name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
IsRealClosed.mk._flat_ctor | Mathlib.FieldTheory.IsRealClosed.Basic | ∀ {R : Type u_1} [inst : Field R],
(∀ {s : R}, IsSumSq s → 1 + s ≠ 0) →
(∀ (x : R), IsSquare x ∨ IsSquare (-x)) → (∀ {f : Polynomial R}, Odd f.natDegree → ∃ x, f.IsRoot x) → IsRealClosed R | null | false |
CategoryTheory.WithTerminal.inclLift._proof_4 | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] {Z : D} (F : CategoryTheory.Functor C D) (M : (x : C) → F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), CategoryTheory.CategoryStruct.comp (F.map f) (M y) = M x) ⦃X Y : C⦄ (f : X ⟶ Y),
Categor... | null | false |
_private.Mathlib.RingTheory.AdicCompletion.Noetherian.0.IsHausdorff.of_le_jacobson._simp_1_1 | Mathlib.RingTheory.AdicCompletion.Noetherian | ∀ {R : Type u_1} [inst : Ring R] {M : Type u_4} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {U : Submodule R M}
{x : M}, (x ≡ 0 [SMOD U]) = (x ∈ U) | null | false |
HasCompactSupport.enorm_le_lintegral_Ici_deriv | Mathlib.MeasureTheory.Integral.IntegralEqImproper | ∀ {F : Type u_2} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {f : ℝ → F},
ContDiff ℝ 1 f → HasCompactSupport f → ∀ (x : ℝ), ‖f x‖ₑ ≤ ∫⁻ (y : ℝ) in Set.Iic x, ‖deriv f y‖ₑ | null | true |
starRingAut._proof_3 | Mathlib.Algebra.Star.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : StarRing R] (x y : R),
starMulAut.toFun (x * y) = starMulAut.toFun x * starMulAut.toFun y | null | false |
RightAddCosetEquivalence | Mathlib.GroupTheory.Coset.Basic | {α : Type u_1} → [Add α] → Set α → α → α → Prop | Equality of two right cosets `s + a` and `s + b`. | true |
CategoryTheory.Adjunction.Quadruple.epi_leftTriple_rightToLeft_iff_mono_rightTriple_leftToRight | Mathlib.CategoryTheory.Adjunction.Quadruple | ∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor D C} {G : CategoryTheory.Functor C D}
{R : CategoryTheory.Functor D C} (q : CategoryTheory.Adjunction.Quadruple L F G R) [inst_2 : F.Fu... | For an adjoint quadruple `L ⊣ F ⊣ G ⊣ R` where `F` (and hence also `R`) is fully faithful and
its domain / codomain has all pushouts resp. pullbacks, the natural transformation `G ⟶ L` is an
epimorphism iff the natural transformation `F ⟶ R` is a monomorphism. | true |
HomologicalComplex.natTransHomologyπ_app | Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | ∀ (C : Type u_1) [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{ι : Type u_2} (c : ComplexShape ι) (i : ι) [inst_2 : CategoryTheory.CategoryWithHomology C]
(K : HomologicalComplex C c), (HomologicalComplex.natTransHomologyπ C c i).app K = K.homologyπ i | null | true |
PSigma.Lex.le | Mathlib.Order.PSigma | {ι : Type u_1} → {α : ι → Type u_2} → [LT ι] → [(i : ι) → LE (α i)] → LE (Σₗ' (i : ι), α i) | The lexicographical `≤` on a sigma type. | true |
CategoryTheory.endofunctorMonoidalCategory_associator_hom_app | Mathlib.CategoryTheory.Monoidal.End | ∀ (C : Type u) [inst : CategoryTheory.Category.{v, u} C] (F G H : CategoryTheory.Functor C C) (X : C),
(CategoryTheory.MonoidalCategoryStruct.associator F G H).hom.app X =
CategoryTheory.CategoryStruct.id
((CategoryTheory.MonoidalCategoryStruct.tensorObj (CategoryTheory.MonoidalCategoryStruct.tensorObj F G)... | null | true |
zero_zpow | Mathlib.Algebra.GroupWithZero.Basic | ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] (n : ℤ), n ≠ 0 → 0 ^ n = 0 | null | true |
ContinuousWithinAt.vsub | Mathlib.Topology.Algebra.Group.Torsor | ∀ {V : Type u_1} {P : Type u_2} {α : Type u_3} [inst : AddGroup V] [inst_1 : TopologicalSpace V]
[inst_2 : AddTorsor V P] [inst_3 : TopologicalSpace P] [IsTopologicalAddTorsor P] [inst_5 : TopologicalSpace α]
{f g : α → P} {x : α} {s : Set α},
ContinuousWithinAt f s x → ContinuousWithinAt g s x → ContinuousWithin... | null | true |
_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule.0.trapezoidal_error_le_of_lt._simp_1_8 | Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [CharZero R] (n : ℕ), (↑n + 1 = 0) = False | null | false |
IsLocalization.mk'_self | Mathlib.RingTheory.Localization.Defs | ∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} (S : Type u_2) [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [inst_3 : IsLocalization M S] {x : R} (hx : x ∈ M), IsLocalization.mk' S x ⟨x, hx⟩ = 1 | null | true |
CategoryTheory.GlueData.f_hasPullback | Mathlib.CategoryTheory.GlueData | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v, u₁} C] (self : CategoryTheory.GlueData C) (i j k : self.J),
CategoryTheory.Limits.HasPullback (self.f i j) (self.f i k) | null | true |
_private.Mathlib.Topology.MetricSpace.Ultra.Basic.0.IsUltrametricDist.isOpen_closedBall._simp_1_2 | Mathlib.Topology.MetricSpace.Ultra.Basic | ∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, (Metric.ball x ε ⊆ Metric.closedBall x ε) = True | null | false |
IsRegularLocalRing.spanFinrank_maximalIdeal | Mathlib.RingTheory.RegularLocalRing.Defs | ∀ {R : Type u_1} {inst : CommRing R} [self : IsRegularLocalRing R],
↑(Submodule.spanFinrank (IsLocalRing.maximalIdeal R)) = ringKrullDim R | null | true |
Pi.mulSingle_eq_mulSingle_iff | Mathlib.Algebra.Notation.Pi.Basic | ∀ {ι : Type u_1} [inst : DecidableEq ι] {i : ι} {M : Type u_7} [inst_1 : One M] {j : ι} {m n : M},
Pi.mulSingle i m = Pi.mulSingle j n ↔ i = j ∧ m = n ∨ m = 1 ∧ n = 1 | null | true |
Std.Ric.Sliceable.mkSlice | Init.Data.Slice.Notation | {α : Type u} → {β : outParam (Type v)} → {γ : outParam (Type w)} → [self : Std.Ric.Sliceable α β γ] → α → Std.Ric β → γ | Slices `carrier` up to `range.upper` \(inclusive\).
| true |
Std.LawfulOrderLeftLeaningMax | Init.Data.Order.Classes | (α : Type u) → [Max α] → [LE α] → Prop | This typeclass states that `max a b = if b ≤ a then a else b` (for any `DecidableLE α` instance).
| true |
_private.Mathlib.Topology.DiscreteSubset.0.isDiscrete_of_codiscreteWithin._simp_1_1 | Mathlib.Topology.DiscreteSubset | ∀ {α : Type u_1} {f : Filter α} {s : Set α}, (f ⊓ Filter.principal sᶜ = ⊥) = (s ∈ f) | null | false |
Mathlib.Tactic._aux_Mathlib_Tactic_Cases___elabRules_Mathlib_Tactic_induction'_1 | Mathlib.Tactic.Cases | Lean.Elab.Tactic.Tactic | `induction' x` applies induction on the variable `x` of the inductive type `t` to the main goal,
producing one goal for each constructor of `t`, in which `x` is replaced by that constructor
applied to newly introduced variables. `induction'` adds an inductive hypothesis for
each recursive argument to the constructor. T... | false |
CategoryTheory.ComposableArrows.Precomp.map_zero_one' | Mathlib.CategoryTheory.ComposableArrows.Basic | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {n : ℕ} (F : CategoryTheory.ComposableArrows C n) {X : C}
(f : X ⟶ F.left), CategoryTheory.ComposableArrows.Precomp.map F f 0 ⟨0 + 1, ⋯⟩ ⋯ = f | null | true |
NonUnitalSeminormedCommRing.mk | Mathlib.Analysis.Normed.Ring.Basic | {α : Type u_5} →
[toNonUnitalSeminormedRing : NonUnitalSeminormedRing α] → (∀ (a b : α), a * b = b * a) → NonUnitalSeminormedCommRing α | null | true |
_private.Mathlib.Control.Fix.0.Part.Fix.approx.match_1.eq_2 | Mathlib.Control.Fix | ∀ (motive : ℕ → Sort u_1) (i : ℕ) (h_1 : Unit → motive 0) (h_2 : (i : ℕ) → motive i.succ),
(match i.succ with
| 0 => h_1 ()
| i.succ => h_2 i) =
h_2 i | null | true |
card_vector | Mathlib.Data.Fintype.BigOperators | ∀ {α : Type u_1} [inst : Fintype α] (n : ℕ), Fintype.card (List.Vector α n) = Fintype.card α ^ n | null | true |
_private.Std.Sat.CNF.Sat.0.Std.Sat.CNF.sat_iff_all_mem_sat._simp_1_1 | Std.Sat.CNF.Sat | ∀ {α : Type u_1} (a : α → Bool) (f : Std.Sat.CNF α), Std.Sat.CNF.Sat a f = (Std.Sat.CNF.eval a f = true) | null | false |
String.intercalate_cons_of_ne_nil | Init.Data.String.Lemmas.Intercalate | ∀ {s t : String} {l : List String}, l ≠ [] → s.intercalate (t :: l) = t ++ s ++ s.intercalate l | null | true |
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' | Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | ∀ {ι : Type u_1} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E} {c : E}
{l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [CompleteSpace E]
[l'.IsMeasurablyGenerated] [Filter.TendstoIxxClass Set.Ioc l l'],
StronglyMeasurableAtFilter f l' μ →
Fil... | **Fundamental theorem of calculus-1**, local version for any measure.
Let filters `l` and `l'` be related by `TendstoIxxClass Ioc`.
If `f` has a finite limit `c` at `l ⊓ ae μ`, where `μ` is a measure
finite at `l`, then `∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both
`u` and `v` tend to `l` so that `v ... | true |
PowerBasis.ofAdjoinEqTop'.congr_simp | Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : Algebra R S]
[inst_4 : IsIntegrallyClosed R] [inst_5 : IsDomain S] [inst_6 : Module.IsTorsionFree R S] {x x_1 : S} (e_x : x = x_1)
(hx : IsIntegral R x) (hx' : R[x] = ⊤), PowerBasis.ofAdjoinEqTop' hx hx' = Powe... | null | true |
_private.Mathlib.NumberTheory.Padics.Measure.Basic.0.AbstractMeasure.contractSnd._proof_3 | Mathlib.NumberTheory.Padics.Measure.Basic | ∀ {X : Type u_3} {Y : Type u_1} {R : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
[inst_2 : CommRing R] [inst_3 : TopologicalSpace R] [inst_4 : IsTopologicalRing R] (c : R) (m : AbstractMeasure Y R R)
(n : C(X × Y, R)), (↑(c • m)).comp n.curry = c • (↑m).comp n.curry | null | false |
AddSubgroup.one_le_sum_inv_index_of_leftCoset_cover | Mathlib.GroupTheory.CosetCover | ∀ {G : Type u_1} [inst : AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι},
⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ → 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹ | Let the additive group `G` be the union
of finitely many left cosets `g i +ᵥ H i`. Then the sum of the inverses of the indexes of the
additive subgroups `H i` is greater than or equal to 1. | true |
CategoryTheory.Comon.tensorObj_comul | Mathlib.CategoryTheory.Monoidal.Comon_ | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
[inst_2 : CategoryTheory.BraidedCategory C] (A B : C) [inst_3 : CategoryTheory.ComonObj A]
[inst_4 : CategoryTheory.ComonObj B],
CategoryTheory.ComonObj.comul =
CategoryTheory.CategoryStruct.comp
(Ca... | The comultiplication on the tensor product of two comonoids is
the tensor product of the comultiplications followed by the tensor strength
(to shuffle the factors back into order).
| true |
Mathlib.Tactic.ClickSuggestions.GrwLemma.mk._flat_ctor | Mathlib.Tactic.ClickSuggestions.GRewrite | Mathlib.Tactic.ClickSuggestions.Premise → Bool → Lean.Name → Mathlib.Tactic.ClickSuggestions.GrwLemma | null | false |
Lean.FromJson.mk.noConfusion | Lean.Data.Json.FromToJson.Basic | {α : Type u} →
{P : Sort u_1} →
{fromJson? fromJson?' : Lean.Json → Except String α} →
{ fromJson? := fromJson? } = { fromJson? := fromJson?' } → (fromJson? ≍ fromJson?' → P) → P | null | false |
_private.Std.Sat.AIG.CachedLemmas.0.Std.Sat.AIG.mkGateCached.go.match_1.splitter | Std.Sat.AIG.CachedLemmas | (motive : Option Bool → Option Bool → Sort u_1) →
(lhsVal rhsVal : Option Bool) →
((x : Option Bool) → motive (some false) x) →
((x : Option Bool) → (x = some false → False) → motive x (some false)) →
((x : Option Bool) → (x = some false → False) → motive (some true) x) →
((x : Option Bool... | null | true |
CategoryTheory.MonoidalCategory.Limits.preservesColimit_curriedTensor | Mathlib.CategoryTheory.Monoidal.Limits.Preserves | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.MonoidalCategory C]
{J : Type u_2} [inst_2 : CategoryTheory.Category.{v_2, u_2} J] (F : CategoryTheory.Functor J C)
[h : ∀ (c : C), CategoryTheory.Limits.PreservesColimit F (CategoryTheory.MonoidalCategory.tensorRight c)],
Cat... | null | true |
SimplexCategory.factorThruImage_eq | Mathlib.AlgebraicTopology.SimplexCategory.Basic | ∀ {Δ Δ'' : SimplexCategory} {φ : Δ ⟶ Δ''} {e : Δ ⟶ CategoryTheory.Limits.image φ} [CategoryTheory.Epi e]
{i : CategoryTheory.Limits.image φ ⟶ Δ''} [CategoryTheory.Mono i],
CategoryTheory.CategoryStruct.comp e i = φ → CategoryTheory.Limits.factorThruImage φ = e | null | true |
_private.Mathlib.LinearAlgebra.Span.Defs.0.Submodule.mem_span_pair._simp_1_3 | Mathlib.LinearAlgebra.Span.Defs | ∀ {α : Sort u_1} {β : Sort u_2} {f : α → β} {p : β → Prop}, (∃ b, (∃ a, f a = b) ∧ p b) = ∃ a, p (f a) | null | false |
Rat.intCast_div_eq_divInt | Mathlib.Data.Rat.Defs | ∀ (n d : ℤ), ↑n / ↑d = Rat.divInt n d | null | true |
Module.End.isSemisimple_iff' | Mathlib.LinearAlgebra.Semisimple | ∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{f : Module.End R M}, f.IsSemisimple ↔ ∀ (p : ↥f.invtSubmodule), ∃ q, IsCompl p q | A linear endomorphism is semisimple if every invariant submodule has in invariant complement.
See also `Module.End.isSemisimple_iff`. | true |
_private.Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk.0.SzemerediRegularity.sum_density_div_card_le_density_add_eps._simp_1_4 | Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
CategoryTheory.MonoidalOpposite.mopMopEquivalence.eq_1 | Mathlib.CategoryTheory.Monoidal.Opposite | ∀ (C : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} C],
CategoryTheory.MonoidalOpposite.mopMopEquivalence C =
(CategoryTheory.MonoidalOpposite.unmopEquiv Cᴹᵒᵖ).trans (CategoryTheory.MonoidalOpposite.unmopEquiv C) | null | true |
TopCat.toSSetObj₁Equiv._proof_3 | Mathlib.Topology.Homotopy.TopCat.ZerothHomotopy | ContinuousMapClass (↑TopCat.I ≃ₜ Convexity.StdSimplex ℝ (Fin 2)) (↑TopCat.I) (Convexity.StdSimplex ℝ (Fin 2)) | null | false |
mul_finsum' | Mathlib.Algebra.BigOperators.Finprod | ∀ {α : Type u_1} {R : Type u_7} [inst : NonUnitalNonAssocSemiring R] (f : α → R) (r : R),
Function.HasFiniteSupport f → r * ∑ᶠ (a : α), f a = ∑ᶠ (a : α), r * f a | For functions with finite support, multiplication commutes with finsums. See `mul_finsum` for a
statement assuming that `R` has no zero divisors.
| true |
Std.Internal.UV.TCP.Socket.getSockName | Std.Internal.UV.TCP | Std.Internal.UV.TCP.Socket → IO Std.Net.SocketAddress | Gets the local address of a bound TCP socket.
| true |
GrpCat.groupObj._aux_16 | Mathlib.Algebra.Category.Grp.Limits | {J : Type u_3} →
[inst : CategoryTheory.Category.{u_2, u_3} J] →
(F : CategoryTheory.Functor J GrpCat) → (j : J) → ZPow ((F.comp (CategoryTheory.forget GrpCat)).obj j) | null | false |
FinBddDistLat.inv_hom_apply | Mathlib.Order.Category.FinBddDistLat | ∀ {X Y : FinBddDistLat} (e : X ≅ Y) (x : ↑X.toDistLat),
(CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x | null | true |
TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ {S : Type u_1} {R : Type u} {M : Type v} [inst : CommSemiring S] [inst_1 : Semiring R] [inst_2 : AddCommMonoid M]
[inst_3 : Algebra S R] [inst_4 : Module S M] [inst_5 : Module R M] [inst_6 : Module Rᵐᵒᵖ M]
[inst_7 : SMulCommClass R Rᵐᵒᵖ M] [inst_8 : IsScalarTower S R M] [inst_9 : IsScalarTower S Rᵐᵒᵖ M],
(TrivS... | null | true |
Exists | Init.Core | {α : Sort u} → (α → Prop) → Prop | Existential quantification. If `p : α → Prop` is a predicate, then `∃ x : α, p x`
asserts that there is some `x` of type `α` such that `p x` holds.
To create an existential proof, use the `exists` tactic,
or the anonymous constructor notation `⟨x, h⟩`.
To unpack an existential, use `cases h` where `h` is a proof of `∃ ... | true |
Lean.Meta.Grind.AC.EqCnstrProof.below | Lean.Meta.Tactic.Grind.AC.Types | {motive_1 : Lean.Meta.Grind.AC.EqCnstr → Sort u} →
{motive_2 : Lean.Meta.Grind.AC.EqCnstrProof → Sort u} → Lean.Meta.Grind.AC.EqCnstrProof → Sort (max 1 u) | null | false |
OrthonormalBasis.mkOfOrthogonalEqBot.congr_simp | Mathlib.Analysis.InnerProductSpace.PiL2 | ∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] {v v_1 : ι → E} (e_v : v = v_1) (hon : Orthonormal 𝕜 v)
(hsp : (Submodule.span 𝕜 (Set.range v))ᗮ = ⊥),
OrthonormalBasis.mkOfOrthogonalEqBot hon hsp = Orthonor... | null | true |
Std.TreeMap.maxKeyD | Std.Data.TreeMap.Basic | {α : Type u} → {β : Type v} → {cmp : α → α → Ordering} → Std.TreeMap α β cmp → α → α | Tries to retrieve the largest key in the tree map, returning `fallback` if the tree map is empty.
| true |
FinTopCat.instCategory._aux_5 | Mathlib.Topology.Category.FinTopCat | {X Y Z : FinTopCat} → (X ⟶ Y) → (Y ⟶ Z) → (X ⟶ Z) | null | false |
Fin.finsetImage_val_Iic | Mathlib.Order.Interval.Finset.Fin | ∀ {n : ℕ} (a : Fin n), Finset.image Fin.val (Finset.Iic a) = Finset.Iic ↑a | null | true |
LinearEquiv.fixedReduce_mkQ | Mathlib.LinearAlgebra.FixedSubmodule | ∀ {R : Type u_4} {V : Type u_5} [inst : Ring R] [inst_1 : AddCommGroup V] [inst_2 : Module R V] (e : V ≃ₗ[R] V) (x : V),
e.fixedReduce ((↑e).fixedSubmodule.mkQ x) = (↑e).fixedSubmodule.mkQ (e x) | null | true |
SaturatedSubmonoid.toSubmonoid | Mathlib.Algebra.Group.Submonoid.Saturation | {M : Type u_1} → [inst : MulOneClass M] → SaturatedSubmonoid M → Submonoid M | null | true |
_private.Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureFibrant.0.HomotopicalAlgebra.FibrantObject.instWeakEquivalenceWWeakEquivalences._simp_1 | Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureFibrant | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C],
HomotopicalAlgebra.WeakEquivalence f = HomotopicalAlgebra.weakEquivalences C f | null | false |
FirstOrder.Language.BoundedFormula.IsQF.isPrenex | Mathlib.ModelTheory.Complexity | ∀ {L : FirstOrder.Language} {α : Type u'} {n : ℕ} {φ : L.BoundedFormula α n}, φ.IsQF → φ.IsPrenex | null | true |
QuaternionGroup.instGroup._proof_10 | Mathlib.GroupTheory.SpecificGroups.Quaternion | ∀ {n : ℕ} (a : QuaternionGroup n), a * 1 = a | null | false |
UniformContinuous.dist | Mathlib.Topology.MetricSpace.Pseudo.Constructions | ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoMetricSpace α] [inst_1 : UniformSpace β] {f g : β → α},
UniformContinuous f → UniformContinuous g → UniformContinuous fun b => dist (f b) (g b) | null | true |
AddEquiv.linearEquiv | Mathlib.Algebra.Module.TransferInstance | (R : Type u_1) →
{α : Type u_2} →
{β : Type u_3} →
[inst : Semiring R] →
[inst_1 : AddCommMonoid α] → [inst_2 : AddCommMonoid β] → [inst_3 : Module R β] → (e : α ≃+ β) → α ≃ₗ[R] β | When `α` is equipped with the `A`-module structure transferred via `e : α ≃+ β`,
this isomorphism is `A`-linear. | true |
Ring.instDirectLimit._proof_24 | Mathlib.Algebra.Colimit.Ring | ∀ {ι : Type u_1} [inst : Preorder ι] (G : ι → Type u_2) [inst_1 : (i : ι) → CommRing (G i)]
(f : (i j : ι) → i ≤ j → G i → G j),
autoParam (∀ (a b : Ring.DirectLimit G f), a - b = a + -b) SubNegMonoid.sub_eq_add_neg._autoParam | null | false |
SSet.Truncated.ι0₂._proof_3 | Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | { len := 2 }.len ≤ 2 | null | false |
finSuccEquiv'_ne_last_apply | Mathlib.Logic.Equiv.Fin.Basic | ∀ {n : ℕ} {i j : Fin (n + 1)} (hi : i ≠ Fin.last n), j ≠ i → (finSuccEquiv' i) j = some ((i.castLT ⋯).predAbove j) | null | true |
List.sym._unary._proof_3 | Mathlib.Data.List.Sym | ∀ {α : Type u_1} (n : ℕ) (x : α) (xs : List α),
(invImage (fun x => PSigma.casesOn x fun n a => (n, a)) Prod.instWellFoundedRelation).1 ⟨n + 1, xs⟩ ⟨n.succ, x :: xs⟩ | null | false |
ChainComplex.cycles₀Iso._proof_3 | Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | IsRightCancelAdd ℕ | null | false |
AddCircle.EndpointIdent.recOn | Mathlib.Topology.Instances.AddCircle.Defs | {𝕜 : Type u_1} →
[inst : AddCommGroup 𝕜] →
[inst_1 : LinearOrder 𝕜] →
[inst_2 : IsOrderedAddMonoid 𝕜] →
{p a : 𝕜} →
[hp : Fact (0 < p)] →
{motive : (a_1 a_2 : ↑(Set.Icc a (a + p))) → AddCircle.EndpointIdent p a a_1 a_2 → Sort u} →
{a_1 a_2 : ↑(Set.Icc a (a + ... | null | false |
FunLike.compMonoidWithZero._proof_2 | Mathlib.Algebra.GroupWithZero.FunLike | ∀ {F : Type u_1} {α : Type u_2} [inst : FunLike F α α] [inst_1 : One F] [inst_2 : Mul F] [IsOneApplyEqSelf F α]
[IsMulApplyEqComp F α] (x : F), 1 * x = x | null | false |
Lean.Meta.Grind.Arith.Cutsat.State.rec | Lean.Meta.Tactic.Grind.Arith.Cutsat.Types | {motive : Lean.Meta.Grind.Arith.Cutsat.State → Sort u} →
((vars : Lean.PArray Lean.Expr) →
(varMap : Lean.PHashMap Lean.Meta.Sym.ExprPtr Int.Internal.Linear.Var) →
(vars' : Lean.PArray Lean.Expr) →
(varMap' : Lean.PHashMap Lean.Meta.Sym.ExprPtr Int.Internal.Linear.Var) →
(natToIntM... | null | false |
BitVec.twoPow_mul_eq_shiftLeft | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} (x : BitVec w) (i : ℕ), BitVec.twoPow w i * x = x <<< i | null | true |
CategoryTheory.Coyoneda.isIso | Mathlib.CategoryTheory.Yoneda | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : Cᵒᵖ} (f : X ⟶ Y)
[CategoryTheory.IsIso (CategoryTheory.coyoneda.map f)], CategoryTheory.IsIso f | If `coyoneda.map f` is an isomorphism, so was `f`.
| true |
EmbeddingLike.map_eq_one_iff._simp_2 | Mathlib.Algebra.Group.Equiv.Defs | ∀ {F : Type u_1} {M : Type u_4} {N : Type u_5} [inst : One M] [inst_1 : One N] [inst_2 : FunLike F M N]
[EmbeddingLike F M N] [OneHomClass F M N] {f : F} {x : M}, (f x = 1) = (x = 1) | null | false |
AddEquiv.toNatLinearEquiv_trans | Mathlib.Algebra.Module.Equiv.Basic | ∀ {M : Type u_5} {M₂ : Type u_7} {M₃ : Type u_8} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₂]
[inst_2 : AddCommMonoid M₃] (e : M ≃+ M₂) (e₂ : M₂ ≃+ M₃),
(e.trans e₂).toNatLinearEquiv = e.toNatLinearEquiv ≪≫ₗ e₂.toNatLinearEquiv | null | true |
_private.Mathlib.Tactic.Tauto.0.Mathlib.Tactic.Tauto.«term_<;>_» | Mathlib.Tactic.Tauto | Lean.TrailingParserDescr | Simulates the `<;>` tactic combinator.
First runs `tac1` and then runs `tac2` on all newly-generated subgoals.
| true |
Colex.instIsRightCancelAdd | Mathlib.Algebra.Order.Group.Synonym | ∀ {α : Type u_1} [inst : Add α] [IsRightCancelAdd α], IsRightCancelAdd (Colex α) | null | true |
CategoryTheory.Adjunction.homEquiv_naturality_left_square_assoc._to_dual_1 | Mathlib.CategoryTheory.Adjunction.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : G ⊣ F) {X' X : C} {Y Y' : D} (f : X ⟶ X')
(g : Y' ⟶ F.obj X) (h : Y ⟶ F.obj X') (k : Y' ⟶ Y),
CategoryTheory.CategoryStru... | null | false |
ContinuousLinearMap.prodMapL._proof_32 | Mathlib.Analysis.Normed.Operator.Prod | ∀ (𝕜 : Type u_1) [inst : NontriviallyNormedField 𝕜] (M₂ : Type u_2) (M₃ : Type u_4) (M₄ : Type u_3)
[inst_1 : SeminormedAddCommGroup M₂] [inst_2 : NormedSpace 𝕜 M₂] [inst_3 : SeminormedAddCommGroup M₃]
[inst_4 : NormedSpace 𝕜 M₃] [inst_5 : SeminormedAddCommGroup M₄] [inst_6 : NormedSpace 𝕜 M₄],
IsNormableSpa... | null | false |
_private.Mathlib.SetTheory.ZFC.PSet.0.PSet.Subset.congr_right.match_1_5 | Mathlib.SetTheory.ZFC.PSet | ∀ («Type» : Type u_1) (Func : «Type» → PSet.{u_1}) (Type_1 : Type u_1) (Func_1 : Type_1 → PSet.{u_1})
(b : { «Type» := Type_1, Func := Func_1 }.Type) (motive : (∃ a, (Func a).Equiv (Func_1 b)) → Prop)
(x : ∃ a, (Func a).Equiv (Func_1 b)), (∀ (a : «Type») (ab : (Func a).Equiv (Func_1 b)), motive ⋯) → motive x | null | false |
_private.Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs.0.isLinearSet_iff._simp_1_2 | Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | ∀ {α : Type u} {p : Finset α → Prop}, (∃ s, p s) = ∃ s, ∃ (hs : s.Finite), p hs.toFinset | null | false |
Function.locallyFinsuppWithin.instAddCommGroup._proof_7 | Mathlib.Topology.LocallyFinsupp | ∀ {X : Type u_1} [inst : TopologicalSpace X] {U : Set X} {Y : Type u_2} [inst_1 : AddCommGroup Y]
(D : Function.locallyFinsuppWithin U Y) (n : ℤ), ⇑(n • D) = n • ⇑D | null | false |
SimpleGraph.Walk.length_ofSupport | Mathlib.Combinatorics.SimpleGraph.Walk.Basic | ∀ {V : Type u} {G : SimpleGraph V} {l : List V} (hne : l ≠ []) (hchain : List.IsChain G.Adj l),
(SimpleGraph.Walk.ofSupport l hne hchain).length = l.length - 1 | null | true |
EuclideanDomain.wellFoundedRelation | Mathlib.Algebra.EuclideanDomain.Defs | {R : Type u} → [EuclideanDomain R] → WellFoundedRelation R | null | true |
UInt64.toUSize_mod_of_dvd_usizeSize | Init.Data.UInt.Lemmas | ∀ (a b : UInt64), b.toNat ∣ USize.size → (a % b).toUSize = a.toUSize % b.toUSize | null | true |
_private.Mathlib.Order.SuccPred.LinearLocallyFinite.0.toZ_neg._proof_1_1 | Mathlib.Order.SuccPred.LinearLocallyFinite | ∀ {ι : Type u_1} [inst : LinearOrder ι] [inst_1 : SuccOrder ι] [inst_2 : IsSuccArchimedean ι] [inst_3 : PredOrder ι]
{i0 i : ι}, i < i0 → ∃ n, Order.pred^[n] i0 = i | null | false |
Lean.Lsp.Location._sizeOf_1 | Lean.Data.Lsp.Basic | Lean.Lsp.Location → ℕ | null | false |
Complex.dist_of_im_eq | Mathlib.Analysis.Complex.Norm | ∀ {z w : ℂ}, z.im = w.im → dist z w = dist z.re w.re | null | true |
Real.tendsto_eulerMascheroniSeq' | Mathlib.NumberTheory.Harmonic.EulerMascheroni | Filter.Tendsto Real.eulerMascheroniSeq' Filter.atTop (nhds Real.eulerMascheroniConstant) | null | true |
Submonoid.mem_saturation_iff | Mathlib.Algebra.Group.Submonoid.Saturation | ∀ {M : Type u_1} [inst : CommMonoid M] {s : Submonoid M} {x : M}, x ∈ s.saturation ↔ ∃ y, x * y ∈ s | null | true |
vadd_mem_nhds_self._simp_1 | Mathlib.Topology.Algebra.ConstMulAction | ∀ {G : Type u_4} [inst : AddGroup G] [inst_1 : TopologicalSpace G] [ContinuousConstVAdd G G] {g : G} {s : Set G},
(g +ᵥ s ∈ nhds g) = (s ∈ nhds 0) | null | false |
Filter.Tendsto.add_atBot | Mathlib.Topology.Order.LeftRightNhds | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : AddCommGroup α] [inst_2 : LinearOrder α]
[IsOrderedAddMonoid α] [OrderTopology α] {l : Filter β} {f g : β → α} {C : α},
Filter.Tendsto f l (nhds C) → Filter.Tendsto g l Filter.atBot → Filter.Tendsto (fun x => f x + g x) l Filter.atBot | In a linearly ordered additive commutative group with the order
topology, if `f` tends to `C` and `g` tends to `atBot` then `f + g` tends to `atBot`. | true |
_private.Aesop.Script.StructureStatic.0.Aesop.Script.StaticStructureM.run._proof_2 | Aesop.Script.StructureStatic | ∀ (script : Aesop.Script.UScript) (i : ℕ) (h : i ∈ [:Array.size script]),
script[i].postGoals.size = 1 → 0 < script[i].postGoals.size | null | false |
Int.floor_lt_ceil_of_lt | Mathlib.Algebra.Order.Floor.Ring | ∀ {R : Type u_2} [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : FloorRing R] {a b : R}, a < b → ⌊a⌋ < ⌈b⌉ | null | true |
_private.Mathlib.Algebra.Lie.Weights.RootSystem.0.LieAlgebra.IsKilling.exists_mem_rootSpace_lie_ne_zero._proof_1_7 | Mathlib.Algebra.Lie.Weights.RootSystem | ∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
[inst_4 : LieAlgebra.IsKilling K L] [inst_5 : FiniteDimensional K L] {H : LieSubalgebra K L}
[inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L] {α β : LieModule.Weight ... | null | false |
MeasureTheory.volume_pi_ball | Mathlib.MeasureTheory.Constructions.Pi | ∀ {ι : Type u_1} {α : ι → Type u_3} [inst : Fintype ι] [inst_1 : (i : ι) → MeasureTheory.MeasureSpace (α i)]
[∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [inst_3 : (i : ι) → MetricSpace (α i)] (x : (i : ι) → α i)
{r : ℝ}, 0 < r → MeasureTheory.volume (Metric.ball x r) = ∏ i, MeasureTheory.volume (Met... | null | true |
Dioph._aux_Mathlib_NumberTheory_Dioph___unexpand_Dioph_eq_dioph_1 | Mathlib.NumberTheory.Dioph | Lean.PrettyPrinter.Unexpander | null | false |
Std.Net.IPv4Addr.octets | Std.Net.Addr | Std.Net.IPv4Addr → Vector UInt8 4 | This structure represents the address: `octets[0].octets[1].octets[2].octets[3]`.
| true |
_private.Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex.0.NumberField.nrComplexPlaces_eq_zero_iff._simp_1_2 | Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | ∀ {α : Sort u_1} (p : α → Prop), IsEmpty (Subtype p) = ∀ (x : α), ¬p x | null | false |
FintypeCat.instFiniteAut | Mathlib.CategoryTheory.FintypeCat | ∀ (X : FintypeCat), Finite (CategoryTheory.Aut X) | null | true |
AdicCompletion.isAdicCauchy_iff | Mathlib.RingTheory.AdicCompletion.Basic | ∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R) (M : Type u_4) [inst_1 : AddCommGroup M] [inst_2 : Module R M]
(f : ℕ → M), AdicCompletion.IsAdicCauchy I M f ↔ ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD I ^ n • ⊤] | The `I`-adic Cauchy condition can be checked on successive `n`. | true |
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