name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.67M | allowCompletion bool 2
classes |
|---|---|---|---|
commutatorElement_def | Mathlib.Algebra.Group.Commutator | ∀ {G : Type u_1} [inst : Group G] (g₁ g₂ : G), ⁅g₁, g₂⁆ = g₁ * g₂ * g₁⁻¹ * g₂⁻¹ | true |
Polynomial.leadingCoeff_X_pow_sub_one | Mathlib.Algebra.Polynomial.Degree.Operations | ∀ {R : Type u} [inst : Ring R] {n : ℕ}, 0 < n → (Polynomial.X ^ n - 1).leadingCoeff = 1 | true |
IsGreatest.nnnorm_cfcₙ_nnreal._auto_1 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | Lean.Syntax | false |
Monotone.of_left_le_map_sup | Mathlib.Order.Lattice | ∀ {α : Type u} {β : Type v} [inst : SemilatticeSup α] [inst_1 : Preorder β] {f : α → β},
(∀ (x y : α), f x ≤ f (x ⊔ y)) → Monotone f | true |
Aesop.NormRuleResult.proved.injEq | Aesop.Search.Expansion.Norm | ∀ (steps? steps?_1 : Option (Array Aesop.Script.LazyStep)),
(Aesop.NormRuleResult.proved steps? = Aesop.NormRuleResult.proved steps?_1) = (steps? = steps?_1) | true |
_private.Lean.Elab.DocString.0.Lean.Doc.commandExpandersForUnsafe | Lean.Elab.DocString | Lean.Ident →
Lean.Elab.TermElabM
(Array
(Lean.Name ×
StateT (Array (Lean.TSyntax `doc_arg)) Lean.Doc.DocM (Lean.Doc.Block Lean.ElabInline Lean.ElabBlock))) | true |
Matroid.IsBasis.isBasis_isRestriction | Mathlib.Combinatorics.Matroid.Minor.Restrict | ∀ {α : Type u_1} {M : Matroid α} {I X : Set α} {N : Matroid α},
M.IsBasis I X → N.IsRestriction M → X ⊆ N.E → N.IsBasis I X | true |
Lean.Meta.SolveByElim.SolveByElimConfig.testPartialSolutions | Lean.Meta.Tactic.SolveByElim | optParam Lean.Meta.SolveByElim.SolveByElimConfig { } →
(List Lean.Expr → Lean.MetaM Bool) → Lean.Meta.SolveByElim.SolveByElimConfig | true |
_private.Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddSound.0.Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.unsat_of_encounteredBoth._proof_1_6 | Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddSound | ∀ {n : ℕ} (assignment : Array Std.Tactic.BVDecide.LRAT.Internal.Assignment)
(l : Std.Sat.Literal (Std.Tactic.BVDecide.LRAT.Internal.PosFin n)),
(if (!l.2) = true then Std.Tactic.BVDecide.LRAT.Internal.ReduceResult.reducedToUnit l
else Std.Tactic.BVDecide.LRAT.Internal.ReduceResult.reducedToEmpty) =
Std.... | false |
CategoryTheory.Functor.sheafPushforwardContinuousComp' | Mathlib.CategoryTheory.Sites.Continuous | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
{E : Type u₃} →
[inst_2 : CategoryTheory.Category.{v₃, u₃} E] →
{F : CategoryTheory.Functor C D} →
{G : CategoryTheory.Functor D E} →
... | true |
Array.merge.go._unsafe_rec | Batteries.Data.Array.Merge | {α : Type u_1} → (α → α → Bool) → Array α → Array α → Array α → ℕ → ℕ → Array α | false |
CategoryTheory.LaxMonoidalFunctor.laxMonoidal._autoParam | Mathlib.CategoryTheory.Monoidal.Functor | Lean.Syntax | false |
Con.ker_rel._simp_2 | Mathlib.GroupTheory.Congruence.Hom | ∀ {M : Type u_1} {N : Type u_2} {F : Type u_4} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]
[inst_3 : MulHomClass F M N] (f : F) {x y : M}, (Con.ker f) x y = (f x = f y) | false |
BitVec.toInt_sub_of_not_ssubOverflow | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {x y : BitVec w}, ¬x.ssubOverflow y = true → (x - y).toInt = x.toInt - y.toInt | true |
_private.Mathlib.Analysis.Analytic.Inverse.0.FormalMultilinearSeries.rightInv_removeZero.match_1_1 | Mathlib.Analysis.Analytic.Inverse | ∀ {𝕜 : Type u_3} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_1} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
(p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
(motive : (n : ℕ) → (∀ m < n, p.removeZero.right... | false |
CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso | Mathlib.CategoryTheory.Limits.IsLimit | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
{C : Type u₃} →
[inst_1 : CategoryTheory.Category.{v₃, u₃} C] →
{F : CategoryTheory.Functor J C} →
{s t : CategoryTheory.Limits.Cone F} →
CategoryTheory.Limits.IsLimit s → CategoryTheory.Limits.IsLimit t → (s.pt ≅ t.... | true |
Lean.Meta.Grind.Arith.CommRing.MonadCanon.noConfusion | Lean.Meta.Tactic.Grind.Arith.CommRing.MonadCanon | {P : Sort u} →
{m : Type → Type} →
{t : Lean.Meta.Grind.Arith.CommRing.MonadCanon m} →
{m' : Type → Type} →
{t' : Lean.Meta.Grind.Arith.CommRing.MonadCanon m'} →
m = m' → t ≍ t' → Lean.Meta.Grind.Arith.CommRing.MonadCanon.noConfusionType P t t' | false |
Cardinal.ofENat_le_lift._simp_1 | Mathlib.SetTheory.Cardinal.ENat | ∀ {x : Cardinal.{v}} {m : ℕ∞}, (↑m ≤ Cardinal.lift.{u, v} x) = (↑m ≤ x) | false |
List.minOn_eq_min | Init.Data.List.MinMaxOn | ∀ {α : Type u_1} {β : Type u_2} [inst : Min α] [inst_1 : LE α] [DecidableLE α] [Std.LawfulOrderLeftLeaningMin α]
[inst_4 : LE β] [inst_5 : DecidableLE β] {f : α → β} {l : List α} {h : l ≠ []},
(∀ (a b : α), f a ≤ f b ↔ a ≤ b) → List.minOn f l h = l.min h | true |
nhds_subtype_eq_comap | Mathlib.Topology.Constructions | ∀ {X : Type u} [inst : TopologicalSpace X] {p : X → Prop} {x : X} {h : p x},
nhds ⟨x, h⟩ = Filter.comap Subtype.val (nhds x) | true |
lt_of_lt_add_of_nonpos_right | Mathlib.Algebra.Order.Monoid.Unbundled.Basic | ∀ {α : Type u_1} [inst : AddZeroClass α] [inst_1 : Preorder α] [AddRightMono α] {a b c : α}, a < b + c → b ≤ 0 → a < c | true |
Std.Internal.IO.Async.System.instInhabitedGroupId.default | Std.Internal.Async.System | Std.Internal.IO.Async.System.GroupId | true |
_private.Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup.0.Algebra.GrothendieckGroup.of_injective._simp_1_1 | Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup | ∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} (x : M), (Localization.monoidOf S) x = Localization.mk x 1 | false |
HomologicalComplex.restrictionMap_id | Mathlib.Algebra.Homology.Embedding.Restriction | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
[inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(K : HomologicalComplex C c') (e : c.Embedding c') [inst_2 : e.IsRelIff],
HomologicalComplex.restrictionMap (CategoryTheory.... | true |
Lean.Meta.AbstractNestedProofs.isNonTrivialProof | Lean.Meta.AbstractNestedProofs | Lean.Expr → Lean.MetaM Bool | true |
strictConcaveOn_of_slope_strict_anti_adjacent | Mathlib.Analysis.Convex.Slope | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜},
Convex 𝕜 s →
(∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)) →
StrictConcaveOn 𝕜 s f | true |
CategoryTheory.Pseudofunctor.DescentData.mk._flat_ctor | Mathlib.CategoryTheory.Sites.Descent.DescentData | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat} →
{ι : Type t} →
{S : C} →
{X : ι → C} →
{f : (i : ι) → X i ⟶ S} →
(obj : (i : ι) → ↑(F.obj { as := Opposite.op (X... | false |
ModularForm.sub_apply | Mathlib.NumberTheory.ModularForms.Basic | ∀ {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ} (f g : ModularForm Γ k) (z : UpperHalfPlane), (f - g) z = f z - g z | true |
_private.Mathlib.GroupTheory.Perm.Cycle.Type.0.Equiv.Perm.IsThreeCycle.nodup_iff_mem_support._proof_1_607 | Mathlib.GroupTheory.Perm.Cycle.Type | ∀ {α : Type u_1} [inst_1 : DecidableEq α] {g : Equiv.Perm α} {a : α} (w w_1 : α),
List.idxOfNth w_1 [] [].length + 1 ≤ (List.filter (fun x => decide (x = w)) [a, g a, g (g a)]).length →
List.idxOfNth w_1 [] [].length < (List.findIdxs (fun x => decide (x = w)) [a, g a, g (g a)]).length | false |
_private.Lean.Meta.Offset.0.Lean.Meta.evalNat._sparseCasesOn_2 | Lean.Meta.Offset | {motive : Lean.Literal → Sort u} →
(t : Lean.Literal) →
((val : ℕ) → motive (Lean.Literal.natVal val)) → (Nat.hasNotBit 1 t.ctorIdx → motive t) → motive t | false |
Lean.Meta.DiscrTree.Trie.ctorIdx | Lean.Meta.DiscrTree.Types | {α : Type} → Lean.Meta.DiscrTree.Trie α → ℕ | false |
_private.Mathlib.LinearAlgebra.QuadraticForm.Signature.0.QuadraticForm.posDef_spanSubset._simp_1_5 | Mathlib.LinearAlgebra.QuadraticForm.Signature | ∀ {α : Sort u_1} {p : α → Prop}, (¬∀ (x : α), p x) = ∃ x, ¬p x | false |
UniformSpace.Completion.instAddMonoid._proof_12 | Mathlib.Topology.Algebra.GroupCompletion | ∀ {α : Type u_1} [inst : UniformSpace α] [inst_1 : AddGroup α] [IsUniformAddGroup α] (n : ℕ)
(a : UniformSpace.Completion α), (n + 1) • a = n • a + a | false |
_private.Mathlib.Topology.Compactness.Lindelof.0.Tendsto.isLindelof_insert_range_of_coLindelof._simp_1_3 | Mathlib.Topology.Compactness.Lindelof | ∀ {α : Type u} {s t : Set α}, (s ∩ t).Nonempty = ¬Disjoint s t | false |
IsLocalRing.ResidueField.map_id_apply | Mathlib.RingTheory.LocalRing.ResidueField.Basic | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] (x : IsLocalRing.ResidueField R),
(IsLocalRing.ResidueField.map (RingHom.id R)) x = x | true |
DomMulAct.instLeftCancelMonoidOfMulOpposite | Mathlib.GroupTheory.GroupAction.DomAct.Basic | {M : Type u_1} → [LeftCancelMonoid Mᵐᵒᵖ] → LeftCancelMonoid Mᵈᵐᵃ | true |
Equiv.Perm.subtypeEquivSubtypePerm_apply_of_not_mem | Mathlib.Algebra.Group.End | ∀ {α : Type u_4} {p : α → Prop} [inst : DecidablePred p] {a : α} (f : Equiv.Perm (Subtype p)),
¬p a → ↑((Equiv.Perm.subtypeEquivSubtypePerm p) f) a = a | true |
Finsupp.sumFinsuppLEquivProdFinsupp_symm_inr | Mathlib.LinearAlgebra.Finsupp.SumProd | ∀ {M : Type u_2} (R : Type u_5) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_7}
{β : Type u_8} (fg : (α →₀ M) × (β →₀ M)) (y : β),
((Finsupp.sumFinsuppLEquivProdFinsupp R).symm fg) (Sum.inr y) = fg.2 y | true |
CategoryTheory.Equivalence.toAdjunction_unit | Mathlib.CategoryTheory.Adjunction.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(e : C ≌ D), e.toAdjunction.unit = e.unit | true |
HomologicalComplex.xNextIso.eq_1 | Mathlib.Algebra.Homology.HomologicalComplex | ∀ {ι : Type u_1} {V : Type u} [inst : CategoryTheory.Category.{v, u} V]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {c : ComplexShape ι} (C : HomologicalComplex V c) {i j : ι}
(r : c.Rel i j), C.xNextIso r = CategoryTheory.eqToIso ⋯ | true |
NonUnitalSeminormedRing.noConfusion | Mathlib.Analysis.Normed.Ring.Basic | {P : Sort u} →
{α : Type u_5} →
{t : NonUnitalSeminormedRing α} →
{α' : Type u_5} →
{t' : NonUnitalSeminormedRing α'} → α = α' → t ≍ t' → NonUnitalSeminormedRing.noConfusionType P t t' | false |
Finset.sdiff_union_erase_cancel | Mathlib.Data.Finset.Basic | ∀ {α : Type u_1} [inst : DecidableEq α] {s t : Finset α} {a : α}, t ⊆ s → a ∈ t → s \ t ∪ t.erase a = s.erase a | true |
GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range | Mathlib.Algebra.Category.Grp.EpiMono | ∀ {A B : GrpCat} (f : A ⟶ B),
∀ x ∈ (GrpCat.Hom.hom f).range,
∀ b ∉ (GrpCat.Hom.hom f).range,
((GrpCat.SurjectiveOfEpiAuxs.h f) x)
(GrpCat.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset ⟨b • ↑(GrpCat.Hom.hom f).range, ⋯⟩) =
GrpCat.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset ⟨(x * b) • ↑(GrpCa... | true |
WittVector.toZModPow_compat | Mathlib.RingTheory.WittVector.Compare | ∀ (p : ℕ) [hp : Fact (Nat.Prime p)] (m n : ℕ) (h : m ≤ n),
(ZMod.castHom ⋯ (ZMod (p ^ m))).comp (WittVector.toZModPow p n) = WittVector.toZModPow p m | true |
AlgebraicGeometry.morphismRestrictRestrictBasicOpen._simp_1 | Mathlib.AlgebraicGeometry.Restrict | ∀ (α : Type u) (p : α → Prop), (∀ (x : (CategoryTheory.forget (Type u)).obj α), p x) = ∀ (x : α), p x | false |
Set.Intersecting.exists_mem_set | Mathlib.Combinatorics.SetFamily.Intersecting | ∀ {α : Type u_1} {𝒜 : Set (Set α)}, 𝒜.Intersecting → ∀ {s t : Set α}, s ∈ 𝒜 → t ∈ 𝒜 → ∃ a ∈ s, a ∈ t | true |
_private.Mathlib.CategoryTheory.EssentiallySmall.0.CategoryTheory.essentiallySmall_iff_of_thin._simp_1_2 | Mathlib.CategoryTheory.EssentiallySmall | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [Quiver.IsThin C],
CategoryTheory.LocallySmall.{w, v, u} C = True | false |
CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor._proof_4 | Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {J : Type u_1} [inst_1 : LinearOrder J]
[inst_2 : OrderBot J] {F : CategoryTheory.Functor J C} (c : CategoryTheory.Limits.Cocone F) {X Y : C} (p : X ⟶ Y)
(f : F.obj ⊥ ⟶ X) (g : c.pt ⟶ Y) {X_1 Y_1 Z : Jᵒᵖ} (f_1 : X_1 ⟶ Y_1) (g_1 : Y_1 ⟶ Z),
(fun sq' =>... | false |
Setoid.mk_eq_bot | Mathlib.Data.Setoid.Basic | ∀ {α : Type u_1} {r : α → α → Prop} (iseqv : Equivalence r), { r := r, iseqv := iseqv } = ⊥ ↔ r = fun x1 x2 => x1 = x2 | true |
Real.sinhOrderIso_symm_apply | Mathlib.Analysis.SpecialFunctions.Arsinh | ⇑(RelIso.symm Real.sinhOrderIso) = Real.arsinh | true |
_private.Mathlib.Algebra.Homology.Monoidal.0.HomologicalComplex.instHasTensorTensorUnit_1._proof_1 | Mathlib.Algebra.Homology.Monoidal | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.Preadditive C] [inst_3 : CategoryTheory.Limits.HasZeroObject C]
[inst_4 : (CategoryTheory.MonoidalCategory.curriedTensor C).Additive]
[inst_5 : ∀ (X₁ : C), ((CategoryTheory.Monoidal... | false |
CategoryTheory.PreOneHypercover.Hom.mk | Mathlib.CategoryTheory.Sites.Hypercover.One | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{S : C} →
{E : CategoryTheory.PreOneHypercover S} →
{F : CategoryTheory.PreOneHypercover S} →
(toHom : E.Hom F.toPreZeroHypercover) →
(s₁ : {i j : E.I₀} → E.I₁ i j → F.I₁ (toHom.s₀ i) (toHom.s₀ j)) →
(h₁ :... | true |
_private.Init.Data.List.Sublist.0.List.infix_filterMap_iff._simp_1_3 | Init.Data.List.Sublist | ∀ {α : Type u_1} {β : Type u_2} {l : List α} {L₁ L₂ : List β} {f : α → Option β},
(List.filterMap f l = L₁ ++ L₂) = ∃ l₁ l₂, l = l₁ ++ l₂ ∧ List.filterMap f l₁ = L₁ ∧ List.filterMap f l₂ = L₂ | false |
CategoryTheory.Limits.Multicoequalizer.instHasCoequalizerFstSigmaMapSndSigmaMap | Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.Limits.MultispanShape}
(I : CategoryTheory.Limits.MultispanIndex J C) [CategoryTheory.Limits.HasMulticoequalizer I]
[inst_2 : CategoryTheory.Limits.HasCoproduct I.left] [inst_3 : CategoryTheory.Limits.HasCoproduct I.right],
CategoryTheor... | true |
_private.Mathlib.Algebra.Ring.Idempotent.0.IsIdempotentElem.sub_iff._simp_1_5 | Mathlib.Algebra.Ring.Idempotent | ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c | false |
FreeMonoid.prodAux.eq_1 | Mathlib.Algebra.FreeMonoid.Basic | ∀ {M : Type u_6} [inst : Monoid M], FreeMonoid.prodAux [] = 1 | true |
_private.Lean.Meta.Tactic.Simp.Types.0.Lean.Meta.Simp.Context.mk.sizeOf_spec | Lean.Meta.Tactic.Simp.Types | ∀ (config : Lean.Meta.Simp.Config) (zetaDeltaSet initUsedZetaDelta : Lean.FVarIdSet)
(metaConfig indexConfig : Lean.Meta.ConfigWithKey) (maxDischargeDepth : UInt32)
(simpTheorems : Lean.Meta.SimpTheoremsArray) (congrTheorems : Lean.Meta.SimpCongrTheorems)
(parent? : Option Lean.Expr) (dischargeDepth : UInt32) (lc... | true |
borel_eq_generateFrom_Ioi | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ (α : Type u_1) [inst : TopologicalSpace α] [SecondCountableTopology α] [inst_2 : LinearOrder α] [OrderTopology α],
borel α = MeasurableSpace.generateFrom (Set.range Set.Ioi) | true |
CommAlgCat.binaryCofanIsColimit._proof_1 | Mathlib.Algebra.Category.CommAlgCat.Monoidal | ∀ {R : Type u_1} [inst : CommRing R] (A B : CommAlgCat R) {T : CommAlgCat R} (f : A ⟶ T) (g : B ⟶ T),
CategoryTheory.CategoryStruct.comp (A.binaryCofan B).inl
((fun {T} f g => CommAlgCat.ofHom (Algebra.TensorProduct.lift (CommAlgCat.Hom.hom f) (CommAlgCat.Hom.hom g) ⋯)) f
g) =
f | false |
iteratedFDeriv_zero | Mathlib.Analysis.Calculus.ContDiff.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : ℕ},
iteratedFDeriv 𝕜 n 0 = 0 | true |
Matrix.cramer | Mathlib.LinearAlgebra.Matrix.Adjugate | {n : Type v} → {α : Type w} → [DecidableEq n] → [Fintype n] → [inst : CommRing α] → Matrix n n α → (n → α) →ₗ[α] n → α | true |
WeierstrassCurve.Jacobian.add_of_Z_eq_zero | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | ∀ {F : Type u} [inst : Field F] {W : WeierstrassCurve.Jacobian F} {P Q : Fin 3 → F},
W.Nonsingular P → W.Nonsingular Q → P 2 = 0 → Q 2 = 0 → W.add P Q = P 0 ^ 2 • ![1, 1, 0] | true |
_private.Mathlib.Order.SupIndep.0.Finset.SupIndep.biUnion._proof_1_1 | Mathlib.Order.SupIndep | ∀ {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [inst : Lattice α] [inst_1 : OrderBot α] [inst_2 : DecidableEq ι]
{s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} ⦃a : Finset ι⦄,
a ⊆ s.biUnion g →
∀ ⦃b : ι⦄, b ∈ s.biUnion g → b ∉ a → ∀ (i' : ι'), a.sup f ≤ ((s.erase i').biUnion g ∪ (g i').erase b).sup f | false |
_private.Lean.Data.Json.FromToJson.Basic.0.Float.toJson.match_1 | Lean.Data.Json.FromToJson.Basic | (motive : String ⊕ Lean.JsonNumber → Sort u_1) →
(x : String ⊕ Lean.JsonNumber) →
((e : String) → motive (Sum.inl e)) → ((n : Lean.JsonNumber) → motive (Sum.inr n)) → motive x | false |
IsAddUnit.eq_add_neg_iff_add_eq | Mathlib.Algebra.Group.Units.Basic | ∀ {α : Type u} [inst : SubtractionMonoid α] {a b c : α}, IsAddUnit c → (a = b + -c ↔ a + c = b) | true |
Batteries.RBNode.upperBound?_of_some | Batteries.Data.RBMap.Lemmas | ∀ {α : Type u_1} {cut : α → Ordering} {y : α} {t : Batteries.RBNode α},
∃ x, Batteries.RBNode.upperBound? cut t (some y) = some x | true |
CompleteSemilatticeSup.toPartialOrder | Mathlib.Order.CompleteLattice.Defs | {α : Type u_8} → [self : CompleteSemilatticeSup α] → PartialOrder α | true |
HasSum.congr_fun | Mathlib.Topology.Algebra.InfiniteSum.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f g : β → α} {a : α}
{L : SummationFilter β}, HasSum f a L → (∀ (x : β), g x = f x) → HasSum g a L | true |
Continuous.matrix_diagonal | Mathlib.Topology.Instances.Matrix | ∀ {X : Type u_1} {n : Type u_5} {R : Type u_8} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace R]
[inst_2 : Zero R] [inst_3 : DecidableEq n] {A : X → n → R}, Continuous A → Continuous fun x => Matrix.diagonal (A x) | true |
Option.some_eq_dite_none_left._simp_1 | Init.Data.Option.Lemmas | ∀ {β : Type u_1} {a : β} {p : Prop} {x : Decidable p} {b : ¬p → Option β},
(some a = if h : p then none else b h) = ∃ (h : ¬p), some a = b h | false |
Bundle.Pretrivialization.symm_trans_symm | Mathlib.Topology.FiberBundle.Trivialization | ∀ {B : Type u_1} {F : Type u_2} {Z : Type u_4} [inst : TopologicalSpace B] [inst_1 : TopologicalSpace F] {proj : Z → B}
(e e' : Bundle.Pretrivialization F proj), (e.symm.trans e'.toPartialEquiv).symm = e'.symm.trans e.toPartialEquiv | true |
Aesop.GoalUnsafe.brecOn_5 | Aesop.Tree.Data | {motive_1 : Aesop.GoalUnsafe → Sort u} →
{motive_2 : Aesop.MVarClusterUnsafe → Sort u} →
{motive_3 : Aesop.RappUnsafe → Sort u} →
{motive_4 : Aesop.GoalData Aesop.RappUnsafe Aesop.MVarClusterUnsafe → Sort u} →
{motive_5 : Aesop.MVarClusterData Aesop.GoalUnsafe Aesop.RappUnsafe → Sort u} →
... | false |
CategoryTheory.IsPullback.hasLiftingProperty | Mathlib.CategoryTheory.LiftingProperties.Limits | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {X Y Z W : C} {f : X ⟶ Y} {s : X ⟶ Z} {g : Z ⟶ W}
{t : Y ⟶ W},
CategoryTheory.IsPullback s f g t →
∀ {X' Y' : C} (f' : X' ⟶ Y') [CategoryTheory.HasLiftingProperty f' g], CategoryTheory.HasLiftingProperty f' f | true |
_private.Mathlib.RingTheory.RootsOfUnity.Complex.0.Complex.mem_rootsOfUnity._simp_1_3 | Mathlib.RingTheory.RootsOfUnity.Complex | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 4] [NeZero 4], (4 = 0) = False | false |
Fin.image_addNat_Ioc | Mathlib.Order.Interval.Set.Fin | ∀ {n : ℕ} (m : ℕ) (i j : Fin n), (fun x => x.addNat m) '' Set.Ioc i j = Set.Ioc (i.addNat m) (j.addNat m) | true |
List.findM?_pure | Init.Data.List.Control | ∀ {α : Type} {m : Type → Type u_1} [inst : Monad m] [LawfulMonad m] (p : α → Bool) (as : List α),
List.findM? (fun x => pure (p x)) as = pure (List.find? p as) | true |
CommRingCat.hom_inv_apply | Mathlib.Algebra.Category.Ring.Basic | ∀ {R S : CommRingCat} (e : R ≅ S) (s : ↑S),
(CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s | true |
_private.Mathlib.Combinatorics.Enumerative.IncidenceAlgebra.0.IncidenceAlgebra.muFun'.eq_def | Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | ∀ (𝕜 : Type u_2) {α : Type u_5} [inst : AddCommGroup 𝕜] [inst_1 : One 𝕜] [inst_2 : Preorder α]
[inst_3 : LocallyFiniteOrder α] [inst_4 : DecidableEq α] (b x : α),
IncidenceAlgebra.muFun'✝ 𝕜 b x =
let a := x;
if a = b then 1
else
-∑ x_1 ∈ (Finset.Ioc a b).attach,
have h := ⋯;
... | true |
CategoryTheory.Bicategory.leftUnitorNatIso | Mathlib.CategoryTheory.Bicategory.Basic | {B : Type u} →
[inst : CategoryTheory.Bicategory B] →
(a b : B) →
(CategoryTheory.Bicategory.precomposing a a b).obj (CategoryTheory.CategoryStruct.id a) ≅
CategoryTheory.Functor.id (a ⟶ b) | true |
_private.Mathlib.CategoryTheory.Sites.Coherent.RegularTopology.0.CategoryTheory.regularTopology.mem_sieves_iff_hasEffectiveEpi._simp_1_1 | Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : Y ⟶ X), ⊤.1 f = True | false |
UniformSpace.mem_ball_comp | Mathlib.Topology.UniformSpace.Defs | ∀ {β : Type ub} {V W : Set (β × β)} {x y z : β},
y ∈ UniformSpace.ball x V → z ∈ UniformSpace.ball y W → z ∈ UniformSpace.ball x (SetRel.comp V W) | true |
Qq.Impl.PatternVar.ctorIdx | Qq.Match | Qq.Impl.PatternVar → ℕ | false |
Std.ExtHashMap.getD_map | Std.Data.ExtHashMap.Lemmas | ∀ {α : Type u} {β : Type v} {γ : Type w} {x : BEq α} {x_1 : Hashable α} {m : Std.ExtHashMap α β} [inst : LawfulBEq α]
{f : α → β → γ} {k : α} {fallback : γ},
(Std.ExtHashMap.map f m).getD k fallback = (Option.map (f k) m[k]?).getD fallback | true |
Lean.Diff.instToStringAction.match_1 | Lean.Util.Diff | (motive : Lean.Diff.Action → Sort u_1) →
(x : Lean.Diff.Action) →
(Unit → motive Lean.Diff.Action.insert) →
(Unit → motive Lean.Diff.Action.delete) → (Unit → motive Lean.Diff.Action.skip) → motive x | false |
EReal.toReal | Mathlib.Data.EReal.Basic | EReal → ℝ | true |
Lean.Sym.Char.eq_eq_false | Init.Sym.Lemmas | ∀ (a b : Char), decide (a = b) = false → (a = b) = False | true |
ZNum.mod_to_int | Mathlib.Data.Num.ZNum | ∀ (n d : ZNum), ↑(n % d) = ↑n % ↑d | true |
Std.Internal.UV.Loop.Options.mk.noConfusion | Std.Internal.UV.Loop | {P : Sort u} →
{accumulateIdleTime blockSigProfSignal accumulateIdleTime' blockSigProfSignal' : Bool} →
{ accumulateIdleTime := accumulateIdleTime, blockSigProfSignal := blockSigProfSignal } =
{ accumulateIdleTime := accumulateIdleTime', blockSigProfSignal := blockSigProfSignal' } →
(accumulateIdleT... | false |
Matrix.vecMulLinear_transpose | Mathlib.LinearAlgebra.Matrix.ToLin | ∀ {R : Type u_1} [inst : CommSemiring R] {m : Type u_4} {n : Type u_5} [inst_1 : Fintype n] (M : Matrix m n R),
M.transpose.vecMulLinear = M.mulVecLin | true |
Std.Tactic.BVDecide.Reflect.BitVec.add_congr | Std.Tactic.BVDecide.Reflect | ∀ (w : ℕ) (lhs rhs lhs' rhs' : BitVec w), lhs' = lhs → rhs' = rhs → lhs' + rhs' = lhs + rhs | true |
SheafOfModules.pushforwardSections_coe | Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D}
{F : CategoryTheory.Functor C D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat}
... | true |
SignType.instBoundedOrder | Mathlib.Data.Sign.Defs | BoundedOrder SignType | true |
_private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_swap.match_1_1 | Mathlib.Topology.MetricSpace.Gluing | ∀ {X : Type u_1} {Y : Type u_2} (motive : X ⊕ Y → X ⊕ Y → Prop) (x x_1 : X ⊕ Y),
(∀ (val val_1 : X), motive (Sum.inl val) (Sum.inl val_1)) →
(∀ (val val_1 : Y), motive (Sum.inr val) (Sum.inr val_1)) →
(∀ (val : X) (val_1 : Y), motive (Sum.inl val) (Sum.inr val_1)) →
(∀ (val : Y) (val_1 : X), motive ... | false |
UniqueAdd.addHom_preimage | Mathlib.Algebra.Group.UniqueProds.Basic | ∀ {G : Type u_1} {H : Type u_2} [inst : Add G] [inst_1 : Add H] (f : G →ₙ+ H) (hf : Function.Injective ⇑f) (a0 b0 : G)
{A B : Finset H}, UniqueAdd A B (f a0) (f b0) → UniqueAdd (A.preimage ⇑f ⋯) (B.preimage ⇑f ⋯) a0 b0 | true |
Set.Nonempty.of_vadd_right | Mathlib.Algebra.Group.Pointwise.Set.Scalar | ∀ {α : Type u_2} {β : Type u_3} [inst : VAdd α β] {s : Set α} {t : Set β}, (s +ᵥ t).Nonempty → t.Nonempty | true |
Prod.normedRing._proof_3 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : NormedRing α] [inst_1 : NormedRing β] (a b c : α × β),
(a + b) * c = a * c + b * c | false |
Polynomial.coeff_eq_zero_of_lt_trailingDegree | Mathlib.Algebra.Polynomial.Degree.TrailingDegree | ∀ {R : Type u} {n : ℕ} [inst : Semiring R] {p : Polynomial R}, ↑n < p.trailingDegree → p.coeff n = 0 | true |
ProofWidgets.Penrose.DiagramProps.maxOptSteps | ProofWidgets.Component.PenroseDiagram | ProofWidgets.Penrose.DiagramProps → ℕ | true |
IsSimpleOrder.instFinite | Mathlib.Order.Atoms.Finite | ∀ {α : Type u_1} [inst : LE α] [inst_1 : BoundedOrder α] [IsSimpleOrder α], Finite α | true |
CompactlySupportedContinuousMap.mk.sizeOf_spec | Mathlib.Topology.ContinuousMap.CompactlySupported | ∀ {α : Type u_5} {β : Type u_6} [inst : TopologicalSpace α] [inst_1 : Zero β] [inst_2 : TopologicalSpace β]
[inst_3 : SizeOf α] [inst_4 : SizeOf β] (toContinuousMap : C(α, β))
(hasCompactSupport' : HasCompactSupport toContinuousMap.toFun),
sizeOf { toContinuousMap := toContinuousMap, hasCompactSupport' := hasComp... | true |
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