name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.NumberTheory.LegendreSymbol.AddCharacter.0.AddChar.val_mem_rootsOfUnity._simp_1_1 | Mathlib.NumberTheory.LegendreSymbol.AddCharacter | ∀ {M : Type u_1} [inst : CommMonoid M] (k : ℕ) (ζ : Mˣ), (ζ ∈ rootsOfUnity k M) = (↑ζ ^ k = 1) | null | false |
Lean.Language.Lean.CommandParsedSnapshot.brecOn_3.eq | Lean.Language.Lean.Types | ∀ {motive_1 : Lean.Language.Lean.CommandParsedSnapshot → Sort u}
{motive_2 : Option (Lean.Language.SnapshotTask Lean.Language.Lean.CommandParsedSnapshot) → Sort u}
{motive_3 : Lean.Language.SnapshotTask Lean.Language.Lean.CommandParsedSnapshot → Sort u}
{motive_4 : Task Lean.Language.Lean.CommandParsedSnapshot → ... | null | true |
_private.Mathlib.Tactic.GRewrite.Core.0.Mathlib.Tactic.GRewrite.GRewriteLemma.noConfusion | Mathlib.Tactic.GRewrite.Core | {P : Sort u} →
{t t' : Mathlib.Tactic.GRewrite.GRewriteLemma✝} →
t = t' → Mathlib.Tactic.GRewrite.GRewriteLemma.noConfusionType✝ P t t' | null | false |
_private.Std.Http.Internal.String.0.Std.Http.Internal.UnquoteState.done | Std.Http.Internal.String | String → Std.Http.Internal.UnquoteState✝ | null | true |
Real.logb_zero | Mathlib.Analysis.SpecialFunctions.Log.Base | ∀ {b : ℝ}, Real.logb b 0 = 0 | null | true |
Inclusion.IntervalDyadicReal.Iic_mem_of_lt._hypothesisExt_1 | Mathlib.Tactic.Inclusion.Extension.IntervalDyadicReal.Hypotheses | Inclusion.HypothesisExt | null | false |
Lean.Compiler.LCNF.NormFVarResult.erased.elim | Lean.Compiler.LCNF.CompilerM | {motive : Lean.Compiler.LCNF.NormFVarResult → Sort u} →
(t : Lean.Compiler.LCNF.NormFVarResult) → t.ctorIdx = 1 → motive Lean.Compiler.LCNF.NormFVarResult.erased → motive t | null | false |
_private.Lean.Meta.Sym.SymM.0.Lean.Meta.Sym.repairShareViolation | Lean.Meta.Sym.SymM | Lean.Expr → Lean.Meta.Sym.SymM Lean.Expr | Restores the `SymM` invariants enabled in the current configuration.
The final `unfoldReducible` is needed because `foldProjs` can introduce fresh reducible
constants: `mkProjection` recovers structure parameters from `inferType`, and those types
come from environment signatures that were never preprocessed.
| true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd.0.SSet.prodStdSimplex.weakRankFunction._proof_12 | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | ∀ {m : ℕ} (k : Fin (m + 1)) (n d : ℕ) (is : Fin (d + 1))
(s :
(CategoryTheory.MonoidalCategoryStruct.tensorObj (SSet.stdSimplex.obj { len := m + 1 })
(SSet.stdSimplex.obj { len := n })).obj
(Opposite.op { len := d + 1 }))
(hs₁ :
s ∈
(CategoryTheory.MonoidalCategoryStruct.tensorObj (SSe... | null | false |
String.Slice.Pattern.ToForwardSearcher.DefaultForwardSearcher.noConfusion | Init.Data.String.Pattern.Basic | {P : Sort u} →
{ρ : Type} →
{pat : ρ} →
{s : String.Slice} →
{t : String.Slice.Pattern.ToForwardSearcher.DefaultForwardSearcher pat s} →
{ρ' : Type} →
{pat' : ρ'} →
{s' : String.Slice} →
{t' : String.Slice.Pattern.ToForwardSearcher.DefaultForwardSe... | null | false |
_private.Lean.Server.FileWorker.RequestHandling.0.Lean.Server.FileWorker.NamespaceEntry.finish.match_1 | Lean.Server.FileWorker.RequestHandling | (motive : Option Lean.Syntax → Sort u_1) →
(endStx : Option Lean.Syntax) → ((endStx : Lean.Syntax) → motive (some endStx)) → (Unit → motive none) → motive endStx | null | false |
ContMDiffAt.eq_1 | Mathlib.Geometry.Manifold.ContMDiff.Defs | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_5} [inst_6 : NormedAddComm... | null | true |
MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff | Mathlib.MeasureTheory.Measure.Real | ∀ {α : Type u_1} {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},
s₁ ⊆ s₂ →
s₂ ⊆ s₃ →
μ.real (s₃ \ s₁) = 0 →
autoParam (μ (s₃ \ s₁) ≠ ⊤) MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_sdiff._auto_1 →
μ.real s₁ = μ.real s₂ | **Alias** of `MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_sdiff`. | true |
_private.Mathlib.Algebra.Group.Monoid.0.npowRec.match_1.eq_2 | Mathlib.Algebra.Group.Monoid | ∀ {M : Type u_2} (motive : ℕ → M → Sort u_1) (n : ℕ) (a : M) (h_1 : (x : M) → motive 0 x)
(h_2 : (n : ℕ) → (a : M) → motive n.succ a),
(match n.succ, a with
| 0, x => h_1 x
| n.succ, a => h_2 n a) =
h_2 n a | null | true |
CategoryTheory.Functor.Accessible.Limits.isColimitMapCocone.injective | Mathlib.CategoryTheory.Presentable.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {K : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} K]
{F : CategoryTheory.Functor K (CategoryTheory.Functor C (Type w'))} (c : CategoryTheory.Limits.Cone F)
(hc : (Y : C) → CategoryTheory.Limits.IsLimit (((CategoryTheory.evaluation C (Type w')).obj Y).m... | null | true |
ArchimedeanClass.FiniteResidueField.ofArchimedean._proof_3 | Mathlib.Algebra.Order.Ring.StandardPart | ∀ {K : Type u_1} [inst : LinearOrder K] [inst_1 : Field K] [inst_2 : IsOrderedRing K] {R : Type u_2}
[inst_3 : LinearOrder R] [inst_4 : CommRing R] [inst_5 : IsStrictOrderedRing R] [inst_6 : Archimedean R]
(f : R →+*o K) (x y : R),
ArchimedeanClass.FiniteResidueField.mk (ArchimedeanClass.FiniteElement.mk (f (x * ... | null | false |
CategoryTheory.SymmetricCategory.toBraidedCategory | Mathlib.CategoryTheory.Monoidal.Braided.Basic | {C : Type u} →
{inst : CategoryTheory.Category.{v, u} C} →
{inst_1 : CategoryTheory.MonoidalCategory C} →
[self : CategoryTheory.SymmetricCategory C] → CategoryTheory.BraidedCategory C | null | true |
WittVector.fromPadicInt | Mathlib.RingTheory.WittVector.Compare | (p : ℕ) → [hp : Fact (Nat.Prime p)] → ℤ_[p] →+* WittVector p (ZMod p) | `fromPadicInt` uses `WittVector.lift` to lift `TruncatedWittVector.zmodEquivTrunc`
composed with `PadicInt.toZModPow` to a ring hom `ℤ_[p] →+* 𝕎 (ZMod p)`.
| true |
_private.Lean.Compiler.LCNF.InferType.0.Lean.Compiler.LCNF.Arg.inferType.match_1 | Lean.Compiler.LCNF.InferType | (motive : (pu : Lean.Compiler.LCNF.Purity) → Lean.Compiler.LCNF.Arg pu → Sort u_1) →
(pu : Lean.Compiler.LCNF.Purity) →
(arg : Lean.Compiler.LCNF.Arg pu) →
((arg : Lean.Compiler.LCNF.Arg Lean.Compiler.LCNF.Purity.pure) → motive Lean.Compiler.LCNF.Purity.pure arg) →
((arg : Lean.Compiler.LCNF.Arg Lea... | null | false |
_private.Mathlib.Basic.Rel.0.SetRel.exists_eq_singleton_of_prod_subset_id._simp_1_3 | Mathlib.Basic.Rel | ∀ {a b c : Prop}, ((a ∧ b) ∧ c) = (a ∧ b ∧ c) | null | false |
TypeVec.toSubtype'._sunfold | Mathlib.Data.TypeVec | {n : ℕ} →
{α : TypeVec.{u} n} →
(p : (α.prod α).Arrow (TypeVec.repeat n Prop)) →
TypeVec.Arrow (fun i => { x // TypeVec.ofRepeat (p i (TypeVec.prod.mk i x.1 x.2)) }) (TypeVec.Subtype_ p) | null | false |
_private.Mathlib.Algebra.Lie.Weights.IsSimple.0.LieAlgebra.IsKilling.chi_not_in_q_aux | Mathlib.Algebra.Lie.Weights.IsSimple | ∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
[inst_4 : FiniteDimensional K L] [inst_5 : LieAlgebra.IsKilling K L] {H : LieSubalgebra K L}
[inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L] (q : Submodule K (Module... | null | true |
Sum.LiftRel.isRight_right | Mathlib.Data.Sum.Basic | ∀ {α : Type u} {β : Type v} {γ : Type u_1} {δ : Type u_2} {r : α → γ → Prop} {s : β → δ → Prop} {y : γ ⊕ δ} {b : β},
Sum.LiftRel r s (Sum.inr b) y → y.isRight = true | null | true |
LinearMap.comap_prod_prod | Mathlib.LinearAlgebra.Prod | ∀ {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module R M₂]
[inst_6 : Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) (p : Submodule R M₂) (q : Submodule R M₃),
Submodule.... | null | true |
Metric.thickening_thickening_subset | Mathlib.Topology.MetricSpace.Thickening | ∀ {α : Type u} [inst : PseudoEMetricSpace α] (ε δ : ℝ) (s : Set α),
Metric.thickening ε (Metric.thickening δ s) ⊆ Metric.thickening (ε + δ) s | For the equality, see `thickening_thickening`. | true |
ENNReal.inv_rpow | Mathlib.Analysis.SpecialFunctions.Pow.NNReal | ∀ (x : ENNReal) (y : ℝ), x⁻¹ ^ y = (x ^ y)⁻¹ | null | true |
MvPowerSeries.instNontrivial | Mathlib.RingTheory.MvPowerSeries.Basic | ∀ {σ : Type u_1} {R : Type u_2} [Nontrivial R], Nontrivial (MvPowerSeries σ R) | null | true |
CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.instCategory | Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic | {A : Type u₁} →
{B : Type u₂} →
{C : Type u₃} →
[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} →
... | null | true |
_private.Lean.Meta.Sym.Canon.0.Lean.Meta.Sym.Canon.instReprShouldCanonResult | Lean.Meta.Sym.Canon | Repr Lean.Meta.Sym.Canon.ShouldCanonResult✝ | null | true |
LieSubalgebra.coe_incl' | Mathlib.Algebra.Lie.Subalgebra | ∀ {R : Type u} {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (L' : LieSubalgebra R L),
⇑L'.incl' = Subtype.val | null | true |
Lean.Grind.OrderConfig.funCC._inherited_default | Init.Grind.Config | Bool | null | false |
Option.instTransOrd | Init.Data.Order.Ord | ∀ {α : Type u_1} [inst : Ord α] [Std.TransOrd α], Std.TransOrd (Option α) | null | true |
HomologicalComplex.HomologySequence.snakeInput._proof_4 | Mathlib.Algebra.Homology.HomologySequence | ∀ {C : Type u_2} {ι : Type u_3} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C]
{c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (i j : ι),
CategoryTheory.CategoryStruct.comp (S.mapNatTrans (HomologicalComplex.natTransOpCyclesToCycles C c i j))
... | null | false |
Function.locallyFinsuppWithin.restrictOrderMonoidHom | Mathlib.Topology.LocallyFinsupp | {X : Type u_1} →
[inst : TopologicalSpace X] →
{U : Set X} →
{Y : Type u_2} →
[inst_1 : AddCommGroup Y] →
[inst_2 : LinearOrder Y] →
{V : Set X} → V ⊆ U → Function.locallyFinsuppWithin U Y →+o Function.locallyFinsuppWithin V Y | Restriction as an ordered group morphism | true |
Topology.IsEmbedding.toPartialHomeomorph_source | Mathlib.Topology.PartialHomeomorph.Basic | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y)
(h : Topology.IsEmbedding f) [inst_2 : Nonempty X], (Topology.IsEmbedding.toPartialHomeomorph f h).source = Set.univ | null | true |
Ring.instIsScalarTowerFractionRingSubtypeAlgebraicClosureMemIntermediateFieldNormalClosure | Mathlib.RingTheory.NormalClosure | ∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : IsDomain S]
[inst_4 : Algebra R S] [inst_5 : Module.IsTorsionFree R S],
IsScalarTower S (FractionRing S)
↥(IntermediateField.normalClosure (FractionRing R) (FractionRing S) (AlgebraicClosure (FractionRing S... | null | true |
Fin.instSub | Init.Data.Fin.Basic | {n : ℕ} → Sub (Fin n) | null | true |
HolderOnWith.ediam_image_le_of_subset_of_le | Mathlib.Topology.MetricSpace.Holder | ∀ {X : Type u_1} {Y : Type u_2} [inst : PseudoEMetricSpace X] [inst_1 : PseudoEMetricSpace Y] {C r : NNReal} {f : X → Y}
{s t : Set X},
HolderOnWith C r f s → t ⊆ s → ∀ {d : ENNReal}, Metric.ediam t ≤ d → Metric.ediam (f '' t) ≤ ↑C * d ^ ↑r | null | true |
CategoryTheory.Pseudofunctor.DescentData'.descentDataEquivalence._proof_11 | Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | ∀ {C : Type u_5} [inst : CategoryTheory.Category.{u_3, u_5} C]
(F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat) {ι : Type u_1} {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) → CategoryT... | null | false |
LipschitzWith.iterate._f | Mathlib.Topology.EMetricSpace.Lipschitz | ∀ {α : Type u} [inst : PseudoEMetricSpace α] {K : NNReal} {f : α → α},
LipschitzWith K f → ∀ (x : ℕ) (f_1 : Nat.below x), LipschitzWith (K ^ x) f^[x] | null | false |
Matrix.separatingRight_toLinearMap₂'_iff_separatingRight_toLinearMap₂ | Mathlib.LinearAlgebra.Matrix.SesquilinearForm | ∀ {R : Type u_1} {M₁ : Type u_6} {M₂ : Type u_7} {n : Type u_11} {m : Type u_12} [inst : CommRing R]
[inst_1 : DecidableEq m] [inst_2 : Fintype m] [inst_3 : DecidableEq n] [inst_4 : Fintype n] {M : Matrix m n R}
[inst_5 : AddCommMonoid M₁] [inst_6 : Module R M₁] [inst_7 : AddCommMonoid M₂] [inst_8 : Module R M₂]
... | null | true |
RingCon.instCommMagmaQuotient | Mathlib.RingTheory.Congruence.Defs | {R : Type u_1} → [inst : Add R] → [inst_1 : CommMagma R] → (c : RingCon R) → CommMagma c.Quotient | null | true |
_private.Mathlib.Analysis.Complex.CanonicalDecomposition.0.Complex.meromorphicNFOn_canonicalFactor._proof_1_1 | Mathlib.Analysis.Complex.CanonicalDecomposition | -1 < 0 | null | false |
ZFSet.image.match_1 | Mathlib.SetTheory.ZFC.Basic | ∀ (f : ZFSet.{u_1} → ZFSet.{u_1}) [inst : ZFSet.Definable₁ f] (x x_1 : PSet.{u_1})
(motive : (∃ z ∈ x, x_1.Equiv (ZFSet.Definable₁.out f z)) → Prop)
(x_2 : ∃ z ∈ x, x_1.Equiv (ZFSet.Definable₁.out f z)),
(∀ (w : PSet.{u_1}) (h1 : w ∈ x) (h2 : x_1.Equiv (ZFSet.Definable₁.out f w)), motive ⋯) → motive x_2 | null | false |
_private.Mathlib.RingTheory.WittVector.InitTail.0.WittVector.init_sub._proof_1_6 | Mathlib.RingTheory.WittVector.InitTail | ∀ (n i : ℕ), i < n → ∀ k < i + 1, k < n | null | false |
_aux_Mathlib_Combinatorics_SimpleGraph_Basic___macroRules_aesop_graph_1 | Mathlib.Combinatorics.SimpleGraph.Basic | Lean.Macro | A variant of the `aesop` tactic for use in the graph library. Changes relative
to standard `aesop`:
- We use the `SimpleGraph` rule set in addition to the default rule sets.
- We instruct Aesop's `intro` rule to unfold with `default` transparency.
- We instruct Aesop to fail if it can't fully solve the goal. This allo... | false |
PadicInt.unitCoeff_spec | Mathlib.NumberTheory.Padics.PadicIntegers | ∀ {p : ℕ} [hp : Fact (Nat.Prime p)] {x : ℤ_[p]} (hx : x ≠ 0), x = ↑(PadicInt.unitCoeff hx) * ↑p ^ x.valuation | null | true |
IsManifold.subset_maximalAtlas | Mathlib.Geometry.Manifold.IsManifold.Basic | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞}
{M : Type u_4} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [IsManifold I n M],
... | null | true |
prod_le_hasProd_of_nonneg | Mathlib.Topology.Algebra.InfiniteSum.Order | ∀ {ι : Type u_1} {α : Type u_3} {L : SummationFilter ι} [inst : CommMonoidWithZero α] [inst_1 : TopologicalSpace α]
[inst_2 : Preorder α] [ZeroLEOneClass α] [PosMulMono α] [ClosedIciTopology α] [L.NeBot] [L.LeAtTop] {f : ι → α}
{a : α} {s : Finset ι}, (∀ i ∈ s, 0 ≤ f i) → (∀ i ∉ s, 1 ≤ f i) → HasProd f a L → ∏ i ∈ ... | null | true |
ULift.instLinearOrder | Mathlib.Order.Lattice | {α : Type u} → [LinearOrder α] → LinearOrder (ULift.{v, u} α) | null | true |
CategoryTheory.Grp.forget₂Mon_map_hom | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{A B : CategoryTheory.Grp C} (f : A ⟶ B), ((CategoryTheory.Grp.forget₂Mon C).map f).hom = f.hom.hom | null | true |
pos_of_right_mul_lt_le | Mathlib.Algebra.Order.Ring.Unbundled.Basic | ∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] {a b c : R} [ExistsAddOfLE R] [PosMulMono R]
[AddRightMono R] [AddRightReflectLE R], a * b < a * c → b ≤ c → 0 < a | null | true |
_private.Mathlib.NumberTheory.BernoulliPolynomials.0.Polynomial.sum_bernoulli._simp_1_4 | Mathlib.NumberTheory.BernoulliPolynomials | ∀ {M : Type u_4} [inst : AddMonoid M] [IsLeftCancelAdd M] {a b : M}, (a + b = a) = (b = 0) | null | false |
String.Slice.isSome_skipPrefix? | Init.Data.String.Lemmas.Pattern.TakeDrop.Basic | ∀ {ρ : Type} {pat : ρ} [inst : String.Slice.Pattern.ForwardPattern pat] [String.Slice.Pattern.LawfulForwardPattern pat]
{s : String.Slice}, (s.skipPrefix? pat).isSome = s.startsWith pat | null | true |
List.getElem_of_append | Init.Data.List.Lemmas | ∀ {α : Type u_1} {l₁ : List α} {a : α} {l₂ : List α} {i : ℕ} {l : List α} (eq : l = l₁ ++ a :: l₂) (h : l₁.length = i),
l[i] = a | null | true |
ZMod.χ₈'._proof_1 | Mathlib.NumberTheory.LegendreSymbol.ZModChar | (match 1 with
| 0 => 0
| 2 => 0
| 4 => 0
| 6 => 0
| 1 => 1
| 3 => 1
| 5 => -1
| 7 => -1) =
match 1 with
| 0 => 0
| 2 => 0
| 4 => 0
| 6 => 0
| 1 => 1
| 3 => 1
| 5 => -1
| 7 => -1 | null | false |
_private.Mathlib.Algebra.Homology.CochainComplexOpposite.0.CochainComplex.homotopyUnop._proof_6 | Mathlib.Algebra.Homology.CochainComplexOpposite | ∀ (n : ℤ), n - 1 + 1 = n | null | false |
AlgebraicGeometry.Scheme.affineOpenCoverOfSpanRangeEqTop | Mathlib.AlgebraicGeometry.Cover.Open | {R : CommRingCat} →
{ι : Type u_1} → (s : ι → ↑R) → Ideal.span (Set.range s) = ⊤ → (AlgebraicGeometry.Spec R).AffineOpenCover | A family of elements spanning the unit ideal of `R` gives an affine open cover of `Spec R`. | true |
Batteries.PairingHeapImp.Heap.toArray | Batteries.Data.PairingHeap | {α : Type u_1} → (α → α → Bool) → Batteries.PairingHeapImp.Heap α → Array α | `O(n log n)`. Convert the heap to an array in increasing order. | true |
CategoryTheory.nerve.edgeMk_id | Mathlib.AlgebraicTopology.SimplicialSet.Nerve | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (x : C),
CategoryTheory.nerve.edgeMk (CategoryTheory.CategoryStruct.id x) = SSet.Edge.id (CategoryTheory.nerveEquiv.symm x) | null | true |
Lean.Lsp.WorkspaceEditClientCapabilities.noConfusion | Lean.Data.Lsp.Capabilities | {P : Sort u} →
{t t' : Lean.Lsp.WorkspaceEditClientCapabilities} →
t = t' → Lean.Lsp.WorkspaceEditClientCapabilities.noConfusionType P t t' | null | false |
Mathlib.Tactic.BehaviorIfUnchanged.error | Mathlib.Util.AtLocation | Mathlib.Tactic.BehaviorIfUnchanged | Throw an error if this action has no effect. | true |
_private.Mathlib.Algebra.Homology.ShortComplex.Homology.0.CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_cyclesMap_of_epi.match_1_2 | Mathlib.Algebra.Homology.ShortComplex.Homology | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology]
(h₁ : CategoryTheory.IsIso (CategoryTheory.ShortComplex.cyclesMap φ))
(motive :
{ l //
... | null | false |
_private.Mathlib.Data.List.TakeDrop.0.List.tail_take_eq_take_tail._proof_1_20 | Mathlib.Data.List.TakeDrop | ∀ {α : Type u_1} (l : List α) (n i : ℕ) (a : α),
((List.take n l).tail[i]? = some a) = ¬(List.take (n - 1) l.tail)[i]? = some a →
¬-1 * ↑l.length + 1 ≤ 0 → -1 * ↑(List.take n l).length + 1 ≤ 0 → ¬-1 * ↑n + 1 ≤ 0 → i + 1 < l.length | null | false |
CategoryTheory.MonoidalCategory.prodMonoidal._proof_21 | Mathlib.CategoryTheory.Monoidal.Category | ∀ (C₁ : Type u_1) [inst : CategoryTheory.Category.{u_3, u_1} C₁] [inst_1 : CategoryTheory.MonoidalCategory C₁]
(C₂ : Type u_2) [inst_2 : CategoryTheory.Category.{u_4, u_2} C₂] [inst_3 : CategoryTheory.MonoidalCategory C₂]
(X Y : C₁ × C₂),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.MonoidalCategoryS... | null | false |
_private.Batteries.Data.Array.Lemmas.0.Array.extract_append_of_size_left_le_start._proof_1_2 | Batteries.Data.Array.Lemmas | ∀ {α : Type u_1} {i j : ℕ} {a b : Array α} (i_1 : ℕ),
i_1 + 1 ≤ (b.extract (i - a.size) (j - a.size)).size → i_1 < (b.extract (i - a.size) (j - a.size)).size | null | false |
CompletePartialOrder.noConfusionType | Mathlib.Order.CompletePartialOrder | Sort u → {α : Type u_4} → CompletePartialOrder α → {α' : Type u_4} → CompletePartialOrder α' → Sort u | null | false |
Option.pfilter_eq_some_iff | Init.Data.Option.Lemmas | ∀ {α : Type u_1} {o : Option α} {p : (a : α) → o = some a → Bool} {a : α},
o.pfilter p = some a ↔ ∃ (ha : o = some a), p a ha = true | null | true |
Affine.Simplex.reflectionCircumcenterWeightsWithCircumcenter.eq_1 | Mathlib.Geometry.Euclidean.Circumcenter | ∀ {n : ℕ} (i₁ i₂ a : Fin (n + 1)),
Affine.Simplex.reflectionCircumcenterWeightsWithCircumcenter i₁ i₂
(Affine.Simplex.PointsWithCircumcenterIndex.pointIndex a) =
if a = i₁ ∨ a = i₂ then 1 else 0 | null | true |
_private.Init.Data.Range.Basic.0.Std.Legacy.Range.forIn'.loop._unary._proof_7 | Init.Data.Range.Basic | ∀ (range : Std.Legacy.Range) (i : ℕ), range.start ≤ i → 0 < range.step → range.start ≤ i + range.step | null | false |
Module.FaithfullyFlat.trans | Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_2) [inst_1 : CommRing S] [inst_2 : Algebra R S] (M : Type u_3)
[inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : Module S M] [IsScalarTower R S M]
[Module.FaithfullyFlat R S] [Module.FaithfullyFlat S M], Module.FaithfullyFlat R M | If `S` is a faithfully flat `R`-algebra, then any faithfully flat `S`-Module is faithfully flat
as an `R`-module. | true |
_private.Mathlib.ModelTheory.Semantics.0.FirstOrder.Language.BoundedFormula.restrictFreeVar.match_1.eq_5 | Mathlib.ModelTheory.Semantics | ∀ {L : FirstOrder.Language} {α : Type u_3} {β : Type u_5} [inst : DecidableEq α]
(motive : (x : ℕ) → (x_1 : L.BoundedFormula α x) → (↥x_1.freeVarFinset → β) → Sort u_4) (_n : ℕ)
(φ : L.BoundedFormula α (_n + 1)) (f : ↥φ.all.freeVarFinset → β)
(h_1 :
(_n : ℕ) →
(_f : ↥FirstOrder.Language.BoundedFormula.f... | null | true |
ArchimedeanClass.mem_closedBallAddSubgroup_iff | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {a : M}
{c : ArchimedeanClass M}, a ∈ c.closedBallAddSubgroup ↔ c ≤ ArchimedeanClass.mk a | null | true |
ZeroHom.instAddCommGroup | Mathlib.Algebra.Group.Hom.Instances | {M : Type uM} → {N : Type uN} → [inst : Zero M] → [inst_1 : AddCommGroup N] → AddCommGroup (ZeroHom M N) | If `G` is an additive commutative group, then so is `ZeroHom M G`. | true |
ExteriorAlgebra.ι_eq_algebraMap_iff._simp_1 | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | ∀ {R : Type u1} [inst : CommRing R] {M : Type u2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (x : M) (r : R),
((ExteriorAlgebra.ι R) x = (algebraMap R (ExteriorAlgebra R M)) r) = (x = 0 ∧ r = 0) | null | false |
_private.Mathlib.Tactic.ErwQuestion.0.Mathlib.Tactic.Erw?._aux_Mathlib_Tactic_ErwQuestion___elabRules_Mathlib_Tactic_Erw?_erw?_1.match_3 | Mathlib.Tactic.ErwQuestion | (motive : Lean.Expr × Lean.Expr → Sort u_1) →
(x : Lean.Expr × Lean.Expr) → ((tgt inferred : Lean.Expr) → motive (tgt, inferred)) → motive x | null | false |
Set.instLawfulMonad | Mathlib.Data.Set.Functor | LawfulMonad Set | null | true |
Std.TreeSet.instInsert | Std.Data.TreeSet.Basic | {α : Type u} → {cmp : α → α → Ordering} → Insert α (Std.TreeSet α cmp) | null | true |
NonnegHomClass.casesOn | Mathlib.Algebra.Order.Hom.Basic | {F : Type u_4} →
{α : Type u_5} →
{β : Type u_6} →
[inst : Zero β] →
[inst_1 : LE β] →
[inst_2 : FunLike F α β] →
{motive : NonnegHomClass F α β → Sort u} →
(t : NonnegHomClass F α β) → ((apply_nonneg : ∀ (f : F) (a : α), 0 ≤ f a) → motive ⋯) → motive t | null | false |
_private.Mathlib.MeasureTheory.VectorMeasure.IntegrationByParts.0.BoundedVariationOn.setIntegral_Icc_rightLim_sub_leftLim_eq._proof_1_6 | Mathlib.MeasureTheory.VectorMeasure.IntegrationByParts | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α},
∀ y ∈ Set.Icc a b, ∀ x ∈ Set.Icc a b, (if a ≤ y ∧ y ≤ x then 1 else 0) = if y ≤ x ∧ x ≤ b then 1 else 0 | null | false |
Subgroup.coe_toSubmonoid | Mathlib.Algebra.Group.Subgroup.Defs | ∀ {G : Type u_1} [inst : Group G] (K : Subgroup G), ↑K.toSubmonoid = ↑K | null | true |
Nat.primeFactorsList_sublist_of_dvd | Mathlib.Data.Nat.Factors | ∀ {n k : ℕ}, n ∣ k → k ≠ 0 → n.primeFactorsList.Sublist k.primeFactorsList | null | true |
Lean.Elab.Do.MonadInfo.noConfusion | Lean.Elab.Do.Basic | {P : Sort u} → {t t' : Lean.Elab.Do.MonadInfo} → t = t' → Lean.Elab.Do.MonadInfo.noConfusionType P t t' | null | false |
_private.Batteries.Data.List.Lemmas.0.List.findIdxs_take._proof_1_3 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {p : α → Bool} (head : α) (tail : List α) {s : ℕ},
List.findIdxs p (List.take 0 (head :: tail)) s =
List.take (List.countP p (List.take 0 (head :: tail))) (List.findIdxs p (head :: tail) s) | null | false |
Mathlib.Meta.FunProp.Mor.getAppArgs | Mathlib.Tactic.FunProp.Mor | Lean.Expr → Lean.MetaM (Array Mathlib.Meta.FunProp.Mor.Arg) | Given `f a₁ a₂ ... aₙ`, returns `#[a₁, ..., aₙ]` where `f` can be bundled morphism. | true |
LinearOrderedAddCommGroup.isAddCyclic_iff_nonempty_equiv_int | Mathlib.GroupTheory.SpecificGroups.Cyclic | ∀ {A : Type u_4} [inst : AddCommGroup A] [inst_1 : LinearOrder A] [IsOrderedAddMonoid A] [Nontrivial A],
IsAddCyclic A ↔ Nonempty (A ≃+o ℤ) | A linearly-ordered additive abelian group is cyclic iff it is isomorphic to `ℤ` as an ordered
additive monoid. | true |
Set.InjOn.ne_iff | Mathlib.Data.Set.Function | ∀ {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {x y : α}, Set.InjOn f s → x ∈ s → y ∈ s → (f x ≠ f y ↔ x ≠ y) | null | true |
_private.Mathlib.RingTheory.Support.0.Module.support_of_isTorsionFree._simp_1_2 | Mathlib.RingTheory.Support | ∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{p : PrimeSpectrum R}, (p ∈ Module.support R M) = ∃ m, ∀ r ∉ p.asIdeal, r • m ≠ 0 | null | false |
Polynomial.instEuclideanDomain | Mathlib.Algebra.Polynomial.FieldDivision | {R : Type u} → [inst : Field R] → EuclideanDomain (Polynomial R) | null | true |
Lean.Meta.RecursorUnivLevelPos.majorType.injEq | Lean.Meta.RecursorInfo | ∀ (idx idx_1 : ℕ),
(Lean.Meta.RecursorUnivLevelPos.majorType idx = Lean.Meta.RecursorUnivLevelPos.majorType idx_1) = (idx = idx_1) | null | true |
StarAlgHom.copy._proof_3 | Mathlib.Algebra.Star.StarAlgHom | ∀ {R : Type u_3} {A : Type u_2} {B : Type u_1} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
[inst_3 : Star A] [inst_4 : Semiring B] [inst_5 : Algebra R B] [inst_6 : Star B] (f : A →⋆ₐ[R] B) (f' : A → B),
f' = ⇑f → f' 0 = 0 | null | false |
Lean.Elab.Term.Do.Code.reassign.inj | Lean.Elab.Do.Legacy | ∀ {xs : Array Lean.Elab.Term.Do.Var} {doElem : Lean.Syntax} {k : Lean.Elab.Term.Do.Code}
{xs_1 : Array Lean.Elab.Term.Do.Var} {doElem_1 : Lean.Syntax} {k_1 : Lean.Elab.Term.Do.Code},
Lean.Elab.Term.Do.Code.reassign xs doElem k = Lean.Elab.Term.Do.Code.reassign xs_1 doElem_1 k_1 →
xs = xs_1 ∧ doElem = doElem_1 ∧... | null | true |
Std.DTreeMap.Internal.Impl.ExplorationStep.lt.injEq | Std.Data.DTreeMap.Internal.Model | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] {k : α → Ordering} (a : α) (a_1 : k a = Ordering.lt) (a_2 : β a)
(a_3 : List ((a : α) × β a)) (a_4 : α) (a_5 : k a_4 = Ordering.lt) (a_6 : β a_4) (a_7 : List ((a : α) × β a)),
(Std.DTreeMap.Internal.Impl.ExplorationStep.lt a a_1 a_2 a_3 =
Std.DTreeMap.Internal.Im... | null | true |
IsOrderedAddMonoid.toIsOrderedCancelAddMonoid | Mathlib.Algebra.Order.Group.Defs | ∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : Preorder α] [IsOrderedAddMonoid α], IsOrderedCancelAddMonoid α | null | true |
Std.Format.nest.elim | Init.Data.Format.Basic | {motive : Std.Format → Sort u} →
(t : Std.Format) → t.ctorIdx = 4 → ((indent : ℤ) → (f : Std.Format) → motive (Std.Format.nest indent f)) → motive t | null | false |
Denumerable.ofEncodableOfInfinite._proof_1 | Mathlib.Basic.Denumerable | ∀ (α : Type u_1) [inst : Encodable α] [Infinite α], Infinite ↑(Set.range Encodable.encode) | null | false |
Std.DTreeMap.Raw.partition.eq_1 | Std.Data.DTreeMap.Raw.WF | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} (f : (a : α) → β a → Bool) (t : Std.DTreeMap.Raw α β cmp),
Std.DTreeMap.Raw.partition f t =
Std.DTreeMap.Raw.foldl
(fun x a b =>
match x with
| (l, r) => if f a b = true then (l.insert a b, r) else (l, r.insert a b))
(∅, ∅) t | null | true |
InnerProductSpace.gramSchmidt_ne_zero_coe | Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{ι : Type u_3} [inst_3 : LinearOrder ι] [inst_4 : LocallyFiniteOrderBot ι] {f : ι → E} (n : ι),
LinearIndependent 𝕜 (f ∘ Subtype.val) → InnerProductSpace.gramSchmidt 𝕜 f n ≠ 0 | null | true |
Affine.Simplex.circumcenter_eq_affineCombination_of_pointsWithCircumcenter | Mathlib.Geometry.Euclidean.Circumcenter | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n),
s.circumcenter =
(Finset.affineCombination ℝ Finset.univ s.pointsWithCircumcenter)
(Affine.Simplex.circumcenterWeightsW... | The circumcenter of a simplex, in terms of `pointsWithCircumcenter`. | true |
cfcₙ_neg | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | ∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommRing R] [inst_1 : Nontrivial R] [inst_2 : StarRing R]
[inst_3 : MetricSpace R] [inst_4 : IsTopologicalRing R] [inst_5 : ContinuousStar R] [inst_6 : TopologicalSpace A]
[inst_7 : NonUnitalRing A] [inst_8 : StarRing A] [inst_9 : Module R A] [inst_10 : IsScala... | null | true |
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