name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
_private.Mathlib.GroupTheory.FreeGroup.Reduce.0.FreeAddGroup.toWord_add_mk_eq._proof_1_23 | Mathlib.GroupTheory.FreeGroup.Reduce | ∀ {α : Type u_1} [inst : DecidableEq α] (a : α) (b : Bool) (x : FreeAddGroup α) (i : ℕ),
i + 1 ≤ (List.map ((fun g => (g.1, !g.2)) ∘ fun g => (g.1, !g.2)) x.toWord ++ [(a, b)]).length →
i < (List.map ((fun g => (g.1, !g.2)) ∘ fun g => (g.1, !g.2)) x.toWord ++ [(a, b)]).length | null | false |
AddChar | Mathlib.Algebra.Group.AddChar | (A : Type u_1) → [AddMonoid A] → (M : Type u_2) → [Monoid M] → Type (max u_1 u_2) | `AddChar A M` is the type of maps `A → M`, for `A` an additive monoid and `M` a multiplicative
monoid, which intertwine addition in `A` with multiplication in `M`.
We only put the typeclasses needed for the definition, although in practice we are usually
interested in much more specific cases (e.g. when `A` is a group... | 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 | null | 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) | The orbit of `a` under a subgroup containing the stabilizer of `a` is a block | true |
SimpleGraph.Subgraph.instCompletelyDistribLattice._proof_2 | Mathlib.Combinatorics.SimpleGraph.Subgraph | ∀ {V : Type u_1} {G : SimpleGraph V} (a a_1 : G.Subgraph) {v w : V}, (∃ G' ∈ {c | c ⊓ a ≤ a_1}, G'.Adj v w) → G.Adj v w | null | 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 | null | 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)... | null | false |
Lean.Meta.Grind.EMatch.State | Lean.Meta.Tactic.Grind.Types | Type | E-matching related fields for the `grind` goal. | 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... | null | 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.NullSingletonClass μ],
MeasureTheory.IntegrableOn f (Set.uIcc a b) μ → ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) (Set.uIcc a b) | null | true |
RingHom.surjective_pullbackSnd_of_surjective | Mathlib.RingTheory.LocalRing.Pullback | ∀ {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : Ring R] [inst_1 : Ring S] [inst_2 : Semiring T] (f : R →+* T)
(g : S →+* T), Function.Surjective ⇑f → Function.Surjective ⇑(f.pullbackSnd g) | null | 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 | In a `p`-th cyclotomic extension of `ℚ`, we have that `ζ - 1` divides `p` in `𝓞 K`. | 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 | null | 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) | null | 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) | null | 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 | null | 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) | null | 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 | A fiber bundle constructed from core is indeed a fiber bundle. | 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 | null | false |
Lean.Syntax.ident.elim | Init.Prelude | {motive_1 : Lean.Syntax → Sort u} →
(t : Lean.Syntax) →
t.ctorIdx = 3 →
((info : Lean.SourceInfo) →
(rawVal : Substring.Raw) →
(val : Lean.Name) →
(preresolved : List Lean.Syntax.Preresolved) → motive_1 (Lean.Syntax.ident info rawVal val preresolved)) →
motive_1 t | null | false |
LinearMap.domRestrict₁₂_apply | Mathlib.LinearAlgebra.BilinearMap | ∀ {R : Type u_1} {R₂ : Type u_2} {S : Type u_3} {S₂ : Type u_4} [inst : Semiring R] [inst_1 : Semiring R₂]
[inst_2 : Semiring S] [inst_3 : Semiring S₂] {M : Type u_5} {N : Type u_7} {P : Type u_9} [inst_4 : AddCommMonoid M]
[inst_5 : AddCommMonoid N] [inst_6 : AddCommMonoid P] [inst_7 : Module R M] [inst_8 : Module... | null | true |
CategoryTheory.Functor.OneHypercoverDenseData.essSurj | Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | ∀ {C₀ : Type u₀} {C : Type u} [inst : CategoryTheory.Category.{v₀, u₀} C₀] [inst_1 : CategoryTheory.Category.{v, u} C]
{F : CategoryTheory.Functor C₀ C} {J₀ : CategoryTheory.GrothendieckTopology C₀}
{J : CategoryTheory.GrothendieckTopology C} (A : Type u') [inst_2 : CategoryTheory.Category.{v', u'} A]
[inst_3 : C... | null | true |
_private.Mathlib.Tactic.GCongr.Core.0.Mathlib.Tactic.GCongr.addGCongrLemma._proof_7 | Mathlib.Tactic.GCongr.Core | ∀ (xs : Array Lean.Expr) (i : Fin xs.size), ¬↑i < xs.size → False | null | false |
HahnEmbedding.Partial.sSup | Mathlib.Algebra.Order.Module.HahnEmbedding | {K : Type u_1} →
[inst : DivisionRing K] →
[inst_1 : LinearOrder K] →
[inst_2 : IsOrderedRing K] →
[inst_3 : Archimedean K] →
{M : Type u_2} →
[inst_4 : AddCommGroup M] →
[inst_5 : LinearOrder M] →
[inst_6 : IsOrderedAddMonoid M] →
... | Promote `HahnEmbedding.Partial.sSupFun` to a `HahnEmbedding.Partial`. | true |
_private.Mathlib.Algebra.Order.Field.Power.0.Mathlib.Meta.Positivity.evalZPow._proof_2 | Mathlib.Algebra.Order.Field.Power | ∀ {u : Lean.Level} {α : Q(Type u)} (zα : Q(Zero «$α»)) (_a : Q(Field «$α»)), «$zα» =Q instMulZeroClassOfSemiring.toZero | null | false |
List.takeD | Batteries.Data.List.Basic | {α : Type u_1} → ℕ → List α → α → List α | Take `n` elements from a list `l`. If `l` has less than `n` elements, append `n - length l`
elements `x`.
| true |
UInt16.lt_add_one | Init.Data.UInt.Lemmas | ∀ {c : UInt16}, c ≠ -1 → c < c + 1 | null | true |
Subring.rec | Mathlib.Algebra.Ring.Subring.Defs | {R : Type u} →
[inst : NonAssocRing R] →
{motive : Subring R → Sort u_1} →
((toSubsemiring : Subsemiring R) →
(neg_mem' : ∀ {x : R}, x ∈ toSubsemiring.carrier → -x ∈ toSubsemiring.carrier) →
motive { toSubsemiring := toSubsemiring, neg_mem' := neg_mem' }) →
(t : Subring R) → mo... | null | false |
UniformEquiv.piCongrLeft | Mathlib.Topology.UniformSpace.Equiv | {ι : Type u_4} →
{ι' : Type u_5} →
{β : ι' → Type u_6} →
[inst : (j : ι') → UniformSpace (β j)] → (e : ι ≃ ι') → ((i : ι) → β (e i)) ≃ᵤ ((j : ι') → β j) | `Equiv.piCongrLeft` as a uniform isomorphism: this is the natural isomorphism
`Π i, β (e i) ≃ᵤ Π j, β j` obtained from a bijection `ι ≃ ι'`. | true |
_private.Mathlib.Order.Filter.ENNReal.0.NNReal.toReal_liminf._simp_1_6 | Mathlib.Order.Filter.ENNReal | ∀ {q : NNReal}, (0 ≤ ↑q) = True | null | false |
Lean.PersistentHashMap.isUnaryEntries | Lean.Data.PersistentHashMap | {α : Type u_1} →
{β : Type u_2} →
Array (Lean.PersistentHashMap.Entry α β (Lean.PersistentHashMap.Node α β)) → ℕ → Option (α × β) → Option (α × β) | null | true |
coe_setBasisOfLinearIndependentOfCardEqFinrank | Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | ∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {s : Set V}
[inst_3 : Nonempty ↑s] [inst_4 : Fintype ↑s] (lin_ind : LinearIndependent K Subtype.val)
(card_eq : s.toFinset.card = Module.finrank K V),
⇑(setBasisOfLinearIndependentOfCardEqFinrank lin_ind card_eq) =... | null | true |
BoxIntegral.BoxAdditiveMap.rec | Mathlib.Analysis.BoxIntegral.Partition.Additive | {ι : Type u_3} →
{M : Type u_4} →
[inst : AddCommMonoid M] →
{I : WithTop (BoxIntegral.Box ι)} →
{motive : BoxIntegral.BoxAdditiveMap ι M I → Sort u} →
((toFun : BoxIntegral.Box ι → M) →
(sum_partition_boxes' :
∀ (J : BoxIntegral.Box ι),
... | null | false |
Lean.Environment.containsOnBranch | Lean.Environment | Lean.Environment → Lean.Name → Bool | Checks whether the given declaration is known on the current branch, in which case `findAsync?` will
not block.
| true |
Std.DTreeMap.Internal.Impl.Balanced.one_le | Std.Data.DTreeMap.Internal.Balanced | ∀ {α : Type u} {β : α → Type v} {sz : ℕ} {k : α} {v : β k} {l r : Std.DTreeMap.Internal.Impl α β},
(Std.DTreeMap.Internal.Impl.inner sz k v l r).Balanced → 1 ≤ sz | null | true |
TwoSidedIdeal.orderIsoRingCon_apply | Mathlib.RingTheory.TwoSidedIdeal.Basic | ∀ {R : Type u_1} [inst : NonUnitalNonAssocRing R] (self : TwoSidedIdeal R),
TwoSidedIdeal.orderIsoRingCon self = self.ringCon | null | true |
CategoryTheory.Limits.HasCountableLimits.mk._flat_ctor | Mathlib.CategoryTheory.Limits.Shapes.Countable | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C],
autoParam
(∀ (J : Type) [inst_1 : CategoryTheory.SmallCategory J] [CategoryTheory.CountableCategory J],
CategoryTheory.Limits.HasLimitsOfShape J C)
CategoryTheory.Limits.HasCountableLimits.out._autoParam →
CategoryTheory.Limits.Ha... | null | false |
_private.Init.Data.Int.DivMod.Bootstrap.0.Int.add_mul_ediv_right.match_1_3 | Init.Data.Int.DivMod.Bootstrap | ∀ (motive : (c : ℤ) → (∃ n, c = ↑n + 1) → ℤ → 0 < c → Prop) (c : ℤ) (x : ∃ n, c = ↑n + 1) (b : ℤ) (H : 0 < c),
(∀ (w a : ℕ) (H : 0 < ↑w + 1), motive (↑w + 1) ⋯ (Int.ofNat a) H) →
(∀ (k n : ℕ) (H : 0 < ↑k + 1), motive (↑k + 1) ⋯ (Int.negSucc n) H) → motive c x b H | null | false |
_private.Mathlib.Order.GaloisConnection.Basic.0.isLUB_image2_of_isLUB_isLUB._simp_1_3 | Mathlib.Order.GaloisConnection.Basic | ∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} {f : α → β → γ} {s : Set α} {t : Set β} {p : γ → Prop},
(∀ z ∈ Set.image2 f s t, p z) = ∀ x ∈ s, ∀ y ∈ t, p (f x y) | null | false |
Matrix.single_apply_of_ne | Mathlib.Data.Matrix.Basis | ∀ {m : Type u_2} {n : Type u_3} {α : Type u_7} [inst : DecidableEq m] [inst_1 : DecidableEq n] [inst_2 : Zero α] (i : m)
(j : n) (c : α) (i' : m) (j' : n), ¬(i = i' ∧ j = j') → Matrix.single i j c i' j' = 0 | null | true |
Lean.Lsp.instFromJsonPrepareRenameParams | Lean.Data.Lsp.LanguageFeatures | Lean.FromJson Lean.Lsp.PrepareRenameParams | null | true |
Cardinal.mk_iio_le_lift | Mathlib.SetTheory.Ordinal.Basic | ∀ (κ : Cardinal.{u}), Cardinal.mk ↑(Set.Iio κ) ≤ Cardinal.lift.{u + 1, u} κ | null | true |
Std.TreeMap.Raw.maxKey?_eq_none_iff._simp_1 | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap.Raw α β cmp} [Std.TransCmp cmp],
t.WF → (t.maxKey? = none) = (t.isEmpty = true) | null | false |
Lean.Elab.Term.ElabElimInfo.ctorIdx | Lean.Elab.App | Lean.Elab.Term.ElabElimInfo → ℕ | null | false |
Polynomial.isSplittingField_C | Mathlib.FieldTheory.SplittingField.IsSplittingField | ∀ {K : Type v} [inst : Field K] (a : K), Polynomial.IsSplittingField K K (Polynomial.C a) | null | true |
Std.DHashMap.size_inter_le_size_right | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.DHashMap α β} [EquivBEq α]
[LawfulHashable α], (m₁ ∩ m₂).size ≤ m₂.size | null | true |
RingHom.liftOfRightInverse._proof_5 | Mathlib.RingTheory.Ideal.Maps | ∀ {A : Type u_3} {B : Type u_1} {C : Type u_2} [inst : Ring A] [inst_1 : Ring B] [inst_2 : Ring C] (f : A →+* B)
(f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (φ : B →+* C),
(fun g => f.liftOfRightInverseAux f_inv hf ↑g ⋯) ((fun φ => ⟨φ.comp f, ⋯⟩) φ) = φ | null | false |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.getValueCast_mem._simp_1_5 | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} {x y : Sigma β}, (x = y) = (x.fst = y.fst ∧ x.snd ≍ y.snd) | null | false |
AlgebraicTopology.DoldKan.HigherFacesVanish.induction | Mathlib.AlgebraicTopology.DoldKan.Faces | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
{X : CategoryTheory.SimplicialObject C} {Y : C} {n q : ℕ} {φ : Y ⟶ X.obj (Opposite.op { len := n + 1 })},
AlgebraicTopology.DoldKan.HigherFacesVanish q φ →
AlgebraicTopology.DoldKan.HigherFacesVanish (q + 1)
... | null | true |
Lean.Parser.Term.open.parenthesizer | Lean.Parser.Command | Lean.PrettyPrinter.Parenthesizer | null | true |
LinearMap.mkContinuous_coe | Mathlib.Analysis.Normed.Operator.ContinuousLinearMap | ∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [inst : Ring 𝕜] [inst_1 : Ring 𝕜₂]
[inst_2 : SeminormedAddCommGroup E] [inst_3 : SeminormedAddCommGroup F] [inst_4 : Module 𝕜 E] [inst_5 : Module 𝕜₂ F]
{σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖), ↑(f.mkContinuous C ... | null | true |
hnot_sup_self | Mathlib.Order.Heyting.Basic | ∀ {α : Type u_2} [inst : CoheytingAlgebra α] (a : α), ￢a ⊔ a = ⊤ | null | true |
CategoryTheory.NatTrans.mk.injEq | Mathlib.CategoryTheory.NatTrans | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C D} (app : (X : C) → F.obj X ⟶ G.obj X)
(naturality :
autoParam
(∀ ⦃X Y : C⦄ (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp (F.map f) (app Y) = Ca... | null | true |
fwdDiff_const | Mathlib.Algebra.Group.ForwardDiff | ∀ {M : Type u_1} {G : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommGroup G] (h : M) (g : G),
(fwdDiff h fun x => g) = fun x => 0 | null | true |
Nat.sSup_mem | Mathlib.Order.Lattice.Nat | ∀ {s : Set ℕ}, s.Nonempty → BddAbove s → sSup s ∈ s | null | true |
Lean.Elab.Term.Do.attachJPs | Lean.Elab.Do.Legacy | Array Lean.Elab.Term.Do.JPDecl → Lean.Elab.Term.Do.Code → Lean.Elab.Term.Do.Code | null | true |
_private.Init.Grind.Ordered.Module.0.Lean.Grind.OrderedAdd.zsmul_le_zsmul._simp_1_1 | Init.Grind.Ordered.Module | ∀ {M : Type u} [inst : LE M] [inst_1 : Std.IsPreorder M] [inst_2 : Lean.Grind.AddCommGroup M] [Lean.Grind.OrderedAdd M]
{a b : M}, (0 ≤ a - b) = (b ≤ a) | null | false |
CategoryTheory.ShortComplex.isoMk._auto_3 | Mathlib.Algebra.Homology.ShortComplex.Basic | Lean.Syntax | null | false |
_private.Mathlib.RepresentationTheory.Coinvariants.0.Representation.Coinvariants.instFinite._proof_1 | Mathlib.RepresentationTheory.Coinvariants | ∀ {k : Type u_1} {G : Type u_3} {V : Type u_2} [inst : CommRing k] [inst_1 : Monoid G] [inst_2 : AddCommGroup V]
[inst_3 : Module k V] (ρ : Representation k G V) [Module.Finite k V], Module.Finite k ρ.Coinvariants | null | false |
SubAddAction.instInhabited.eq_1 | Mathlib.GroupTheory.GroupAction.SubMulAction | ∀ {R : Type u} {M : Type v} [inst : VAdd R M], SubAddAction.instInhabited = { default := ⊥ } | null | true |
Std.DTreeMap.Raw.Const.ofList._auto_1 | Std.Data.DTreeMap.Raw.Basic | Lean.Syntax | null | false |
RingHom.map_iterate_frobenius | Mathlib.Algebra.CharP.Frobenius | ∀ {R : Type u_1} [inst : CommSemiring R] {S : Type u_2} [inst_1 : CommSemiring S] (g : R →+* S) (p : ℕ)
[inst_2 : ExpChar R p] [inst_3 : ExpChar S p] (x : R) (n : ℕ),
g ((⇑(frobenius R p))^[n] x) = (⇑(frobenius S p))^[n] (g x) | null | true |
MeasureTheory.VectorMeasure.Integrable._proof_1 | Mathlib.MeasureTheory.VectorMeasure.Integral | ∀ {F : Type u_1} [inst : NormedAddCommGroup F], T2Space F | null | false |
PFunctor.Idx | Mathlib.Data.PFunctor.Univariate.Basic | PFunctor.{uA, uB} → Type (max uA uB) | `Idx` identifies a location inside the application of a polynomial functor. For `F : PFunctor`,
`x : F α` and `i : F.Idx`, `i` can designate one part of `x` or is invalid, if `i.1 ≠ x.1`. | true |
Lean.Meta.Try.Collector.OrdSet.insert | Lean.Meta.Tactic.Try.Collect | {α : Type} → {x : Hashable α} → {x_1 : BEq α} → Lean.Meta.Try.Collector.OrdSet α → α → Lean.Meta.Try.Collector.OrdSet α | null | true |
_private.Mathlib.Data.Multiset.DershowitzManna.0.Multiset.transGen_oneStep_of_isDershowitzMannaLT._simp_1_1 | Mathlib.Data.Multiset.DershowitzManna | ∀ {α : Type u_1} {p : α → Prop} [inst : DecidablePred p] {a : α} {s : Multiset α},
(a ∈ Multiset.filter p s) = (a ∈ s ∧ p a) | null | false |
_private.Lean.Meta.Tactic.Grind.Main.0.Lean.Meta.Grind.discharge?.match_1 | Lean.Meta.Tactic.Grind.Main | (motive : Option Lean.Expr → Sort u_1) →
(__do_lift : Option Lean.Expr) →
((p : Lean.Expr) → motive (some p)) → ((x : Option Lean.Expr) → motive x) → motive __do_lift | null | false |
ContinuousMap.instNonUnitalCommCStarAlgebra._proof_6 | Mathlib.Analysis.CStarAlgebra.ContinuousMap | ∀ {α : Type u_1} {A : Type u_2} [inst : TopologicalSpace α] [inst_1 : CompactSpace α]
[inst_2 : NonUnitalCommCStarAlgebra A] (a b c : C(α, A)), a * (b + c) = a * b + a * c | null | false |
Finset.sdiv_empty | Mathlib.Algebra.Group.Pointwise.Finset.Scalar | ∀ {α : Type u_1} {β : Type u_2} [inst : SDiv α β] [inst_1 : DecidableEq α] (s : Finset β), s /ₛ ∅ = ∅ | null | true |
instContMDiffConstSMulOfSomeENatTopOfLEInfty | Mathlib.Geometry.Manifold.Algebra.SMul | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
[inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_6}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {Γ : Type u_8} [inst_6 : SMul Γ M] {n :... | null | true |
Mathlib.Tactic.Monoidal.instMkEvalWhiskerLeftMonoidalM.match_1 | Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | (ctx : Mathlib.Tactic.Monoidal.Context) →
(motive : Option Q(CategoryTheory.MonoidalCategory unknown_1) → Sort u_1) →
(x : Option Q(CategoryTheory.MonoidalCategory unknown_1)) →
((_monoidal : Q(CategoryTheory.MonoidalCategory unknown_1)) → motive (some _monoidal)) →
((x : Option Q(CategoryTheory.Mon... | null | false |
isClosed_Ioo_iff | Mathlib.Topology.Order.DenselyOrdered | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α},
IsClosed (Set.Ioo a b) ↔ b ≤ a | `Set.Ioo a b` is only closed if it is empty. | true |
_private.Mathlib.Algebra.Module.Submodule.Union.0.Submodule.iUnion_ssubset_of_forall_ne_top_of_card_lt._simp_1_1 | Mathlib.Algebra.Module.Submodule.Union | ∀ {α : Type u} {s : Set α}, (s ⊂ Set.univ) = (s ≠ Set.univ) | null | false |
Submonoid.smulDistribClass | Mathlib.Algebra.Group.Submonoid.MulAction | ∀ {M' : Type u_1} {α : Type u_2} {β : Type u_4} {S : Type u_5} [inst : SMul M' α] [inst_1 : SMul M' β]
[inst_2 : SMul α β] [inst_3 : SetLike S M'] [h : SMulDistribClass M' α β] (N' : S), SMulDistribClass (↥N') α β | null | true |
ZeroAtInftyContinuousMap.toOnePointAddMonoidHom_apply | Mathlib.Topology.ContinuousMap.ZeroAtInftyUnitization | ∀ (X : Type u_1) (R : Type u_2) [inst : TopologicalSpace X] [inst_1 : R1Space X] [inst_2 : TopologicalSpace R]
[inst_3 : AddMonoid R] [inst_4 : ContinuousAdd R] (f : ZeroAtInftyContinuousMap X R),
(ZeroAtInftyContinuousMap.toOnePointAddMonoidHom X R) f = f.toOnePoint | null | true |
UInt8.mod_eq_of_lt | Init.Data.UInt.Lemmas | ∀ {a b : UInt8}, a < b → a % b = a | null | true |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.List.0.List.reduceReplicate._regBuiltin.List.reduceReplicate.declare_1._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.List.4079200032._hygCtx._hyg.25 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.List | IO Unit | null | false |
Aesop.RuleBuilderInput.noConfusion | Aesop.Builder.Basic | {P : Sort u} → {t t' : Aesop.RuleBuilderInput} → t = t' → Aesop.RuleBuilderInput.noConfusionType P t t' | null | false |
_private.Std.Data.DHashMap.Internal.AssocList.Lemmas.0.Std.DHashMap.Internal.AssocList.replace.eq_2 | Std.Data.DHashMap.Internal.AssocList.Lemmas | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] (a : α) (b : β a) (a_1 : α) (b_1 : β a_1)
(es : Std.DHashMap.Internal.AssocList α β),
Std.DHashMap.Internal.AssocList.replace a b (Std.DHashMap.Internal.AssocList.cons a_1 b_1 es) =
bif a_1 == a then Std.DHashMap.Internal.AssocList.cons a b es
else Std.DHashMap... | null | true |
CategoryTheory.Dial.rightUnitorImpl._proof_1 | Mathlib.CategoryTheory.Dialectica.Monoidal | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Limits.HasFiniteProducts C]
[inst_2 : CategoryTheory.Limits.HasPullbacks C] (X : CategoryTheory.Dial C),
(X.tensorObjImpl CategoryTheory.Dial.tensorUnitImpl).rel =
(CategoryTheory.Subobject.pullback
(CategoryTheory... | null | false |
NumberField.mixedEmbedding.convexBodySum_volume_eq_zero_of_le_zero | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | ∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] {B : ℝ},
B ≤ 0 → MeasureTheory.volume (NumberField.mixedEmbedding.convexBodySum K B) = 0 | null | true |
iUnion_Iic_eq_Iio_of_lt_of_tendsto | Mathlib.Topology.Order.OrderClosed | ∀ {α : Type u} {ι : Type u_1} {F : Filter ι} [F.NeBot] [inst : ConditionallyCompleteLinearOrder α]
[inst_1 : TopologicalSpace α] [ClosedIicTopology α] {a : α} {f : ι → α},
(∀ (i : ι), f i < a) → Filter.Tendsto f F (nhds a) → ⋃ i, Set.Iic (f i) = Set.Iio a | null | true |
CategoryTheory.Limits.HasBiproductsOfShape.colimIsoLim_inv_app | Mathlib.CategoryTheory.Limits.Shapes.Biproducts | ∀ {J : Type w} {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] [inst_2 : CategoryTheory.Limits.HasBiproductsOfShape J C]
(X : CategoryTheory.Functor (CategoryTheory.Discrete J) C),
CategoryTheory.Limits.HasBiproductsOfShape.colimIsoLim.inv.app X =
Cat... | null | true |
_private.Lean.Elab.Match.0.Lean.Elab.Term.reportMatcherResultErrors.match_1 | Lean.Elab.Match | (motive : Array Lean.Meta.Match.CounterExample × Bool → Sort u_1) →
(x : Array Lean.Meta.Match.CounterExample × Bool) →
((shown : Array Lean.Meta.Match.CounterExample) → (truncated : Bool) → motive (shown, truncated)) → motive x | null | false |
MonadControlT | Init.Control.Basic | (Type u → Type v) → (Type u → Type w) → Type (max (max (u + 1) v) w) | A way to lift a computation from one monad to another while providing the lifted computation with a
means of interpreting computations from the outer monad. This provides a means of lifting
higher-order operations automatically.
Clients should typically use `control` or `controlAt`, which request an instance of `Monad... | true |
Matrix.conjTranspose_reindex | Mathlib.LinearAlgebra.Matrix.ConjTranspose | ∀ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [inst : Star α] (eₘ : m ≃ l) (eₙ : n ≃ o)
(M : Matrix m n α), ((Matrix.reindex eₘ eₙ) M).conjTranspose = (Matrix.reindex eₙ eₘ) M.conjTranspose | null | true |
Lean.Parser.Attr.tactic_alt.parenthesizer | Lean.Parser.Attr | Lean.PrettyPrinter.Parenthesizer | null | true |
Sum.map_surjective | Mathlib.Data.Sum.Basic | ∀ {α : Type u} {β : Type v} {γ : Type u_1} {δ : Type u_2} {f : α → γ} {g : β → δ},
Function.Surjective (Sum.map f g) ↔ Function.Surjective f ∧ Function.Surjective g | null | true |
LinearMap.BilinForm.Equivalent.symm | Mathlib.LinearAlgebra.BilinearForm.IsometryEquiv | ∀ {R : Type u_1} {M₁ : Type u_3} {M₂ : Type u_4} [inst : CommSemiring R] [inst_1 : AddCommMonoid M₁]
[inst_2 : AddCommMonoid M₂] [inst_3 : Module R M₁] [inst_4 : Module R M₂] {B₁ : LinearMap.BilinForm R M₁}
{B₂ : LinearMap.BilinForm R M₂}, B₁.Equivalent B₂ → B₂.Equivalent B₁ | null | true |
_private.Mathlib.RingTheory.Valuation.ValuativeRel.Basic.0.ValuativeRel.ext.match_1 | Mathlib.RingTheory.Valuation.ValuativeRel.Basic | ∀ {R : Type u_1} {inst : Semiring R} (motive : ValuativeRel R → Prop) (h : ValuativeRel R),
(∀ (vle : R → R → Prop) (vle_total : ∀ (x y : R), vle x y ∨ vle y x)
(vle_trans : ∀ {z y x : R}, vle x y → vle y z → vle x z)
(vle_add : ∀ {x y z : R}, vle x z → vle y z → vle (x + y) z)
(mul_vle_mul_left : ∀... | null | false |
Module.Basis.prod_apply_inl_fst | Mathlib.LinearAlgebra.Basis.Prod | ∀ {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_4} {M' : Type u_5} [inst : Semiring R]
[inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M']
(b : Module.Basis ι R M) (b' : Module.Basis ι' R M') (i : ι), ((b.prod b') (Sum.inl i)).1 = b i | null | true |
UInt8.toInt8_ofNat' | Init.Data.SInt.Lemmas | ∀ {n : ℕ}, (UInt8.ofNat n).toInt8 = Int8.ofNat n | null | true |
GradeOrder.wellFoundedGT | Mathlib.Order.Grade | ∀ {α : Type u_3} [inst : Preorder α] (𝕆 : Type u_5) [inst_1 : Preorder 𝕆] [GradeOrder 𝕆 α] [WellFoundedGT 𝕆],
WellFoundedGT α | null | true |
WeierstrassCurve.Affine.instDecidableEqPoint.decEq._proof_6 | Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | ∀ {R : Type u_1} {inst : CommRing R} {W' : WeierstrassCurve.Affine R} (a a_1 : R) (a_2 : W'.Nonsingular a a_1)
(b b_1 : R) (b_2 : W'.Nonsingular b b_1),
¬a = b → ¬WeierstrassCurve.Affine.Point.some a a_1 a_2 = WeierstrassCurve.Affine.Point.some b b_1 b_2 | null | false |
_private.Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit.0.CategoryTheory.coherentTopology.preimageDiagram.eq_1 | Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preregular C]
[inst_2 : CategoryTheory.FinitaryExtensive C]
{F : CategoryTheory.Functor ℕᵒᵖ (CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) (Type v))}
(hF : ∀ (n : ℕ), CategoryTheory.Sheaf.IsLocallySurjective (F.map (Categ... | null | true |
IsSimpleRing.of_surjective | Mathlib.RingTheory.SimpleRing.Congr | ∀ {R : Type u_1} {S : Type u_2} [inst : NonAssocRing R] [inst_1 : NonAssocRing S] [Nontrivial S] (f : R →+* S),
IsSimpleRing R → Function.Surjective ⇑f → IsSimpleRing S | null | true |
Std.HashSet.getD_union_of_not_mem_left | Std.Data.HashSet.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {m₁ m₂ : Std.HashSet α} [EquivBEq α] [LawfulHashable α] {k fallback : α},
k ∉ m₁ → (m₁ ∪ m₂).getD k fallback = m₂.getD k fallback | null | true |
Lean.Parser.Term.set_option.parenthesizer | Lean.Parser.Command | Lean.PrettyPrinter.Parenthesizer | null | true |
CircularPartialOrder.toCircularPreorder | Mathlib.Order.Circular | {α : Type u_1} → [self : CircularPartialOrder α] → CircularPreorder α | null | true |
ModuleCon.mk.noConfusion | Mathlib.Algebra.Module.Congruence.Defs | {S : Type u_2} →
{M : Type u_3} →
{inst : Add M} →
{inst_1 : SMul S M} →
{P : Sort u} →
{toAddCon : AddCon M} →
{smul : ∀ (s : S) {x y : M}, toAddCon.toSetoid x y → toAddCon.toSetoid (s • x) (s • y)} →
{toAddCon' : AddCon M} →
{smul' : ∀ (s : S) {x... | null | false |
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