name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.67M | allowCompletion bool 2
classes |
|---|---|---|---|
Matroid.exists_isBasis_union_inter_isBasis._auto_3 | Mathlib.Combinatorics.Matroid.Basic | Lean.Syntax | false |
_private.Mathlib.Logic.Equiv.Fin.Rotate.0.Fin.snoc_eq_cons_rotate._simp_1_1 | Mathlib.Logic.Equiv.Fin.Rotate | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a < b) = (b ≤ a) | false |
Matrix.PosDef.fromBlocks₂₂ | Mathlib.LinearAlgebra.Matrix.PosDef | ∀ {m : Type u_1} {n : Type u_2} {R' : Type u_4} [inst : CommRing R'] [inst_1 : PartialOrder R'] [inst_2 : StarRing R']
[inst_3 : Fintype n] [StarOrderedRing R'] [Finite m] [inst_6 : DecidableEq n] (A : Matrix m m R') (B : Matrix m n R')
{D : Matrix n n R'},
D.PosDef →
∀ [Invertible D], (Matrix.fromBlocks A B B.conjTranspose D).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef | true |
Real.arcosh_lt_arcosh | Mathlib.Analysis.SpecialFunctions.Arcosh | ∀ {x y : ℝ}, 0 < x → 0 < y → (Real.arcosh x < Real.arcosh y ↔ x < y) | true |
CategoryTheory.ComposableArrows.IsComplex.mk | Mathlib.Algebra.Homology.ExactSequence | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{n : ℕ} {S : CategoryTheory.ComposableArrows C n},
(∀ (i : ℕ) (hi : autoParam (i + 2 ≤ n) CategoryTheory.ComposableArrows.IsComplex._auto_1),
CategoryTheory.CategoryStruct.comp (S.map' i (i + 1) ⋯ ⋯) (S.map' (i + 1) (i + 2) ⋯ hi) = 0) →
S.IsComplex | true |
_private.Lean.Elab.Tactic.Do.ProofMode.Cases.0.Lean.Elab.Tactic.Do.ProofMode.mCasesExists.match_3 | Lean.Elab.Tactic.Do.ProofMode.Cases | (motive : Lean.Name × Lean.Syntax → Sort u_1) →
(__discr : Lean.Name × Lean.Syntax) → ((name : Lean.Name) → (ref : Lean.Syntax) → motive (name, ref)) → motive __discr | false |
ContDiffAt.csin | Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {f : E → ℂ} {x : E} {n : WithTop ℕ∞},
ContDiffAt ℂ n f x → ContDiffAt ℂ n (fun x => Complex.sin (f x)) x | true |
Std.TreeMap.getElem?_eq_some_iff_exists_compare_eq_eq_and_mem_toList | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [Std.TransCmp cmp] {k : α} {v : β},
t[k]? = some v ↔ ∃ k', cmp k k' = Ordering.eq ∧ (k', v) ∈ t.toList | true |
enorm_add_le | Mathlib.Analysis.Normed.Group.Basic | ∀ {E : Type u_8} [inst : TopologicalSpace E] [inst_1 : ESeminormedAddMonoid E] (a b : E), ‖a + b‖ₑ ≤ ‖a‖ₑ + ‖b‖ₑ | true |
Array.isEmpty_toList | Init.Data.Array.Lemmas | ∀ {α : Type u_1} {xs : Array α}, xs.toList.isEmpty = xs.isEmpty | true |
Profinite.NobelingProof.spanCone_isLimit | Mathlib.Topology.Category.Profinite.Nobeling.Basic | {I : Type u} →
{C : Set (I → Bool)} →
[inst : (s : Finset I) → (i : I) → Decidable (i ∈ s)] →
(hC : IsCompact C) → CategoryTheory.Limits.IsLimit (Profinite.NobelingProof.spanCone hC) | true |
_private.Lean.Meta.DiscrTree.Basic.0.Lean.Meta.DiscrTree.keysAsPattern.mkApp | Lean.Meta.DiscrTree.Basic | Lean.MessageData → Array Lean.MessageData → Bool → Lean.CoreM Lean.MessageData | true |
LieAlgebra.SemiDirectSum.smul_eq_mk | Mathlib.Algebra.Lie.SemiDirect | ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : LieRing K] [inst_2 : LieAlgebra R K] {L : Type u_3}
[inst_3 : LieRing L] [inst_4 : LieAlgebra R L] (ψ : L →ₗ⁅R⁆ LieDerivation R K K) (t : R) (x : K ⋊⁅ψ⁆ L),
t • x = { left := t • x.left, right := t • x.right } | true |
AddUnits.instCoeHead | Mathlib.Algebra.Group.Units.Defs | {α : Type u} → [inst : AddMonoid α] → CoeHead (AddUnits α) α | true |
_private.Init.Data.Nat.Lemmas.0.Nat.mul_add_mod.match_1_1 | Init.Data.Nat.Lemmas | ∀ (motive : ℕ → Prop) (x : ℕ), (∀ (a : Unit), motive 0) → (∀ (x : ℕ), motive x.succ) → motive x | false |
CategoryTheory.Limits.ChosenPullback₃.p₁₂_p_assoc | Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ S : C} {f₁ : X₁ ⟶ S} {f₂ : X₂ ⟶ S} {f₃ : X₃ ⟶ S}
{h₁₂ : CategoryTheory.Limits.ChosenPullback f₁ f₂} {h₂₃ : CategoryTheory.Limits.ChosenPullback f₂ f₃}
{h₁₃ : CategoryTheory.Limits.ChosenPullback f₁ f₃} (h : CategoryTheory.Limits.ChosenPullback₃ h₁₂ h₂₃ h₁₃) {Z : C}
(h_1 : S ⟶ Z),
CategoryTheory.CategoryStruct.comp h.p₁₂ (CategoryTheory.CategoryStruct.comp h₁₂.p h_1) =
CategoryTheory.CategoryStruct.comp h.p h_1 | true |
NumberField.Units.dirichletUnitTheorem.map_logEmbedding_sup_torsion | Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | ∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] (s : AddSubgroup (Additive (NumberField.RingOfIntegers K)ˣ)),
AddSubgroup.map (NumberField.Units.logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (NumberField.Units.torsion K)) =
AddSubgroup.map (NumberField.Units.logEmbedding K) s | true |
FintypeCat.homMk_eq_comp_iff | Mathlib.CategoryTheory.FintypeCat | ∀ {X Y Z : FintypeCat} (f : X.obj → Y.obj) (g : Y.obj → Z.obj) (h : X.obj → Z.obj),
FintypeCat.homMk h = CategoryTheory.CategoryStruct.comp (FintypeCat.homMk f) (FintypeCat.homMk g) ↔ h = g ∘ f | true |
_private.Mathlib.Order.WithBot.0.WithBot.ofDual_le_iff._simp_1_1 | Mathlib.Order.WithBot | ∀ {α : Type u_1} [inst : LE α] {a : αᵒᵈ} {b : α}, (OrderDual.toDual b ≤ a) = (OrderDual.ofDual a ≤ b) | false |
PFunctor.Approx.CofixA.recOn | Mathlib.Data.PFunctor.Univariate.M | {F : PFunctor.{uA, uB}} →
{motive : (a : ℕ) → PFunctor.Approx.CofixA F a → Sort u} →
{a : ℕ} →
(t : PFunctor.Approx.CofixA F a) →
motive 0 PFunctor.Approx.CofixA.continue →
({n : ℕ} →
(a : F.A) →
(a_1 : F.B a → PFunctor.Approx.CofixA F n) →
((a : F.B a) → motive n (a_1 a)) → motive n.succ (PFunctor.Approx.CofixA.intro a a_1)) →
motive a t | false |
Std.DTreeMap.Internal.Impl.SizedBalancedTree.toBalancedTree | Std.Data.DTreeMap.Internal.Operations | {α : Type u} →
{β : α → Type v} →
{lb ub : ℕ} → Std.DTreeMap.Internal.Impl.SizedBalancedTree α β lb ub → Std.DTreeMap.Internal.Impl.BalancedTree α β | true |
_private.Lean.Server.Watchdog.0.Lean.Server.Watchdog.forwardMessages | Lean.Server.Watchdog | Lean.Server.Watchdog.FileWorker → Lean.Server.Watchdog.ServerM (Lean.Server.ServerTask Lean.Server.Watchdog.WorkerEvent) | true |
IntermediateField.sup_toSubalgebra_of_isAlgebraic_left | Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | ∀ {K : Type u_3} {L : Type u_4} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L]
(E1 E2 : IntermediateField K L) [Algebra.IsAlgebraic K ↥E1],
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra | true |
Rat.commRing._proof_4 | Mathlib.Algebra.Ring.Rat | ∀ (n : ℕ), ↑(Int.negSucc n) = -↑(n + 1) | false |
derivWithin_pow | Mathlib.Analysis.Calculus.Deriv.Pow | ∀ {𝕜 : Type u_1} {𝔸 : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedCommRing 𝔸]
[inst_2 : NormedAlgebra 𝕜 𝔸] {f : 𝕜 → 𝔸} {x : 𝕜} {s : Set 𝕜},
DifferentiableWithinAt 𝕜 f s x → ∀ (n : ℕ), derivWithin (f ^ n) s x = ↑n * f x ^ (n - 1) * derivWithin f s x | true |
Char.card_pow_card | Mathlib.NumberTheory.GaussSum | ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] {F' : Type u_2} [inst_2 : Field F'] [inst_3 : Fintype F']
{χ : MulChar F F'},
χ ≠ 1 →
χ.IsQuadratic →
ringChar F' ≠ ringChar F →
ringChar F' ≠ 2 → (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F') | true |
BitVec.getElem_neg | Init.Data.BitVec.Bitblast | ∀ {w i : ℕ} {x : BitVec w} (h : i < w), (-x)[i] = (x[i] ^^ decide (∃ j < i, x.getLsbD j = true)) | true |
Lean.Elab.addPreDefInfo | Lean.Elab.PreDefinition.Basic | Lean.Elab.PreDefinition → Lean.Elab.TermElabM Unit | true |
HahnSeries.map.congr_simp | Mathlib.RingTheory.HahnSeries.Basic | ∀ {Γ : Type u_1} {R : Type u_3} {S : Type u_4} [inst : PartialOrder Γ] [inst_1 : Zero R] [inst_2 : Zero S]
(x x_1 : HahnSeries Γ R),
x = x_1 →
∀ {F : Type u_5} [inst_3 : FunLike F R S] [inst_4 : ZeroHomClass F R S] (f f_1 : F), f = f_1 → x.map f = x_1.map f_1 | true |
_private.Mathlib.Probability.Independence.Integration.0.ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator._simp_1_2 | Mathlib.Probability.Independence.Integration | ∀ {α : Type u_1} {m : MeasurableSpace α}, MeasurableSet Set.univ = True | false |
_private.Mathlib.CategoryTheory.Monoidal.Cartesian.Basic.0.CategoryTheory.CartesianMonoidalCategory.associator_hom_snd_fst._simp_1_1 | Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.MonoidalCategory C] (X Y : C) {Z : C}
(h : CategoryTheory.MonoidalCategoryStruct.tensorObj X Y ⟶ Z),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.associator X Y (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).inv
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.rightUnitor (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)).hom
h) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft X (CategoryTheory.MonoidalCategoryStruct.rightUnitor Y).hom) h | false |
Int32.ofNat_add | Init.Data.SInt.Lemmas | ∀ (a b : ℕ), Int32.ofNat (a + b) = Int32.ofNat a + Int32.ofNat b | true |
Sat.Literal.noConfusion | Mathlib.Tactic.Sat.FromLRAT | {P : Sort u} → {t t' : Sat.Literal} → t = t' → Sat.Literal.noConfusionType P t t' | false |
MonoidWithZeroHom.instGroupWithZeroSubtypeMemSubmonoidMrange | Mathlib.Algebra.GroupWithZero.Submonoid.Instances | {G : Type u_1} →
{H : Type u_2} →
[inst : GroupWithZero G] → [inst_1 : GroupWithZero H] → (f : G →*₀ H) → GroupWithZero ↥(MonoidHom.mrange f) | true |
Lean.Elab.Tactic.evalExposeNames | Lean.Elab.Tactic.ExposeNames | Lean.Elab.Tactic.Tactic | true |
Lean.TagDeclarationExtension | Lean.EnvExtension | Type | true |
openSegment_subset_segment | Mathlib.Analysis.Convex.Segment | ∀ (𝕜 : Type u_1) {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : SMul 𝕜 E] (x y : E), openSegment 𝕜 x y ⊆ segment 𝕜 x y | true |
Lean.Compiler.LCNF.Phase.toNat | Lean.Compiler.LCNF.PassManager | Lean.Compiler.LCNF.Phase → ℕ | true |
Units.isOpenEmbedding_val | Mathlib.Analysis.Normed.Ring.Units | ∀ {R : Type u_1} [inst : NormedRing R] [HasSummableGeomSeries R], Topology.IsOpenEmbedding Units.val | true |
AlgebraicGeometry.Scheme.instIsOverMapStalkSpecializesCommRingCatPresheaf | Mathlib.AlgebraicGeometry.Stalk | ∀ {X : AlgebraicGeometry.Scheme} {x y : ↥X} (h : x ⤳ y),
AlgebraicGeometry.Scheme.Hom.IsOver (AlgebraicGeometry.Spec.map (X.presheaf.stalkSpecializes h)) X | true |
Algebra.Extension.infinitesimal.eq_1 | Mathlib.RingTheory.Smooth.Kaehler | ∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
(P : Algebra.Extension R S),
P.infinitesimal =
{ Ring := P.Ring ⧸ P.ker ^ 2, commRing := Ideal.Quotient.commRing (P.ker ^ 2),
algebra₁ := Ideal.instAlgebraQuotient R (P.ker ^ 2), algebra₂ := Ideal.Algebra.kerSquareLift, isScalarTower := ⋯,
σ := ⇑(Ideal.Quotient.mk (P.ker ^ 2)) ∘ P.σ, algebraMap_σ := ⋯ } | true |
HSpaces._aux_Mathlib_Topology_Homotopy_HSpaces___unexpand_HSpace_hmul_1 | Mathlib.Topology.Homotopy.HSpaces | Lean.PrettyPrinter.Unexpander | false |
List.getRest._sunfold | Batteries.Data.List.Basic | {α : Type u_1} → [DecidableEq α] → List α → List α → Option (List α) | false |
PredOrder.pred | Mathlib.Order.SuccPred.Basic | {α : Type u_3} → {inst : Preorder α} → [self : PredOrder α] → α → α | true |
Stream'.Seq.mem_cons_of_mem | Mathlib.Data.Seq.Defs | ∀ {α : Type u} (y : α) {a : α} {s : Stream'.Seq α}, a ∈ s → a ∈ Stream'.Seq.cons y s | true |
_private.Mathlib.Analysis.FunctionalSpaces.SobolevInequality.0.«_aux_Mathlib_Analysis_FunctionalSpaces_SobolevInequality___macroRules__private_Mathlib_Analysis_FunctionalSpaces_SobolevInequality_0_term#__1» | Mathlib.Analysis.FunctionalSpaces.SobolevInequality | Lean.Macro | false |
_private.Mathlib.Probability.ProductMeasure.0.MeasureTheory.Measure.infinitePi_pi_of_countable._simp_1_5 | Mathlib.Probability.ProductMeasure | ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsZeroOrProbabilityMeasure μ]
{s : Set α}, (μ s ≤ 1) = True | false |
_private.Mathlib.LinearAlgebra.PiTensorProduct.Basis.0.Basis.piTensorProduct_apply._simp_1_1 | Mathlib.LinearAlgebra.PiTensorProduct.Basis | ∀ {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x}, (∀ (x : α), f x = g x) = (f = g) | false |
Filter.Germ.liftPred_const | Mathlib.Order.Filter.Germ.Basic | ∀ {α : Type u_1} {β : Type u_2} {l : Filter α} {p : β → Prop} {x : β}, p x → Filter.Germ.LiftPred p ↑x | true |
SummationFilter.NeBot.ne_bot | Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | ∀ {β : Type u_2} {L : SummationFilter β} [self : L.NeBot], L.filter.NeBot | true |
Mathlib.Tactic.TFAE.Parser.impFrom.formatter | Mathlib.Tactic.TFAE | Lean.PrettyPrinter.Formatter | true |
Set.Nonempty.right | Mathlib.Data.Set.Basic | ∀ {α : Type u} {s t : Set α}, (s ∩ t).Nonempty → t.Nonempty | true |
CategoryTheory.Retract.projective | Mathlib.CategoryTheory.Preadditive.Projective.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (h : CategoryTheory.Retract X Y)
[p : CategoryTheory.Projective Y], CategoryTheory.Projective X | true |
Filter.map₂_inf_subset_left | Mathlib.Order.Filter.NAry | ∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} {m : α → β → γ} {f₁ f₂ : Filter α} {g : Filter β},
Filter.map₂ m (f₁ ⊓ f₂) g ≤ Filter.map₂ m f₁ g ⊓ Filter.map₂ m f₂ g | true |
StandardEtalePair.lift.eq_1 | Mathlib.RingTheory.Etale.StandardEtale | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
(P : StandardEtalePair R) (x : S) (h : P.HasMap x),
P.lift x h =
Ideal.Quotient.liftₐ (Ideal.span {Polynomial.C P.f, Polynomial.X * Polynomial.C P.g - 1})
(Polynomial.aevalAeval x ↑⋯.unit⁻¹) ⋯ | true |
MeasureTheory.Filtration.piFinset | Mathlib.Probability.Process.Filtration | {ι : Type u_4} →
{X : ι → Type u_5} → [inst : (i : ι) → MeasurableSpace (X i)] → MeasureTheory.Filtration (Finset ι) MeasurableSpace.pi | true |
CpltSepUniformSpace.coe_of | Mathlib.Topology.Category.UniformSpace | ∀ (X : Type u) [inst : UniformSpace X] [inst_1 : CompleteSpace X] [inst_2 : T0Space X], (CpltSepUniformSpace.of X).α = X | true |
Std.HashSet.Raw.get!_diff | Std.Data.HashSet.RawLemmas | ∀ {α : Type u} [inst : BEq α] [inst_1 : Hashable α] {m₁ m₂ : Std.HashSet.Raw α} [EquivBEq α] [LawfulHashable α]
[inst_4 : Inhabited α], m₁.WF → m₂.WF → ∀ {k : α}, (m₁ \ m₂).get! k = if k ∈ m₂ then default else m₁.get! k | true |
_private.Mathlib.RingTheory.LocalProperties.InjectiveDimension.0.ModuleCat.localizedModule_hasInjectiveDimensionLE._simp_1_1 | Mathlib.RingTheory.LocalProperties.InjectiveDimension | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] {X : C},
CategoryTheory.HasInjectiveDimensionLT X 1 = CategoryTheory.Injective X | false |
Std.DTreeMap.Internal.Impl.empty | Std.Data.DTreeMap.Internal.Operations | {α : Type u} → {β : α → Type v} → Std.DTreeMap.Internal.Impl α β | true |
ContinuousMapZero.nonUnitalStarAlgHom_precomp | Mathlib.Topology.ContinuousMap.ContinuousMapZero | {X : Type u_1} →
{Y : Type u_2} →
(R : Type u_4) →
[inst : Zero X] →
[inst_1 : Zero Y] →
[inst_2 : TopologicalSpace X] →
[inst_3 : TopologicalSpace Y] →
[inst_4 : TopologicalSpace R] →
[inst_5 : CommSemiring R] →
[inst_6 : StarRing R] →
[inst_7 : IsTopologicalSemiring R] →
[inst_8 : ContinuousStar R] →
ContinuousMapZero X Y → ContinuousMapZero Y R →⋆ₙₐ[R] ContinuousMapZero X R | true |
_private.Mathlib.SetTheory.Cardinal.NatCount.0.Nat.count_le_setENCard._simp_1_1 | Mathlib.SetTheory.Cardinal.NatCount | ∀ {c : Cardinal.{u}} {n : ℕ}, (↑n ≤ Cardinal.toENat c) = (↑n ≤ c) | false |
CategoryTheory.ObjectProperty.limitsClosure.below.of_mem | Mathlib.CategoryTheory.ObjectProperty.LimitsClosure | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {P : CategoryTheory.ObjectProperty C} {α : Type t}
{J : α → Type u'} [inst_1 : (a : α) → CategoryTheory.Category.{v', u'} (J a)]
{motive : (a : C) → P.limitsClosure J a → Prop} (X : C) (hX : P X),
CategoryTheory.ObjectProperty.limitsClosure.below ⋯ | true |
Set.Icc_top | Mathlib.Order.Interval.Set.Basic | ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : OrderTop α] {a : α}, Set.Icc a ⊤ = Set.Ici a | true |
RatFunc._sizeOf_inst | Mathlib.FieldTheory.RatFunc.Defs | (K : Type u) → {inst : CommRing K} → [SizeOf K] → SizeOf (RatFunc K) | false |
Lean.Doc.PostponedName.mk.noConfusion | Lean.Elab.DocString.Builtin.Postponed | {P : Sort u} → {name name' : Lean.Name} → { name := name } = { name := name' } → (name = name' → P) → P | false |
RingCat.instConcreteCategoryRingHomCarrier._proof_4 | Mathlib.Algebra.Category.Ring.Basic | ∀ {X Y Z : RingCat} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↑X),
(CategoryTheory.CategoryStruct.comp f g).hom' x = g.hom' (f.hom' x) | false |
Finset.iSup_singleton | Mathlib.Order.CompleteLattice.Finset | ∀ {α : Type u_2} {β : Type u_3} [inst : CompleteLattice β] (a : α) (s : α → β), ⨆ x ∈ {a}, s x = s a | true |
Std.ExtTreeMap.minKeyD_insertIfNew_of_isEmpty | Std.Data.ExtTreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.ExtTreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
{v : β}, t.isEmpty = true → ∀ {fallback : α}, (t.insertIfNew k v).minKeyD fallback = k | true |
minpoly.dvd_iff | Mathlib.FieldTheory.Minpoly.Field | ∀ {A : Type u_1} {B : Type u_2} [inst : Field A] [inst_1 : Ring B] [inst_2 : Algebra A B] {x : B} {p : Polynomial A},
minpoly A x ∣ p ↔ (Polynomial.aeval x) p = 0 | true |
_private.Mathlib.GroupTheory.Perm.Cycle.Type.0.Equiv.Perm.IsThreeCycle.nodup_iff_mem_support._proof_1_310 | Mathlib.GroupTheory.Perm.Cycle.Type | ∀ {α : Type u_1} [inst_1 : DecidableEq α] {g : Equiv.Perm α} {a : α} (w w_1 : α),
List.findIdxNth (fun x => decide (x = w)) [g (g a)] {g (g a)}.card + 1 ≤
(List.filter (fun x => decide (x = w_1)) [g (g a)]).length →
List.findIdxNth (fun x => decide (x = w)) [g (g a)] {g (g a)}.card <
(List.findIdxs (fun x => decide (x = w_1)) [g (g a)]).length | false |
CategoryTheory.GradedObject.ι_mapBifunctorComp₂₃MapObjIso_hom_assoc | Mathlib.CategoryTheory.GradedObject.Trifunctor | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6}
[inst : CategoryTheory.Category.{v_1, u_1} C₁] [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂]
[inst_2 : CategoryTheory.Category.{v_3, u_3} C₃] [inst_3 : CategoryTheory.Category.{v_4, u_4} C₄]
[inst_4 : CategoryTheory.Category.{v_6, u_6} C₂₃] (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂₃ C₄))
(G₂₃ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₂₃)) {I₁ : Type u_7} {I₂ : Type u_8} {I₃ : Type u_9}
{J : Type u_10} {r : I₁ × I₂ × I₃ → J} (ρ₂₃ : CategoryTheory.GradedObject.BifunctorComp₂₃IndexData r)
(X₁ : CategoryTheory.GradedObject I₁ C₁) (X₂ : CategoryTheory.GradedObject I₂ C₂)
(X₃ : CategoryTheory.GradedObject I₃ C₃)
[inst_5 : (((CategoryTheory.GradedObject.mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p]
[inst_6 :
(((CategoryTheory.GradedObject.mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj
(CategoryTheory.GradedObject.mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap
ρ₂₃.q]
[H : CategoryTheory.GradedObject.HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃]
[inst_7 :
((((CategoryTheory.GradedObject.mapTrifunctor (CategoryTheory.bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj
X₃).HasMap
r]
(i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (j : J) (h : r (i₁, i₂, i₃) = j) {Z : C₄}
(h_1 :
CategoryTheory.GradedObject.mapBifunctorMapObj F ρ₂₃.q X₁
(CategoryTheory.GradedObject.mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃) j ⟶
Z),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.GradedObject.ιMapTrifunctorMapObj (CategoryTheory.bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ i₁ i₂ i₃ j h)
(CategoryTheory.CategoryStruct.comp
((CategoryTheory.GradedObject.mapBifunctorComp₂₃MapObjIso F G₂₃ ρ₂₃ X₁ X₂ X₃).hom j) h_1) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.GradedObject.ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h) h_1 | true |
Derivation.mk'._proof_2 | Mathlib.RingTheory.Derivation.Basic | ∀ {R : Type u_3} [inst : CommSemiring R] {A : Type u_2} [inst_1 : CommSemiring A] [inst_2 : Algebra R A] {M : Type u_1}
[inst_3 : AddCancelCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M] (D : A →ₗ[R] M),
(∀ (a b : A), D (a * b) = a • D b + b • D a) → D 1 = 0 | false |
Set.biUnion_diff_biUnion_subset | Mathlib.Data.Set.Lattice | ∀ {α : Type u_1} {β : Type u_2} (t : α → Set β) (s₁ s₂ : Set α), (⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x ⊆ ⋃ x ∈ s₁ \ s₂, t x | true |
SemiRingCat.limitSemiring._proof_10 | Mathlib.Algebra.Category.Ring.Limits | ∀ {J : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} J] (F : CategoryTheory.Functor J SemiRingCat)
[inst_1 : Small.{u_2, max u_2 u_3} ↑(F.comp (CategoryTheory.forget SemiRingCat)).sections],
autoParam
(∀ (x : (CategoryTheory.Limits.Types.Small.limitCone (F.comp (CategoryTheory.forget SemiRingCat))).pt),
SemiRingCat.limitSemiring._aux_8 F (SemiRingCat.sectionsSubsemiring F).toSemiring 0 x = 0)
AddMonoid.nsmul_zero._autoParam | false |
ContinuousLinearMap.module._proof_1 | Mathlib.Topology.Algebra.Module.LinearMap | ∀ {R : Type u_1} {R₃ : Type u_2} {S₃ : Type u_5} [inst : Semiring R] [inst_1 : Semiring R₃] [inst_2 : Semiring S₃]
{M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : AddCommMonoid M] [inst_5 : Module R M] {M₃ : Type u_4}
[inst_6 : TopologicalSpace M₃] [inst_7 : AddCommMonoid M₃] [inst_8 : Module R₃ M₃] [inst_9 : Module S₃ M₃]
[inst_10 : SMulCommClass R₃ S₃ M₃] [inst_11 : ContinuousConstSMul S₃ M₃] {σ₁₃ : R →+* R₃} [inst_12 : ContinuousAdd M₃]
(x x_1 : S₃) (x_2 : M →SL[σ₁₃] M₃), (x + x_1) • x_2 = x • x_2 + x_1 • x_2 | false |
CategoryTheory.Limits.coconeOpEquiv_counitIso | Mathlib.CategoryTheory.Limits.Cones | ∀ {J : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} J] {C : Type u₃} [inst_1 : CategoryTheory.Category.{v₃, u₃} C]
{F : CategoryTheory.Functor J C},
CategoryTheory.Limits.coconeOpEquiv.counitIso =
CategoryTheory.Iso.refl
({ obj := fun c => Opposite.op c.unop, map := fun {x x_1} f => Opposite.op { hom := f.hom.unop, w := ⋯ },
map_id := ⋯, map_comp := ⋯ }.comp
{ obj := fun c => (Opposite.unop c).op, map := fun {x x_1} f => { hom := f.unop.hom.op, w := ⋯ }, map_id := ⋯,
map_comp := ⋯ }) | true |
Lean.Server.FileWorker.handleRpcKeepAlive | Lean.Server.FileWorker | Lean.Lsp.RpcKeepAliveParams → Lean.Server.FileWorker.WorkerM Unit | true |
Function.Periodic.map_vadd_multiples | Mathlib.Algebra.Ring.Periodic | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {c : α} [inst : AddCommMonoid α],
Function.Periodic f c → ∀ (a : ↥(AddSubmonoid.multiples c)) (x : α), f (a +ᵥ x) = f x | true |
integral_log_sin_zero_pi | Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric | ∫ (x : ℝ) in 0..Real.pi, Real.log (Real.sin x) = -Real.log 2 * Real.pi | true |
AddMonoidHom.coprod_inl_inr | Mathlib.Algebra.Group.Prod | ∀ {M : Type u_6} {N : Type u_7} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid N],
(AddMonoidHom.inl M N).coprod (AddMonoidHom.inr M N) = AddMonoidHom.id (M × N) | true |
ArithmeticFunction.sigma_one | Mathlib.NumberTheory.ArithmeticFunction.Misc | ∀ (k : ℕ), (ArithmeticFunction.sigma k) 1 = 1 | true |
AddChar | Mathlib.Algebra.Group.AddChar | (A : Type u_1) → [AddMonoid A] → (M : Type u_2) → [Monoid M] → Type (max u_1 u_2) | true |
CategoryTheory.Pretriangulated.Triangle.rotate_mor₃ | Mathlib.CategoryTheory.Triangulated.Rotate | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.HasShift C ℤ] (T : CategoryTheory.Pretriangulated.Triangle C),
T.rotate.mor₃ = -(CategoryTheory.shiftFunctor C 1).map T.mor₁ | true |
MulAction.IsBlock.of_orbit | Mathlib.GroupTheory.GroupAction.Blocks | ∀ {G : Type u_1} [inst : Group G] {X : Type u_2} [inst_1 : MulAction G X] {H : Subgroup G} {a : X},
MulAction.stabilizer G a ≤ H → MulAction.IsBlock G (MulAction.orbit (↥H) a) | true |
SimpleGraph.Subgraph.instCompletelyDistribLattice._proof_2 | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u_1} {G : SimpleGraph V} (s : Set G.Subgraph), IsGLB s (sInf s) | false |
mul_neg_geom_sum | Mathlib.Algebra.Ring.GeomSum | ∀ {R : Type u_1} [inst : Ring R] (x : R) (n : ℕ), (1 - x) * ∑ i ∈ Finset.range n, x ^ i = 1 - x ^ n | true |
CompletelyDistribLattice.mk._flat_ctor | Mathlib.Order.CompleteBooleanAlgebra | {α : Type u} →
(le lt : α → α → Prop) →
(le_refl : ∀ (a : α), le a a) →
(le_trans : ∀ (a b c : α), le a b → le b c → le a c) →
(lt_iff_le_not_ge : autoParam (∀ (a b : α), lt a b ↔ le a b ∧ ¬le b a) Preorder.lt_iff_le_not_ge._autoParam) →
(le_antisymm : ∀ (a b : α), le a b → le b a → a = b) →
(sup : α → α → α) →
(le_sup_left : ∀ (a b : α), le a (sup a b)) →
(le_sup_right : ∀ (a b : α), le b (sup a b)) →
(sup_le : ∀ (a b c : α), le a c → le b c → le (sup a b) c) →
(inf : α → α → α) →
(inf_le_left : ∀ (a b : α), le (inf a b) a) →
(inf_le_right : ∀ (a b : α), le (inf a b) b) →
(le_inf : ∀ (a b c : α), le a b → le a c → le a (inf b c)) →
(sSup : Set α → α) →
(∀ (s : Set α), IsLUB s (sSup s)) →
(sInf : Set α → α) →
(∀ (s : Set α), IsGLB s (sInf s)) →
(top : α) →
(∀ (a : α), le a top) →
(bot : α) →
(∀ (a : α), le bot a) →
(himp : α → α → α) →
(∀ (a b c : α), le a (himp b c) ↔ le (a ⊓ b) c) →
(compl : α → α) →
(∀ (a : α), himp a bot = compl a) →
(sdiff : α → α → α) →
(hnot : α → α) →
(∀ (a b c : α), le (sdiff a b) c ↔ le a (b ⊔ c)) →
(∀ (a : α), sdiff top a = hnot a) →
(∀ {ι : Type u} {κ : ι → Type u} (f : (a : ι) → κ a → α),
⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)) →
CompletelyDistribLattice α | false |
Lean.Meta.Grind.EMatch.State | Lean.Meta.Tactic.Grind.Types | Type | true |
Configuration.HasLines.rec | Mathlib.Combinatorics.Configuration | {P : Type u_1} →
{L : Type u_2} →
[inst : Membership P L] →
{motive : Configuration.HasLines P L → Sort u} →
([toNondegenerate : Configuration.Nondegenerate P L] →
(mkLine : {p₁ p₂ : P} → p₁ ≠ p₂ → L) →
(mkLine_ax : ∀ {p₁ p₂ : P} (h : p₁ ≠ p₂), p₁ ∈ mkLine h ∧ p₂ ∈ mkLine h) →
motive { toNondegenerate := toNondegenerate, mkLine := mkLine, mkLine_ax := mkLine_ax }) →
(t : Configuration.HasLines P L) → motive t | false |
CategoryTheory.Mod_.noConfusionType | Mathlib.CategoryTheory.Monoidal.Mod_ | Sort u →
{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] →
{A : C} →
[inst_4 : CategoryTheory.MonObj A] →
CategoryTheory.Mod_ D A →
{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'] →
{A' : C'} → [inst'_4 : CategoryTheory.MonObj A'] → CategoryTheory.Mod_ D' A' → Sort u | false |
intervalIntegral.continuousOn_primitive_interval_left | Mathlib.MeasureTheory.Integral.DominatedConvergence | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ}
{f : ℝ → E} [MeasureTheory.NoAtoms μ],
MeasureTheory.IntegrableOn f (Set.uIcc a b) μ → ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) (Set.uIcc a b) | true |
IsPrimitiveRoot.toInteger_sub_one_dvd_prime' | Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | ∀ {p : ℕ} {K : Type u} [inst : Field K] {ζ : K} [hp : Fact (Nat.Prime p)] [inst_1 : CharZero K]
[hcycl : IsCyclotomicExtension {p} ℚ K] (hζ : IsPrimitiveRoot ζ p), hζ.toInteger - 1 ∣ ↑p | true |
MulEquiv.coprodAssoc_apply_inl_inl | Mathlib.GroupTheory.Coprod.Basic | ∀ {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] [inst_2 : Monoid P] (x : M),
(MulEquiv.coprodAssoc M N P) (Monoid.Coprod.inl (Monoid.Coprod.inl x)) = Monoid.Coprod.inl x | true |
ModuleCat.projectiveResolution._proof_1 | Mathlib.Algebra.Category.ModuleCat.LeftResolution | ∀ (R : Type u_1) [inst : Ring R] (X : ModuleCat R),
CategoryTheory.isProjective (ModuleCat R) (((CategoryTheory.forget (ModuleCat R)).comp (ModuleCat.free R)).obj X) | false |
NumberField.mixedEmbedding.euclidean.instNontrivialMixedSpace | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | ∀ (K : Type u_1) [inst : Field K] [NumberField K], Nontrivial (NumberField.mixedEmbedding.euclidean.mixedSpace K) | true |
Matrix.detp_smul_add_adjp | Mathlib.LinearAlgebra.Matrix.SemiringInverse | ∀ {n : Type u_1} {R : Type u_3} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommSemiring R]
{A B : Matrix n n R}, A * B = 1 → Matrix.detp 1 B • A + Matrix.adjp (-1) B = Matrix.detp (-1) B • A + Matrix.adjp 1 B | true |
Ordinal.preOmega_max | Mathlib.SetTheory.Cardinal.Aleph | ∀ (o₁ o₂ : Ordinal.{u_1}), Ordinal.preOmega (max o₁ o₂) = max (Ordinal.preOmega o₁) (Ordinal.preOmega o₂) | true |
FiberBundleCore.fiberBundle | Mathlib.Topology.FiberBundle.Basic | {ι : Type u_1} →
{B : Type u_2} →
{F : Type u_3} →
[inst : TopologicalSpace B] → [inst_1 : TopologicalSpace F] → (Z : FiberBundleCore ι B F) → FiberBundle F Z.Fiber | true |
_private.Init.Data.List.TakeDrop.0.List.take_left.match_1_1 | Init.Data.List.TakeDrop | ∀ {α : Type u_1} (motive : List α → List α → Prop) (x x_1 : List α),
(∀ (x : List α), motive [] x) → (∀ (a : α) (tail x : List α), motive (a :: tail) x) → motive x x_1 | false |
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