : Type u_2} [inst : Semigroup M] (a b c : l.Germ M), a * b * c = a * (b * c) |
Int.mul_tmod_left | Init.Data.Int.DivMod.Lemmas | ∀ (a b : ℤ), (a * b).tmod b = 0 |
Lean.Lsp.ReferenceContext.rec | Lean.Data.Lsp.LanguageFeatures | {motive : Lean.Lsp.ReferenceContext → Sort u} →
((includeDeclaration : Bool) → motive { includeDeclaration := includeDeclaration }) →
(t : Lean.Lsp.ReferenceContext) → motive t |
Mathlib.Meta.Positivity.evalFinsetSum | Mathlib.Tactic.Positivity.Finset | Mathlib.Meta.Positivity.PositivityExt |
_private.Init.Grind.Offset.0.Lean.Grind.Nat.le_offset._proof_1_1 | Init.Grind.Offset | ∀ (a k : ℕ), ¬k ≤ a + k → False |
CategoryTheory.CommGrp.forget₂Grp_obj_one | Mathlib.CategoryTheory.Monoidal.CommGrp_ | ∀ (C : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] (A : CategoryTheory.CommGrp C),
CategoryTheory.MonObj.one = CategoryTheory.MonObj.one |
AffineBasis.tot | Mathlib.LinearAlgebra.AffineSpace.Basis | ∀ {ι : Type u_1} {k : Type u_5} {V : Type u_6} {P : Type u_7} [inst : AddCommGroup V] [inst_1 : AddTorsor V P]
[inst_2 : Ring k] [inst_3 : Module k V] (b : AffineBasis ι k P), affineSpan k (Set.range ⇑b) = ⊤ |
_private.Mathlib.Algebra.Order.Group.Pointwise.Interval.0.Set.inv_Ioc._simp_1_1 | Mathlib.Algebra.Order.Group.Pointwise.Interval | ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, Set.Ioc a b = Set.Ioi a ∩ Set.Iic b |
AddCommGrpCat.forget_commGrp_preserves_epi | Mathlib.Algebra.Category.Grp.EpiMono | (CategoryTheory.forget AddCommGrpCat).PreservesEpimorphisms |
IsAdicComplete.toIsPrecomplete | Mathlib.RingTheory.AdicCompletion.Basic | ∀ {R : Type u_1} {inst : CommRing R} {I : Ideal R} {M : Type u_4} {inst_1 : AddCommGroup M} {inst_2 : Module R M}
[self : IsAdicComplete I M], IsPrecomplete I M |
CategoryTheory.RegularMono._sizeOf_1 | Mathlib.CategoryTheory.Limits.Shapes.RegularMono | {C : Type u₁} →
{inst : CategoryTheory.Category.{v₁, u₁} C} → {X Y : C} → {f : X ⟶ Y} → [SizeOf C] → CategoryTheory.RegularMono f → ℕ |
FirstOrder.Language.PartialEquiv.mk.inj | Mathlib.ModelTheory.PartialEquiv | ∀ {L : FirstOrder.Language} {M : Type w} {N : Type w'} {inst : L.Structure M} {inst_1 : L.Structure N}
{dom : L.Substructure M} {cod : L.Substructure N} {toEquiv : L.Equiv ↥dom ↥cod} {dom_1 : L.Substructure M}
{cod_1 : L.Substructure N} {toEquiv_1 : L.Equiv ↥dom_1 ↥cod_1},
{ dom := dom, cod := cod, toEquiv := toEquiv } = { dom := dom_1, cod := cod_1, toEquiv := toEquiv_1 } →
dom = dom_1 ∧ cod = cod_1 ∧ toEquiv ≍ toEquiv_1 |
Condensed.locallyConstantIsoFinYoneda | Mathlib.Condensed.Discrete.Colimit | (F : CategoryTheory.Functor Profiniteᵒᵖ (Type (u + 1))) →
FintypeCat.toProfinite.op.comp
(Condensed.locallyConstantPresheaf
(F.obj (FintypeCat.toProfinite.op.obj (Opposite.op { carrier := PUnit.{u + 1}, str := PUnit.fintype })))) ≅
Condensed.finYoneda F |
CategoryTheory.Limits.Concrete.initial_of_empty_of_reflects | Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type w}
[inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC]
[CategoryTheory.Limits.ReflectsColimit (CategoryTheory.Functor.empty C) (CategoryTheory.forget C)] (X : C),
IsEmpty (CategoryTheory.ToType X) → Nonempty (CategoryTheory.Limits.IsInitial X) |
_private.Mathlib.Combinatorics.SimpleGraph.Matching.0.SimpleGraph.IsCycles.other_adj_of_adj._simp_1_1 | Mathlib.Combinatorics.SimpleGraph.Matching | ∀ {V : Type u} (G : SimpleGraph V) (v w : V), G.Adj v w = (w ∈ G.neighborSet v) |
AddCommGroup.toDistribLattice.eq_1 | Mathlib.Algebra.Order.Group.Lattice | ∀ (α : Type u_2) [inst : Lattice α] [inst_1 : AddCommGroup α] [inst_2 : AddLeftMono α],
AddCommGroup.toDistribLattice α = { toLattice := inst, le_sup_inf := ⋯ } |
Aesop.instInhabitedMVarClusterData.default | Aesop.Tree.Data | {a a_1 : Type} → Aesop.MVarClusterData a a_1 |
Lean.Elab.Term.SyntheticMVarKind.typeClass.inj | Lean.Elab.Term.TermElabM | ∀ {extraErrorMsg? extraErrorMsg?_1 : Option Lean.MessageData},
Lean.Elab.Term.SyntheticMVarKind.typeClass extraErrorMsg? =
Lean.Elab.Term.SyntheticMVarKind.typeClass extraErrorMsg?_1 →
extraErrorMsg? = extraErrorMsg?_1 |
Lean.PrefixTreeNode.below_2 | Lean.Data.PrefixTree | {α : Type u} →
{β : Type v} →
{cmp : α → α → Ordering} →
{motive_1 : Lean.PrefixTreeNode α β cmp → Sort u_1} →
{motive_2 : Std.TreeMap.Raw α (Lean.PrefixTreeNode α β cmp) cmp → Sort u_1} →
{motive_3 : Std.DTreeMap.Raw α (fun x => Lean.PrefixTreeNode α β cmp) cmp → Sort u_1} →
{motive_4 : (Std.DTreeMap.Internal.Impl α fun x => Lean.PrefixTreeNode α β cmp) → Sort u_1} →
Std.DTreeMap.Raw α (fun x => Lean.PrefixTreeNode α β cmp) cmp →
Sort (max (max (u + 1) ((max u v) + 1)) u_1) |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Equiv.entryAtIdx_eq._simp_1_2 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {a b : α}, (compare a b ≠ Ordering.eq) = ((a == b) = false) |
Mathlib.Notation3.MatchState.noConfusion | Mathlib.Util.Notation3 | {P : Sort u} → {t t' : Mathlib.Notation3.MatchState} → t = t' → Mathlib.Notation3.MatchState.noConfusionType P t t' |
_private.Mathlib.Algebra.Polynomial.Degree.Units.0.Polynomial.isUnit_iff.match_1_1 | Mathlib.Algebra.Polynomial.Degree.Units | ∀ {R : Type u_1} [inst : Semiring R] {p : Polynomial R} (motive : (∃ r, IsUnit r ∧ Polynomial.C r = p) → Prop)
(x : ∃ r, IsUnit r ∧ Polynomial.C r = p), (∀ (w : R) (hr : IsUnit w) (hrp : Polynomial.C w = p), motive ⋯) → motive x |
_private.Mathlib.Tactic.Linter.FindDeprecations.0.Mathlib.Tactic.DeprecationInfo.mk.noConfusion | Mathlib.Tactic.Linter.FindDeprecations | {P : Sort u} →
{module decl : Lean.Name} →
{rgStart rgStop : Lean.Position} →
{since : String} →
{module' decl' : Lean.Name} →
{rgStart' rgStop' : Lean.Position} →
{since' : String} →
{ module := module, decl := decl, rgStart := rgStart, rgStop := rgStop, since := since } =
{ module := module', decl := decl', rgStart := rgStart', rgStop := rgStop', since := since' } →
(module = module' → decl = decl' → rgStart = rgStart' → rgStop = rgStop' → since = since' → P) → P |
CategoryTheory.Functor.IsRepresentedBy.representableBy | Mathlib.CategoryTheory.RepresentedBy | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{F : CategoryTheory.Functor Cᵒᵖ (Type w)} →
{X : C} → {x : F.obj (Opposite.op X)} → F.IsRepresentedBy x → F.RepresentableBy X |
integrable_cexp_quadratic' | Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | ∀ {b : ℂ},
b.re < 0 → ∀ (c d : ℂ), MeasureTheory.Integrable (fun x => Complex.exp (b * ↑x ^ 2 + c * ↑x + d)) MeasureTheory.volume |
Lean.Doc.Inline.emph.injEq | Lean.DocString.Types | ∀ {i : Type u} (content content_1 : Array (Lean.Doc.Inline i)),
(Lean.Doc.Inline.emph content = Lean.Doc.Inline.emph content_1) = (content = content_1) |
CategoryTheory.Functor.toPrefunctor | Mathlib.CategoryTheory.Functor.Basic | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} → [inst_1 : CategoryTheory.Category.{v₂, u₂} D] → CategoryTheory.Functor C D → C ⥤q D |
Lean.Meta.saveState | Lean.Meta.Basic | Lean.MetaM Lean.Meta.SavedState |
AffineSubspace.map_bot | Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | ∀ {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} {V₂ : Type u_4} {P₂ : Type u_5} [inst : Ring k]
[inst_1 : AddCommGroup V₁] [inst_2 : Module k V₁] [inst_3 : AddTorsor V₁ P₁] [inst_4 : AddCommGroup V₂]
[inst_5 : Module k V₂] [inst_6 : AddTorsor V₂ P₂] (f : P₁ →ᵃ[k] P₂), AffineSubspace.map f ⊥ = ⊥ |
Lean.Constructor.mk.injEq | Lean.Declaration | ∀ (name : Lean.Name) (type : Lean.Expr) (name_1 : Lean.Name) (type_1 : Lean.Expr),
({ name := name, type := type } = { name := name_1, type := type_1 }) = (name = name_1 ∧ type = type_1) |
Filter.eventually_bind._simp_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → Filter β} {p : β → Prop},
(∀ᶠ (y : β) in f.bind m, p y) = ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in m x, p y |
_private.Init.Data.Iterators.Lemmas.Combinators.Monadic.FlatMap.0.Std.IterM.step_flattenAfter.match_1.eq_1 | Init.Data.Iterators.Lemmas.Combinators.Monadic.FlatMap | ∀ {α α₂ β : Type u_1} {m : Type u_1 → Type u_2} [inst : Std.Iterator α m (Std.IterM m β)]
{it₁ : Std.IterM m (Std.IterM m β)} (motive : it₁.Step → Sort u_3) (it₁' : Std.IterM m (Std.IterM m β))
(it₂' : Std.IterM m β) (h : it₁.IsPlausibleStep (Std.IterStep.yield it₁' it₂'))
(h_1 :
(it₁' : Std.IterM m (Std.IterM m β)) →
(it₂' : Std.IterM m β) →
(h : it₁.IsPlausibleStep (Std.IterStep.yield it₁' it₂')) → motive ⟨Std.IterStep.yield it₁' it₂', h⟩)
(h_2 :
(it₁' : Std.IterM m (Std.IterM m β)) →
(h : it₁.IsPlausibleStep (Std.IterStep.skip it₁')) → motive ⟨Std.IterStep.skip it₁', h⟩)
(h_3 : (h : it₁.IsPlausibleStep Std.IterStep.done) → motive ⟨Std.IterStep.done, h⟩),
(match ⟨Std.IterStep.yield it₁' it₂', h⟩ with
| ⟨Std.IterStep.yield it₁' it₂', h⟩ => h_1 it₁' it₂' h
| ⟨Std.IterStep.skip it₁', h⟩ => h_2 it₁' h
| ⟨Std.IterStep.done, h⟩ => h_3 h) =
h_1 it₁' it₂' h |
Irrational.of_mul_ratCast | Mathlib.NumberTheory.Real.Irrational | ∀ (q : ℚ) {x : ℝ}, Irrational (x * ↑q) → Irrational x |
Set.Finite.cast_ncard_eq | Mathlib.Data.Set.Card | ∀ {α : Type u_1} {s : Set α}, s.Finite → ↑s.ncard = s.encard |
LinearMap.transvection.comp_of_left_eq | Mathlib.LinearAlgebra.Transvection | ∀ {R : Type u_1} {V : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid V] [inst_2 : Module R V]
{f : Module.Dual R V} {v w : V},
f w = 0 → LinearMap.transvection f v ∘ₗ LinearMap.transvection f w = LinearMap.transvection f (v + w) |
AlgebraicGeometry.instIsLocallyArtinianFiberOfLocallyQuasiFinite | Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.LocallyQuasiFinite f] (y : ↥Y),
AlgebraicGeometry.IsLocallyArtinian (AlgebraicGeometry.Scheme.Hom.fiber f y) |
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