name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
AddCommGrpCat.isFinite | Mathlib.Algebra.Category.Grp.IsFinite | CategoryTheory.ObjectProperty AddCommGrpCat | The Serre class of finite abelian groups
in the category of abelian groups. | true |
instRingObjOppositeOpensCarrierOfPresheafSmoothSheaf._proof_18 | Mathlib.Geometry.Manifold.Sheaf.Smooth | ∀ {𝕜 : Type u_2} [inst : NontriviallyNormedField 𝕜] {EM : Type u_3} [inst_1 : NormedAddCommGroup EM]
[inst_2 : NormedSpace 𝕜 EM] {HM : Type u_4} [inst_3 : TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM)
{E : Type u_5} [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace 𝕜 E] {H : Type u_6} [inst_6 : Topo... | null | false |
NonUnitalRingHom.range._proof_2 | Mathlib.RingTheory.NonUnitalSubring.Basic | ∀ {R : Type u_2} {S : Type u_1} [inst : NonUnitalNonAssocRing R] [inst_1 : NonUnitalNonAssocRing S] (f : R →ₙ+* S),
Set.range ⇑f = ⇑f '' Set.univ | null | false |
_private.Mathlib.Tactic.ClickSuggestions.TryPremises.0.Mathlib.Tactic.ClickSuggestions.getImportCandidates | Mathlib.Tactic.ClickSuggestions.TryPremises | Lean.Expr →
Lean.Expr →
Array Mathlib.Tactic.ClickSuggestions.GrwPos →
Mathlib.Tactic.ClickSuggestions.RwKind →
Option Lean.Expr →
(String → BaseIO Unit) →
Mathlib.Tactic.ClickSuggestions.ClickSuggestionsM (Array Mathlib.Tactic.ClickSuggestions.Candidates✝) | Get the candidate theorems from imported files. | true |
AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point_assoc | Mathlib.Geometry.RingedSpace.LocallyRingedSpace | ∀ {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x x' : ↑X.toTopCat) (hxx' : x = x') {Z : CommRingCat}
(h : X.presheaf.stalk x' ⟶ Z),
CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x)
(CategoryTheory.CategoryStruct.comp (X.presheaf.stalkSpecializes ⋯) h) =
... | null | true |
FractionalIdeal.mul_induction_on | Mathlib.RingTheory.FractionalIdeal.Basic | ∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
{I J : FractionalIdeal S P} {C : P → Prop} {r : P},
r ∈ I * J → (∀ i ∈ I, ∀ j ∈ J, C (i * j)) → (∀ (x y : P), C x → C y → C (x + y)) → C r | null | true |
Set.op_vadd_set_subset_add | Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | ∀ {α : Type u_2} [inst : Add α] {s t : Set α} {a : α}, a ∈ t → AddOpposite.op a +ᵥ s ⊆ s + t | null | true |
MeasureTheory.SignedMeasure.totalVariation.eq_1 | Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | ∀ {α : Type u_1} [inst : MeasurableSpace α] (s : MeasureTheory.SignedMeasure α),
s.totalVariation = s.toJordanDecomposition.posPart + s.toJordanDecomposition.negPart | null | true |
MeasureTheory.IsStoppingTime.measurableSet_le | Mathlib.Probability.Process.Stopping | ∀ {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [inst : Preorder ι] {f : MeasureTheory.Filtration ι m}
{τ : Ω → WithTop ι}, MeasureTheory.IsStoppingTime f τ → ∀ (i : ι), MeasurableSet {ω | τ ω ≤ ↑i} | null | true |
ClosedSubmodule.map_le_iff_le_comap | Mathlib.Topology.Algebra.Module.ClosedSubmodule | ∀ {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : TopologicalSpace M] [inst_3 : Module R M] [inst_4 : AddCommMonoid N] [inst_5 : TopologicalSpace N]
[inst_6 : Module R N] [inst_7 : ContinuousAdd N] [inst_8 : ContinuousConstSMul R N] {f : M →L[R] N}
{s : Closed... | null | true |
Lean.Lsp.HighlightMatchesOptions._sizeOf_inst | Lean.Data.Lsp.Extra | SizeOf Lean.Lsp.HighlightMatchesOptions | null | false |
ULift.commMonoid.eq_1 | Mathlib.Algebra.Group.ULift | ∀ {α : Type u} [inst : CommMonoid α], ULift.commMonoid = Function.Injective.commMonoid ⇑Equiv.ulift ⋯ ⋯ ⋯ ⋯ | null | true |
_private.Mathlib.Analysis.SpecialFunctions.Pow.Real.0.Real.rpow_zpow_comm._simp_1_1 | Mathlib.Analysis.SpecialFunctions.Pow.Real | ∀ (x : ℝ) (n : ℤ), x ^ n = x ^ ↑n | null | false |
Std.DTreeMap.Internal.Impl.erase._sunfold | Std.Data.DTreeMap.Internal.Operations | {α : Type u} →
{β : α → Type v} →
[Ord α] →
α →
(t : Std.DTreeMap.Internal.Impl α β) →
t.Balanced → Std.DTreeMap.Internal.Impl.SizedBalancedTree α β (t.size - 1) t.size | null | false |
Matrix.transpose_reindex | Mathlib.LinearAlgebra.Matrix.Defs | ∀ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} (eₘ : m ≃ l) (eₙ : n ≃ o) (M : Matrix m n α),
((Matrix.reindex eₘ eₙ) M).transpose = (Matrix.reindex eₙ eₘ) M.transpose | null | true |
_private.Mathlib.Algebra.Homology.Localization.0.HomotopyCategory.quotient_map_mem_quasiIso_iff._simp_1_3 | Mathlib.Algebra.Homology.Localization | ∀ {ι : Type u_1} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (f : K ⟶ L)
[inst_2 : ∀ (i : ι), K.HasHomology i] [inst_3 : ∀ (i : ι), L.HasHomology i], QuasiIso f = ∀ (i : ι), QuasiIsoAt f i | null | false |
CategoryTheory.ShortComplex.SnakeInput.Hom.comp_f₂ | Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C]
{S₁ S₂ S₃ : CategoryTheory.ShortComplex.SnakeInput C} (f : S₁.Hom S₂) (g : S₂.Hom S₃),
(f.comp g).f₂ = CategoryTheory.CategoryStruct.comp f.f₂ g.f₂ | null | true |
AlgebraicGeometry.Scheme.IdealSheafData.glueDataT'Aux.congr_simp | Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | ∀ {X : AlgebraicGeometry.Scheme} (I : X.IdealSheafData) (U V W U₀ : ↑X.affineOpens) (hU₀ : ↑U ⊓ ↑W ≤ ↑U₀),
I.glueDataT'Aux U V W U₀ hU₀ = I.glueDataT'Aux U V W U₀ hU₀ | null | true |
_private.Mathlib.Tactic.Linter.TextBased.0.Mathlib.Linter.TextBased.UnicodeLinter.findBadUnicodeAux.match_1.eq_1 | Mathlib.Tactic.Linter.TextBased | ∀ (motive : Option Char → Sort u_1) (h_1 : Unit → motive none) (h_2 : (cₙ : Char) → motive (some cₙ)),
(match none with
| none => h_1 ()
| some cₙ => h_2 cₙ) =
h_1 () | null | true |
Polynomial.hasFDerivAt_aeval | Mathlib.Analysis.Calculus.Deriv.Polynomial | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {R : Type u_1} [inst_1 : CommSemiring R] [inst_2 : Algebra R 𝕜]
(q : Polynomial R) (x : 𝕜),
HasFDerivAt (fun x => (Polynomial.aeval x) q)
(ContinuousLinearMap.smulRight 1 ((Polynomial.aeval x) (Polynomial.derivative q))) x | null | true |
Sum.Ioc_inr_inr | Mathlib.Data.Sum.Interval | ∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : LocallyFiniteOrder α]
[inst_3 : LocallyFiniteOrder β] (b₁ b₂ : β),
Finset.Ioc (Sum.inr b₁) (Sum.inr b₂) = Finset.map Function.Embedding.inr (Finset.Ioc b₁ b₂) | null | true |
Multiset.sum_map_singleton | Mathlib.Algebra.BigOperators.Group.Multiset.Basic | ∀ {M : Type u_5} (s : Multiset M), (Multiset.map (fun a => {a}) s).sum = s | null | true |
RingCat.moduleCatRestrictScalarsPseudofunctor._proof_1 | Mathlib.Algebra.Category.ModuleCat.Pseudofunctor | ∀ {b₀ b₁ b₂ b₃ : RingCatᵒᵖ} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (h : b₂ ⟶ b₃),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Cat.Hom.isoMk
(ModuleCat.restrictScalarsComp (RingCat.Hom.hom h.unop)
(RingCat.Hom.hom (CategoryTheory.CategoryStruct.comp f g).unop))).hom
(CategoryTheory.Catego... | null | false |
BialgHom.instOneWithConv | Mathlib.RingTheory.Bialgebra.Convolution | {R : Type u_1} →
{A : Type u_2} →
{C : Type u_4} →
[inst : CommSemiring R] →
[inst_1 : CommSemiring A] →
[inst_2 : Semiring C] → [inst_3 : Bialgebra R A] → [inst_4 : Bialgebra R C] → One (WithConv (C →ₐc[R] A)) | null | true |
ModelWithCorners.Boundaryless.mk | 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},
Set.range ↑I = Set.univ → I.Boundaryless | null | true |
_private.Mathlib.Algebra.Polynomial.Lifts.0.Polynomial.exists_support_eq_of_mem_lifts._simp_1_6 | Mathlib.Algebra.Polynomial.Lifts | ∀ {a b : Prop}, (¬a → ¬b) = (b → a) | null | false |
_private.Mathlib.NumberTheory.Primorial.0.«_aux_Mathlib_NumberTheory_Primorial___macroRules__private_Mathlib_NumberTheory_Primorial_0_term_#_1» | Mathlib.NumberTheory.Primorial | Lean.Macro | null | false |
_private.Lean.Util.SortExprs.0.Lean.sortExprs.match_3 | Lean.Util.SortExprs | (motive : ℕ × Lean.Perm → Sort u_1) → (x : ℕ × Lean.Perm) → ((i : ℕ) → (perm : Lean.Perm) → motive (i, perm)) → motive x | null | false |
MeasureTheory.AEEqFun.comp_comp | Mathlib.MeasureTheory.Function.AEEqFun | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α}
[inst_1 : TopologicalSpace δ] [inst_2 : TopologicalSpace β] [inst_3 : TopologicalSpace γ] (g : γ → δ) (g' : β → γ)
(hg : Continuous g) (hg' : Continuous g') (f : α →ₘ[μ] β),
MeasureTheory.AEEqFun... | null | true |
Field.absoluteGaloisGroup | Mathlib.FieldTheory.AbsoluteGaloisGroup | (K : Type u_1) → [Field K] → Type u_1 | The absolute Galois group of `K`, defined as the Galois group of the field extension `K^al/K`,
where `K^al` is an algebraic closure of `K`. | true |
Submodule.mem_span_finset | Mathlib.LinearAlgebra.Finsupp.LinearCombination | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {s : Finset M}
{x : M}, x ∈ Submodule.span R ↑s ↔ ∃ f, Function.support f ⊆ ↑s ∧ ∑ a ∈ s, f a • a = x | null | true |
Asymptotics.isLittleOTVS_iff | Mathlib.Analysis.Asymptotics.TVS | ∀ (𝕜 : Type u_1) {α : Type u_2} {E : Type u_3} {F : Type u_4} [inst : ENorm 𝕜] [inst_1 : TopologicalSpace E]
[inst_2 : TopologicalSpace F] [inst_3 : Zero E] [inst_4 : Zero F] [inst_5 : SMul 𝕜 E] [inst_6 : SMul 𝕜 F]
(l : Filter α) (f : α → E) (g : α → F),
f =o[𝕜; l] g ↔
∀ U ∈ nhds 0, ∃ V ∈ nhds 0, ∀ (ε : ... | null | true |
Lean.Expr.ProdTree.getType._f | Mathlib.Tactic.ProdAssoc | (x : Lean.Expr.ProdTree) → Lean.Expr.ProdTree.below x → Lean.Expr | null | false |
NormedRing.induced._proof_9 | Mathlib.Analysis.Normed.Ring.Basic | ∀ {F : Type u_3} (R : Type u_1) (S : Type u_2) [inst : FunLike F R S] [inst_1 : Ring R] [inst_2 : NormedRing S]
[NonUnitalRingHomClass F R S] (f : F) (a b : R), ‖a * b‖ ≤ ‖a‖ * ‖b‖ | null | false |
UniformCauchySeqOn.fun_neg | Mathlib.Topology.Algebra.IsUniformGroup.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : UniformSpace α] [inst_1 : AddGroup α] [IsUniformAddGroup α] {ι : Type u_3}
{l : Filter ι} {f : ι → β → α} {s : Set β}, UniformCauchySeqOn f l s → UniformCauchySeqOn (fun i i_1 => -f i i_1) l s | Eta-expanded form of `UniformCauchySeqOn.neg` | true |
Matrix.IsHermitian.submatrix | Mathlib.LinearAlgebra.Matrix.Hermitian | ∀ {α : Type u_1} {m : Type u_3} {n : Type u_4} [inst : Star α] {A : Matrix n n α},
A.IsHermitian → ∀ (f : m → n), (A.submatrix f f).IsHermitian | null | true |
_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule.0.trapezoidal_error_le_of_lt._simp_1_7 | Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀} (n : ℤ), a ≠ 0 → (a ^ n = 0) = False | null | false |
_private.Lean.Compiler.Specialize.0.Lean.Compiler.initFn._@.Lean.Compiler.Specialize.149776412._hygCtx._hyg.2 | Lean.Compiler.Specialize | IO (Lean.ParametricAttribute (Array ℕ)) | null | false |
Std.IteratorLoop.WithWF.mk.injEq | Init.Data.Iterators.Consumers.Monadic.Loop | ∀ {α : Type w} {m : Type w → Type w'} {β : Type w} [inst : Std.Iterator α m β] {γ : Type x}
{PlausibleForInStep : β → γ → ForInStep γ → Prop} {hwf : Std.IteratorLoop.WellFounded α m PlausibleForInStep}
(it : Std.IterM m β) (acc : γ) (it_1 : Std.IterM m β) (acc_1 : γ),
({ it := it, acc := acc } = { it := it_1, acc... | null | true |
SimpleGraph.regularityReduced_anti | Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform | ∀ {α : Type u_1} {𝕜 : Type u_2} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [inst_2 : DecidableEq α] {A : Finset α}
{P : Finpartition A} {G : SimpleGraph α} [inst_3 : DecidableRel G.Adj] {ε δ₁ δ₂ : 𝕜},
δ₁ ≤ δ₂ → SimpleGraph.regularityReduced P G ε δ₂ ≤ SimpleGraph.regularityReduced P G ε δ₁ | null | true |
monotone_vecCons | Mathlib.Order.Fin.Tuple | ∀ {α : Type u_1} [inst : Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α},
Monotone (Matrix.vecCons a f) ↔ a ≤ f 0 ∧ Monotone f | null | true |
Dense.exists_ge | Mathlib.Topology.Order.OrderClosed | ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [ClosedIicTopology α] [NoMaxOrder α] {s : Set α},
Dense s → ∀ (x : α), ∃ y ∈ s, x ≤ y | null | true |
Std.TreeSet.Raw.size_filter_eq_size_iff | Std.Data.TreeSet.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet.Raw α cmp} [Std.TransCmp cmp] {f : α → Bool},
t.WF → ((Std.TreeSet.Raw.filter f t).size = t.size ↔ ∀ (k : α) (h : k ∈ t), f (t.get k h) = true) | null | true |
OrderDual.instConditionallyCompletePartialOrder | Mathlib.Order.ConditionallyCompletePartialOrder.Basic | {α : Type u_1} → [ConditionallyCompletePartialOrder α] → ConditionallyCompletePartialOrder αᵒᵈ | null | true |
CategoryTheory.AddMon.monMonoidal._proof_7 | Mathlib.CategoryTheory.Monoidal.Mon | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] (X Y : CategoryTheory.AddMon C),
CategoryTheory.MonoidalCategoryStruct.whiskerLeft X (CategoryTheory.CategoryStruct.id Y) =
CategoryTheory.CategoryStruct.id (Ca... | null | false |
_private.Mathlib.CategoryTheory.Monoidal.Braided.Basic.0._auto_154 | Mathlib.CategoryTheory.Monoidal.Braided.Basic | Lean.Syntax | null | false |
Std.Internal.Do.Spec.Iter.forIn_filterMapWithPostcondition | Std.Internal.Do.Triple.SpecLemmas | ∀ {α β β₂ γ : Type w} {n : Type w → Type w'} {o : Type w → Type w''} {Pred EPred : Type w} [inst : Std.Iterator α Id β]
[inst_1 : Monad n] [LawfulMonad n] [inst_3 : Monad o] [LawfulMonad o] [inst_5 : Std.Internal.Do.Assertion Pred]
[inst_6 : Std.Internal.Do.Assertion EPred] [inst_7 : Std.Internal.Do.WPMonad o Pred ... | null | true |
birkhoffSum_zero | Mathlib.Dynamics.BirkhoffSum.Basic | ∀ {α : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (f : α → α) (g : α → M) (x : α), birkhoffSum f g 0 x = 0 | null | true |
NNRat.cast_eq_zero | Mathlib.Data.Rat.Cast.CharZero | ∀ {α : Type u_3} [inst : DivisionSemiring α] [CharZero α] {q : ℚ≥0}, ↑q = 0 ↔ q = 0 | null | true |
RingCat.ofHom | Mathlib.Algebra.Category.Ring.Basic | {R S : Type u} → [inst : Ring R] → [inst_1 : Ring S] → (R →+* S) → (RingCat.of R ⟶ RingCat.of S) | Typecheck a `RingHom` as a morphism in `RingCat`. | true |
Lean.Parser.symbolNoAntiquot | Lean.Parser.Basic | String → Lean.Parser.Parser | null | true |
Int8.reduceOfIntLE | Lean.Meta.Tactic.Simp.BuiltinSimprocs.SInt | Lean.Meta.Simp.DSimproc | null | true |
HomologicalComplex.truncLEXIso | Mathlib.Algebra.Homology.Embedding.TruncLE | {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.Embeddi... | The isomorphism `(K.truncLE e).X i' ≅ K.X i'` when `e.f i = i'`
and `e.BoundaryLE i` does not hold. | true |
_private.Mathlib.Topology.Constructions.SumProd.0.Prod.instNeBotNhdsWithinIio.match_1 | Mathlib.Topology.Constructions.SumProd | ∀ {X : Type u_1} {Y : Type u_2} [inst : Preorder X] [inst_1 : Preorder Y] {x : X × Y} (x_1 : X × Y)
(motive : x_1 ∈ Set.Iio x.1 ×ˢ Set.Iio x.2 → Prop) (x_2 : x_1 ∈ Set.Iio x.1 ×ˢ Set.Iio x.2),
(∀ (h₁ : x_1.1 ∈ Set.Iio x.1) (h₂ : x_1.2 ∈ Set.Iio x.2), motive ⋯) → motive x_2 | null | false |
sub_right_injective | Mathlib.Algebra.Group.Basic | ∀ {G : Type u_3} [inst : AddGroup G] {b : G}, Function.Injective fun a => b - a | null | true |
_private.Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints.0.UpperHalfPlane.gl_smul_eq_iff_num_eq._simp_1_2 | Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a b c : G₀}, b ≠ 0 → (a / b = c) = (a = c * b) | null | false |
_private.Mathlib.Topology.UniformSpace.Separation.0.t0Space_iff_ker_uniformity._simp_1_3 | Mathlib.Topology.UniformSpace.Separation | ∀ {α : Type u_1} {s : Set (α × α)}, (Set.diagonal α ⊆ s) = ∀ (x : α), (x, x) ∈ s | null | false |
MeasureTheory.memLp_of_memLp_trim | Mathlib.MeasureTheory.Function.LpSeminorm.Trim | ∀ {α : Type u_1} {ε : Type u_3} {m m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α}
[inst : TopologicalSpace ε] [inst_1 : ContinuousENorm ε] (hm : m ≤ m0) {f : α → ε},
MeasureTheory.MemLp f p (μ.trim hm) → MeasureTheory.MemLp f p μ | null | true |
Int.Linear.le_of_le_cert._sunfold | Init.Data.Int.Linear | Int.Linear.Poly → Int.Linear.Poly → Bool | null | false |
Std.Iterators.PostconditionT | Init.Data.Iterators.PostconditionMonad | (Type w → Type w') → Type w → Type (max w w') | `PostconditionT m α` represents an operation in the monad `m` together with a
intrinsic proof that some postcondition holds for the `α` valued monadic result.
It consists of a predicate `P` about `α` and an element of `m ({ a // P a })` and is a helpful tool
for intrinsic verification, notably termination proofs, in th... | true |
AddMonoid.Coprod.clift._proof_2 | Mathlib.GroupTheory.Coprod.Basic | ∀ {M : Type u_1} {N : Type u_2} {P : Type u_3} [inst : AddZeroClass M] [inst_1 : AddZeroClass N]
[inst_2 : AddZeroClass P] (f : FreeAddMonoid (M ⊕ N) →+ P),
f (FreeAddMonoid.of (Sum.inl 0)) = 0 →
f (FreeAddMonoid.of (Sum.inr 0)) = 0 →
(∀ (x y : M),
f (FreeAddMonoid.of (Sum.inl (x + y))) = f (Fre... | null | false |
instWeakPseudoEMetricSpaceSubtype | Mathlib.Topology.EMetricSpace.Defs | {α : Type u_2} →
{p : α → Prop} → [inst : TopologicalSpace α] → [WeakPseudoEMetricSpace α] → WeakPseudoEMetricSpace (Subtype p) | Weak pseudo-emetric space instance on subsets of weak pseudo-emetric spaces | true |
_private.Lean.Meta.Sym.Simp.DiscrTree.0.Lean.Meta.Sym.getMatchLoop._unsafe_rec | Lean.Meta.Sym.Simp.DiscrTree | {α : Type} → Array Lean.Expr → Lean.Meta.DiscrTree.Trie α → Array α → Array α | null | false |
UniformSpace.Completion.toComplₗᵢ._proof_1 | Mathlib.Analysis.Normed.Module.Completion | ∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] (x y : E),
(↑UniformSpace.Completion.toCompl).toFun (x + y) =
(↑UniformSpace.Completion.toCompl).toFun x + (↑UniformSpace.Completion.toCompl).toFun y | null | false |
instMulSemiringActionSubtypeMemSubalgebraIntegralClosure._proof_1 | Mathlib.RingTheory.IntegralClosure.Algebra.Basic | ∀ {G : Type u_3} {R : Type u_2} {K : Type u_1} [inst : CommRing R] [inst_1 : CommRing K] [inst_2 : Algebra R K]
[inst_3 : Group G] [inst_4 : MulSemiringAction G K] [inst_5 : SMulCommClass G R K] (g h : G)
(x : ↥(integralClosure R K)), (g * h) • x = g • h • x | null | false |
CategoryTheory.Limits.WalkingMulticospan.noConfusion | Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {P : Sort u} →
{J : CategoryTheory.Limits.MulticospanShape} →
{t : CategoryTheory.Limits.WalkingMulticospan J} →
{J' : CategoryTheory.Limits.MulticospanShape} →
{t' : CategoryTheory.Limits.WalkingMulticospan J'} →
J = J' → t ≍ t' → CategoryTheory.Limits.WalkingMulticospan.noConfusionType P... | null | false |
ContinuousMap.realToRCLikeStarAlgHom_apply | Mathlib.Analysis.RCLike.ContinuousMap | ∀ {X : Type u_1} (𝕜 : Type u_2) [inst : TopologicalSpace X] [inst_1 : RCLike 𝕜] (f : C(X, ℝ)),
(ContinuousMap.realToRCLikeStarAlgHom X 𝕜) f = ContinuousMap.realToRCLike 𝕜 f | null | true |
IsAddQuantale.recOn | Mathlib.Algebra.Order.Quantale | {α : Type u_1} →
[inst : AddSemigroup α] →
[inst_1 : CompleteLattice α] →
{motive : IsAddQuantale α → Sort u} →
(t : IsAddQuantale α) →
((add_sSup_distrib : ∀ (x : α) (s : Set α), x + sSup s = ⨆ y ∈ s, x + y) →
(sSup_add_distrib : ∀ (s : Set α) (y : α), sSup s + y = ⨆ x ∈ s, ... | null | false |
_private.Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital.0._auto_325 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | Lean.Syntax | null | false |
_private.Mathlib.Analysis.MellinInversion.0.mellin_eq_fourier._simp_1_2 | Mathlib.Analysis.MellinInversion | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 3] [NeZero 3], (3 = 0) = False | null | false |
ContinuousAlgEquiv.coe_inj | Mathlib.Topology.Algebra.Algebra.Equiv | ∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : TopologicalSpace A] [inst_3 : Semiring B] [inst_4 : TopologicalSpace B] [inst_5 : Algebra R A]
[inst_6 : Algebra R B] {f g : A ≃A[R] B}, ↑f = ↑g ↔ f = g | null | true |
Std.DTreeMap.getD_alter_self | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp]
[inst : Std.LawfulEqCmp cmp] {k : α} {fallback : β k} {f : Option (β k) → Option (β k)},
(t.alter k f).getD k fallback = (f (t.get? k)).getD fallback | null | true |
SeparationQuotient.instAddCommMonoid.eq_1 | Mathlib.Topology.Algebra.SeparationQuotient.Basic | ∀ {M : Type u_1} [inst : TopologicalSpace M] [inst_1 : AddCommMonoid M] [inst_2 : ContinuousAdd M],
SeparationQuotient.instAddCommMonoid = { toAddMonoid := SeparationQuotient.instAddMonoid, add_comm := ⋯ } | null | true |
Affine.Simplex.height_pos | Mathlib.Geometry.Euclidean.Altitude | ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] {n : ℕ} [inst_4 : NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)),
0 < s.height i | null | true |
Matroid.eq_loopyOn_iff_loops | Mathlib.Combinatorics.Matroid.Loop | ∀ {α : Type u_1} {M : Matroid α} {E : Set α}, M = Matroid.loopyOn E ↔ M.loops = E ∧ M.E = E | null | true |
Std.Http.Status.conflict.elim | Std.Http.Data.Status | {motive : Std.Http.Status → Sort u} →
(t : Std.Http.Status) → t.ctorIdx = 32 → motive Std.Http.Status.conflict → motive t | null | false |
CategoryTheory.PreOneHypercover.instCategory._proof_4 | Mathlib.CategoryTheory.Sites.Hypercover.One | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} C] {S : C} {X Y : CategoryTheory.PreOneHypercover S}
(f : X.Hom Y), f.comp (CategoryTheory.PreOneHypercover.Hom.id Y) = f | null | false |
_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic.0.CFC.sqrt_ringInverse._proof_1_13 | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | ∀ {A : Type u_1} [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A] [inst_5 : Algebra ℝ A]
[inst_6 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint], NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint | null | false |
BitVec.smtSDiv.eq_1 | Init.Data.BitVec.Lemmas | ∀ {n : ℕ} (x y : BitVec n),
x.smtSDiv y =
match x.msb, y.msb with
| false, false => x.smtUDiv y
| false, true => (x.smtUDiv y.neg).neg
| true, false => (x.neg.smtUDiv y).neg
| true, true => x.neg.smtUDiv y.neg | null | true |
IsFractionRing.num_mul_den_eq_num_iff_eq | Mathlib.RingTheory.Localization.NumDen | ∀ {A : Type u_1} [inst : CommRing A] [inst_1 : IsDomain A] [inst_2 : UniqueFactorizationMonoid A] {K : Type u_2}
[inst_3 : Field K] [inst_4 : Algebra A K] [inst_5 : IsFractionRing A K] {x y : K},
x * (algebraMap A K) ↑(IsFractionRing.den A y) = (algebraMap A K) (IsFractionRing.num A y) ↔ x = y | null | true |
AddMonoid.neZero_exponent_of_finite | Mathlib.GroupTheory.Exponent | ∀ {G : Type u} [inst : AddLeftCancelMonoid G] [Finite G], NeZero (AddMonoid.exponent G) | null | true |
_private.Lean.Elab.Tactic.BuiltinTactic.0.Lean.Elab.Tactic.evalFirst.loop | Lean.Elab.Tactic.BuiltinTactic | Array Lean.Syntax → ℕ → Lean.Elab.Tactic.TacticM Unit | null | true |
Lean.Order.meet_le_left | Std.Internal.Do.Assertion | ∀ {α : Type u} [inst : Lean.Order.CompleteLattice α] (x y : α), Lean.Order.PartialOrder.rel (Lean.Order.meet x y) x | null | true |
_private.Mathlib.Topology.AlexandrovDiscrete.0.Prod.instAlexandrovDiscrete._simp_3 | Mathlib.Topology.AlexandrovDiscrete | ∀ {α : Type u_3} [inst : TopologicalSpace α] [AlexandrovDiscrete α] (a : α), nhds a = Filter.principal (nhdsKer {a}) | null | false |
Configuration.ProjectivePlane.instDual._proof_1 | Mathlib.Combinatorics.Configuration | ∀ (P : Type u_2) (L : Type u_1) [inst : Membership P L] [inst_1 : Configuration.ProjectivePlane P L]
{p₁ p₂ : Configuration.Dual L} (h : p₁ ≠ p₂),
p₁ ∈ Configuration.HasLines.mkLine h ∧ p₂ ∈ Configuration.HasLines.mkLine h | null | false |
Real.normedCommRing._proof_14 | Mathlib.Analysis.Normed.Ring.Basic | ∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1) | null | false |
Std.HashMap.Raw.getKey!_unitOfList_of_contains_eq_false | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α] [LawfulHashable α] [inst_4 : Inhabited α] {l : List α}
{k : α}, l.contains k = false → (Std.HashMap.Raw.unitOfList l).getKey! k = default | null | true |
_private.Mathlib.Geometry.Euclidean.Incenter.0.Affine.Simplex.sum_excenterWeights._simp_1_7 | Mathlib.Geometry.Euclidean.Incenter | ∀ {a : Prop}, (¬¬a) = a | null | false |
Lean.Grind.Semiring.add_comm | Init.Grind.Ring.Basic | ∀ {α : Type u} [self : Lean.Grind.Semiring α] (a b : α), a + b = b + a | Addition is commutative. | true |
exists_Ioc_subset_of_mem_nhds | Mathlib.Topology.Order.Basic | ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] {a : α} {s : Set α},
s ∈ nhds a → (∃ l, l < a) → ∃ l < a, Set.Ioc l a ⊆ s | null | true |
Lean.Meta.InjectionResult.recOn | Lean.Meta.Tactic.Injection | {motive : Lean.Meta.InjectionResult → Sort u} →
(t : Lean.Meta.InjectionResult) →
motive Lean.Meta.InjectionResult.solved →
((mvarId : Lean.MVarId) →
(newEqs : Array Lean.FVarId) →
(remainingNames : List Lean.Name) →
motive (Lean.Meta.InjectionResult.subgoal mvarId newEqs... | null | false |
CliffordAlgebra.even.lift_ι | Mathlib.LinearAlgebra.CliffordAlgebra.Even | ∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
(Q : QuadraticForm R M) {A : Type u_3} [inst_3 : Ring A] [inst_4 : Algebra R A] (f : CliffordAlgebra.EvenHom Q A)
(m₁ m₂ : M), ((CliffordAlgebra.even.lift Q) f) (((CliffordAlgebra.even.ι Q).bilin m₁) m₂) = (f.bilin m... | null | true |
ContIntertwiningMap.toFun_injective | Mathlib.RepresentationTheory.Continuous.Basic | ∀ {R : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [inst : Monoid G] [inst_1 : Ring R]
[inst_2 : AddCommGroup V] [inst_3 : TopologicalSpace V] [inst_4 : IsTopologicalAddGroup V] [inst_5 : Module R V]
[inst_6 : AddCommGroup W] [inst_7 : TopologicalSpace W] [inst_8 : IsTopologicalAddGroup W] [inst_9 : Modu... | null | true |
CategoryTheory.Limits.DiagramOfCocones.coconePoints | Mathlib.CategoryTheory.Limits.Fubini | {J : Type u_1} →
{K : Type u_2} →
[inst : CategoryTheory.Category.{v_1, u_1} J] →
[inst_1 : CategoryTheory.Category.{v_2, u_2} K] →
{C : Type u_3} →
[inst_2 : CategoryTheory.Category.{v_3, u_3} C] →
{F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} →
Cat... | Extract the functor `J ⥤ C` consisting of the cocone points and the maps between them,
from a `DiagramOfCocones`.
| true |
_private.Mathlib.Tactic.Linter.FlexibleLinter.0.Mathlib.Linter.Flexible.stoppers | Mathlib.Tactic.Linter.FlexibleLinter | Std.HashSet Lean.Name | `SyntaxNodeKind`s that are mostly "formatting": mostly they are ignored
because we do not want the linter to spend time on them.
The nodes that they contain will be visited by the linter anyway.
The nodes that *follow* them, though, will *not* be visited by the linter.
| true |
CompactIccSpace.isCompact_Icc | Mathlib.Topology.Order.Compact | ∀ {α : Type u_1} {inst : TopologicalSpace α} {inst_1 : Preorder α} [self : CompactIccSpace α] {a b : α},
IsCompact (Set.Icc a b) | A closed interval `Set.Icc a b` is a compact set for all `a` and `b`. | true |
StrictAnti.prodMap | Mathlib.Order.Monotone.Defs | ∀ {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_2} [inst : PartialOrder α] [inst_1 : PartialOrder β]
[inst_2 : Preorder γ] [inst_3 : Preorder δ] {f : α → γ} {g : β → δ},
StrictAnti f → StrictAnti g → StrictAnti (Prod.map f g) | null | true |
SupPrime.le_sup | Mathlib.Order.Irreducible | ∀ {α : Type u_2} [inst : SemilatticeSup α] {a b c : α}, SupPrime a → (a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c) | null | true |
ENNReal.continuous_rpow_const | Mathlib.Analysis.SpecialFunctions.Pow.Continuity | ∀ {y : ℝ}, Continuous fun a => a ^ y | null | true |
Int.Linear.Poly.collectVars._f | Lean.Meta.Tactic.Grind.Arith.Cutsat.VarRename | (p : Int.Linear.Poly) → Int.Linear.Poly.below p → Lean.Meta.Grind.VarCollector | null | false |
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