name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Homotopy.nullHomotopicMap'_f_eq_zero | Mathlib.Algebra.Homology.Homotopy | ∀ {ι : Type u_1} {V : Type u} [inst : CategoryTheory.Category.{v, u} V] [inst_1 : CategoryTheory.Preadditive V]
{c : ComplexShape ι} {C D : HomologicalComplex V c} {k₀ : ι},
(∀ (l : ι), ¬c.Rel k₀ l) →
(∀ (l : ι), ¬c.Rel l k₀) → ∀ (h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)), (Homotopy.nullHomotopicMap' h).f k₀... | null | true |
IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe | Mathlib.RingTheory.Localization.AtPrime.Basic | ∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [inst_3 : IsLocalization.AtPrime S I] (a : { p // p.IsPrime ∧ p ≤ I }),
↑↑((RelIso.symm (IsLocalization.AtPrime.orderIsoOfPrime S I)) a) =
⋂ s,
⋂ (_ :
↑↑((Set.ord... | null | true |
List.subset_dedup | Mathlib.Data.List.Dedup | ∀ {α : Type u_1} [inst : DecidableEq α] (l : List α), l ⊆ l.dedup | null | true |
_private.Std.Tactic.BVDecide.LRAT.Internal.LRATCheckerSound.0.Std.Tactic.BVDecide.LRAT.Internal.addEmptyCaseSound._proof_1_3 | Std.Tactic.BVDecide.LRAT.Internal.LRATCheckerSound | ∀ {α : Type u_2} {β : Type u_1} {σ : Type u_3} [inst : Std.Tactic.BVDecide.LRAT.Internal.Clause α β]
[inst_1 : Std.Tactic.BVDecide.LRAT.Internal.Entails α σ] [inst_2 : Std.Tactic.BVDecide.LRAT.Internal.Formula α β σ]
(f : σ),
Std.Tactic.BVDecide.LRAT.Internal.Clause.empty ∈
Std.Tactic.BVDecide.LRAT.Internal.F... | null | false |
Subring.topologicalClosure._proof_3 | Mathlib.Topology.Algebra.Ring.Basic | ∀ {R : Type u_1} [inst : TopologicalSpace R] [inst_1 : Ring R] [inst_2 : IsSemitopologicalRing R] (S : Subring R)
{a b : R}, a ∈ S.topologicalClosure.carrier → b ∈ S.topologicalClosure.carrier → a * b ∈ S.topologicalClosure.carrier | null | false |
USize.ofFin_mk | Init.Data.UInt.Lemmas | ∀ {n : ℕ} (hn : n < USize.size), USize.ofFin ⟨n, hn⟩ = USize.ofNatLT n hn | null | true |
LawfulBitraversable.binaturality' | Mathlib.Control.Bitraversable.Basic | ∀ {t : Type u → Type u → Type u} {inst : Bitraversable t} [self : LawfulBitraversable t] {F G : Type u → Type u}
[inst_1 : Applicative F] [inst_2 : Applicative G] [LawfulApplicative F] [LawfulApplicative G]
(η : ApplicativeTransformation F G) {α α' β β' : Type u} (f : α → F β) (f' : α' → F β'),
(fun {α} => η.app ... | null | true |
Localization.eq_1 | Mathlib.GroupTheory.MonoidLocalization.Basic | ∀ {M : Type u_1} [inst : CommMonoid M] (S : Submonoid M), Localization S = OreLocalization S M | null | true |
Inclusion.instMonadBacktrackProdSavedStateStateInclusionM | Mathlib.Tactic.Inclusion.Core.Types | Lean.MonadBacktrack (Lean.Meta.SavedState × Inclusion.InclusionM.State) Inclusion.InclusionM | null | true |
Finsupp.fst_sumFinsuppAddEquivProdFinsupp | Mathlib.Data.Finsupp.Basic | ∀ {M : Type u_5} [inst : AddMonoid M] {α : Type u_12} {β : Type u_13} (f : α ⊕ β →₀ M) (x : α),
(Finsupp.sumFinsuppAddEquivProdFinsupp f).1 x = f (Sum.inl x) | null | true |
List.scanrM_map | Init.Data.List.Scan.Lemmas | ∀ {m : Type u_1 → Type u_2} {α₁ : Type u_3} {α₂ : Type u_4} {β : Type u_1} {init : β} [inst : Monad m] [LawfulMonad m]
{f : α₁ → α₂} {g : α₂ → β → m β} {as : List α₁},
List.scanrM g init (List.map f as) = List.scanrM (fun a b => g (f a) b) init as | null | true |
Std.TreeMap.Raw.instCoeWFWFInner | Std.Data.TreeMap.Raw.Basic | {α : Type u} → {β : Type v} → {cmp : α → α → Ordering} → {t : Std.TreeMap.Raw α β cmp} → Coe t.WF t.inner.WF | null | true |
MonomialOrder.leadingTerm_add_of_lt | Mathlib.RingTheory.MvPolynomial.MonomialOrder | ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommSemiring R] {f g : MvPolynomial σ R},
m.toSyn (m.degree g) < m.toSyn (m.degree f) → m.leadingTerm (f + g) = m.leadingTerm f | null | true |
Representation.linearize._proof_2 | Mathlib.RepresentationTheory.Action | ∀ (k : Type u_1) (G : Type u_3) [inst : Monoid G] [inst_1 : Semiring k] (X : Action (Type u_2) G) (x x_1 : G),
MonoidAlgebra.mapDomainLinearMap k k ⇑(CategoryTheory.ConcreteCategory.hom (X.ρ (x * x_1))) =
MonoidAlgebra.mapDomainLinearMap k k ⇑(CategoryTheory.ConcreteCategory.hom (X.ρ x)) *
MonoidAlgebra.map... | null | false |
_private.Init.Data.String.Lemmas.Pattern.Memcmp.0.String.Slice.Pattern.Internal.memcmpStr_eq_true_iff._proof_1_8 | Init.Data.String.Lemmas.Pattern.Memcmp | ∀ {lhs rhs : String} {lstart rstart : String.Pos.Raw} {len : String.Pos.Raw} (curr : String.Pos.Raw),
¬len.byteIdx < curr.byteIdx → ¬len.byteIdx = curr.byteIdx + (len.byteIdx - curr.byteIdx) → False | null | false |
NNReal.iSup_div | Mathlib.Basic.NNReal.Basic | ∀ {ι : Sort u_3} (f : ι → NNReal) (a : NNReal), (⨆ i, f i) / a = ⨆ i, f i / a | null | true |
SeminormedAddCommGroup.toIsTopologicalAddGroup | Mathlib.Analysis.Normed.Group.Uniform | ∀ {E : Type u_2} [inst : SeminormedAddCommGroup E], IsTopologicalAddGroup E | null | true |
List.dropWhile_eq_self_iff | Mathlib.Data.List.TakeWhile | ∀ {α : Type u_1} {p : α → Bool} {l : List α}, List.dropWhile p l = l ↔ ∀ (hl : 0 < l.length), ¬p l[0] = true | null | true |
FirstOrder.Language.BoundedFormula.realize_relabel._simp_1 | Mathlib.ModelTheory.Semantics | ∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {α : Type u'} {β : Type v'} {m n : ℕ}
{φ : L.BoundedFormula α n} {g : α → β ⊕ Fin m} {v : β → M} {xs : Fin (m + n) → M},
(FirstOrder.Language.BoundedFormula.relabel g φ).Realize v xs =
φ.Realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAd... | null | false |
map_mem_separableClosure_iff | Mathlib.FieldTheory.SeparableClosure | ∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {K : Type w} [inst_3 : Field K]
[inst_4 : Algebra F K] (i : E →ₐ[F] K) {x : E}, i x ∈ separableClosure F K ↔ x ∈ separableClosure F E | If `i` is an `F`-algebra homomorphism from `E` to `K`, then `i x` is contained in
`separableClosure F K` if and only if `x` is contained in `separableClosure F E`. | true |
UInt32.toBitVec_toUInt8 | Init.Data.UInt.Lemmas | ∀ (n : UInt32), n.toUInt8.toBitVec = BitVec.setWidth 8 n.toBitVec | null | true |
Aesop.PostponedSafeRule.mk._flat_ctor | Aesop.Tree.UnsafeQueue | Aesop.SafeRule → Aesop.RuleTacOutput → Aesop.PostponedSafeRule | null | false |
AddGroupSeminorm.toSeminormedAddGroup._proof_1 | Mathlib.Analysis.Normed.Group.Defs | ∀ {E : Type u_1} [inst : AddGroup E] (f : AddGroupSeminorm E) (x : E), f (-x + x) = 0 | null | false |
CategoryTheory.instHasLimitsInd | Mathlib.CategoryTheory.Limits.Indization.Category | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C],
CategoryTheory.Limits.HasLimits (CategoryTheory.Ind C) | null | true |
Float.Model.rec.eq._@.Mathlib.Util.CompileInductive.3310122970._hygCtx._hyg.315 | Mathlib.Util.CompileInductive | @Float.Model.rec = @Float.Model.rec✝ | null | false |
List.map_zip_eq_zipWith | Init.Data.List.Zip | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α × β → γ} {l : List α} {l' : List β},
List.map f (l.zip l') = List.zipWith (Function.curry f) l l' | null | true |
Matrix.kroneckerTMulAlgEquiv_symm_apply | Mathlib.RingTheory.MatrixAlgebra | ∀ {m : Type u_2} {n : Type u_3} (R : Type u_5) (S : Type u_6) {A : Type u_7} {B : Type u_8} [inst : CommSemiring R]
[inst_1 : Semiring A] [inst_2 : Semiring B] [inst_3 : Algebra R A] [inst_4 : Algebra R B] [inst_5 : Fintype n]
[inst_6 : DecidableEq n] [inst_7 : CommSemiring S] [inst_8 : Algebra R S] [inst_9 : Algeb... | null | true |
tendsto_fib_succ_div_fib_atTop | Mathlib.Analysis.SpecificLimits.Fibonacci | Filter.Tendsto (fun n => ↑(Nat.fib (n + 1)) / ↑(Nat.fib n)) Filter.atTop (nhds Real.goldenRatio) | The limit of `fib (n + 1) / fib n` as `n → ∞` is the golden ratio. | true |
CommGroup.subgroupOrderIsoSubgroupMonoidHom.eq_1 | Mathlib.GroupTheory.FiniteAbelian.Duality | ∀ (G : Type u_1) (M : Type u_2) [inst : CommGroup G] [inst_1 : Finite G] [inst_2 : CommMonoid M]
[hM : HasEnoughRootsOfUnity M (Monoid.exponent G)],
CommGroup.subgroupOrderIsoSubgroupMonoidHom G M =
{ toFun := fun H => OrderDual.toDual (MonoidHom.domRestrictHom H Mˣ).ker,
invFun := fun Φ =>
(CommG... | null | true |
Nat.lcm_eq_mul_iff | Init.Data.Nat.Lcm | ∀ {m n : ℕ}, m.lcm n = m * n ↔ m = 0 ∨ n = 0 ∨ m.gcd n = 1 | null | true |
Nat.choose_lt_pow | Mathlib.Data.Nat.Choose.Bounds | ∀ {n k : ℕ}, n ≠ 0 → 2 ≤ k → n.choose k < n ^ k | null | true |
String.Slice.SplitIterator.instIteratorIdSubslice.eq_1 | Init.Data.String.Slice | ∀ {ρ : Type} {σ : String.Slice → Type}
[inst : (s : String.Slice) → Std.Iterator (σ s) Id (String.Slice.Pattern.SearchStep s)] {pat : ρ}
[inst_1 : String.Slice.Pattern.ToForwardSearcher pat σ] {s : String.Slice},
String.Slice.SplitIterator.instIteratorIdSubslice =
{
IsPlausibleStep := fun x x_1 =>
... | null | true |
CategoryTheory.Limits.WidePushoutShape.struct._proof_5 | Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | ∀ {J : Type u_1} {Z : CategoryTheory.Limits.WidePushoutShape J}, Z = Z | null | false |
CategoryTheory.Functor.copyObj.eq_1 | Mathlib.CategoryTheory.NatIso | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(F : CategoryTheory.Functor C D) (obj : C → D) (e : (X : C) → F.obj X ≅ obj X),
F.copyObj obj e =
{ obj := obj,
map := fun {X Y} f =>
CategoryTheory.CategoryStruct.comp (e X).inv... | null | true |
Basis.le_span'' | Mathlib.LinearAlgebra.Dimension.StrongRankCondition | ∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [RankCondition R]
{ι : Type u_2} [inst_4 : Fintype ι] (b : Module.Basis ι R M) {w : Set M} [inst_5 : Fintype ↑w],
Submodule.span R w = ⊤ → Fintype.card ι ≤ Fintype.card ↑w | An auxiliary lemma for `Basis.le_span`.
If `R` satisfies the rank condition,
then for any finite basis `b : Basis ι R M`,
and any finite spanning set `w : Set M`,
the cardinality of `ι` is bounded by the cardinality of `w`.
| true |
Module.Baer.extensionOfMaxAdjoin._proof_2 | Mathlib.Algebra.Module.Injective | ∀ {R : Type u_2} [inst : Ring R] {Q : Type u_1} [inst_1 : AddCommGroup Q] [inst_2 : Module R Q] {M : Type u_3}
{N : Type u_4} [inst_3 : AddCommGroup M] [inst_4 : AddCommGroup N] [inst_5 : Module R M] [inst_6 : Module R N]
(i : M →ₗ[R] N) (f : M →ₗ[R] Q) [inst_7 : Fact (Function.Injective ⇑i)] (h : Module.Baer R Q) ... | null | false |
BitVec.forall_zero_iff | Init.Data.BitVec.Decidable | ∀ {P : BitVec 0 → Prop}, (∀ (v : BitVec 0), P v) ↔ P 0#0 | null | true |
_private.Mathlib.Topology.DiscreteSubset.0.isClosed_and_discrete_iff._simp_1_1 | Mathlib.Topology.DiscreteSubset | ∀ {α : Type u_1} [inst : SemilatticeInf α] [inst_1 : OrderBot α] {a b : α}, Disjoint a b = (a ⊓ b = ⊥) | null | false |
Subspace.orderIsoFiniteCodimDim._proof_2 | Mathlib.LinearAlgebra.Dual.Lemmas | ∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : AddCommGroup V] [inst_2 : Module K V]
(W : { W // FiniteDimensional K ↥W }ᵒᵈ), FiniteDimensional K (V ⧸ Submodule.dualCoannihilator ↑(OrderDual.ofDual W)) | null | false |
Lean.Compiler.LCNF.instBEqParam.beq | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} → Lean.Compiler.LCNF.Param pu → Lean.Compiler.LCNF.Param pu → Bool | null | true |
List.Vector.get_replicate | Mathlib.Data.Vector.Basic | ∀ {α : Type u_1} {n : ℕ} (a : α) (i : Fin n), (List.Vector.replicate n a).get i = a | null | true |
_private.Mathlib.Data.Seq.Parallel.0.Computation.parallel.aux1.eq_1 | Mathlib.Data.Seq.Parallel | ∀ {α : Type u} (l : List (Computation α)) (S : Stream'.WSeq (Computation α)),
Computation.parallel.aux1✝ (l, S) =
Computation.rmap
(fun l' =>
match Stream'.Seq.destruct S with
| none => (l', Stream'.Seq.nil)
| some (none, S') => (l', S')
| some (some c, S') => (c :: l', S'))
... | null | true |
Int.sign_eq_neg_one_iff_neg | Init.Data.Int.Order | ∀ {a : ℤ}, a.sign = -1 ↔ a < 0 | null | true |
ULift.seminormedRing | Mathlib.Analysis.Normed.Ring.Basic | {α : Type u_2} → [SeminormedRing α] → SeminormedRing (ULift.{u_5, u_2} α) | null | true |
AddCircle.equivAddCircle.congr_simp | Mathlib.Topology.Instances.AddCircle.Defs | ∀ {𝕜 : Type u_1} [inst : Field 𝕜] (p q : 𝕜) (hp : p ≠ 0) (hq : q ≠ 0),
AddCircle.equivAddCircle p q hp hq = AddCircle.equivAddCircle p q hp hq | null | true |
TensorProduct.AlgebraTensorModule.congr_symm_tmul | Mathlib.LinearAlgebra.TensorProduct.Tower | ∀ {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [inst : CommSemiring R]
[inst_1 : Semiring A] [inst_2 : Algebra R A] [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M]
[inst_6 : IsScalarTower R A M] [inst_7 : AddCommMonoid N] [inst_8 : Module R N] [inst_9 : ... | null | true |
CategoryTheory.Triangulated.SpectralObject.recOn | Mathlib.CategoryTheory.Triangulated.SpectralObject | {C : Type u_1} →
{ι : Type u_2} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] →
[inst_2 : CategoryTheory.Limits.HasZeroObject C] →
[inst_3 : CategoryTheory.HasShift C ℤ] →
[inst_4 : CategoryTheory.Preadditive C] →
... | null | false |
_private.Mathlib.Tactic.HigherOrder.0.Mathlib.Tactic.higherOrderGetParam.match_1 | Mathlib.Tactic.HigherOrder | (motive : Lean.FVarId × Lean.MVarId → Sort u_1) →
(x : Lean.FVarId × Lean.MVarId) → ((fst : Lean.FVarId) → (mvarId : Lean.MVarId) → motive (fst, mvarId)) → motive x | null | false |
ISize.toInt_inj | Init.Data.SInt.Lemmas | ∀ {x y : ISize}, x.toInt = y.toInt ↔ x = y | null | true |
CategoryTheory.ChosenPullbacksAlong.isoInv_pullback_obj_left | Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {Y X : C} (f : Y ≅ X) (Z : CategoryTheory.Over Y),
((CategoryTheory.ChosenPullbacksAlong.pullback f.inv).obj Z).left = Z.left | null | true |
List.min?_eq_none_iff._simp_1 | Init.Data.List.MinMax | ∀ {α : Type u_1} {xs : List α} [inst : Min α], (xs.min? = none) = (xs = []) | null | false |
Lean.Elab.Term.wrapAsyncAsSnapshot | Lean.Elab.Term.TermElabM | {α : Type} →
(α → Lean.Elab.TermElabM Unit) →
Option IO.CancelToken →
autoParam String Lean.Elab.Term.wrapAsyncAsSnapshot._auto_1 →
Lean.Elab.TermElabM (α → BaseIO Lean.Language.SnapshotTree) | Wraps the given action for use in `BaseIO.asTask` etc., discarding its final state except for
`logSnapshotTask` tasks, which are reported as part of the returned tree. The given cancellation
token, if any, should be stored in a `SnapshotTask` for the server to trigger it when the result is
no longer needed.
| true |
_private.Mathlib.Order.Cover.0.WithTop.coe_wcovBy_top._simp_1_2 | Mathlib.Order.Cover | ∀ {α : Type u_1} [inst : Preorder α] {a : α}, Set.Ioo ↑a ⊤ = WithTop.some '' Set.Ioi a | null | false |
Std.DTreeMap.Internal.Impl.balanceLErase._proof_14 | Std.Data.DTreeMap.Internal.Balancing | ∀ {α : Type u_1} {β : α → Type u_2} (rs : ℕ) (k : α) (v : β k) (l r : Std.DTreeMap.Internal.Impl α β),
(Std.DTreeMap.Internal.Impl.inner rs k v l r).Balanced →
∀ (ls : ℕ) (lk : α) (lv : β lk),
Std.DTreeMap.Internal.delta * rs < ls →
∀ (x : Std.DTreeMap.Internal.Impl α β),
(Std.DTreeMap.Int... | null | false |
Mathlib.Tactic.Sat.Clause.expr | Mathlib.Tactic.Sat.FromLRAT | Mathlib.Tactic.Sat.Clause → Lean.Expr | The clause expression of type `Clause` | true |
_private.Mathlib.LinearAlgebra.Prod.0.Submodule.map_inl._simp_1_7 | Mathlib.LinearAlgebra.Prod | ∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_8}
[inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] {p : Submodule R M} {q : Submodule R M'} {x : M × M'},
(x ∈ p.prod q) = (x.1 ∈ p ∧ x.2 ∈ q) | null | false |
AddMonoidHom.ext_nat | Mathlib.Algebra.Group.Nat.Hom | ∀ {A : Type u_2} [inst : AddZeroClass A] {f g : ℕ →+ A}, f 1 = g 1 → f = g | null | true |
_private.Mathlib.Algebra.ContinuedFractions.Computation.Translations.0.Option.getD.match_1.eq_2 | Mathlib.Algebra.ContinuedFractions.Computation.Translations | ∀ {α : Type u_1} (motive : Option α → Sort u_2) (h_1 : (x : α) → motive (some x)) (h_2 : Unit → motive none),
(match none with
| some x => h_1 x
| none => h_2 ()) =
h_2 () | null | true |
Real.Angle.abs_toReal_coe_eq_self_iff | Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | ∀ {θ : ℝ}, |(↑θ).toReal| = θ ↔ 0 ≤ θ ∧ θ ≤ Real.pi | null | true |
_private.Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy.0.HomotopicalAlgebra.FibrantObject.instHasQuotientWeakEquivalencesHomRel._simp_1 | Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C],
HomotopicalAlgebra.weakEquivalences C f = HomotopicalAlgebra.WeakEquivalence f | null | false |
Complex.measurableEquivRealProd | Mathlib.MeasureTheory.Measure.Lebesgue.Complex | ℂ ≃ᵐ ℝ × ℝ | Measurable equivalence between `ℂ` and `ℝ × ℝ`. | true |
_private.Mathlib.Data.ENat.Pow.0.ENat.epow_top._proof_1_3 | Mathlib.Data.ENat.Pow | ¬⊤ < ⊤ | null | false |
Std.TreeSet.getD_max? | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp] {km fallback : α},
t.max? = some km → t.getD km fallback = km | null | true |
Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Init.Tactics | Lean.Macro | `letI` behaves like `let`, but inlines the value instead of producing a `let` term. | false |
CategoryTheory.Under.instHasColimitsOfSizeOfHasWidePushouts | Mathlib.CategoryTheory.Limits.Constructions.Over.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {B : C} [CategoryTheory.Limits.HasWidePushouts C],
CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, max u v} (CategoryTheory.Under B) | null | true |
Polynomial.divModByMonicAux.congr_simp | Mathlib.Algebra.Polynomial.Div | ∀ {R : Type u} [inst : Ring R] (_p _p_1 : Polynomial R),
_p = _p_1 → ∀ {q q_1 : Polynomial R} (e_q : q = q_1) (a : q.Monic), _p.divModByMonicAux a = _p_1.divModByMonicAux ⋯ | null | true |
List.infix_append' | Init.Data.List.Sublist | ∀ {α : Type u_1} (l₁ l₂ l₃ : List α), l₂ <:+: l₁ ++ (l₂ ++ l₃) | null | true |
_private.Mathlib.Probability.Kernel.IonescuTulcea.Maps.0.IocProdIoc_preimage._proof_1_15 | Mathlib.Probability.Kernel.IonescuTulcea.Maps | ∀ {ι : Type u_1} [inst : LinearOrder ι] [inst_1 : LocallyFiniteOrder ι] {a b c : ι} (hab : a ≤ b) (w : ι) (w_1 : b < w)
(w_2 : w ≤ c), ↑⟨w, ⋯⟩ ∈ Finset.Ioc b c | null | false |
summable_of_hasFiniteSupport | Mathlib.Topology.Algebra.InfiniteSum.Basic | ∀ {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst_1 : TopologicalSpace α] {f : β → α}
{L : SummationFilter β} [L.HasSupport], Function.HasFiniteSupport f → Summable f L | null | true |
SetLike.GradeZero.instRing._aux_6 | Mathlib.Algebra.DirectSum.Internal | {ι : Type u_3} →
{σ : Type u_2} →
{R : Type u_1} →
[inst : Ring R] →
[inst_1 : AddMonoid ι] →
[inst_2 : SetLike σ R] → [AddSubgroupClass σ R] → (A : ι → σ) → [SetLike.GradedMonoid A] → ℤ → ↥(A 0) | null | false |
LinearMap.re_inner_adjoint_mul_self_nonneg | Mathlib.Analysis.InnerProductSpace.Adjoint | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
[inst_3 : FiniteDimensional 𝕜 E] (T : E →ₗ[𝕜] E) (x : E), 0 ≤ RCLike.re (inner 𝕜 x ((LinearMap.adjoint T * T) x)) | The Gram operator T†T is a positive operator. | true |
Algebra.norm_eq_of_ringEquiv | Mathlib.RingTheory.Norm.Basic | ∀ {A : Type u_8} {B : Type u_9} {C : Type u_10} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Ring C]
[inst_3 : Algebra A C] [inst_4 : Algebra B C] (e : A ≃+* B),
(algebraMap B C).comp ↑e = algebraMap A C → ∀ (x : C), e ((Algebra.norm A) x) = (Algebra.norm B) x | null | true |
Std.Time.Modifier.ctorElimType | Std.Time.Format.Modifier | {motive : Std.Time.Modifier → Sort u} → ℕ → Sort (max 1 u) | null | false |
Lean.IR.Decl.ctorElim | Lean.Compiler.IR.Basic | {motive : Lean.IR.Decl → Sort u} →
(ctorIdx : ℕ) → (t : Lean.IR.Decl) → ctorIdx = t.ctorIdx → Lean.IR.Decl.ctorElimType ctorIdx → motive t | null | false |
instIsPredArchimedeanOrderDualOfIsSuccArchimedean | Mathlib.Order.SuccPred.Archimedean | ∀ {α : Type u_1} [inst : Preorder α] [inst_1 : SuccOrder α] [IsSuccArchimedean α], IsPredArchimedean αᵒᵈ | null | true |
MulEquiv.val_piUnits_apply | Mathlib.Algebra.Group.Pi.Units | ∀ {ι : Type u_1} {M : ι → Type u_2} [inst : (i : ι) → Monoid (M i)] (f : ((i : ι) → M i)ˣ) (i : ι),
↑(MulEquiv.piUnits f i) = ↑f i | null | true |
_private.Lean.Elab.PatternVar.0.Lean.Elab.Term.CollectPatternVars.collect.processId.match_1 | Lean.Elab.PatternVar | (motive : Option Lean.ConstantInfo → Sort u_1) →
(x : Option Lean.ConstantInfo) →
((val : Lean.ConstructorVal) → motive (some (Lean.ConstantInfo.ctorInfo val))) →
((val : Lean.ConstantInfo) → motive (some val)) → (Unit → motive none) → motive x | null | false |
Equiv.isCancelAdd | Mathlib.Algebra.Group.TransferInstance | ∀ {α : Type u_2} {β : Type u_3} (e : α ≃ β) [inst : Add β] [IsCancelAdd β], IsCancelAdd α | Transfer `IsCancelAdd` across an `Equiv` | true |
_private.Mathlib.Topology.UniformSpace.Closeds.0.UniformSpace.hausdorff.uniformContinuous_union.match_1_1 | Mathlib.Topology.UniformSpace.Closeds | ∀ {α : Type u_1} (U : Set (α × α)) (a : (Set α × Set α) × Set α × Set α)
(motive : a ∈ entourageProd (hausdorffEntourage U) (hausdorffEntourage U) → Prop)
(h : a ∈ entourageProd (hausdorffEntourage U) (hausdorffEntourage U)),
(∀ (h₁ : (a.1.1, a.2.1) ∈ hausdorffEntourage U) (h₂ : (a.1.2, a.2.2) ∈ hausdorffEntourag... | null | false |
_private.Mathlib.Tactic.Linter.FindDeprecations.0.Mathlib.Tactic.rewriteOneFile.match_7 | Mathlib.Tactic.Linter.FindDeprecations | (motive : List String.Pos.Raw → Sort u_1) →
(x : List String.Pos.Raw) →
((a b _c : String.Pos.Raw) → motive [a, b, _c]) → ((x : List String.Pos.Raw) → motive x) → motive x | null | false |
_private.Lean.Meta.Transform.0.Lean.Meta.unfoldIfArgIsAppOf._sparseCasesOn_1 | Lean.Meta.Transform | {motive : Lean.ConstantInfo → Sort u} →
(t : Lean.ConstantInfo) →
((val : Lean.TheoremVal) → motive (Lean.ConstantInfo.thmInfo val)) →
(Nat.hasNotBit 4 t.ctorIdx → motive t) → motive t | null | false |
AlgebraicGeometry.one_spec_asOver_spec | Mathlib.AlgebraicGeometry.Group.Affine | ∀ {R A : CommRingCat} [inst : Bialgebra ↑R ↑A],
CategoryTheory.MonObj.one =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.ε (AlgebraicGeometry.algSpec R))
(CategoryTheory.Over.homMk (AlgebraicGeometry.Spec.map (CommRingCat.ofHom ↑(Bialgebra.counitAlgHom ↑R ↑A))) ⋯) | null | true |
CategoryTheory.Limits.HasWideCoequalizer | Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {J : Type w} → {C : Type u} → [inst : CategoryTheory.Category.{v, u} C] → {X Y : C} → (J → (X ⟶ Y)) → Prop | A family `f` of parallel morphisms has a wide coequalizer if the diagram `parallelFamily f` has
a colimit. | true |
_private.Batteries.Data.Array.Scan.0.Array.getElem_succ_scanl._proof_1_4 | Batteries.Data.Array.Scan | ∀ {β : Type u_1} {α : Type u_2} {i : ℕ} {b : β} {as : Array α} {f : β → α → β} (h : i + 1 < (Array.scanl f b as).size),
(List.scanl f b as.toList)[i + 1] = f (List.scanl f b as.toList)[i] as[i] | null | false |
MinTropical.instDecidableLT.eq_1 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u} [inst : LT R] [inst_1 : DecidableLT R], MinTropical.instDecidableLT = MinTropical.instDecidableLT._aux_1 | null | true |
IsCompact.of_subset_of_specializes | Mathlib.Topology.Inseparable | ∀ {X : Type u_1} [inst : TopologicalSpace X] {s t : Set X},
IsCompact s → t ⊆ s → (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) → IsCompact t | null | true |
RootPairing.RootPositiveForm.posForm._proof_5 | Mathlib.LinearAlgebra.RootSystem.RootPositive | ∀ {S : Type u_1} [inst : CommRing S], RingHomSurjective (RingHom.id S) | null | false |
_private.Init.Data.String.Basic.0.String.Pos.ofSliceTo_sliceTo._simp_1_1 | Init.Data.String.Basic | ∀ {s : String} {x y : s.Pos}, (x = y) = (x.offset = y.offset) | null | false |
_private.Mathlib.Analysis.SpecificLimits.Normed.0.TFAE_exists_lt_isLittleO_pow.match_1_7 | Mathlib.Analysis.SpecificLimits.Normed | ∀ (f : ℕ → ℝ) (R : ℝ) (motive : (∃ a ∈ Set.Ioo 0 R, f =O[Filter.atTop] fun x => a ^ x) → Prop)
(x : ∃ a ∈ Set.Ioo 0 R, f =O[Filter.atTop] fun x => a ^ x),
(∀ (a : ℝ) (ha : a ∈ Set.Ioo 0 R) (H : f =O[Filter.atTop] fun x => a ^ x), motive ⋯) → motive x | null | false |
_private.Lean.Meta.Tactic.Grind.PropagateInj.0.Lean.Meta.Grind.getInvFor? | Lean.Meta.Tactic.Grind.PropagateInj | Lean.Expr → Lean.Expr → Lean.Meta.Grind.GoalM (Option (Lean.Expr × Lean.Expr)) | null | true |
Std.DHashMap.Const.get!_inter_of_not_mem_right | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m₁ m₂ : Std.DHashMap α fun x => β} [EquivBEq α]
[LawfulHashable α] [inst : Inhabited β] {k : α}, k ∉ m₂ → Std.DHashMap.Const.get! (m₁.inter m₂) k = default | null | true |
Std.TreeMap.minKey_le_minKey_erase | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [inst : Std.TransCmp cmp] {k : α}
{he : (t.erase k).isEmpty = false}, (cmp (t.minKey ⋯) ((t.erase k).minKey he)).isLE = true | null | true |
CategoryTheory.Abelian.SpectralObject.homologyDataIdId._auto_1 | Mathlib.Algebra.Homology.SpectralObject.Page | Lean.Syntax | null | false |
_private.Batteries.Util.ProofWanted.0.elabWanted.match_11 | Batteries.Util.ProofWanted | (motive : Option Lean.LocalDecl → Sort u_1) →
(x : Option Lean.LocalDecl) →
((d : Lean.LocalDecl) → motive (some d)) → ((x : Option Lean.LocalDecl) → motive x) → motive x | null | false |
Quaternion.imI_sub | Mathlib.Algebra.Quaternion | ∀ {R : Type u_3} [inst : CommRing R] (a b : Quaternion R), (a - b).imI = a.imI - b.imI | null | true |
contDiffGroupoid_prod | Mathlib.Geometry.Manifold.IsManifold.Basic | ∀ {n : WithTop ℕ∞} {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] {E' : Type u_5} {H' : Type u_6}
[inst_4 : NormedAddCommGroup E'] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : TopologicalSpace H']
... | The product of two `C^n` open partial homeomorphisms is `C^n`. | true |
LinOrd.hom_comp | Mathlib.Order.Category.LinOrd | ∀ {X Y Z : LinOrd} (f : X ⟶ Y) (g : Y ⟶ Z),
LinOrd.Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (LinOrd.Hom.hom g).comp (LinOrd.Hom.hom f) | null | true |
isOpen_setOfPred_eventually_nhdsWithin | Mathlib.Topology.Separation.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {p : X → Prop},
IsOpen {x | ∀ᶠ (y : X) in nhdsWithin x {x}ᶜ, p y} | null | true |
_private.Mathlib.Data.Num.Basic.0.PosNum.sqrtAux._unsafe_rec | Mathlib.Data.Num.Basic | PosNum → Num → Num → Num | null | false |
LE.le.lt_or_eq_dec | Mathlib.Order.Basic | ∀ {α : Type u_1} [inst : PartialOrder α] {a b : α} [DecidableLE α], a ≤ b → a < b ∨ a = b | **Alias** of `Decidable.lt_or_eq_of_le`. | true |
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