name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Lean.Compiler.ExternAttr.0.Lean.expandExternPatternAux._unary.eq_def | Lean.Compiler.ExternAttr | ∀ (args : List String) (pattern : String) (_x : (_ : pattern.Pos) ×' String),
Lean.expandExternPatternAux._unary args pattern _x =
PSigma.casesOn _x fun it r =>
if h : it.IsAtEnd then r
else
have c := it.get h;
if c ≠ '#' then Lean.expandExternPatternAux._unary args pattern ⟨it.next h,... | null | false |
Graph.IsEdgeCut.inter_edgeSet_of_le | Mathlib.Combinatorics.Graph.Connected.EdgeCut | ∀ {α : Type u_1} {β : Type u_2} {G H : Graph α β} {F : Set β}, H ≤ G → G.IsEdgeCut F → H.IsEdgeCut (H.edgeSet ∩ F) | null | true |
Algebra.Extension.algebraBaseChange._proof_3 | Mathlib.RingTheory.Extension.Basic | ∀ {R : Type u_1} [inst : CommRing R] (T : Type u_2) [inst_1 : CommRing T] [inst_2 : Algebra R T], SMulCommClass R R T | null | false |
MeasureTheory.ae_mem_iff_measure_eq | Mathlib.MeasureTheory.Measure.Typeclasses.Finite | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {s : Set α},
MeasureTheory.NullMeasurableSet s μ → ((∀ᵐ (a : α) ∂μ, a ∈ s) ↔ μ s = μ Set.univ) | null | true |
Filter.HasBasis.inf_neBot_iff | Mathlib.Order.Filter.Bases.Basic | ∀ {α : Type u_1} {ι : Sort u_3} {l l' : Filter α} {p : ι → Prop} {s : ι → Set α},
l.HasBasis p s → ((l ⊓ l').NeBot ↔ ∀ ⦃i : ι⦄, p i → ∀ ⦃s' : Set α⦄, s' ∈ l' → (s i ∩ s').Nonempty) | null | true |
BialgEquiv.ofAlgEquiv._proof_7 | Mathlib.RingTheory.Bialgebra.Equiv | ∀ {R : Type u_3} {A : Type u_1} {B : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
[inst_3 : Bialgebra R A] [inst_4 : Bialgebra R B] (f : A ≃ₐ[R] B), Function.LeftInverse f.invFun f.toFun | null | false |
_private.Mathlib.NumberTheory.LSeries.Nonvanishing.0.DirichletCharacter.zetaMul_prime_pow_nonneg._simp_1_2 | Mathlib.NumberTheory.LSeries.Nonvanishing | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {a : M₀} {n : ℕ} [IsReduced M₀] [Nontrivial M₀],
(a ^ n = 0) = (a = 0 ∧ n ≠ 0) | null | false |
Filter.mem_sup | Mathlib.Order.Filter.Basic | ∀ {α : Type u} {f g : Filter α} {s : Set α}, s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g | null | true |
CategoryTheory.Functor.PreservesRightHomologyOf.mk._flat_ctor | Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | ∀ {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] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] {F : CategoryTheory.Functor C D}
[inst_4 : F.PreservesZeroMorphisms] {S : CategoryTh... | null | false |
_private.Aesop.Forward.State.0.Aesop.VariableMap.modifyM.match_3 | Aesop.Forward.State | (motive : Option Aesop.InstMap → Sort u_1) →
(x : Option Aesop.InstMap) → (Unit → motive none) → ((m : Aesop.InstMap) → motive (some m)) → motive x | null | false |
Matroid.mapSetEmbedding_indep_iff' | Mathlib.Combinatorics.Matroid.Map | ∀ {α : Type u_1} {β : Type u_2} {M : Matroid α} {f : ↑M.E ↪ β} {I : Set β},
(M.mapSetEmbedding f).Indep I ↔ ∃ I₀, M.Indep (Subtype.val '' I₀) ∧ I = ⇑f '' I₀ | null | true |
CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso._proof_1 | Mathlib.Algebra.Homology.ShortComplex.RightHomology | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C]
[inst_4 : CategoryTheory.Limits.HasKernels Cᵒᵖ] [inst_5 : CategoryTheory.Limits.HasCokernels Cᵒᵖ]
{X Y : ... | null | false |
AffineMap.instFunLike | Mathlib.LinearAlgebra.AffineSpace.AffineMap | (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] →
... | null | true |
Topology.IsEmbedding.comapUniformSpace | Mathlib.Topology.UniformSpace.UniformEmbedding | {α : Type u_1} →
{β : Type u_2} →
[inst : TopologicalSpace α] → [u : UniformSpace β] → (f : α → β) → Topology.IsEmbedding f → UniformSpace α | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. | true |
_private.Mathlib.NumberTheory.DirichletCharacter.Orthogonality.0.DirichletCharacter.sum_char_inv_mul_char_eq._simp_1_1 | Mathlib.NumberTheory.DirichletCharacter.Orthogonality | ∀ {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) | null | false |
Nat.add_mod_add_ite | Mathlib.Data.Nat.ModEq | ∀ (a b c : ℕ), ((a + b) % c + if c ≤ a % c + b % c then c else 0) = a % c + b % c | null | true |
continuousAt_nsmul | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : AddMonoid M] [ContinuousAdd M] (x : M) (n : ℕ),
ContinuousAt (fun x => n • x) x | null | true |
List.idxOf_cons_ne | Mathlib.Data.List.Basic | ∀ {α : Type u} [inst : BEq α] [LawfulBEq α] {a b : α} (l : List α),
b ≠ a → List.idxOf a (b :: l) = (List.idxOf a l).succ | null | true |
_private.Mathlib.RingTheory.Nullstellensatz.0.MvPolynomial.eq_vanishingIdeal_singleton_of_isMaximal._simp_1_1 | Mathlib.RingTheory.Nullstellensatz | ∀ {α : Type u} [inst : Semiring α] {I J : Ideal α}, (I = J) = ∀ (x : α), x ∈ I ↔ x ∈ J | null | false |
Algebra.idealMap._proof_1 | Mathlib.RingTheory.Ideal.Maps | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : Semiring S] [inst_2 : Algebra R S] (I : Ideal R),
∀ x ∈ I, (algebraMap R S) x ∈ Ideal.map (algebraMap R S) I | null | false |
WeierstrassCurve.variableChange_a₂ | Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange | ∀ {R : Type u} [inst : CommRing R] (W : WeierstrassCurve R) (C : WeierstrassCurve.VariableChange R),
(C • W).a₂ = ↑C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2) | null | true |
_private.Mathlib.NumberTheory.EulerProduct.DirichletLSeries.0.DirichletCharacter.eulerProduct_log_eq_LSeries._simp_1_4 | Mathlib.NumberTheory.EulerProduct.DirichletLSeries | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False | null | false |
_private.Mathlib.RingTheory.WittVector.TeichmullerSeries.0.WittVector._aux_Mathlib_RingTheory_WittVector_TeichmullerSeries___unexpand_WittVector_1 | Mathlib.RingTheory.WittVector.TeichmullerSeries | Lean.PrettyPrinter.Unexpander | null | false |
Finset.toRight_union | Mathlib.Data.Finset.Sum | ∀ {α : Type u_1} {β : Type u_2} {u v : Finset (α ⊕ β)} [inst : DecidableEq α] [inst_1 : DecidableEq β],
(u ∪ v).toRight = u.toRight ∪ v.toRight | null | true |
CategoryTheory.Span.id.congr_simp | Mathlib.CategoryTheory.Bicategory.Span.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {Wₗ Wᵣ : CategoryTheory.MorphismProperty C}
[inst_1 : Wₗ.ContainsIdentities] [inst_2 : Wᵣ.ContainsIdentities] (c : C),
CategoryTheory.Span.id c = CategoryTheory.Span.id c | null | true |
Lean.JsonRpc.MessageMetaData.response.elim | Lean.Data.JsonRpc | {motive : Lean.JsonRpc.MessageMetaData → Sort u} →
(t : Lean.JsonRpc.MessageMetaData) →
t.ctorIdx = 2 → ((id : Lean.JsonRpc.RequestID) → motive (Lean.JsonRpc.MessageMetaData.response id)) → motive t | null | false |
CategoryTheory.Limits.imageSubobject_arrow_comp | Mathlib.CategoryTheory.Subobject.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : CategoryTheory.Limits.HasImage f],
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.factorThruImageSubobject f)
(CategoryTheory.Limits.imageSubobject f).arrow =
f | null | true |
HasSum.mul_of_nonarchimedean | Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | ∀ {α : Type u_1} {β : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : UniformSpace R] [IsUniformAddGroup R]
[NonarchimedeanRing R] {f : α → R} {g : β → R} {a b : R},
HasSum f a → HasSum g b → HasSum (fun i => f i.1 * g i.2) (a * b) | Let `R` be a nonarchimedean ring, let `f : α → R` be a function that sums to `a : R`,
and let `g : β → R` be a function that sums to `b : R`. Then `fun i : α × β ↦ f i.1 * g i.2`
sums to `a * b`. | true |
RootPairing.EmbeddedG2.longRoot | Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {ι : Type u_1} →
{R : Type u_2} →
{M : Type u_3} →
{N : Type u_4} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
[inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (P : RootPairing ι R M N) → [P.EmbeddedG2] → M | The long root `β`. | true |
Ordinal.exists_lsub_cof | Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | ∀ (o : Ordinal.{u}), ∃ ι f, Ordinal.lsub f = o ∧ Cardinal.mk ι = o.cof | null | true |
_private.Lean.Elab.Command.0.Lean.Elab.Command.runLinters.match_3 | Lean.Elab.Command | (motive : Option (IO.Promise Lean.Elab.InfoTree) → Sort u_1) →
(promise : Option (IO.Promise Lean.Elab.InfoTree)) →
((promise : IO.Promise Lean.Elab.InfoTree) → motive (some promise)) →
((x : Option (IO.Promise Lean.Elab.InfoTree)) → motive x) → motive promise | null | false |
Asymptotics.isEquivalent_of_tendsto_one | Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | ∀ {α : Type u_1} {β : Type u_2} [inst : NormedField β] {u v : α → β} {l : Filter α},
Filter.Tendsto (u / v) l (nhds 1) → Asymptotics.IsEquivalent l u v | null | true |
Int.ediv_of_neg_of_pos | Mathlib.Data.Int.Init | ∀ {a b : ℤ}, a < 0 → 0 < b → a.ediv b = -((-a - 1) / b + 1) | null | true |
TopologicalSpace.Clopens.coe_inf | Mathlib.Topology.Sets.Closeds | ∀ {α : Type u_2} [inst : TopologicalSpace α] (s t : TopologicalSpace.Clopens α), ↑(s ⊓ t) = ↑s ∩ ↑t | null | true |
Aesop.Queue.mk._flat_ctor | Aesop.Search.Queue.Class | {Q : Type} → BaseIO Q → (Q → Array Aesop.GoalRef → BaseIO Q) → (Q → BaseIO (Option Aesop.GoalRef × Q)) → Aesop.Queue Q | null | false |
Complex.norm_natCast_cpow_of_pos | Mathlib.Analysis.SpecialFunctions.Pow.Real | ∀ {n : ℕ}, 0 < n → ∀ (s : ℂ), ‖↑n ^ s‖ = ↑n ^ s.re | null | true |
_private.Mathlib.Algebra.Homology.DerivedCategory.TStructure.0.DerivedCategory.isGE_Q_obj_iff._simp_1_2 | Mathlib.Algebra.Homology.DerivedCategory.TStructure | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(K : CochainComplex C ℤ) (n : ℤ), K.IsGE n = ∀ i < n, HomologicalComplex.ExactAt K i | null | false |
PresheafOfModules.homMk._proof_1 | Mathlib.Algebra.Category.ModuleCat.Presheaf | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat}
{M₁ M₂ : PresheafOfModules R} (φ : M₁.presheaf ⟶ M₂.presheaf) (X : Cᵒᵖ) (x y : ↑(M₁.1 X)),
(CategoryTheory.ConcreteCategory.hom (φ.app X)) (x + y) =
(CategoryTheory.ConcreteCategory.hom (φ.app X)) x + (Catego... | null | false |
WType.brecOn | Mathlib.Data.W.Basic | {α : Type u_1} →
{β : α → Type u_2} →
{motive : WType β → Sort u} → (t : WType β) → ((t : WType β) → WType.below t → motive t) → motive t | null | false |
SimpleGraph.center_top | Mathlib.Combinatorics.SimpleGraph.Diam | ∀ {α : Type u_1}, ⊤.center = Set.univ | null | true |
Char.any | Batteries.Data.Char.Basic | (Char → Bool) → Bool | Returns `true` if `p` returns true for some `Char`. | true |
CategoryTheory.RetractArrow.map_i_left | Mathlib.CategoryTheory.Retract | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
{X Y Z W : C} {f : X ⟶ Y} {g : Z ⟶ W} (h : CategoryTheory.RetractArrow f g) (F : CategoryTheory.Functor C D),
(h.map F).i.left = F.map (CategoryTheory.Arrow.Hom.left h.i) | null | true |
IsScalarTower.of_compHom | Mathlib.Algebra.Algebra.Tower | ∀ (R : Type u) (A : Type w) (M : Type v₁) [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
[inst_3 : MulAction A M], IsScalarTower R A M | null | true |
ContDiffMapSupportedIn.integralAgainstBilinLM_eq_integral | Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace ℝ E] {n : ℕ∞} {K : TopologicalSpace.Compacts E} {m : MeasurableSpace E}
[inst_3 : OpensMeasurableSpace E] {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [inst_4 : NormedAddCommGroup F₁]
[ins... | null | true |
OrthonormalBasis.repr_injective | Mathlib.Analysis.InnerProductSpace.PiL2 | ∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι], Function.Injective OrthonormalBasis.repr | null | true |
_private.Mathlib.Data.Int.Interval.0.Finset.Ioc_succ_succ._simp_1_2 | Mathlib.Data.Int.Interval | ∀ {α : Type u_1} [inst : DecidableEq α] {s : Finset α} {a b : α}, (a ∈ insert b s) = (a = b ∨ a ∈ s) | null | false |
ShiftRight.recOn | Init.Prelude | {α : Type u} →
{motive : ShiftRight α → Sort u_1} →
(t : ShiftRight α) → ((shiftRight : α → α → α) → motive { shiftRight := shiftRight }) → motive t | null | false |
Finset.filter_subset._simp_1 | Mathlib.Data.Finset.Filter | ∀ {α : Type u_1} (p : α → Prop) [inst : DecidablePred p] (s : Finset α), (Finset.filter p s ⊆ s) = True | null | false |
Matrix.transpose_hadamard | Mathlib.LinearAlgebra.Matrix.Hadamard | ∀ {α : Type u_1} {m : Type u_2} {n : Type u_3} [inst : Mul α] (A B : Matrix m n α),
(A.hadamard B).transpose = A.transpose.hadamard B.transpose | null | true |
CategoryTheory.ModObj.rec | Mathlib.CategoryTheory.Monoidal.Mod | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
{D : Type u₂} →
[inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
[inst_3 : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] →
{M : C} →
[inst_4 : Cat... | null | false |
RingHom.closure_preimage_le | Mathlib.Algebra.Ring.Subring.Basic | ∀ {R : Type u} {S : Type v} [inst : NonAssocRing R] [inst_1 : NonAssocRing S] (f : R →+* S) (s : Set S),
Subring.closure (⇑f ⁻¹' s) ≤ Subring.comap f (Subring.closure s) | null | true |
Finset.add_subset_add_left | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : Add α] {s t₁ t₂ : Finset α}, t₁ ⊆ t₂ → s + t₁ ⊆ s + t₂ | null | true |
SheafOfModules.hom_ext_iff | Mathlib.Algebra.Category.ModuleCat.Sheaf | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C}
{R : CategoryTheory.Sheaf J RingCat} {X Y : SheafOfModules R} {f g : X ⟶ Y}, f = g ↔ f.val = g.val | null | true |
Turing.TM1.stmts | Mathlib.Computability.TuringMachine.PostTuringMachine | {Γ : Type u_1} →
{Λ : Type u_2} → {σ : Type u_3} → (Λ → Turing.TM1.Stmt Γ Λ σ) → Finset Λ → Finset (Option (Turing.TM1.Stmt Γ Λ σ)) | The set of all statements in a Turing machine, plus one extra value `none` representing the
halt state. This is used in the TM1 to TM0 reduction. | true |
DiscreteUniformity.mk._flat_ctor | Mathlib.Topology.UniformSpace.DiscreteUniformity | ∀ {X : Type u_1} [u : UniformSpace X], u = ⊥ → DiscreteUniformity X | null | false |
FinPartOrd.dualEquiv_unitIso | Mathlib.Order.Category.FinPartOrd | FinPartOrd.dualEquiv.unitIso =
CategoryTheory.NatIso.ofComponents (fun X => FinPartOrd.Iso.mk (OrderIso.dualDual ↑X.toPartOrd))
@FinPartOrd.dualEquiv._proof_1 | null | true |
algebraMap_smul | Mathlib.Algebra.Algebra.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (A : Type u_2) [inst_1 : Semiring A] [inst_2 : Algebra R A] {M : Type u_3}
[inst_3 : AddCommMonoid M] [inst_4 : Module A M] [inst_5 : Module R M] [IsScalarTower R A M] (r : R) (m : M),
(algebraMap R A) r • m = r • m | null | true |
DifferentiableOn.inverse | Mathlib.Analysis.Calculus.FDeriv.Mul | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {R : Type u_4} [inst_3 : NormedRing R] [HasSummableGeomSeries R]
[inst_5 : NormedAlgebra 𝕜 R] {h : E → R} {S : Set E},
DifferentiableOn 𝕜 h S → (∀ x ∈ S, IsUnit (h x)) → Differentiabl... | null | true |
SymAlg.instNonAssocRingOfInvertibleOfNat._proof_14 | Mathlib.Algebra.Symmetrized | ∀ {α : Type u_1} [inst : Ring α] [inst_1 : Invertible 2] (a : αˢʸᵐ), a * 1 = a | null | false |
pi_generateFrom_eq_finite | Mathlib.Topology.Constructions | ∀ {ι : Type u_2} {X : ι → Type u_6} {g : (a : ι) → Set (Set (X a))} [Finite ι],
(∀ (a : ι), ⋃₀ g a = Set.univ) →
Pi.topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = Set.univ.pi s} | null | true |
_private.Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper.0.AlgebraicGeometry.Proj.valuativeCriterion_existence_aux._simp_1_11 | Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | ∀ (A : Type u) [inst : CommRing A] (K : Type v) [inst_1 : Field K] [inst_2 : Algebra A K] [inst_3 : IsDomain A]
[inst_4 : ValuationRing A] [inst_5 : IsFractionRing A K] (x : K),
(x ∈ (ValuationRing.valuation A K).integer) = ∃ a, (algebraMap A K) a = x | null | false |
TwoSidedIdeal.asIdealOpposite | Mathlib.RingTheory.TwoSidedIdeal.Operations | {R : Type u_1} → [inst : Ring R] → TwoSidedIdeal R →o Ideal Rᵐᵒᵖ | Every two-sided ideal is also a right ideal. | true |
Mathlib.Meta.NormNum.evalIsSquareRat | Mathlib.Tactic.NormNum.IsSquare | Mathlib.Meta.NormNum.NormNumExt | `norm_num` extension for `IsSquare` on `ℚ`. | true |
BddOrd.mk.injEq | Mathlib.Order.Category.BddOrd | ∀ (toPartOrd : PartOrd) [isBoundedOrder : BoundedOrder ↑toPartOrd] (toPartOrd_1 : PartOrd)
(isBoundedOrder_1 : BoundedOrder ↑toPartOrd_1),
({ toPartOrd := toPartOrd, isBoundedOrder := isBoundedOrder } =
{ toPartOrd := toPartOrd_1, isBoundedOrder := isBoundedOrder_1 }) =
(toPartOrd = toPartOrd_1 ∧ isBounde... | null | true |
Std.MaxEqOr | Init.Data.Order.Classes | (α : Type u) → [Max α] → Prop | This typeclass states that `Max.max a b` returns one of its arguments, either `a` or `b`.
| true |
BitVec.ofInt_iSizeToInt | Init.Data.SInt.Lemmas | ∀ (x : ISize), BitVec.ofInt System.Platform.numBits x.toInt = x.toBitVec | null | true |
_private.Mathlib.LinearAlgebra.LinearIndependent.Lemmas.0.exists_linearIndepOn_extension.match_1_1 | Mathlib.LinearAlgebra.LinearIndependent.Lemmas | ∀ {ι : Type u_2} {V : Type u_1} {v : ι → V} {t : Set ι} (x : V) (motive : x ∈ v '' t → Prop) (x_1 : x ∈ v '' t),
(∀ (x_2 : ι) (hx : x_2 ∈ t) (hvx : v x_2 = x), motive ⋯) → motive x_1 | null | false |
Btw.rec | Mathlib.Order.Circular | {α : Type u_1} → {motive : Btw α → Sort u} → ((btw : α → α → α → Prop) → motive { btw := btw }) → (t : Btw α) → motive t | null | false |
QuadraticAlgebra.changeGeneratorEquiv.congr_simp | Mathlib.Algebra.QuadraticAlgebra.Discriminant | ∀ {R : Type u_2} [inst : CommRing R] (a b : R) (u u_1 : Rˣ) (e_u : u = u_1) (k k_1 : R) (e_k : k = k_1) {a' b' : R}
(ha : a' = ↑u ^ 2 * a - ↑u * b * k - k ^ 2) (hb : b' = ↑u * b + 2 * k),
QuadraticAlgebra.changeGeneratorEquiv a b u k ha hb = QuadraticAlgebra.changeGeneratorEquiv a b u_1 k_1 ⋯ ⋯ | null | true |
AddAut.vadd_def | Mathlib.Algebra.Group.Action.End | ∀ {M : Type u_2} [inst : AddMonoid M] (f : AddAut M) (a : M), f +ᵥ a = f a | null | true |
Metric.nonneg_of_mem_closedBall | Mathlib.Topology.MetricSpace.Pseudo.Defs | ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ Metric.closedBall x ε → 0 ≤ ε | null | true |
CategoryTheory.Precoverage.over | Mathlib.CategoryTheory.Sites.Over | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(X : C) → CategoryTheory.Precoverage C → CategoryTheory.Precoverage (CategoryTheory.Over X) | The precoverage on `Over X` for any `X : C` that is induced by a precoverage on `C`. | true |
MeasurableSpace.DynkinSystem.instPartialOrder._proof_3 | Mathlib.MeasureTheory.PiSystem | ∀ {α : Type u_1} (x x_1 x_2 : MeasurableSpace.DynkinSystem α), x ≤ x_1 → x_1 ≤ x_2 → x ≤ x_2 | null | false |
Std.DTreeMap.Internal.Impl.getKey._sunfold | Std.Data.DTreeMap.Internal.Queries | {α : Type u} →
{β : α → Type v} →
[inst : Ord α] → (t : Std.DTreeMap.Internal.Impl α β) → (k : α) → Std.DTreeMap.Internal.Impl.contains k t = true → α | null | false |
LinearMap.addMonoid._proof_3 | Mathlib.Algebra.Module.LinearMap.Defs | ∀ {R₁ : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [inst : Semiring R₁] [inst_1 : Semiring R₂]
[inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R₁ M] [inst_5 : Module R₂ M₂]
{σ₁₂ : R₁ →+* R₂} (a : M →ₛₗ[σ₁₂] M₂), a + 0 = a | null | false |
BddLat.Iso.mk._proof_4 | Mathlib.Order.Category.BddLat | ∀ {α β : BddLat} (e : ↑α.toLat ≃o ↑β.toLat) (a b : ↑α.1), e (a ⊓ b) = e a ⊓ e b | null | false |
OneHom.comp_apply | Mathlib.Algebra.Group.Hom.Defs | ∀ {M : Type u_4} {N : Type u_5} {P : Type u_6} [inst : One M] [inst_1 : One N] [inst_2 : One P] (g : OneHom N P)
(f : OneHom M N) (x : M), (g.comp f) x = g (f x) | null | true |
Ideal.powQuotPowSuccLinearEquivMapMkPowSuccPow._proof_3 | Mathlib.RingTheory.Ideal.Quotient.Operations | ∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R) (n : ℕ), IsScalarTower R (R ⧸ I ^ (n + 1)) (R ⧸ I ^ (n + 1)) | null | false |
SimpleGraph.Copy.ext | Mathlib.Combinatorics.SimpleGraph.Copy | ∀ {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {H : SimpleGraph W} {f g : H.Copy G}, (∀ (a : W), f a = g a) → f = g | null | true |
Lean.Meta.Grind.AC.DiseqCnstrProof.erase_dup | Lean.Meta.Tactic.Grind.AC.Types | Lean.Meta.Grind.AC.DiseqCnstr → Lean.Meta.Grind.AC.DiseqCnstrProof | null | true |
BitVec.getElem?_zero_ofNat_zero | Init.Data.BitVec.Lemmas | ∀ {w : ℕ}, (0#(w + 1))[0]? = some false | null | true |
_private.Init.Data.Range.Polymorphic.RangeIterator.0.Std.Rxi.Iterator.instIteratorLoop.loop.wf._unary._proof_2 | Init.Data.Range.Polymorphic.RangeIterator | ∀ {α : Type u_1} [inst : Std.PRange.UpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerable α] (LargeEnough : α → Prop),
(∀ (a b : α), Std.PRange.UpwardEnumerable.LE a b → LargeEnough a → LargeEnough b) →
∀ (next : α), LargeEnough next → ∀ (next' : α), Std.PRange.succ? next = some next' → LargeEnough next' | null | false |
Filter.pureAddHom._proof_1 | Mathlib.Order.Filter.Pointwise | ∀ {α : Type u_1} [inst : Add α] (x x_1 : α), pure (x + x_1) = pure x + pure x_1 | null | false |
FreeMonoid.lift.eq_1 | Mathlib.Algebra.FreeMonoid.Basic | ∀ {α : Type u_1} {M : Type u_4} [inst : Monoid M],
FreeMonoid.lift =
{
toFun := fun f =>
{ toFun := fun l => FreeMonoid.prodAux (List.map f (FreeMonoid.toList l)), map_one' := ⋯, map_mul' := ⋯ },
invFun := fun f x => f (FreeMonoid.of x), left_inv := ⋯, right_inv := ⋯ } | null | true |
_private.Mathlib.Analysis.Normed.Module.Seminorm.Norm.0.closedBall_normSeminorm._simp_1_1 | Mathlib.Analysis.Normed.Module.Seminorm.Norm | ∀ {𝕜 : Type u_3} {E : Type u_7} [inst : SeminormedRing 𝕜] [inst_1 : AddCommGroup E] [inst_2 : SMul 𝕜 E]
(p : Seminorm 𝕜 E) {x y : E} {r : ℝ}, (y ∈ p.closedBall x r) = (p (y - x) ≤ r) | null | false |
Sum.swap_swap_eq | Init.Data.Sum.Lemmas | ∀ {α : Type u_1} {β : Type u_2}, Sum.swap ∘ Sum.swap = id | null | true |
_private.Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital.0._auto_382 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | Lean.Syntax | null | false |
Finset.singleton_sdiv | Mathlib.Algebra.Group.Pointwise.Finset.Scalar | ∀ {α : Type u_1} {β : Type u_2} [inst : SDiv α β] [inst_1 : DecidableEq α] {t : Finset β} (a : β),
{a} /ₛ t = Finset.image (fun x => a /ₛ x) t | null | true |
UniformSpace.hausdorff | Mathlib.Topology.UniformSpace.Closeds | (α : Type u_1) → [UniformSpace α] → UniformSpace (Set α) | The Hausdorff uniformity on the powerset of a uniform space. Used for defining the uniformities
on `Closeds`, `Compacts` and `NonemptyCompacts`.
See note [reducible non-instances]. | true |
CovariantDerivative.difference | Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.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} →
{M : Type u_4} →
... | The difference of two covariant derivatives, as a one-form taking values in the
endomorphisms of `V`. | true |
CategoryTheory.MorphismProperty.HasPushoutsAgainst.casesOn | Mathlib.CategoryTheory.MorphismProperty.Limits | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{P P' : CategoryTheory.MorphismProperty C} →
{motive : P.HasPushoutsAgainst P' → Sort u_1} →
(t : P.HasPushoutsAgainst P') →
((hasPushoutsAlong : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P' f → P.HasPushoutsAlong f) → motive ⋯) → motive t | null | false |
Sum.getRight_eq_getRight? | Mathlib.Data.Sum.Basic | ∀ {α : Type u} {β : Type v} {x : α ⊕ β} (h₁ : x.isRight = true) (h₂ : x.getRight?.isSome = true),
x.getRight h₁ = x.getRight?.get h₂ | null | true |
Lean.Meta.Sym.getInt16Value? | Lean.Meta.Sym.LitValues | Lean.Expr → OptionT Id Int16 | null | true |
VectorPrebundle.totalSpaceTopology | Mathlib.Topology.VectorBundle.Basic | {R : Type u_1} →
{B : Type u_2} →
{F : Type u_3} →
{E : B → Type u_4} →
[inst : NontriviallyNormedField R] →
[inst_1 : (x : B) → AddCommMonoid (E x)] →
[inst_2 : (x : B) → Module R (E x)] →
[inst_3 : NormedAddCommGroup F] →
[inst_4 : NormedSpace R ... | Topology on the total space that will make the prebundle into a bundle. | true |
Submodule.instDiv._proof_1 | Mathlib.Algebra.Algebra.Operations | ∀ {R : Type u_2} [inst : CommSemiring R] {A : Type u_1} [inst_1 : CommSemiring A] [inst_2 : Algebra R A]
(I J : Submodule R A) {a b : A},
a ∈ {x | ∀ y ∈ J, x * y ∈ I} → b ∈ {x | ∀ y ∈ J, x * y ∈ I} → ∀ y ∈ J, (a + b) * y ∈ I | null | false |
PowerSeries.HasSubst | Mathlib.RingTheory.PowerSeries.Substitution | {τ : Type u_3} → {S : Type u_4} → [CommRing S] → MvPowerSeries τ S → Prop | (Possibly multivariate) power series which can be substituted in a `PowerSeries`. | true |
SeminormedCommGroup | Mathlib.Analysis.Normed.Group.Defs | Type u_4 → Type u_4 | A seminormed group is a group endowed with a norm for which `dist x y = ‖x⁻¹ * y‖`
defines a pseudometric space structure. | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.0.UInt8.reduceBin.match_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.2231631932._hygCtx._hyg.9 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | (motive : Option UInt8 → Sort u_1) →
(__x : Option UInt8) → ((m : UInt8) → motive (some m)) → ((x : Option UInt8) → motive x) → motive __x | null | false |
_private.Lean.Elab.Tactic.Grind.Config.0.Lean.Elab.Tactic.instEvalExprConfig | Lean.Elab.Tactic.Grind.Config | Lean.Elab.ConfigEval.EvalExpr Lean.Grind.Config | null | true |
Topology.IsQuotientMap.trivializationOfVAddDisjoint._proof_8 | Mathlib.Topology.Covering.Quotient | ∀ {E : Type u_2} {X : Type u_1} {f : E → X} {G : Type u_3} [inst : AddGroup G] [inst_1 : AddAction G E] (U : Set E),
(∀ (g : G) (e : E), f (g +ᵥ e) = f e) → ∀ (g : G) ⦃x : X⦄, x ∈ f '' U → x ∈ f '' (fun x => g +ᵥ x) ⁻¹' U | null | false |
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