name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
AddHom.inverse._proof_2 | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_1} {N : Type u_2} [inst : Add M] [inst_1 : Add N] (f : M →ₙ+ N) (g : N → M) (x y : N),
g (f (g x) + f (g y)) = g (f (g x + g y)) |
ENat.lift_coe | Mathlib.Data.ENat.Basic | ∀ (n : ℕ), (↑n).lift ⋯ = n |
Lean.Elab.expandOptNamedPrio | Lean.Elab.Util | Lean.Syntax → Lean.MacroM ℕ |
_private.Lean.Environment.0.Lean.Environment.instTypeNameRealizeConstResult | Lean.Environment | TypeName Lean.Environment.RealizeConstResult✝ |
_private.Mathlib.Algebra.Order.Floor.Semiring.0.Nat.ceil_intCast._simp_1_2 | Mathlib.Algebra.Order.Floor.Semiring | ∀ {m : ℤ} {n : ℕ}, (m.toNat ≤ n) = (m ≤ ↑n) |
ContinuousConstVAdd.toMeasurableConstVAdd | Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | ∀ {M : Type u_7} {α : Type u_8} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] [BorelSpace α]
[inst_3 : VAdd M α] [ContinuousConstVAdd M α], MeasurableConstVAdd M α |
_private.Qq.Match.0.Qq._aux_Qq_Match___macroRules_Lean_Parser_Term_doMatch_1.match_5 | Qq.Match | (motive : Array (Lean.TSyntax `term) × Lean.TSyntax `Lean.Parser.Term.doSeq → Sort u_1) →
(x : Array (Lean.TSyntax `term) × Lean.TSyntax `Lean.Parser.Term.doSeq) →
((patss : Array (Lean.TSyntax `term)) → (rhss : Lean.TSyntax `Lean.Parser.Term.doSeq) → motive (patss, rhss)) →
motive x |
Finset.neg_mem_neg | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : Neg α] {s : Finset α} {a : α}, a ∈ s → -a ∈ -s |
Filter.IsBoundedUnder.sup | Mathlib.Order.Filter.IsBounded | ∀ {α : Type u_1} {β : Type u_2} [inst : SemilatticeSup α] {f : Filter β} {u v : β → α},
Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u →
Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v → Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f fun a => u a ⊔ v a |
_private.Lean.Meta.Constructions.SparseCasesOn.0.Lean.Meta.SparseCasesOnKey.isPrivate | Lean.Meta.Constructions.SparseCasesOn | Lean.Meta.SparseCasesOnKey✝ → Bool |
Filter.HasBasis.inv | Mathlib.Order.Filter.Pointwise | ∀ {α : Type u_2} [inst : InvolutiveInv α] {f : Filter α} {ι : Sort u_7} {p : ι → Prop} {s : ι → Set α},
f.HasBasis p s → f⁻¹.HasBasis p fun i => (s i)⁻¹ |
Finset.add_univ_of_zero_mem | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : AddMonoid α] {s : Finset α} [inst_2 : Fintype α],
0 ∈ s → s + Finset.univ = Finset.univ |
ContinuousOpenMap.casesOn | Mathlib.Topology.Hom.Open | {α : Type u_6} →
{β : Type u_7} →
[inst : TopologicalSpace α] →
[inst_1 : TopologicalSpace β] →
{motive : (α →CO β) → Sort u} →
(t : α →CO β) →
((toContinuousMap : C(α, β)) →
(map_open' : IsOpenMap toContinuousMap.toFun) →
motive { toContinuousMap := toContinuousMap, map_open' := map_open' }) →
motive t |
mem_balancedHull_iff | Mathlib.Analysis.LocallyConvex.BalancedCoreHull | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : SeminormedRing 𝕜] [inst_1 : SMul 𝕜 E] {s : Set E} {x : E},
x ∈ balancedHull 𝕜 s ↔ ∃ r, ‖r‖ ≤ 1 ∧ x ∈ r • s |
IsValuativeTopology.hasBasis_nhds | Mathlib.Topology.Algebra.Valued.ValuativeRel | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : ValuativeRel R] [inst_2 : TopologicalSpace R] [IsValuativeTopology R]
(x : R), (nhds x).HasBasis (fun x => True) fun γ => {z | (ValuativeRel.valuation R) (z - x) < ↑γ} |
UniqueFactorizationMonoid.associated_iff_normalizedFactors_eq_normalizedFactors | Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors | ∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
{x y : α},
x ≠ 0 →
y ≠ 0 →
(Associated x y ↔ UniqueFactorizationMonoid.normalizedFactors x = UniqueFactorizationMonoid.normalizedFactors y) |
Matroid.cRk_map_image | Mathlib.Combinatorics.Matroid.Rank.Cardinal | ∀ {α β : Type u} {f : α → β} (M : Matroid α) (hf : Set.InjOn f M.E) (X : Set α),
autoParam (X ⊆ M.E) Matroid.cRk_map_image._auto_1 → (M.map f hf).cRk (f '' X) = M.cRk X |
Std.Internal.Parsec.ByteArray.octDigit | Std.Internal.Parsec.ByteArray | Std.Internal.Parsec.ByteArray.Parser Char |
Computation.recOn | Mathlib.Data.Seq.Computation | {α : Type u} →
{motive : Computation α → Sort v} →
(s : Computation α) → ((a : α) → motive (Computation.pure a)) → ((s : Computation α) → motive s.think) → motive s |
_private.Lean.Compiler.LCNF.MonoTypes.0.Lean.Compiler.LCNF.toMonoType.visitApp._sparseCasesOn_8 | Lean.Compiler.LCNF.MonoTypes | {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 |
_private.Mathlib.CategoryTheory.Filtered.Small.0.CategoryTheory.IsCofiltered.CofilteredClosureSmall.InductiveStep.min.noConfusion | Mathlib.CategoryTheory.Filtered.Small | {C : Type u} →
{inst : CategoryTheory.Category.{v, u} C} →
{n : ℕ} →
{X : (k : ℕ) → k < n → (t : Type (max v w)) × (t → C)} →
{P : Sort u_1} →
{k k' : ℕ} →
{hk : k < n} →
{hk' : k' < n} →
{a : (X k hk).fst} →
{a_1 : (X k' hk').fst} →
{k'_1 k'' : ℕ} →
{hk'_1 : k'_1 < n} →
{hk'' : k'' < n} →
{a' : (X k'_1 hk'_1).fst} →
{a'_1 : (X k'' hk'').fst} →
CategoryTheory.IsCofiltered.CofilteredClosureSmall.InductiveStep.min✝ hk hk' a a_1 =
CategoryTheory.IsCofiltered.CofilteredClosureSmall.InductiveStep.min✝¹ hk'_1 hk'' a'
a'_1 →
(k = k'_1 → k' = k'' → a ≍ a' → a_1 ≍ a'_1 → P) → P |
CategoryTheory.Discrete._aux_Mathlib_CategoryTheory_Discrete_Basic___macroRules_CategoryTheory_Discrete_tacticDiscrete_cases_1 | Mathlib.CategoryTheory.Discrete.Basic | Lean.Macro |
_private.Mathlib.Tactic.Linter.FlexibleLinter.0.Mathlib.Linter.Flexible.Stained.name | Mathlib.Tactic.Linter.FlexibleLinter | Lean.Name → Mathlib.Linter.Flexible.Stained✝ |
norm_inner_eq_norm_iff | Mathlib.Analysis.InnerProductSpace.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{x y : E}, x ≠ 0 → y ≠ 0 → (‖inner 𝕜 x y‖ = ‖x‖ * ‖y‖ ↔ ∃ r, r ≠ 0 ∧ y = r • x) |
CategoryTheory.Functor.LeibnizAdjunction.adj_unit_app_right | 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]
(F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₃))
(G : CategoryTheory.Functor C₁ᵒᵖ (CategoryTheory.Functor C₃ C₂)) (adj₂ : F ⊣₂ G) (X₁ : CategoryTheory.Arrow C₁)
[inst_3 : CategoryTheory.Limits.HasPullbacks C₂] [inst_4 : CategoryTheory.Limits.HasPushouts C₃]
(X₂ : CategoryTheory.Arrow C₂),
((CategoryTheory.Functor.LeibnizAdjunction.adj F G adj₂ X₁).unit.app X₂).right =
CategoryTheory.Limits.pullback.lift
(adj₂.homEquiv
(CategoryTheory.Limits.pushout.inr ((F.map X₁.hom).app ((CategoryTheory.Functor.id C₂).obj X₂.left))
((F.obj ((CategoryTheory.Functor.id C₁).obj X₁.left)).map X₂.hom)))
(adj₂.homEquiv
(CategoryTheory.CategoryStruct.id
((F.obj ((CategoryTheory.Functor.id C₁).obj (Opposite.unop (Opposite.op X₁)).right)).obj
((CategoryTheory.Functor.id (CategoryTheory.Arrow C₂)).obj X₂).right)))
⋯ |
autAdjoinRootXPowSubCEquiv.eq_1 | Mathlib.FieldTheory.KummerExtension | ∀ {K : Type u} [inst : Field K] {n : ℕ} (hζ : (primitiveRoots n K).Nonempty) {a : K}
(H : Irreducible (Polynomial.X ^ n - Polynomial.C a)) [inst_1 : NeZero n],
autAdjoinRootXPowSubCEquiv hζ H =
{ toFun := (↑(autAdjoinRootXPowSubC n a)).toFun, invFun := AdjoinRootXPowSubCEquivToRootsOfUnity hζ H,
left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ } |
Char.val_ofOrdinal._proof_2 | Init.Data.Char.Ordinal | ∀ {f : Fin Char.numCodePoints}, ↑f + Char.numSurrogates < UInt32.size |
Polynomial.hasseDeriv_apply_one | Mathlib.Algebra.Polynomial.HasseDeriv | ∀ {R : Type u_1} [inst : Semiring R] (k : ℕ), 0 < k → (Polynomial.hasseDeriv k) 1 = 0 |
Std.ExtDTreeMap.minKey?_mem | Std.Data.ExtDTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.ExtDTreeMap α β cmp} [inst : Std.TransCmp cmp]
{km : α}, t.minKey? = some km → km ∈ t |
Finset.coe_inj | Mathlib.Data.Finset.Defs | ∀ {α : Type u_1} {s₁ s₂ : Finset α}, ↑s₁ = ↑s₂ ↔ s₁ = s₂ |
legendreSym.quadratic_reciprocity | Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | ∀ {p q : ℕ} [inst : Fact (Nat.Prime p)] [inst_1 : Fact (Nat.Prime q)],
p ≠ 2 → q ≠ 2 → p ≠ q → legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2)) |
Finset.op_vadd_finset_vadd_eq_vadd_vadd_finset | Mathlib.Algebra.Group.Pointwise.Finset.Scalar | ∀ {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : DecidableEq β] [inst_1 : DecidableEq γ] [inst_2 : VAdd αᵃᵒᵖ β]
[inst_3 : VAdd β γ] [inst_4 : VAdd α γ] (a : α) (s : Finset β) (t : Finset γ),
(∀ (a : α) (b : β) (c : γ), (AddOpposite.op a +ᵥ b) +ᵥ c = b +ᵥ a +ᵥ c) → (AddOpposite.op a +ᵥ s) +ᵥ t = s +ᵥ a +ᵥ t |
_private.Mathlib.Analysis.Convex.Function.0.neg_strictConvexOn_iff._simp_1_2 | Mathlib.Analysis.Convex.Function | ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LT α] [AddRightStrictMono α] {a b c : α}, (a + -b < c) = (a < c + b) |
_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.span_singleton_mul_eq_span_singleton_mul._simp_1_2 | Mathlib.RingTheory.Ideal.Operations | ∀ {R : Type u} [inst : CommSemiring R] {x y : R} {I J : Ideal R},
(Ideal.span {x} * I ≤ Ideal.span {y} * J) = ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ |
Real.range_arctan | Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | Set.range Real.arctan = Set.Ioo (-(Real.pi / 2)) (Real.pi / 2) |
Lean.Elab.Term.elabCoeSortNotation._regBuiltin.Lean.Elab.Term.elabCoeSortNotation.declRange_3 | Lean.Elab.BuiltinNotation | IO Unit |
Lean.Meta.Grind.Arith.CommRing.EqCnstrProof.brecOn.eq | Lean.Meta.Tactic.Grind.Arith.CommRing.Types | ∀ {motive_1 : Lean.Meta.Grind.Arith.CommRing.EqCnstr → Sort u}
{motive_2 : Lean.Meta.Grind.Arith.CommRing.EqCnstrProof → Sort u} (t : Lean.Meta.Grind.Arith.CommRing.EqCnstrProof)
(F_1 : (t : Lean.Meta.Grind.Arith.CommRing.EqCnstr) → t.below → motive_1 t)
(F_2 : (t : Lean.Meta.Grind.Arith.CommRing.EqCnstrProof) → t.below → motive_2 t),
t.brecOn F_1 F_2 = F_2 t (Lean.Meta.Grind.Arith.CommRing.EqCnstrProof.brecOn.go t F_1 F_2).2 |
AddMonoidHom.transfer_def | Mathlib.GroupTheory.Transfer | ∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G} {A : Type u_2} [inst_1 : AddCommGroup A] (ϕ : ↥H →+ A)
(T : H.LeftTransversal) [inst_2 : H.FiniteIndex] (g : G),
ϕ.transfer g = AddSubgroup.leftTransversals.diff ϕ T (g +ᵥ T) |
_private.Mathlib.LinearAlgebra.AffineSpace.Pointwise.0.AffineSubspace.pointwise_vadd_top._simp_1_1 | Mathlib.LinearAlgebra.AffineSpace.Pointwise | ∀ {α : Type u_5} {β : Type u_6} [inst : AddGroup α] [inst_1 : AddAction α β] (g : α) {x y : β},
(g +ᵥ x = y) = (x = -g +ᵥ y) |
Std.HashMap.getKey?_union | Std.Data.HashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m₁ m₂ : Std.HashMap α β} [EquivBEq α] [LawfulHashable α]
{k : α}, (m₁ ∪ m₂).getKey? k = (m₂.getKey? k).or (m₁.getKey? k) |
WithTop.continuousOn_untopD | Mathlib.Topology.Order.WithTop | ∀ {ι : Type u_1} [inst : LinearOrder ι] [inst_1 : TopologicalSpace ι] [inst_2 : OrderTopology ι] (d : ι),
ContinuousOn (WithTop.untopD d) {a | a ≠ ⊤} |
_private.Init.Data.String.Lemmas.Pattern.Split.0.String.Slice.Pattern.Model.splitFromSteps.match_1.eq_2 | Init.Data.String.Lemmas.Pattern.Split | ∀ {s : String.Slice} (motive : List (String.Slice.Pattern.SearchStep s) → Sort u_1) (startPos endPos : s.Pos)
(l : List (String.Slice.Pattern.SearchStep s)) (h_1 : Unit → motive [])
(h_2 :
(startPos endPos : s.Pos) →
(l : List (String.Slice.Pattern.SearchStep s)) →
motive (String.Slice.Pattern.SearchStep.rejected startPos endPos :: l))
(h_3 :
(p q : s.Pos) →
(l : List (String.Slice.Pattern.SearchStep s)) → motive (String.Slice.Pattern.SearchStep.matched p q :: l)),
(match String.Slice.Pattern.SearchStep.rejected startPos endPos :: l with
| [] => h_1 ()
| String.Slice.Pattern.SearchStep.rejected startPos endPos :: l => h_2 startPos endPos l
| String.Slice.Pattern.SearchStep.matched p q :: l => h_3 p q l) =
h_2 startPos endPos l |
ENNReal.orderIsoUnitIntervalBirational | Mathlib.Data.ENNReal.Inv | ENNReal ≃o ↑(Set.Icc 0 1) |
Homeomorph.Set.prod._proof_1 | Mathlib.Topology.Homeomorph.Lemmas | ∀ {X : Type u_2} {Y : Type u_1} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (s : Set X) (t : Set Y),
Continuous fun x => (⟨(↑x).1, ⋯⟩, ⟨(↑x).2, ⋯⟩) |
_private.Lean.Compiler.IR.SimpleGroundExpr.0.Lean.IR.compileToSimpleGroundExpr.compileArgs | Lean.Compiler.IR.SimpleGroundExpr | Array Lean.IR.Arg → Lean.IR.M✝ (Array Lean.IR.SimpleGroundArg) |
DividedPowers.SubDPIdeal.mk.congr_simp | Mathlib.RingTheory.DividedPowers.SubDPIdeal | ∀ {A : Type u_1} [inst : CommSemiring A] {I : Ideal A} {hI : DividedPowers I} (carrier carrier_1 : Ideal A)
(e_carrier : carrier = carrier_1) (isSubideal : carrier ≤ I)
(dpow_mem : ∀ (n : ℕ), n ≠ 0 → ∀ j ∈ carrier, hI.dpow n j ∈ carrier),
{ carrier := carrier, isSubideal := isSubideal, dpow_mem := dpow_mem } =
{ carrier := carrier_1, isSubideal := ⋯, dpow_mem := ⋯ } |
SSet.PtSimplex | Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | (X : SSet) → ℕ → X.obj (Opposite.op (SimplexCategory.mk 0)) → Type u |
_private.Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone.0.NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_eq_iff.match_1_6 | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
(a b : ↑(NumberField.mixedEmbedding.fundamentalCone.integerSet K)) (motive : (∃ ζ, ↑ζ • ↑a = ↑b) → Prop)
(x : ∃ ζ, ↑ζ • ↑a = ↑b),
(∀ (u : (NumberField.RingOfIntegers K)ˣ) (property : u ∈ NumberField.Units.torsion K) (h : ↑⟨u, property⟩ • ↑a = ↑b),
motive ⋯) →
motive x |
AlgebraicTopology.map_alternatingFaceMapComplex | Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {D : Type u_2}
[inst_2 : CategoryTheory.Category.{v_2, u_2} D] [inst_3 : CategoryTheory.Preadditive D]
(F : CategoryTheory.Functor C D) [inst_4 : F.Additive],
(AlgebraicTopology.alternatingFaceMapComplex C).comp (F.mapHomologicalComplex (ComplexShape.down ℕ)) =
((CategoryTheory.SimplicialObject.whiskering C D).obj F).comp (AlgebraicTopology.alternatingFaceMapComplex D) |
iteratedDerivWithin_of_isOpen_eq_iterate | Mathlib.Analysis.Calculus.IteratedDeriv.Defs | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜},
IsOpen s → Set.EqOn (iteratedDerivWithin n f s) (deriv^[n] f) s |
Lean.PersistentHashMap.Stats._sizeOf_inst | Lean.Data.PersistentHashMap | SizeOf Lean.PersistentHashMap.Stats |
IsometryEquiv.mk.noConfusion | Mathlib.Topology.MetricSpace.Isometry | {α : Type u} →
{β : Type v} →
{inst : PseudoEMetricSpace α} →
{inst_1 : PseudoEMetricSpace β} →
{P : Sort u_1} →
{toEquiv : α ≃ β} →
{isometry_toFun : Isometry toEquiv.toFun} →
{toEquiv' : α ≃ β} →
{isometry_toFun' : Isometry toEquiv'.toFun} →
{ toEquiv := toEquiv, isometry_toFun := isometry_toFun } =
{ toEquiv := toEquiv', isometry_toFun := isometry_toFun' } →
(toEquiv ≍ toEquiv' → P) → P |
StateCpsT.runK_bind_pure | Init.Control.StateCps | ∀ {α σ : Type u} {m : Type u → Type v} {β γ : Type u} (a : α) (f : α → StateCpsT σ m β) (s : σ) (k : β → σ → m γ),
(pure a >>= f).runK s k = (f a).runK s k |
LieHom.toLinearMap_comp._simp_1 | Mathlib.Algebra.Lie.Basic | ∀ {R : Type u} {L₁ : Type v} {L₂ : Type w} {L₃ : Type w₁} [inst : CommRing R] [inst_1 : LieRing L₁]
[inst_2 : LieAlgebra R L₁] [inst_3 : LieRing L₂] [inst_4 : LieAlgebra R L₂] [inst_5 : LieRing L₃]
[inst_6 : LieAlgebra R L₃] (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂), ↑f ∘ₗ ↑g = ↑(f.comp g) |
Std.Do.SPred.Tactic.instIsPureImpPureForall | Std.Do.SPred.DerivedLaws | ∀ {φ ψ : Prop} (σs : List (Type u_1)), Std.Do.SPred.Tactic.IsPure spred(⌜φ⌝ → ⌜ψ⌝) (φ → ψ) |
Equiv.pointReflection_midpoint_right | Mathlib.LinearAlgebra.AffineSpace.Midpoint | ∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Ring R] [inst_1 : Invertible 2] [inst_2 : AddCommGroup V]
[inst_3 : Module R V] [inst_4 : AddTorsor V P] (x y : P), (Equiv.pointReflection (midpoint R x y)) y = x |
BiheytingAlgebra.ctorIdx | Mathlib.Order.Heyting.Basic | {α : Type u_4} → BiheytingAlgebra α → ℕ |
_private.Mathlib.Analysis.Calculus.FDeriv.Const.0.differentiableAt_of_fderiv_injective._simp_1_2 | Mathlib.Analysis.Calculus.FDeriv.Const | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
[inst_6 : TopologicalSpace F] {f : E → F}, fderiv 𝕜 f = fderivWithin 𝕜 f Set.univ |
AlgebraicGeometry.Scheme.residueFieldCongr | Mathlib.AlgebraicGeometry.ResidueField | {X : AlgebraicGeometry.Scheme} → {x y : ↥X} → x = y → (X.residueField x ≅ X.residueField y) |
CategoryTheory.MorphismProperty.HasLocalization.noConfusionType | Mathlib.CategoryTheory.Localization.HasLocalization | Sort u_1 →
{C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{W : CategoryTheory.MorphismProperty C} →
W.HasLocalization →
{C' : Type u} →
[inst' : CategoryTheory.Category.{v, u} C'] →
{W' : CategoryTheory.MorphismProperty C'} → W'.HasLocalization → Sort u_1 |
Lean.Server.MonadCancellable.noConfusionType | Lean.Server.RequestCancellation | Sort u →
{m : Type → Type v} → Lean.Server.MonadCancellable m → {m' : Type → Type v} → Lean.Server.MonadCancellable m' → Sort u |
LinearMap.prod_eq_inf_comap | Mathlib.LinearAlgebra.Prod | ∀ {R : Type u} {M : Type v} {M₂ : Type w} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]
[inst_3 : Module R M] [inst_4 : Module R M₂] (p : Submodule R M) (q : Submodule R M₂),
p.prod q = Submodule.comap (LinearMap.fst R M M₂) p ⊓ Submodule.comap (LinearMap.snd R M M₂) q |
Std.TreeSet.getD_eq_fallback | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp] {a fallback : α},
a ∉ t → t.getD a fallback = fallback |
LSeriesHasSum.smul | Mathlib.NumberTheory.LSeries.Linearity | ∀ {f : ℕ → ℂ} (c : ℂ) {s a : ℂ}, LSeriesHasSum f s a → LSeriesHasSum (c • f) s (c * a) |
CategoryTheory.WithTerminal.comp.match_1 | Mathlib.CategoryTheory.WithTerminal.Basic | {C : Type u_1} →
(motive : CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → CategoryTheory.WithTerminal C → Sort u_2) →
(x x_1 x_2 : CategoryTheory.WithTerminal C) →
((_X _Y _Z : C) →
motive (CategoryTheory.WithTerminal.of _X) (CategoryTheory.WithTerminal.of _Y)
(CategoryTheory.WithTerminal.of _Z)) →
((_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive (CategoryTheory.WithTerminal.of _X) x CategoryTheory.WithTerminal.star) →
((_X : C) →
(x : CategoryTheory.WithTerminal C) →
motive CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _X) x) →
((x : CategoryTheory.WithTerminal C) →
(_Y : C) → motive x CategoryTheory.WithTerminal.star (CategoryTheory.WithTerminal.of _Y)) →
(Unit →
motive CategoryTheory.WithTerminal.star CategoryTheory.WithTerminal.star
CategoryTheory.WithTerminal.star) →
motive x x_1 x_2 |
Lean.Lsp.InitializationOptions.hasWidgets? | Lean.Data.Lsp.InitShutdown | Lean.Lsp.InitializationOptions → Option Bool |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point.0.WeierstrassCurve.Jacobian.Point.toAffine_some._simp_1_1 | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | ∀ {R : Type r} (a b c : R), ![a, b, c] 0 = a |
Filter.Tendsto.atTop_of_add_le_const | Mathlib.Order.Filter.AtTopBot.Monoid | ∀ {α : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : Preorder M] [IsOrderedCancelAddMonoid M]
{l : Filter α} {f g : α → M},
(∃ C, ∀ (x : α), g x ≤ C) → Filter.Tendsto (fun x => f x + g x) l Filter.atTop → Filter.Tendsto f l Filter.atTop |
groupCohomology.coboundaries₁_le_cocycles₁ | Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | ∀ {k G : Type u} [inst : CommRing k] [inst_1 : Group G] (A : Rep k G),
groupCohomology.coboundaries₁ A ≤ groupCohomology.cocycles₁ A |
Padic.limSeq | Mathlib.NumberTheory.Padics.PadicNumbers | {p : ℕ} → [inst : Fact (Nat.Prime p)] → CauSeq ℚ_[p] ⇑padicNormE → ℕ → ℚ |
Vector.eq_empty | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {xs : Vector α 0}, xs = #v[] |
_private.Mathlib.LinearAlgebra.ExteriorAlgebra.Grading.0.ExteriorAlgebra.ιMulti_span.match_1_1 | Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | ∀ (R : Type u_1) (M : Type u_2) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {i : ℕ}
(motive : ↥(⋀[R]^i M) → Prop) (hm : ↥(⋀[R]^i M)),
(∀ (m : ExteriorAlgebra R M) (hm : m ∈ ⋀[R]^i M), motive ⟨m, hm⟩) → motive hm |
_private.Lean.Elab.PreDefinition.WF.GuessLex.0.Lean.Elab.WF.GuessLex.explainMutualFailure.match_1 | Lean.Elab.PreDefinition.WF.GuessLex | (motive : Array (Array String) × String → Sort u_1) →
(__discr : Array (Array String) × String) →
((headerss : Array (Array String)) → (footer : String) → motive (headerss, footer)) → motive __discr |
Lean.Meta.instReduceEvalUInt64_qq | Qq.ForLean.ReduceEval | Lean.Meta.ReduceEval UInt64 |
Std.Iterators.Types.Zip.right | Std.Data.Iterators.Combinators.Monadic.Zip | {α₁ : Type w} →
{m : Type w → Type w'} →
{β₁ : Type w} →
[inst : Std.Iterator α₁ m β₁] → {α₂ β₂ : Type w} → Std.Iterators.Types.Zip α₁ m α₂ β₂ → Std.IterM m β₂ |
Vector.eraseIdx_append_of_lt_size._proof_2 | Init.Data.Vector.Erase | ∀ {n k : ℕ}, k < n → n - 1 + n = n + n - 1 |
_private.Lean.Meta.Sym.Simp.EvalGround.0.Lean.Meta.Sym.Simp.evalUnaryBitVec'.match_1 | Lean.Meta.Sym.Simp.EvalGround | (motive : OptionT Id Lean.Meta.Sym.BitVecValue → Sort u_1) →
(x : OptionT Id Lean.Meta.Sym.BitVecValue) →
((a : Lean.Meta.Sym.BitVecValue) → motive (some a)) →
((x : OptionT Id Lean.Meta.Sym.BitVecValue) → motive x) → motive x |
Matrix.IsAdjMatrix.toGraph_adj | Mathlib.Combinatorics.SimpleGraph.AdjMatrix | ∀ {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [inst : MulZeroOneClass α] [inst_1 : Nontrivial α]
(h : A.IsAdjMatrix) (i j : V), h.toGraph.Adj i j = (A i j = 1) |
Function.Surjective.moduleLeft._proof_3 | Mathlib.Algebra.Module.RingHom | ∀ {R : Type u_3} {S : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : Semiring S] [inst_4 : SMul S M] (f : R →+* S) (hf : Function.Surjective ⇑f)
(hsmul : ∀ (c : R) (x : M), f c • x = c • x) (y₁ y₂ : S) (x : M), (y₁ + y₂) • x = y₁ • x + y₂ • x |
Std.HashMap.getKeyD_alter_self | Std.Data.HashMap.Lemmas | ∀ {α : Type u} {β : Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.HashMap α β} [EquivBEq α] [LawfulHashable α]
[Inhabited α] {k fallback : α} {f : Option β → Option β},
(m.alter k f).getKeyD k fallback = if (f m[k]?).isSome = true then k else fallback |
TopModuleCat.isColimitCoker._proof_4 | Mathlib.Algebra.Category.ModuleCat.Topology.Homology | ∀ {R : Type u_1} [inst : Ring R] [inst_1 : TopologicalSpace R] {M N : TopModuleCat R} (φ : M ⟶ N)
(s : CategoryTheory.Limits.CokernelCofork φ), ContinuousSMul R ↑s.1.toModuleCat |
ValuationRing.iff_dvd_total | Mathlib.RingTheory.Valuation.ValuationRing | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R], ValuationRing R ↔ Std.Total fun x1 x2 => x1 ∣ x2 |
disjointed_add_one | Mathlib.Algebra.Order.Disjointed | ∀ {α : Type u_1} {ι : Type u_2} [inst : GeneralizedBooleanAlgebra α] [inst_1 : LinearOrder ι]
[inst_2 : LocallyFiniteOrderBot ι] [inst_3 : Add ι] [inst_4 : One ι] [SuccAddOrder ι] [NoMaxOrder ι] (f : ι → α)
(i : ι), disjointed f (i + 1) = f (i + 1) \ (partialSups f) i |
MulEquiv.mapSubgroup.eq_1 | Mathlib.Algebra.Group.Subgroup.Map | ∀ {G : Type u_1} [inst : Group G] {H : Type u_6} [inst_1 : Group H] (f : G ≃* H),
f.mapSubgroup =
{ toFun := Subgroup.map ↑f, invFun := Subgroup.map ↑f.symm, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ } |
Lean.Lsp.ShowDocumentClientCapabilities | Lean.Data.Lsp.Capabilities | Type |
_private.Mathlib.Order.Partition.Finpartition.0.Finpartition.exists_le_of_le._proof_1_2 | Mathlib.Order.Partition.Finpartition | ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : OrderBot α] {a b : α} {P Q : Finpartition a},
(∀ p ∈ P.parts, ∃ q ∈ Q.parts.erase b, p ≤ q) → P.parts.sup id ≤ (Q.parts.erase b).sup id |
_private.Mathlib.LinearAlgebra.Dimension.Finite.0.Module.finite_finsupp_iff._simp_1_4 | Mathlib.LinearAlgebra.Dimension.Finite | ∀ {α : Type u_1}, (¬Subsingleton α) = Nontrivial α |
MeasureTheory.Measure.isOpenPosMeasure_smul | Mathlib.MeasureTheory.Measure.OpenPos | ∀ {X : Type u_1} [inst : TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure]
{c : ENNReal}, c ≠ 0 → (c • μ).IsOpenPosMeasure |
SimpleGraph.Finsubgraph.coe_compl._simp_1 | Mathlib.Combinatorics.SimpleGraph.Finsubgraph | ∀ {V : Type u} {G : SimpleGraph V} [inst : Finite V] (G' : G.Finsubgraph), (↑G')ᶜ = ↑G'ᶜ |
_private.Mathlib.CategoryTheory.NatTrans.0.CategoryTheory.NatTrans.ext.match_1 | Mathlib.CategoryTheory.NatTrans | ∀ {C : Type u_3} {inst : CategoryTheory.Category.{u_1, u_3} C} {D : Type u_4}
{inst_1 : CategoryTheory.Category.{u_2, u_4} D} {F G : CategoryTheory.Functor C D}
(motive : CategoryTheory.NatTrans F G → Prop) (h : CategoryTheory.NatTrans F G),
(∀ (app : (X : C) → F.obj X ⟶ G.obj X)
(naturality :
autoParam
(∀ ⦃X Y : C⦄ (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp (F.map f) (app Y) = CategoryTheory.CategoryStruct.comp (app X) (G.map f))
CategoryTheory.NatTrans.naturality._autoParam),
motive { app := app, naturality := naturality }) →
motive h |
Lean.IR.ExpandResetReuse.removeSelfSet | Lean.Compiler.IR.ExpandResetReuse | Lean.IR.ExpandResetReuse.Context → Lean.IR.FnBody → Lean.IR.FnBody |
CategoryTheory.Localization.instHasSmallLocalizedHomObjShiftFunctor | Mathlib.CategoryTheory.Localization.SmallShiftedHom | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] (W : CategoryTheory.MorphismProperty C) {M : Type w'}
[inst_1 : AddMonoid M] [inst_2 : CategoryTheory.HasShift C M] (X Y : C)
[CategoryTheory.Localization.HasSmallLocalizedShiftedHom W M X Y] (m : M),
CategoryTheory.Localization.HasSmallLocalizedHom W X ((CategoryTheory.shiftFunctor C m).obj Y) |
Lean.Meta.Hint.Suggestion | Lean.Meta.Hint | Type |
_private.Mathlib.MeasureTheory.Function.LpSpace.Basic.0.MeasureTheory.Lp.instNormedAddCommGroup._simp_3 | Mathlib.MeasureTheory.Function.LpSpace.Basic | ∀ {r q : NNReal}, (↑r ≤ ↑q) = (r ≤ q) |
OpenPartialHomeomorph.subtypeRestr_source | Mathlib.Topology.OpenPartialHomeomorph.Constructions | ∀ {X : Type u_1} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
(e : OpenPartialHomeomorph X Y) {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s),
(e.subtypeRestr hs).source = Subtype.val ⁻¹' e.source |
AddMonoidAlgebra.mapDomain.eq_1 | Mathlib.Algebra.MonoidAlgebra.MapDomain | ∀ {R : Type u_2} {M : Type u_5} {N : Type u_6} [inst : Semiring R] (f : M → N) (x : AddMonoidAlgebra R M),
AddMonoidAlgebra.mapDomain f x = Finsupp.mapDomain f x |
_private.Lean.Elab.PatternVar.0.Lean.Elab.Term.CollectPatternVars.collect.processCtorApp._sparseCasesOn_1 | Lean.Elab.PatternVar | {motive : Lean.Elab.Term.Arg → Sort u} →
(t : Lean.Elab.Term.Arg) →
((val : Lean.Syntax) → motive (Lean.Elab.Term.Arg.stx val)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t |
Int.Linear.orOver_one | Init.Data.Int.Linear | ∀ {p : ℕ → Prop}, Int.Linear.OrOver 1 p → p 0 |
CategoryTheory.GlueData.mapGlueData._proof_6 | Mathlib.CategoryTheory.GlueData | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] {C' : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} C'] (D : CategoryTheory.GlueData C) (F : CategoryTheory.Functor C C')
[inst_2 : ∀ (i j k : D.J), CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan (D.f i j) (D.f i k)) F]
(i j k : D.J),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesPullback.iso F (D.f i j) (D.f i k)).inv
(CategoryTheory.CategoryStruct.comp (F.map (D.t' i j k))
(CategoryTheory.Limits.PreservesPullback.iso F (D.f j k) (D.f j i)).hom))
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesPullback.iso F (D.f j k) (D.f j i)).inv
(CategoryTheory.CategoryStruct.comp (F.map (D.t' j k i))
(CategoryTheory.Limits.PreservesPullback.iso F (D.f k i) (D.f k j)).hom))
(CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesPullback.iso F (D.f k i) (D.f k j)).inv
(CategoryTheory.CategoryStruct.comp (F.map (D.t' k i j))
(CategoryTheory.Limits.PreservesPullback.iso F (D.f i j) (D.f i k)).hom))) =
CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.pullback (F.map (D.f i j)) (F.map (D.f i k))) |
_private.Mathlib.Order.Fin.Tuple.0.Fin.preimage_insertNth_Icc_of_notMem._simp_1_1 | Mathlib.Order.Fin.Tuple | ∀ {α : Type u} {β : Type v} {f : α → β} {s : Set β} {a : α}, (a ∈ f ⁻¹' s) = (f a ∈ s) |
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