name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
instContinuousMulULift | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} [inst : TopologicalSpace M] [inst_1 : Mul M] [ContinuousMul M], ContinuousMul (ULift.{u, u_3} M) |
WithZero.mapAddHom_injective | Mathlib.Algebra.Group.WithOne.Basic | ∀ {α : Type u} {β : Type v} [inst : Add α] [inst_1 : Add β] {f : α →ₙ+ β},
Function.Injective ⇑f → Function.Injective ⇑(WithZero.mapAddHom f) |
_private.Mathlib.Data.Nat.Fib.Zeckendorf.0.Nat.zeckendorf_sum_fib._simp_1_15 | Mathlib.Data.Nat.Fib.Zeckendorf | ∀ {α : Type u} [inst : AddZeroClass α] [inst_1 : LE α] [CanonicallyOrderedAdd α] (a : α), (0 ≤ a) = True |
CategoryTheory.MonoidalCategory.MonoidalLeftAction.curriedActionActionOfMonoidalFunctorToEndofunctorMopIso | Mathlib.CategoryTheory.Monoidal.Action.End | {C : Type u_1} →
{D : Type u_2} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.MonoidalCategory C] →
[inst_2 : CategoryTheory.Category.{v_2, u_2} D] →
(F : CategoryTheory.Functor C (CategoryTheory.Functor D D)ᴹᵒᵖ) →
[inst_3 : F.Monoidal] → CategoryTheory.MonoidalCategory.MonoidalLeftAction.curriedActionMop C D ≅ F |
NormedRing.inverse_add_norm | Mathlib.Analysis.Normed.Ring.Units | ∀ {R : Type u_1} [inst : NormedRing R] [HasSummableGeomSeries R] (x : Rˣ),
(fun t => Ring.inverse (↑x + t)) =O[nhds 0] fun _t => 1 |
nonempty_subtype | Mathlib.Logic.Nonempty | ∀ {α : Sort u_3} {p : α → Prop}, Nonempty (Subtype p) ↔ ∃ a, p a |
CategoryTheory.Over.pullback.congr_simp | Mathlib.CategoryTheory.Comma.Over.Pullback | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f f_1 : X ⟶ Y) (e_f : f = f_1)
[inst_1 : CategoryTheory.Limits.HasPullbacksAlong f],
CategoryTheory.Over.pullback f = CategoryTheory.Over.pullback f_1 |
SSet.stdSimplex.instFunLikeObjOppositeSimplexCategoryMkOpFinHAddNatOfNat | Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | (n i : ℕ) →
FunLike ((SSet.stdSimplex.obj (SimplexCategory.mk n)).obj (Opposite.op (SimplexCategory.mk i))) (Fin (i + 1))
(Fin (n + 1)) |
NumberField.nrRealPlaces_eq_zero_iff | Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | ∀ {K : Type u_2} [inst : Field K] [inst_1 : NumberField K],
NumberField.InfinitePlace.nrRealPlaces K = 0 ↔ NumberField.IsTotallyComplex K |
_private.Init.Data.List.Sort.Impl.0.List.MergeSort.Internal.mergeTR.go.eq_2 | Init.Data.List.Sort.Impl | ∀ {α : Type u_1} (le : α → α → Bool) (x x_1 : List α),
(x = [] → False) → List.MergeSort.Internal.mergeTR.go✝ le x [] x_1 = x_1.reverseAux x |
HasCompactMulSupport.comp_homeomorph | Mathlib.Topology.Algebra.Support | ∀ {X : Type u_9} {Y : Type u_10} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {M : Type u_11}
[inst_2 : One M] {f : Y → M}, HasCompactMulSupport f → ∀ (φ : X ≃ₜ Y), HasCompactMulSupport (f ∘ ⇑φ) |
CategoryTheory.Limits.MultispanShape._sizeOf_1 | Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | CategoryTheory.Limits.MultispanShape → ℕ |
differentiableOn_intCast | Mathlib.Analysis.Calculus.FDeriv.Const | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
[inst_6 : TopologicalSpace F] {s : Set E} [inst_7 : IntCast F] (z : ℤ), DifferentiableOn 𝕜 (↑z) s |
Std.Tactic.BVDecide.LRAT.Internal.Assignment.ctorElim | Std.Tactic.BVDecide.LRAT.Internal.Assignment | {motive : Std.Tactic.BVDecide.LRAT.Internal.Assignment → Sort u} →
(ctorIdx : ℕ) →
(t : Std.Tactic.BVDecide.LRAT.Internal.Assignment) →
ctorIdx = t.ctorIdx → Std.Tactic.BVDecide.LRAT.Internal.Assignment.ctorElimType ctorIdx → motive t |
String.valid_toSubstring | Batteries.Data.String.Lemmas | ∀ (s : String), s.toRawSubstring.Valid |
OrderIso.setIsotypicComponents_apply | Mathlib.RingTheory.SimpleModule.Isotypic | ∀ {R : Type u_2} {M : Type u} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
[inst_3 : IsSemisimpleModule R M] (s : Set ↑(isotypicComponents R M)),
OrderIso.setIsotypicComponents s = ⨆ c ∈ s, ⟨↑c, ⋯⟩ |
_private.Lean.Elab.MutualInductive.0.Lean.Elab.Command.addAuxRecs.match_1 | Lean.Elab.MutualInductive | (motive : Option Lean.ConstantInfo → Sort u_1) →
(x : Option Lean.ConstantInfo) →
((const : Lean.ConstantInfo) → motive (some const)) → ((x : Option Lean.ConstantInfo) → motive x) → motive x |
PSigma.Lex.recOn | Init.WF | ∀ {α : Sort u} {β : α → Sort v} {r : α → α → Prop} {s : (a : α) → β a → β a → Prop}
{motive : (a a_1 : PSigma β) → PSigma.Lex r s a a_1 → Prop} {a a_1 : PSigma β} (t : PSigma.Lex r s a a_1),
(∀ {a₁ : α} (b₁ : β a₁) {a₂ : α} (b₂ : β a₂) (a : r a₁ a₂), motive ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⋯) →
(∀ (a : α) {b₁ b₂ : β a} (a_2 : s a b₁ b₂), motive ⟨a, b₁⟩ ⟨a, b₂⟩ ⋯) → motive a a_1 t |
finsum_eq_if | Mathlib.Algebra.BigOperators.Finprod | ∀ {M : Type u_2} [inst : AddCommMonoid M] {p : Prop} [inst_1 : Decidable p] {x : M}, ∑ᶠ (_ : p), x = if p then x else 0 |
_private.Init.Grind.Ring.CommSolver.0.Lean.Grind.CommRing.instBEqPoly.beq.match_1.splitter | Init.Grind.Ring.CommSolver | (motive : Lean.Grind.CommRing.Poly → Lean.Grind.CommRing.Poly → Sort u_1) →
(x x_1 : Lean.Grind.CommRing.Poly) →
((a b : ℤ) → motive (Lean.Grind.CommRing.Poly.num a) (Lean.Grind.CommRing.Poly.num b)) →
((a : ℤ) →
(a_1 : Lean.Grind.CommRing.Mon) →
(a_2 : Lean.Grind.CommRing.Poly) →
(b : ℤ) →
(b_1 : Lean.Grind.CommRing.Mon) →
(b_2 : Lean.Grind.CommRing.Poly) →
motive (Lean.Grind.CommRing.Poly.add a a_1 a_2) (Lean.Grind.CommRing.Poly.add b b_1 b_2)) →
((x x_2 : Lean.Grind.CommRing.Poly) →
(∀ (a b : ℤ), x = Lean.Grind.CommRing.Poly.num a → x_2 = Lean.Grind.CommRing.Poly.num b → False) →
(∀ (a : ℤ) (a_1 : Lean.Grind.CommRing.Mon) (a_2 : Lean.Grind.CommRing.Poly) (b : ℤ)
(b_1 : Lean.Grind.CommRing.Mon) (b_2 : Lean.Grind.CommRing.Poly),
x = Lean.Grind.CommRing.Poly.add a a_1 a_2 → x_2 = Lean.Grind.CommRing.Poly.add b b_1 b_2 → False) →
motive x x_2) →
motive x x_1 |
_private.Lean.Meta.Tactic.Grind.Attr.0.Lean.Meta.Grind.Extension.addFunCCAttr | Lean.Meta.Tactic.Grind.Attr | Lean.Meta.Grind.Extension → Lean.Name → Lean.AttributeKind → Lean.CoreM Unit |
Nat.recDiagAux_succ_succ | Batteries.Data.Nat.Lemmas | ∀ {motive : ℕ → ℕ → Sort u_1} (zero_left : (n : ℕ) → motive 0 n) (zero_right : (m : ℕ) → motive m 0)
(succ_succ : (m n : ℕ) → motive m n → motive (m + 1) (n + 1)) (m n : ℕ),
Nat.recDiagAux zero_left zero_right succ_succ (m + 1) (n + 1) =
succ_succ m n (Nat.recDiagAux zero_left zero_right succ_succ m n) |
CategoryTheory.Equivalence.changeFunctor._proof_2 | Mathlib.CategoryTheory.Equivalence | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] (e : C ≌ D) {G : CategoryTheory.Functor C D} (iso : e.functor ≅ G)
(X : C),
CategoryTheory.CategoryStruct.comp
(G.map ((e.unitIso ≪≫ CategoryTheory.Functor.isoWhiskerRight iso e.inverse).hom.app X))
((e.inverse.isoWhiskerLeft iso.symm ≪≫ e.counitIso).hom.app (G.obj X)) =
CategoryTheory.CategoryStruct.id (G.obj X) |
CategoryTheory.PreZeroHypercover.hom_inv_h₀._proof_1 | Mathlib.CategoryTheory.Sites.Hypercover.Zero | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] {S : C} {E F : CategoryTheory.PreZeroHypercover S}
(e : E ≅ F) (i : E.I₀), E.X i = E.X (e.inv.s₀ (e.hom.s₀ i)) |
_private.Mathlib.RingTheory.LittleWedderburn.0.LittleWedderburn.InductionHyp.field._proof_11 | Mathlib.RingTheory.LittleWedderburn | ∀ {D : Type u_1} [inst : DivisionRing D] {R : Subring D} [inst_1 : Fintype D] [inst_2 : DecidableEq D]
[inst_3 : DecidablePred fun x => x ∈ R] (q : ℚ≥0) (a : ↥R), DivisionRing.nnqsmul q a = ↑q * a |
_private.Mathlib.Topology.QuasiSeparated.0.QuasiSeparatedSpace.isCompact_sInter_of_nonempty._proof_1_8 | Mathlib.Topology.QuasiSeparated | ∀ {α : Type u_1} [inst : TopologicalSpace α] {s : Set (Set α)},
(∀ t ∈ s, IsCompact t) → ∀ t_1 ∈ {t | t ∈ s ∧ IsOpen t}, IsCompact t_1 |
_private.Mathlib.CategoryTheory.Limits.Opposites.0.CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι._simp_1_1 | Mathlib.CategoryTheory.Limits.Opposites | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z},
(CategoryTheory.CategoryStruct.comp α.hom g = f) = (g = CategoryTheory.CategoryStruct.comp α.inv f) |
Lean.Elab.Command.Structure.checkValidFieldModifier | Lean.Elab.Structure | Lean.Elab.Modifiers → Lean.Elab.TermElabM Unit |
LipschitzWith.compLp | Mathlib.MeasureTheory.Function.LpSpace.Basic | {α : Type u_1} →
{E : Type u_4} →
{F : Type u_5} →
{m : MeasurableSpace α} →
{p : ENNReal} →
{μ : MeasureTheory.Measure α} →
[inst : NormedAddCommGroup E] →
[inst_1 : NormedAddCommGroup F] →
{g : E → F} →
{c : NNReal} → LipschitzWith c g → g 0 = 0 → ↥(MeasureTheory.Lp E p μ) → ↥(MeasureTheory.Lp F p μ) |
FormalMultilinearSeries.leftInv._proof_30 | Mathlib.Analysis.Analytic.Inverse | ∀ {F : Type u_1} [inst : NormedAddCommGroup F], ContinuousAdd F |
_private.Init.Data.Array.Lemmas.0.Array.range.eq_1 | Init.Data.Array.Lemmas | ∀ (n : ℕ), Array.range n = Array.ofFn fun i => ↑i |
List.merge_of_le | Init.Data.List.Sort.Lemmas | ∀ {α : Type u_1} {le : α → α → Bool} {xs ys : List α},
(∀ (a b : α), a ∈ xs → b ∈ ys → le a b = true) → xs.merge ys le = xs ++ ys |
Std.TreeMap.Raw.Equiv.getEntryLT?_eq | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ t₂ : Std.TreeMap.Raw α β cmp} [Std.TransCmp cmp] {k : α},
t₁.WF → t₂.WF → t₁.Equiv t₂ → t₁.getEntryLT? k = t₂.getEntryLT? k |
CategoryTheory.StrictlyUnitaryLaxFunctorCore.map₂_comp | Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | ∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
(self : CategoryTheory.StrictlyUnitaryLaxFunctorCore B C) {a b : B} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h),
self.map₂ (CategoryTheory.CategoryStruct.comp η θ) = CategoryTheory.CategoryStruct.comp (self.map₂ η) (self.map₂ θ) |
Lean.Parser.Tactic.quot | Lean.Parser.Term | Lean.Parser.Parser |
ArchimedeanClass.FiniteResidueField.instField._proof_1 | Mathlib.Algebra.Order.Ring.StandardPart | ∀ {K : Type u_1} [inst : LinearOrder K] [inst_1 : Field K] [inst_2 : IsOrderedRing K]
(a b c : ArchimedeanClass.FiniteResidueField K), a + b + c = a + (b + c) |
ContinuousMultilinearMap.currySumEquiv._proof_10 | Mathlib.Analysis.Normed.Module.Multilinear.Curry | ∀ (𝕜 : Type u_1) (G' : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup G']
[inst_2 : NormedSpace 𝕜 G'], ContinuousConstSMul 𝕜 G' |
Std.TreeSet.Raw.toList_roc | Std.Data.TreeSet.Raw.Slice | ∀ {α : Type u} (cmp : autoParam (α → α → Ordering) Std.TreeSet.Raw.toList_roc._auto_1) [Std.TransCmp cmp]
{t : Std.TreeSet.Raw α cmp},
t.WF →
∀ {lowerBound upperBound : α},
Std.Slice.toList (Std.Roc.Sliceable.mkSlice t lowerBound<...=upperBound) =
List.filter (fun e => decide ((cmp e lowerBound).isGT = true ∧ (cmp e upperBound).isLE = true)) t.toList |
contMDiffOn_zero_iff | 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 : NormedAddCommGroup E']
[inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
{M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {s : Set M},
ContMDiffOn I I' 0 f s ↔ ContinuousOn f s |
LibraryNote.foundational_algebra_order_theory | Mathlib.Data.Nat.Init | Batteries.Util.LibraryNote |
Fintype | Mathlib.Data.Fintype.Defs | Type u_4 → Type u_4 |
Subalgebra.val._proof_5 | Mathlib.Algebra.Algebra.Subalgebra.Basic | ∀ {R : Type u_2} {A : Type u_1} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A]
(S : Subalgebra R A) (x : R), ↑((algebraMap R ↥S) x) = ↑((algebraMap R ↥S) x) |
_private.Lean.PrettyPrinter.Delaborator.TopDownAnalyze.0.Lean.PrettyPrinter.Delaborator.isType2Type._sparseCasesOn_2 | Lean.PrettyPrinter.Delaborator.TopDownAnalyze | {motive : Lean.Expr → Sort u} →
(t : Lean.Expr) → ((u : Lean.Level) → motive (Lean.Expr.sort u)) → (Nat.hasNotBit 8 t.ctorIdx → motive t) → motive t |
Rat.instNormedField | Mathlib.Analysis.Normed.Field.Lemmas | NormedField ℚ |
sqrt_one_add_norm_sq_le | Mathlib.Analysis.SpecialFunctions.JapaneseBracket | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] (x : E), √(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖ |
instAddUInt32 | Init.Data.UInt.BasicAux | Add UInt32 |
CategoryTheory.GrothendieckTopology.Point.over | Mathlib.CategoryTheory.Sites.Point.Over | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
{J : CategoryTheory.GrothendieckTopology C} →
[CategoryTheory.LocallySmall.{w, v, u} C] → (Φ : J.Point) → {X : C} → Φ.fiber.obj X → (J.over X).Point |
CategoryTheory.GrothendieckTopology.Point.skyscraperSheafAdjunction_homEquiv_apply_val | Mathlib.CategoryTheory.Sites.Point.Skyscraper | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} (Φ : J.Point)
{A : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} A] [inst_2 : CategoryTheory.Limits.HasProducts A]
[inst_3 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] {F : CategoryTheory.Sheaf J A} {M : A}
(f : Φ.presheafFiber.obj F.obj ⟶ M),
((Φ.skyscraperSheafAdjunction.homEquiv F M) f).hom = Φ.skyscraperPresheafHomEquiv f |
_private.Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact.0.AlgebraicGeometry.instHasAffinePropertyQuasiCompactCompactSpaceCarrierCarrierCommRingCat._simp_2 | Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | ∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X}, IsCompact s = CompactSpace ↑s |
div_right_injective | Mathlib.Algebra.Group.Basic | ∀ {G : Type u_3} [inst : Group G] {b : G}, Function.Injective fun a => b / a |
_private.Init.Data.Nat.Bitwise.Lemmas.0.Nat.testBit_two_pow._proof_1_3 | Init.Data.Nat.Bitwise.Lemmas | ∀ {n m : ℕ}, m < n → ¬m ≤ n → False |
Prod.mk_le_mk._simp_1 | Mathlib.Order.Basic | ∀ {α : Type u_2} {β : Type u_3} [inst : LE α] [inst_1 : LE β] {a₁ a₂ : α} {b₁ b₂ : β},
((a₁, b₁) ≤ (a₂, b₂)) = (a₁ ≤ a₂ ∧ b₁ ≤ b₂) |
_private.Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups.0.alternatingGroup.exists_mem_stabilizer_smul_eq._proof_1_3 | Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | ∀ {α : Type u_1} [inst : DecidableEq α] {t : Set α},
∀ a ∈ t, ∀ b ∈ t, ∀ c ∈ t, ∀ ⦃b_1 : α⦄, b_1 ∈ t → (⇑(Equiv.swap c a) ∘ ⇑(Equiv.swap a b)) b_1 ∈ t |
_private.Init.Data.UInt.Lemmas.0.UInt32.ofNat_mul._simp_1_1 | Init.Data.UInt.Lemmas | ∀ (a : ℕ) (b : UInt32), (UInt32.ofNat a = b) = (a % 2 ^ 32 = b.toNat) |
Lean.FileMap.lineStart | Lean.Data.Position | Lean.FileMap → ℕ → String.Pos.Raw |
SimpleGraph.isNIndepSet_iff | Mathlib.Combinatorics.SimpleGraph.Clique | ∀ {α : Type u_1} (G : SimpleGraph α) (n : ℕ) (s : Finset α), G.IsNIndepSet n s ↔ G.IsIndepSet ↑s ∧ s.card = n |
_private.Mathlib.Order.Interval.Finset.Fin.0.Fin.finsetImage_natAdd_Icc._simp_1_1 | Mathlib.Order.Interval.Finset.Fin | ∀ {α : Type u_1} {s₁ s₂ : Finset α}, (s₁ = s₂) = (↑s₁ = ↑s₂) |
CategoryTheory.Functor.LaxMonoidal.μ_whiskerRight_comp_μ_assoc | Mathlib.CategoryTheory.Monoidal.Functor | ∀ {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 D]
(F : CategoryTheory.Functor C D) [inst_4 : F.LaxMonoidal] (X Y Z : C) {Z_1 : D}
(h :
F.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) Z) ⟶
Z_1),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Functor.LaxMonoidal.μ F X Y) (F.obj Z))
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.Functor.LaxMonoidal.μ F (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) Z) h) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.associator (F.obj X) (F.obj Y) (F.obj Z)).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft (F.obj X) (CategoryTheory.Functor.LaxMonoidal.μ F Y Z))
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.Functor.LaxMonoidal.μ F X (CategoryTheory.MonoidalCategoryStruct.tensorObj Y Z))
(CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv) h))) |
Nat.gcd_sub_right_right_of_dvd | Init.Data.Nat.Gcd | ∀ {m k : ℕ} (n : ℕ), k ≤ m → n ∣ k → n.gcd (m - k) = n.gcd m |
FundamentalGroupoid.instIsEmpty | Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | ∀ (X : Type u_3) [IsEmpty X], IsEmpty (FundamentalGroupoid X) |
signedDist_vadd_right_swap | Mathlib.Geometry.Euclidean.SignedDist | ∀ {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] (v w : V) (p q : P), ((signedDist v) p) (w +ᵥ q) = ((signedDist v) (-w +ᵥ p)) q |
CategoryTheory.Lax.OplaxTrans.Hom._sizeOf_1 | Mathlib.CategoryTheory.Bicategory.Modification.Lax | {B : Type u₁} →
{inst : CategoryTheory.Bicategory B} →
{C : Type u₂} →
{inst_1 : CategoryTheory.Bicategory C} →
{F G : CategoryTheory.LaxFunctor B C} →
{η θ : F ⟶ G} → [SizeOf B] → [SizeOf C] → CategoryTheory.Lax.OplaxTrans.Hom η θ → ℕ |
hasFDerivAt_inv | Mathlib.Analysis.Calculus.Deriv.Inv | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜},
x ≠ 0 → HasFDerivAt (fun x => x⁻¹) (ContinuousLinearMap.toSpanSingleton 𝕜 (-(x ^ 2)⁻¹)) x |
DenselyOrdered.rec | Mathlib.Order.Basic | {α : Type u_5} →
[inst : LT α] →
{motive : DenselyOrdered α → Sort u} →
((dense : ∀ (a₁ a₂ : α), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂) → motive ⋯) → (t : DenselyOrdered α) → motive t |
_private.Mathlib.Lean.Expr.Basic.0.Lean.Name.fromComponents.go._unsafe_rec | Mathlib.Lean.Expr.Basic | Lean.Name → List Lean.Name → Lean.Name |
Turing.ToPartrec.Cfg.ctorIdx | Mathlib.Computability.TuringMachine.Config | Turing.ToPartrec.Cfg → ℕ |
Nat.shiftLeft'._unsafe_rec | Mathlib.Data.Nat.Bits | Bool → ℕ → ℕ → ℕ |
_private.Init.Data.SInt.Lemmas.0.Int64.lt_iff_le_and_ne._simp_1_2 | Init.Data.SInt.Lemmas | ∀ {x y : Int64}, (x ≤ y) = (x.toInt ≤ y.toInt) |
Mathlib.Tactic.BicategoryLike.AtomIso.mk.sizeOf_spec | Mathlib.Tactic.CategoryTheory.Coherence.Datatypes | ∀ (e : Lean.Expr) (src tgt : Mathlib.Tactic.BicategoryLike.Mor₁),
sizeOf { e := e, src := src, tgt := tgt } = 1 + sizeOf e + sizeOf src + sizeOf tgt |
CategoryTheory.Bicategory.RightLift.mk | Mathlib.CategoryTheory.Bicategory.Extension | {B : Type u} →
[inst : CategoryTheory.Bicategory B] →
{a b c : B} →
{f : b ⟶ a} →
{g : c ⟶ a} →
(h : c ⟶ b) → (CategoryTheory.CategoryStruct.comp h f ⟶ g) → CategoryTheory.Bicategory.RightLift f g |
Submodule.mem_adjoint_iff | Mathlib.Analysis.InnerProductSpace.LinearPMap | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace 𝕜 F]
(g : Submodule 𝕜 (E × F)) (x : F × E),
x ∈ g.adjoint ↔ ∀ (a : E) (b : F), (a, b) ∈ g → inner 𝕜 b x.1 - inner 𝕜 a x.2 = 0 |
CategoryTheory.Functor.pointwiseLeftKanExtensionCompIsoOfPreserves_fac_app | Mathlib.CategoryTheory.Functor.KanExtension.Preserves | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} A]
[inst_1 : CategoryTheory.Category.{v_2, u_2} B] [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
[inst_3 : CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B)
(L : CategoryTheory.Functor A C) [inst_4 : G.PreservesPointwiseLeftKanExtension F L]
[inst_5 : L.HasPointwiseLeftKanExtension F] (a : A),
CategoryTheory.CategoryStruct.comp ((L.pointwiseLeftKanExtensionUnit (F.comp G)).app a)
((G.pointwiseLeftKanExtensionCompIsoOfPreserves F L).inv.app (L.obj a)) =
G.map ((L.pointwiseLeftKanExtensionUnit F).app a) |
_private.Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme.0.AlgebraicGeometry.termProj | Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | Lean.ParserDescr |
_private.Mathlib.Data.Fin.SuccPred.0.Fin.succAbove_succAbove_succAbove_predAbove._proof_1_12 | Mathlib.Data.Fin.SuccPred | ∀ {n : ℕ} (i : Fin (n + 2)) (j : Fin (n + 1)) (k : Fin n),
↑j < ↑i → ¬↑k + 1 < ↑j → ↑k < ↑j → ¬↑k < ↑i → ↑k + 1 + 1 = ↑k + 1 |
_private.Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic.0.tacticEval_simp | Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | Lean.ParserDescr |
Stream'.WSeq.ofList_cons | Mathlib.Data.WSeq.Basic | ∀ {α : Type u} (a : α) (l : List α), ↑(a :: l) = Stream'.WSeq.cons a ↑l |
_private.Mathlib.NumberTheory.Divisors.0.Int.mul_mem_zero_one_two_three_four_iff._simp_1_1 | Mathlib.NumberTheory.Divisors | ∀ {x y z : ℤ}, z ≠ 0 → (x * y = z) = ((x, y) ∈ z.divisorsAntidiag) |
CompareReals.compareEquiv | Mathlib.Topology.UniformSpace.CompareReals | CompareReals.Bourbakiℝ ≃ᵤ ℝ |
Lean.Options.getInPattern | Lean.Data.Options | Lean.Options → Bool |
_private.Mathlib.Combinatorics.SimpleGraph.Walks.Decomp.0.SimpleGraph.Walk.takeUntil.match_1.eq_1 | Mathlib.Combinatorics.SimpleGraph.Walks.Decomp | ∀ {V : Type u_1} {G : SimpleGraph V} {v : V}
(motive : (w : V) → (x : G.Walk v w) → (x_1 : V) → x_1 ∈ x.support → Sort u_2) (u : V)
(h : u ∈ SimpleGraph.Walk.nil.support)
(h_1 : (u : V) → (h : u ∈ SimpleGraph.Walk.nil.support) → motive v SimpleGraph.Walk.nil u h)
(h_2 :
(w v_1 : V) →
(r : G.Adj v v_1) →
(p : G.Walk v_1 w) →
(u : V) → (h : u ∈ (SimpleGraph.Walk.cons r p).support) → motive w (SimpleGraph.Walk.cons r p) u h),
(match v, SimpleGraph.Walk.nil, u, h with
| .(v), SimpleGraph.Walk.nil, u, h => h_1 u h
| w, SimpleGraph.Walk.cons r p, u, h => h_2 w v_1 r p u h) =
h_1 u h |
StandardEtalePair.instEtaleRing | Mathlib.RingTheory.Etale.StandardEtale | ∀ {R : Type u_1} [inst : CommRing R] (P : StandardEtalePair R), Algebra.Etale R P.Ring |
CategoryTheory.Equivalence.counitInv.eq_1 | Mathlib.CategoryTheory.Equivalence | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(e : C ≌ D), e.counitInv = e.counitIso.inv |
MulSemiringActionHom.map_mul' | Mathlib.GroupTheory.GroupAction.Hom | ∀ {M : Type u_1} [inst : Monoid M] {N : Type u_2} [inst_1 : Monoid N] {φ : M →* N} {R : Type u_10} [inst_2 : Semiring R]
[inst_3 : MulSemiringAction M R] {S : Type u_12} [inst_4 : Semiring S] [inst_5 : MulSemiringAction N S]
(self : R →ₑ+*[φ] S) (x y : R), self.toFun (x * y) = self.toFun x * self.toFun y |
EstimatorData.improve | Mathlib.Deprecated.Estimator | {α : Type u_1} → (a : Thunk α) → {ε : Type u_3} → [self : EstimatorData a ε] → ε → Option ε |
_private.Lean.Server.ProtocolOverview.0.Lean.Server.Overview.ProtocolExtensionKind.ctorIdx | Lean.Server.ProtocolOverview | Lean.Server.Overview.ProtocolExtensionKind✝ → ℕ |
_private.Mathlib.Algebra.MvPolynomial.SchwartzZippel.0.MvPolynomial.schwartz_zippel_sup_sum._simp_1_5 | Mathlib.Algebra.MvPolynomial.SchwartzZippel | ∀ {a b c d : Prop}, ((a ∧ b) ∧ c ∧ d) = ((a ∧ c) ∧ b ∧ d) |
NonUnitalStarAlgHom.mk | Mathlib.Algebra.Star.StarAlgHom | {R : Type u_1} →
{A : Type u_2} →
{B : Type u_3} →
[inst : Monoid R] →
[inst_1 : NonUnitalNonAssocSemiring A] →
[inst_2 : DistribMulAction R A] →
[inst_3 : Star A] →
[inst_4 : NonUnitalNonAssocSemiring B] →
[inst_5 : DistribMulAction R B] →
[inst_6 : Star B] →
(toNonUnitalAlgHom : A →ₙₐ[R] B) →
(∀ (a : A), toNonUnitalAlgHom.toFun (star a) = star (toNonUnitalAlgHom.toFun a)) → A →⋆ₙₐ[R] B |
ContinuousOrderHom._sizeOf_inst | Mathlib.Topology.Order.Hom.Basic | (α : Type u_6) →
(β : Type u_7) →
{inst : Preorder α} →
{inst_1 : Preorder β} →
{inst_2 : TopologicalSpace α} → {inst_3 : TopologicalSpace β} → [SizeOf α] → [SizeOf β] → SizeOf (α →Co β) |
Std.DTreeMap.isEmpty_toList | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp}, t.toList.isEmpty = t.isEmpty |
_private.Mathlib.Data.Int.Init.0.Int.le_induction_down._proof_1_3 | Mathlib.Data.Int.Init | ∀ {m : ℤ} {motive : (n : ℤ) → n ≤ m → Prop} (k : ℤ), m ≤ k → ∀ (hle' : k + 1 ≤ m), motive (k + 1) hle' |
_private.Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite.0.SimpleGraph.TripartiteFromTriangles.toTriangle._simp_5 | Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | ∀ {α : Type u_1} [inst : DecidableEq α] {s : Finset α} {a b : α}, (a ∈ insert b s) = (a = b ∨ a ∈ s) |
Real.geom_mean_le_arith_mean3_weighted | Mathlib.Analysis.MeanInequalities | ∀ {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ},
0 ≤ w₁ →
0 ≤ w₂ →
0 ≤ w₃ → 0 ≤ p₁ → 0 ≤ p₂ → 0 ≤ p₃ → w₁ + w₂ + w₃ = 1 → p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ |
AddMonCat.HasLimits.limitConeIsLimit._proof_5 | Mathlib.Algebra.Category.MonCat.Limits | ∀ {J : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} J] (F : CategoryTheory.Functor J AddMonCat)
(s : CategoryTheory.Limits.Cone F) (x y : ↑s.1) {j j' : J} (f : j ⟶ j'),
CategoryTheory.CategoryStruct.comp (((CategoryTheory.forget AddMonCat).mapCone s).π.app j)
((F.comp (CategoryTheory.forget AddMonCat)).map f) (x + y) =
((CategoryTheory.forget AddMonCat).mapCone s).π.app j' (x + y) |
AddMonoidHom.mulOp._proof_4 | Mathlib.Algebra.Group.Equiv.Opposite | ∀ {M : Type u_1} {N : Type u_2} [inst : AddZeroClass M] [inst_1 : AddZeroClass N] (f : M →+ N) (x y : Mᵐᵒᵖ),
(MulOpposite.op ∘ ⇑f ∘ MulOpposite.unop) (x + y) =
(MulOpposite.op ∘ ⇑f ∘ MulOpposite.unop) x + (MulOpposite.op ∘ ⇑f ∘ MulOpposite.unop) y |
CategoryTheory.comp_eqToHom_iff | Mathlib.CategoryTheory.EqToHom | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y Y' : C} (p : Y = Y') (f : X ⟶ Y) (g : X ⟶ Y'),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.eqToHom p) = g ↔
f = CategoryTheory.CategoryStruct.comp g (CategoryTheory.eqToHom ⋯) |
_private.Init.Data.Format.Basic.0.Std.Format.SpaceResult.foundLine | Init.Data.Format.Basic | Std.Format.SpaceResult✝ → Bool |
Ordinal.isNormal_veblen_zero | Mathlib.SetTheory.Ordinal.Veblen | Order.IsNormal fun x => Ordinal.veblen x 0 |
instContinuousSMulTangentSpace | 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) {M : Type u_4}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] (_x : M), ContinuousSMul 𝕜 (TangentSpace I _x) |
Cardinal.lift_sSup | Mathlib.SetTheory.Cardinal.Basic | ∀ {s : Set Cardinal.{u_1}}, BddAbove s → Cardinal.lift.{u, u_1} (sSup s) = sSup (Cardinal.lift.{u, u_1} '' s) |
_private.Mathlib.Order.ModularLattice.0.strictMono_inf_prod_sup.match_1_1 | Mathlib.Order.ModularLattice | ∀ {α : Type u_1} [inst : Lattice α] {z : α} (_x _y : α)
(motive : (fun x => (x ⊓ z, x ⊔ z)) _y ≤ (fun x => (x ⊓ z, x ⊔ z)) _x → Prop)
(x : (fun x => (x ⊓ z, x ⊔ z)) _y ≤ (fun x => (x ⊓ z, x ⊔ z)) _x),
(∀ (inf_le : ((fun x => (x ⊓ z, x ⊔ z)) _y).1 ≤ ((fun x => (x ⊓ z, x ⊔ z)) _x).1)
(sup_le : ((fun x => (x ⊓ z, x ⊔ z)) _y).2 ≤ ((fun x => (x ⊓ z, x ⊔ z)) _x).2), motive ⋯) →
motive x |
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