name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
_private.Batteries.Data.UnionFind.Basic.0.Batteries.UnionFind.setParentBump_rankD_lt._proof_1 | Batteries.Data.UnionFind.Basic | ∀ {arr' : Array Batteries.UFNode} {arr : Array Batteries.UFNode} {x : Fin arr.size} {y : Fin arr.size} {i : ℕ},
¬↑x < arr.size → False |
Polynomial.Monic.add_of_left | Mathlib.Algebra.Polynomial.Monic | ∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R}, p.Monic → q.degree < p.degree → (p + q).Monic |
setOf_riemannianEDist_lt_subset_nhds' | Mathlib.Geometry.Manifold.Riemannian.Basic | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
(I : ModelWithCorners ℝ E H) {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
[inst_5 : Bundle.RiemannianBundle fun x => TangentSpace I x] [inst_6 : IsManifold I 1 M]
[IsContinuousRiemannianBundle E fun x => TangentSpace I x] [RegularSpace M] {x : M} {s : Set M},
s ∈ nhds x → ∃ c > 0, {y | Manifold.riemannianEDist I x y < c} ⊆ s |
_private.Mathlib.GroupTheory.Perm.Cycle.Type.0.Equiv.Perm.IsThreeCycle.nodup_iff_mem_support._proof_1_725 | Mathlib.GroupTheory.Perm.Cycle.Type | ∀ {α : Type u_1} [inst_1 : DecidableEq α] {g : Equiv.Perm α} {a : α} (w_1 : α),
List.findIdxNth (fun x => decide (x = w_1)) [g a, g (g a)] [g (g a)].length + 1 ≤
(List.findIdxs (fun x => decide (x = w_1)) [g a, g (g a)]).length →
List.findIdxNth (fun x => decide (x = w_1)) [g a, g (g a)] [g (g a)].length <
(List.findIdxs (fun x => decide (x = w_1)) [g a, g (g a)]).length |
Bundle.Pullback.lift | Mathlib.Data.Bundle | {B : Type u_1} →
{F : Type u_2} →
{E : B → Type u_3} → {B' : Type u_4} → (f : B' → B) → Bundle.TotalSpace F (f *ᵖ E) → Bundle.TotalSpace F E |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic.0.WeierstrassCurve.Projective.Y_eq_of_equiv._simp_1_2 | Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | ∀ {R : Type r} [inst : CommRing R] (P : Fin 3 → R) (u : R), (u • P) 1 = u * P 1 |
Matrix.diagonalLinearMap._proof_1 | Mathlib.Data.Matrix.Basic | ∀ (n : Type u_2) (α : Type u_1) [inst : DecidableEq n] [inst_1 : AddCommMonoid α] (x y : n → α),
(↑(Matrix.diagonalAddMonoidHom n α)).toFun (x + y) =
(↑(Matrix.diagonalAddMonoidHom n α)).toFun x + (↑(Matrix.diagonalAddMonoidHom n α)).toFun y |
Turing.TM1.SupportsStmt.eq_2 | Mathlib.Computability.PostTuringMachine | ∀ {Γ : Type u_1} {Λ : Type u_2} {σ : Type u_3} (S : Finset Λ) (a : Γ → σ → Γ) (q : Turing.TM1.Stmt Γ Λ σ),
Turing.TM1.SupportsStmt S (Turing.TM1.Stmt.write a q) = Turing.TM1.SupportsStmt S q |
Ideal.exists_finset_card_eq_height_of_isNoetherianRing | Mathlib.RingTheory.Ideal.KrullsHeightTheorem | ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) [p.IsPrime],
∃ s, p ∈ (Ideal.span ↑s).minimalPrimes ∧ ↑s.card = p.height |
_private.Mathlib.Data.Fin.Tuple.Finset.0.Fin.mem_piFinset_iff_last_init._simp_1_2 | Mathlib.Data.Fin.Tuple.Finset | ∀ {n : ℕ} {P : Fin (n + 1) → Prop}, (∀ (i : Fin (n + 1)), P i) = ((∀ (i : Fin n), P i.castSucc) ∧ P (Fin.last n)) |
Mathlib.Tactic.Widget.StringDiagram.PenroseVar.indices | Mathlib.Tactic.Widget.StringDiagram | Mathlib.Tactic.Widget.StringDiagram.PenroseVar → List ℕ |
Lean.Meta.NormCast.NormCastExtension.up | Lean.Meta.Tactic.NormCast | Lean.Meta.NormCast.NormCastExtension → Lean.Meta.SimpExtension |
CategoryTheory.Localization.Lifting | Mathlib.CategoryTheory.Localization.Predicate | {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] →
{E : Type u_3} →
[inst_2 : CategoryTheory.Category.{v_3, u_3} E] →
CategoryTheory.Functor C D →
CategoryTheory.MorphismProperty C →
CategoryTheory.Functor C E → CategoryTheory.Functor D E → Type (max u_1 v_3) |
CategoryTheory.MorphismProperty.HasRightCalculusOfFractions.exists_rightFraction | Mathlib.CategoryTheory.Localization.CalculusOfFractions | ∀ {C : Type u_1} {inst : CategoryTheory.Category.{v_1, u_1} C} {W : CategoryTheory.MorphismProperty C}
[self : W.HasRightCalculusOfFractions] ⦃X Y : C⦄ (φ : W.LeftFraction X Y),
∃ ψ, CategoryTheory.CategoryStruct.comp ψ.s φ.f = CategoryTheory.CategoryStruct.comp ψ.f φ.s |
CategoryTheory.Limits.prod.inl.eq_1 | Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C)
[inst_2 : CategoryTheory.Limits.HasBinaryProduct X Y],
CategoryTheory.Limits.prod.inl X Y = CategoryTheory.Limits.prod.lift (CategoryTheory.CategoryStruct.id X) 0 |
Nat.succ_injective | Mathlib.Data.Nat.Basic | Function.Injective Nat.succ |
_private.Mathlib.RingTheory.Localization.Away.Basic.0.IsLocalization.Away.map_injective_iff._simp_1_1 | Mathlib.RingTheory.Localization.Away.Basic | ∀ {M : Type u_1} [inst : Monoid M] (x z : M), (x ∈ Submonoid.powers z) = ∃ n, z ^ n = x |
LinearMap.isBigOTVS_rev_comp | Mathlib.Analysis.Asymptotics.TVS | ∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
[inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E] [inst_3 : Module 𝕜 E] [inst_4 : AddCommGroup F]
[inst_5 : TopologicalSpace F] [inst_6 : Module 𝕜 F] {l : Filter α} {f : α → E} (g : E →ₗ[𝕜] F),
Filter.comap (⇑g) (nhds 0) ≤ nhds 0 → f =O[𝕜; l] (⇑g ∘ f) |
CommMonCat.FilteredColimits.forget_preservesFilteredColimits | Mathlib.Algebra.Category.MonCat.FilteredColimits | CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget CommMonCat) |
_private.Lean.Elab.PatternVar.0.Lean.Elab.Term.CollectPatternVars.collect.processId._sparseCasesOn_1 | Lean.Elab.PatternVar | {motive : Lean.ConstantInfo → Sort u} →
(t : Lean.ConstantInfo) →
((val : Lean.ConstructorVal) → motive (Lean.ConstantInfo.ctorInfo val)) →
(Nat.hasNotBit 64 t.ctorIdx → motive t) → motive t |
Complex.im_mul_ofReal | Mathlib.Data.Complex.Basic | ∀ (z : ℂ) (r : ℝ), (z * ↑r).im = z.im * r |
ConvexOn.exists_lipschitzOnWith_of_isBounded | Mathlib.Analysis.Convex.Continuous | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {x₀ : E} {r r' : ℝ},
ConvexOn ℝ (Metric.ball x₀ r) f →
r' < r → Bornology.IsBounded (f '' Metric.ball x₀ r) → ∃ K, LipschitzOnWith K f (Metric.ball x₀ r') |
_private.Mathlib.Analysis.Convex.Function.0.OrderIso.strictConvexOn_symm._simp_1_1 | Mathlib.Analysis.Convex.Function | ∀ {α : Type u_1} [inst : LT α] {x y : α}, (x > y) = (y < x) |
CategoryTheory.Triangulated.SpectralObject.ω₂_map_hom₃ | Mathlib.CategoryTheory.Triangulated.SpectralObject | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Limits.HasZeroObject C]
[inst_3 : CategoryTheory.HasShift C ℤ] [inst_4 : CategoryTheory.Preadditive C]
[inst_5 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_6 : CategoryTheory.Pretriangulated C]
(X : CategoryTheory.Triangulated.SpectralObject C ι) {X_1 Y : CategoryTheory.ComposableArrows ι 2} (φ : X_1 ⟶ Y),
(X.ω₂.map φ).hom₃ = X.ω₁.map (CategoryTheory.ComposableArrows.homMk₁ (φ.app 1) (φ.app ⟨2, ⋯⟩) ⋯) |
CategoryTheory.PreOneHypercover.map_f | Mathlib.CategoryTheory.Sites.Continuous | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{X : C} (E : CategoryTheory.PreOneHypercover X) (F : CategoryTheory.Functor C D) (i : E.I₀),
(E.map F).f i = F.map (E.f i) |
Pi.instBiheytingAlgebra._proof_1 | Mathlib.Order.Heyting.Basic | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → BiheytingAlgebra (α i)] (a b c : (i : ι) → α i),
a \ b ≤ c ↔ a ≤ b ⊔ c |
Interval.commMonoid._proof_5 | Mathlib.Algebra.Order.Interval.Basic | ∀ {α : Type u_1} [inst : CommMonoid α] [inst_1 : Preorder α] [inst_2 : IsOrderedMonoid α] (a : Interval α), a * 1 = a |
jacobiSum_one_one | Mathlib.NumberTheory.JacobiSum.Basic | ∀ {F : Type u_1} {R : Type u_2} [inst : Field F] [inst_1 : Fintype F] [inst_2 : CommRing R],
jacobiSum 1 1 = ↑(Fintype.card F) - 2 |
Nat.decidableForallFin._proof_1 | Init.Data.Nat.Lemmas | ∀ {n : ℕ} (P : Fin n → Prop), (∀ (k : ℕ) (h : k < n), P ⟨k, h⟩) ↔ ∀ (i : Fin n), P i |
ContinuousMap.compactConvergenceUniformSpace | Mathlib.Topology.UniformSpace.CompactConvergence | {α : Type u₁} → {β : Type u₂} → [inst : TopologicalSpace α] → [inst_1 : UniformSpace β] → UniformSpace C(α, β) |
IsStarNormal.neg | Mathlib.Algebra.Star.SelfAdjoint | ∀ {R : Type u_1} [inst : NonUnitalNonAssocRing R] [inst_1 : StarAddMonoid R] {x : R} [IsStarNormal x], IsStarNormal (-x) |
Std.DTreeMap.Internal.Impl.insertMin.match_3.congr_eq_1 | Std.Data.DTreeMap.Internal.WF.Lemmas | ∀ {α : Type u_1} {β : α → Type u_2} (motive : (t : Std.DTreeMap.Internal.Impl α β) → t.Balanced → Sort u_3)
(t : Std.DTreeMap.Internal.Impl α β) (hr : t.Balanced)
(h_1 : (hr : Std.DTreeMap.Internal.Impl.leaf.Balanced) → motive Std.DTreeMap.Internal.Impl.leaf hr)
(h_2 :
(sz : ℕ) →
(k' : α) →
(v' : β k') →
(l' r' : Std.DTreeMap.Internal.Impl α β) →
(hr : (Std.DTreeMap.Internal.Impl.inner sz k' v' l' r').Balanced) →
motive (Std.DTreeMap.Internal.Impl.inner sz k' v' l' r') hr)
(hr_1 : Std.DTreeMap.Internal.Impl.leaf.Balanced),
t = Std.DTreeMap.Internal.Impl.leaf →
hr ≍ hr_1 →
(match t, hr with
| Std.DTreeMap.Internal.Impl.leaf, hr => h_1 hr
| Std.DTreeMap.Internal.Impl.inner sz k' v' l' r', hr => h_2 sz k' v' l' r' hr) ≍
h_1 hr_1 |
_private.Mathlib.Topology.MetricSpace.Infsep.0.Set.einfsep_insert._simp_1_1 | Mathlib.Topology.MetricSpace.Infsep | ∀ {α : Type u_1} [inst : EDist α] {s : Set α} {d : ENNReal}, (d ≤ s.einfsep) = ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ edist x y |
NNRat.cast_div_of_ne_zero | Mathlib.Data.Rat.Cast.Defs | ∀ {α : Type u_3} [inst : DivisionSemiring α] {q r : ℚ≥0}, ↑q.den ≠ 0 → ↑r.num ≠ 0 → ↑(q / r) = ↑q / ↑r |
List.minOn_append._proof_1 | Init.Data.List.MinMaxOn | ∀ {α : Type u_1} {xs ys : List α}, xs ≠ [] → xs ++ ys ≠ [] |
norm_deriv_eq_norm_fderiv | Mathlib.Analysis.Calculus.Deriv.Basic | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}, ‖deriv f x‖ = ‖fderiv 𝕜 f x‖ |
FreeGroup.Red.decidableRel._proof_3 | Mathlib.GroupTheory.FreeGroup.Reduce | ∀ {α : Type u_1} (x : α) (b : Bool) (tl : List (α × Bool)), FreeGroup.Red tl [(x, !b)] → FreeGroup.Red ((x, b) :: tl) [] |
RCLike.norm_coe_norm | Mathlib.Analysis.Normed.Module.RCLike.Basic | ∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] {z : E}, ‖↑‖z‖‖ = ‖z‖ |
FreeAddGroup.Red.Step.sublist | Mathlib.GroupTheory.FreeGroup.Basic | ∀ {α : Type u} {L₁ L₂ : List (α × Bool)}, FreeAddGroup.Red.Step L₁ L₂ → L₂.Sublist L₁ |
_private.Std.Data.DHashMap.Basic.0.Std.DHashMap.Const.modify._proof_1 | Std.Data.DHashMap.Basic | ∀ {α : Type u_1} {x : BEq α} {x_1 : Hashable α} {β : Type u_2} (m : Std.DHashMap α fun x => β) (a : α) (f : β → β),
(↑(Std.DHashMap.Internal.Raw₀.Const.modify ⟨m.inner, ⋯⟩ a f)).WF |
MvPowerSeries.eq_inv_iff_mul_eq_one | Mathlib.RingTheory.MvPowerSeries.Inverse | ∀ {σ : Type u_1} {k : Type u_3} [inst : Field k] {φ ψ : MvPowerSeries σ k},
MvPowerSeries.constantCoeff ψ ≠ 0 → (φ = ψ⁻¹ ↔ φ * ψ = 1) |
Set.singleton_inter_of_notMem | Mathlib.Data.Set.Insert | ∀ {α : Type u} {s : Set α} {a : α}, a ∉ s → {a} ∩ s = ∅ |
Lean.Parser.Term.optIdent.parenthesizer | Lean.Parser.Term.Basic | Lean.PrettyPrinter.Parenthesizer |
NumberField.mixedEmbedding.fundamentalCone.expMapBasis_nonneg | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] (x : NumberField.mixedEmbedding.realSpace K)
(w : NumberField.InfinitePlace K), 0 ≤ ↑NumberField.mixedEmbedding.fundamentalCone.expMapBasis x w |
Batteries.BinomialHeap.Imp.FindMin.recOn | Batteries.Data.BinomialHeap.Basic | {α : Type u_1} →
{motive : Batteries.BinomialHeap.Imp.FindMin α → Sort u} →
(t : Batteries.BinomialHeap.Imp.FindMin α) →
((before : Batteries.BinomialHeap.Imp.Heap α → Batteries.BinomialHeap.Imp.Heap α) →
(val : α) →
(node : Batteries.BinomialHeap.Imp.HeapNode α) →
(next : Batteries.BinomialHeap.Imp.Heap α) →
motive { before := before, val := val, node := node, next := next }) →
motive t |
_private.Mathlib.Combinatorics.Enumerative.Partition.Glaisher.0.Nat.Partition.aux_mul_one_sub_X_pow._proof_1_2 | Mathlib.Combinatorics.Enumerative.Partition.Glaisher | ∀ (R : Type u_1) [inst : CommRing R] {m : ℕ},
0 < m → ∀ (i : ↑(Function.mulSupport fun i => 1 - (PowerSeries.X ^ (i + 1)) ^ m)), (↑i + 1) * m - 1 + 1 = (↑i + 1) * m |
Lean.Grind.CommRing.Poly.cancelVar | Init.Grind.Ring.CommSolver | ℤ → Lean.Grind.CommRing.Var → Lean.Grind.CommRing.Poly → Lean.Grind.CommRing.Poly |
CategoryTheory.Pseudofunctor.DescentData'.instCategory._proof_2 | Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C]
{F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat} {ι : Type u_5} {S : C}
{X : ι → C} {f : (i : ι) → X i ⟶ S} {sq : (i j : ι) → CategoryTheory.Limits.ChosenPullback (f i) (f j)}
{sq₃ : (i₁ i₂ i₃ : ι) → CategoryTheory.Limits.ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)}
(x : F.DescentData' sq sq₃) (i₁ i₂ : ι),
CategoryTheory.CategoryStruct.comp
((F.map (sq i₁ i₂).p₁.op.toLoc).toFunctor.map (CategoryTheory.CategoryStruct.id (x.obj i₁))) (x.hom i₁ i₂) =
CategoryTheory.CategoryStruct.comp (x.hom i₁ i₂)
((F.map (sq i₁ i₂).p₂.op.toLoc).toFunctor.map (CategoryTheory.CategoryStruct.id (x.obj i₂))) |
ContMDiffWithinAt.change_section_trivialization | Mathlib.Geometry.Manifold.VectorBundle.Basic | ∀ {n : WithTop ℕ∞} {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6}
[inst : NontriviallyNormedField 𝕜] {EB : Type u_7} [inst_1 : NormedAddCommGroup EB] [inst_2 : NormedSpace 𝕜 EB]
{HB : Type u_8} [inst_3 : TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [inst_4 : TopologicalSpace B]
[inst_5 : ChartedSpace HB B] {EM : Type u_9} [inst_6 : NormedAddCommGroup EM] [inst_7 : NormedSpace 𝕜 EM]
{HM : Type u_10} [inst_8 : TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [inst_9 : TopologicalSpace M]
[inst_10 : ChartedSpace HM M] [inst_11 : (x : B) → AddCommMonoid (E x)] [inst_12 : (x : B) → Module 𝕜 (E x)]
[inst_13 : NormedAddCommGroup F] [inst_14 : NormedSpace 𝕜 F] [inst_15 : TopologicalSpace (Bundle.TotalSpace F E)]
[inst_16 : (x : B) → TopologicalSpace (E x)] [inst_17 : FiberBundle F E] [inst_18 : VectorBundle 𝕜 F E]
[ContMDiffVectorBundle n F E IB] {e e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e]
[MemTrivializationAtlas e'] {s : Set M} {x : M} {f : M → Bundle.TotalSpace F E},
ContMDiffWithinAt IM IB n (Bundle.TotalSpace.proj ∘ f) s x →
ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 F) n (fun y => (↑e (f y)).2) s x →
f x ∈ e.source → f x ∈ e'.source → ContMDiffWithinAt IM (modelWithCornersSelf 𝕜 F) n (fun y => (↑e' (f y)).2) s x |
MvPolynomial.monomial_zero' | Mathlib.Algebra.MvPolynomial.Basic | ∀ {R : Type u} {σ : Type u_1} [inst : CommSemiring R], ⇑(MvPolynomial.monomial 0) = ⇑MvPolynomial.C |
WittVector.succNthValUnits.congr_simp | Mathlib.RingTheory.WittVector.DiscreteValuationRing | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] {k : Type u_1} [inst : CommRing k] [inst_1 : CharP k p] (n : ℕ) (a a_1 : kˣ),
a = a_1 →
∀ (A A_1 : WittVector p k),
A = A_1 →
∀ (bs bs_1 : Fin (n + 1) → k),
bs = bs_1 → WittVector.succNthValUnits n a A bs = WittVector.succNthValUnits n a_1 A_1 bs_1 |
Finset.measure_zero | Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms | ∀ {α : Type u_1} {m0 : MeasurableSpace α} (s : Finset α) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ],
μ ↑s = 0 |
Lean.Meta.RefinedDiscrTree.Key.bvar | Mathlib.Lean.Meta.RefinedDiscrTree.Basic | ℕ → ℕ → Lean.Meta.RefinedDiscrTree.Key |
Lean.Grind.AC.diseq_simp_rhs_ac | Init.Grind.AC | ∀ {α : Sort u_1} (ctx : Lean.Grind.AC.Context α) {inst₁ : Std.Associative ctx.op} {inst₂ : Std.Commutative ctx.op}
(c lhs₁ rhs₁ lhs₂ rhs₂ rhs₂' : Lean.Grind.AC.Seq),
Lean.Grind.AC.simp_ac_cert c lhs₁ rhs₁ rhs₂ rhs₂' = true →
Lean.Grind.AC.Seq.denote ctx lhs₁ = Lean.Grind.AC.Seq.denote ctx rhs₁ →
Lean.Grind.AC.Seq.denote ctx lhs₂ ≠ Lean.Grind.AC.Seq.denote ctx rhs₂ →
Lean.Grind.AC.Seq.denote ctx lhs₂ ≠ Lean.Grind.AC.Seq.denote ctx rhs₂' |
Lean.isInstanceReducibleCore | Lean.ReducibilityAttrs | Lean.Environment → Lean.Name → Bool |
MulEquiv.withOneCongr._proof_2 | Mathlib.Algebra.Group.WithOne.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : Mul α] [inst_1 : Mul β] (e : α ≃* β) (x : WithOne α),
(WithOne.mapMulHom e.symm.toMulHom) ((WithOne.mapMulHom e.toMulHom) x) = x |
ONote.zero.elim | Mathlib.SetTheory.Ordinal.Notation | {motive : ONote → Sort u} → (t : ONote) → t.ctorIdx = 0 → motive ONote.zero → motive t |
Con.lift | Mathlib.GroupTheory.Congruence.Hom | {M : Type u_1} →
{P : Type u_3} →
[inst : MulOneClass M] → [inst_1 : MulOneClass P] → (c : Con M) → (f : M →* P) → c ≤ Con.ker f → c.Quotient →* P |
TopologicalSpace.Closeds.noncompactSpace_iff._simp_1 | Mathlib.Topology.UniformSpace.Closeds | ∀ {α : Type u_1} [inst : UniformSpace α], NoncompactSpace (TopologicalSpace.Closeds α) = NoncompactSpace α |
_private.Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Lemmas.Operations.RotateLeft.0.Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateLeft.go_get_aux._proof_1_2 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Lemmas.Operations.RotateLeft | ∀ {w : ℕ} (distance curr idx : ℕ), idx < curr → ¬idx < curr + 1 → False |
Int.mul_fmod_right | Init.Data.Int.DivMod.Lemmas | ∀ (a b : ℤ), (a * b).fmod a = 0 |
Lean.Parser.OrElseOnAntiquotBehavior.rec | Lean.Parser.Basic | {motive : Lean.Parser.OrElseOnAntiquotBehavior → Sort u} →
motive Lean.Parser.OrElseOnAntiquotBehavior.acceptLhs →
motive Lean.Parser.OrElseOnAntiquotBehavior.takeLongest →
motive Lean.Parser.OrElseOnAntiquotBehavior.merge → (t : Lean.Parser.OrElseOnAntiquotBehavior) → motive t |
_private.Std.Data.DHashMap.Internal.WF.0.Std.DHashMap.Internal.Raw₀.alterₘ.match_1.eq_2 | Std.Data.DHashMap.Internal.WF | ∀ {α : Type u_3} {β : α → Type u_1} (a : α) (motive : Option (β a) → Sort u_2) (b : β a) (h_1 : Unit → motive none)
(h_2 : (b : β a) → motive (some b)),
(match some b with
| none => h_1 ()
| some b => h_2 b) =
h_2 b |
Std.HashMap.getKey!_eq_getKeyD_default | 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]
[inst : Inhabited α] {a : α}, m.getKey! a = m.getKeyD a default |
_private.Init.Data.List.Count.0.List.count_erase.match_1_1 | Init.Data.List.Count | ∀ {α : Type u_1} (motive : List α → Prop) (x : List α),
(∀ (a : Unit), motive []) → (∀ (c : α) (l : List α), motive (c :: l)) → motive x |
LieModule.toEnd_eq_iff | Mathlib.Algebra.Lie.OfAssociative | ∀ (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] [inst_6 : LieModule R L M]
[LieModule.IsFaithful R L M] {x y : L}, (LieModule.toEnd R L M) x = (LieModule.toEnd R L M) y ↔ x = y |
SSet.Subcomplex.PairingCore.index | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PairingCore | {X : SSet} → {A : X.Subcomplex} → (self : A.PairingCore) → (s : self.ι) → Fin (self.dim s + 2) |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.minKey!_eq_default._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) |
Int.two_dvd_mul_add_one | Mathlib.Algebra.Ring.Int.Parity | ∀ (k : ℤ), 2 ∣ k * (k + 1) |
_private.Mathlib.Topology.EMetricSpace.Basic.0.EMetric.totallyBounded_iff'.match_1_1 | Mathlib.Topology.EMetricSpace.Basic | ∀ {α : Type u_1} [inst : PseudoEMetricSpace α] {s : Set α} (ε : ENNReal)
(motive : (∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.eball y ε) → Prop)
(x : ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.eball y ε),
(∀ (t : Set α) (left : t ⊆ s) (ft : t.Finite) (h : s ⊆ ⋃ y ∈ t, Metric.eball y ε), motive ⋯) → motive x |
Lean.Elab.Tactic.Omega.MetaProblem.ctorIdx | Lean.Elab.Tactic.Omega.Frontend | Lean.Elab.Tactic.Omega.MetaProblem → ℕ |
_private.Mathlib.RingTheory.RootsOfUnity.Basic.0.IsCyclic.monoidHomMulEquivRootsOfUnityOfGenerator._simp_2 | Mathlib.RingTheory.RootsOfUnity.Basic | ∀ {G : Type u_1} [inst : Monoid G] {x : G} {n : ℕ}, (orderOf x ∣ n) = (x ^ n = 1) |
Order.krullDim_eq_iSup_length | Mathlib.Order.KrullDimension | ∀ {α : Type u_1} [inst : Preorder α] [Nonempty α], Order.krullDim α = ↑(⨆ p, ↑p.length) |
Std.DTreeMap.Internal.Impl.Const.insertManyIfNewUnit._proof_2 | Std.Data.DTreeMap.Internal.Operations | ∀ {α : Type u_1} [inst : Ord α] (t : Std.DTreeMap.Internal.Impl α fun x => Unit) (h : t.Balanced) (a : α)
(r : Std.DTreeMap.Internal.Impl.Const.IteratedUnitInsertionInto t)
{P : (Std.DTreeMap.Internal.Impl α fun x => Unit) → Prop},
P t →
(∀ (t'' : Std.DTreeMap.Internal.Impl α fun x => Unit) (a : α) (h : t''.Balanced),
P t'' → P (Std.DTreeMap.Internal.Impl.insertIfNew a () t'' h).impl) →
P (Std.DTreeMap.Internal.Impl.insertIfNew a () ↑r ⋯).impl |
Vector.getElem_zero_flatten | Init.Data.Vector.Find | ∀ {α : Type u_1} {m n : ℕ} {xss : Vector (Vector α m) n} (h : 0 < n * m),
xss.flatten[0] = (Vector.findSome? (fun xs => xs[0]?) xss).get ⋯ |
MonoidHom.coeToMulHom.eq_1 | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} [inst : MulOne M] [inst_1 : MulOne N],
MonoidHom.coeToMulHom = { coe := MonoidHom.toMulHom } |
String.Pos.lt_of_lt_of_le | Init.Data.String.Basic | ∀ {s : String} {p q r : s.Pos}, p < q → q ≤ r → p < r |
HomotopyCategory.instPretriangulatedIntUp | Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | (C : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] →
[CategoryTheory.Limits.HasBinaryBiproducts C] →
[inst_3 : CategoryTheory.Limits.HasZeroObject C] →
CategoryTheory.Pretriangulated (HomotopyCategory C (ComplexShape.up ℤ)) |
_private.Lean.Meta.Basic.0.Lean.Meta.setInlineAttribute.match_1 | Lean.Meta.Basic | (motive : Except String Lean.Environment → Sort u_1) →
(x : Except String Lean.Environment) →
((env : Lean.Environment) → motive (Except.ok env)) → ((msg : String) → motive (Except.error msg)) → motive x |
_private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.inner_._proof_1 | Mathlib.Analysis.InnerProductSpace.OfNorm | (3 + 1).AtLeastTwo |
summable_sigma_of_nonneg | Mathlib.Topology.Algebra.InfiniteSum.Real | ∀ {α : Type u_4} {β : α → Type u_3} {f : (x : α) × β x → ℝ},
(∀ (x : (x : α) × β x), 0 ≤ f x) →
(Summable f ↔ (∀ (x : α), Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' (y : β x), f ⟨x, y⟩) |
Set.toFinset_empty | Mathlib.Data.Fintype.Sets | ∀ {α : Type u_1} [inst : Fintype ↑∅], ∅.toFinset = ∅ |
Computability.instDecidableEqΓ'.decEq._proof_3 | Mathlib.Computability.Encoding | ∀ (b : Bool), ¬Computability.Γ'.blank = Computability.Γ'.bit b |
CompleteBooleanAlgebra.himp._inherited_default | Mathlib.Order.CompleteBooleanAlgebra | {α : Type u_1} →
(le lt : α → α → Prop) →
(∀ (a : α), le a a) →
(∀ (a b c : α), le a b → le b c → le a c) →
(∀ (a b : α), lt a b ↔ le a b ∧ ¬le b a) →
(∀ (a b : α), le a b → le b a → a = b) →
(sup : α → α → α) →
(∀ (a b : α), le a (sup a b)) →
(∀ (a b : α), le b (sup a b)) → (∀ (a b c : α), le a c → le b c → le (sup a b) c) → (α → α) → α → α → α |
ClosedAddSubgroup | Mathlib.Topology.Algebra.Group.ClosedSubgroup | (G : Type u) → [AddGroup G] → [TopologicalSpace G] → Type u |
Std.Tactic.BVDecide.BVExpr.bitblast.blastExtract.go | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.Extract | {α : Type} →
[inst : Hashable α] →
[inst_1 : DecidableEq α] →
{newWidth : ℕ} →
{aig : Std.Sat.AIG α} →
{w : ℕ} → aig.RefVec w → ℕ → (curr : ℕ) → curr ≤ newWidth → aig.RefVec curr → aig.RefVec newWidth |
_private.Mathlib.RingTheory.TensorProduct.Quotient.0.Algebra.TensorProduct.quotIdealMapEquivTensorQuot._simp_1 | Mathlib.RingTheory.TensorProduct.Quotient | ∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]
[MulHomClass F M N] (f : F) (x y : M), f x * f y = f (x * y) |
Commute.tsum_left | Mathlib.Topology.Algebra.InfiniteSum.Ring | ∀ {ι : Type u_1} {α : Type u_3} {L : SummationFilter ι} [inst : NonUnitalNonAssocSemiring α]
[inst_1 : TopologicalSpace α] [IsTopologicalSemiring α] {f : ι → α} [T2Space α] [L.NeBot] (a : α),
(∀ (i : ι), Commute (f i) a) → Commute (∑'[L] (i : ι), f i) a |
Submodule.dualCoannihilator | Mathlib.LinearAlgebra.Dual.Defs | {R : Type u_1} →
{M : Type u_2} →
[inst : CommSemiring R] →
[inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R (Module.Dual R M) → Submodule R M |
_private.Lean.Elab.BuiltinEvalCommand.0.Lean.Elab.Command.elabEvalCoreUnsafe.match_3 | Lean.Elab.BuiltinEvalCommand | (motive : Option Lean.Elab.Command.EvalAction✝ → Sort u_1) →
(__do_lift : Option Lean.Elab.Command.EvalAction✝¹) →
((act : Lean.Elab.Command.EvalAction✝²) → motive (some act)) →
((x : Option Lean.Elab.Command.EvalAction✝³) → motive x) → motive __do_lift |
MeasurableEmbedding.measurableSet_range | Mathlib.MeasureTheory.MeasurableSpace.Embedding | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β},
MeasurableEmbedding f → MeasurableSet (Set.range f) |
_private.Mathlib.Order.Filter.Map.0.Filter.comap_neBot_iff_frequently._simp_1_1 | Mathlib.Order.Filter.Map | ∀ {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β}, (Filter.comap m f).NeBot = ∀ t ∈ f, ∃ a, m a ∈ t |
_private.Mathlib.CategoryTheory.Category.Pairwise.0.CategoryTheory.instFintypePairwise.match_5.eq_2 | Mathlib.CategoryTheory.Category.Pairwise | ∀ (ι : Type u_1) (motive : CategoryTheory.Pairwise ι → Sort u_2) (a a_1 : ι)
(h_1 : (a : ι) → motive (CategoryTheory.Pairwise.single a))
(h_2 : (a a_2 : ι) → motive (CategoryTheory.Pairwise.pair a a_2)),
(match CategoryTheory.Pairwise.pair a a_1 with
| CategoryTheory.Pairwise.single a => h_1 a
| CategoryTheory.Pairwise.pair a a_2 => h_2 a a_2) =
h_2 a a_1 |
«_aux_ImportGraph_Tools_FindHome___elabRules_command#find_home!__1» | ImportGraph.Tools.FindHome | Lean.Elab.Command.CommandElab |
Option.mem_pmem | Mathlib.Data.Option.Basic | ∀ {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (x : Option α) {a : α} (h : ∀ a ∈ x, p a)
(ha : a ∈ x), f a ⋯ ∈ Option.pmap f x h |
_private.Batteries.Data.List.Lemmas.0.List.getElem_idxOf_eq_idxOfNth_add._proof_1_34 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {x : α} [inst : BEq α] (head : α) (tail : List α) {n s : ℕ}
{h : n < (List.idxsOf x (head :: tail) s).length}, 0 < (List.filter (fun x_1 => x_1 == x) (head :: tail)).length |
_private.Lean.Elab.DocString.0.Lean.Doc.suggestionName.match_1 | Lean.Elab.DocString | (motive : Option Lean.Name → Sort u_1) →
(resolved? : Option Lean.Name) →
((resolved : Lean.Name) → motive (some resolved)) → (Unit → motive none) → motive resolved? |
Topology.IsConstructible.preimage | Mathlib.Topology.Constructible | ∀ {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y} {s : Set Y},
Continuous f →
(∀ (s : Set Y), IsOpen s → IsRetrocompact s → IsRetrocompact (f ⁻¹' s)) →
Topology.IsConstructible s → Topology.IsConstructible (f ⁻¹' s) |
Lean.Grind.CommRing.Expr.toPolyC_nc.go | Init.Grind.Ring.CommSolver | ℕ → Lean.Grind.CommRing.Expr → Lean.Grind.CommRing.Poly |
_private.Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple.0.RootPairing.GeckConstruction.instIsIrreducible_aux₂ | Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | ∀ {ι : Type u_1} {K : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Field K] [inst_1 : CharZero K]
[inst_2 : DecidableEq ι] [inst_3 : Fintype ι] [inst_4 : AddCommGroup M] [inst_5 : Module K M]
[inst_6 : AddCommGroup N] [inst_7 : Module K N] {P : RootPairing ι K M N} [inst_8 : P.IsCrystallographic] {b : P.Base}
[P.IsReduced] [P.IsIrreducible]
{U : LieSubmodule K (↥(RootPairing.GeckConstruction.lieAlgebra b)) (↥b.support ⊕ ι → K)} {i : ι},
RootPairing.GeckConstruction.v b i ∈ U → U = ⊤ |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.