name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
GT.gt.lt | Mathlib.Order.Basic | ∀ {α : Type u_2} [inst : LT α] {a b : α}, a > b → b < a | null | true |
_private.Init.Data.List.Lemmas.0.List.flatMap_eq_nil_iff._simp_1_2 | Init.Data.List.Lemmas | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
MulChar.commGroup._proof_5 | Mathlib.NumberTheory.MulChar.Basic | ∀ {R : Type u_1} [inst : CommMonoid R] {R' : Type u_2} [inst_1 : CommMonoidWithZero R'] (a : MulChar R R'),
zpowRec npowRec 0 a = 1 | null | false |
AlgEquiv.lift_trdeg_eq | Mathlib.RingTheory.AlgebraicIndependent.Basic | ∀ {R : Type u_2} {A : Type v} {A' : Type v'} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : CommRing A']
[inst_3 : Algebra R A] [inst_4 : Algebra R A'] (e : A ≃ₐ[R] A'),
Cardinal.lift.{v', v} (Algebra.trdeg R A) = Cardinal.lift.{v, v'} (Algebra.trdeg R A') | null | true |
CochainComplex.mappingCone.shiftTriangleIso._proof_2 | Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
[inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Pretriangulated.Triangle.shiftFunctor (CochainComple... | null | false |
_private.Mathlib.GroupTheory.Perm.Support.0.Equiv.Perm.support_inv._simp_1_2 | Mathlib.GroupTheory.Perm.Support | ∀ {a b : Prop}, (¬a ↔ ¬b) = (a ↔ b) | null | false |
instDecidableEqSum.decEq | Init.Core | {α : Type u_1} → {β : Type u_2} → [DecidableEq α] → [DecidableEq β] → (x x_1 : α ⊕ β) → Decidable (x = x_1) | null | true |
Lean.Meta.AuxLemmaKey._sizeOf_inst | Lean.Meta.Tactic.AuxLemma | SizeOf Lean.Meta.AuxLemmaKey | null | false |
Differentiable.sub._simp_2 | Mathlib.Analysis.Calculus.FDeriv.Add | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f g : E → F},
Differentiable 𝕜 f → Differentiable 𝕜 g → Differentiable 𝕜 (f - g) = True | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.Basic.0.SimpleGraph.mk'._simp_4 | Mathlib.Combinatorics.SimpleGraph.Basic | ∀ {a b : Bool}, (a = true ↔ b = true) = (a = b) | null | false |
Vector.map_inj_left | Init.Data.Vector.Lemmas | ∀ {α : Type u_1} {β : Type u_2} {n : ℕ} {xs : Vector α n} {f g : α → β},
Vector.map f xs = Vector.map g xs ↔ ∀ a ∈ xs, f a = g a | null | true |
Lean.Elab.Tactic.TacticParsedSnapshot.ctorIdx | Lean.Elab.Term.TermElabM | Lean.Elab.Tactic.TacticParsedSnapshot → ℕ | null | false |
commMonTypeEquivalenceCommMon._proof_1 | Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic | ∀ {X Y : CommMonCat} (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp
((CommMonTypeEquivalenceCommMon.inverse.comp CommMonTypeEquivalenceCommMon.functor).map f)
(let __src := Equiv.refl (CommMonTypeEquivalenceCommMon.inverse.obj Y).X;
{ toEquiv := __src, map_mul' := ⋯ }).toCommMonCatIso.hom =
... | null | false |
_private.Init.Data.Array.BasicAux.0.Array.mapM'.go._unary.eq_def | Init.Data.Array.BasicAux | ∀ {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [inst : Monad m] (f : α → m β) (as : Array α)
(_x : (i : ℕ) ×' (_ : { bs // bs.size = i }) ×' i ≤ as.size),
Array.mapM'.go._unary✝ f as _x =
PSigma.casesOn _x fun i acc =>
PSigma.casesOn acc fun acc hle =>
if h : i = as.size then pure (h ▸ ... | null | false |
EquicontinuousWithinAt.eq_1 | Mathlib.Topology.UniformSpace.Equicontinuity | ∀ {ι : Type u_1} {X : Type u_3} {α : Type u_6} [tX : TopologicalSpace X] [uα : UniformSpace α] (F : ι → X → α)
(S : Set X) (x₀ : X),
EquicontinuousWithinAt F S x₀ = ∀ U ∈ uniformity α, ∀ᶠ (x : X) in nhdsWithin x₀ S, ∀ (i : ι), (F i x₀, F i x) ∈ U | null | true |
Std.DTreeMap.Internal.Impl.minKey!_insertIfNew_le_minKey! | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {instOrd : Ord α} {t : Std.DTreeMap.Internal.Impl α β} [Std.TransOrd α]
[inst : Inhabited α] (h : t.WF),
t.isEmpty = false →
∀ {k : α} {v : β k}, (compare (Std.DTreeMap.Internal.Impl.insertIfNew k v t ⋯).impl.minKey! t.minKey!).isLE = true | null | true |
Filter.liminf_eq_iSup_iInf | Mathlib.Order.LiminfLimsup | ∀ {α : Type u_1} {β : Type u_2} [inst : CompleteLattice α] {f : Filter β} {u : β → α},
Filter.liminf u f = ⨆ s ∈ f, ⨅ a ∈ s, u a | In a complete lattice, the liminf of a function is the infimum over sets `s` in the filter
of the supremum of the function over `s` | true |
_private.Lean.Environment.0.Lean.Kernel.Environment.Diagnostics.recordUnfold.match_1 | Lean.Environment | (motive : Option ℕ → Sort u_1) → (x : Option ℕ) → ((c : ℕ) → motive (some c)) → ((x : Option ℕ) → motive x) → motive x | null | false |
AddAction.vadd_zsmul_movedBy_eq_of_addCommute | Mathlib.GroupTheory.GroupAction.FixedPoints | ∀ {α : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α] {g h : G},
AddCommute g h → ∀ (j : ℤ), j • h +ᵥ (AddAction.fixedBy α g)ᶜ = (AddAction.fixedBy α g)ᶜ | If `g` and `h` commute, then `g` moves `(j • h) +ᵥ x` iff `g` moves `x`. | true |
IsLocalMax.add | Mathlib.Topology.Order.LocalExtr | ∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : AddCommMonoid β] [inst_2 : PartialOrder β]
[IsOrderedAddMonoid β] {f g : α → β} {a : α}, IsLocalMax f a → IsLocalMax g a → IsLocalMax (fun x => f x + g x) a | null | true |
_private.Mathlib.Algebra.Order.SuccPred.WithBot.0.WithBot.one_le_iff_pos._simp_1_1 | Mathlib.Algebra.Order.SuccPred.WithBot | ∀ {α : Type u_1} {x : α} [inst : PartialOrder α] [inst_1 : AddMonoidWithOne α] [ZeroLEOneClass α] [NeZero 1]
[SuccAddOrder α], (1 ≤ x) = (0 < x) | null | false |
List.filterMap_append | Init.Data.List.Lemmas | ∀ {α : Type u_1} {β : Type u_2} {l l' : List α} {f : α → Option β},
List.filterMap f (l ++ l') = List.filterMap f l ++ List.filterMap f l' | null | true |
AlgebraicGeometry.Scheme.Hom.copyBase._proof_1 | Mathlib.AlgebraicGeometry.Scheme | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) (g : ↥X → ↥Y) (h : ⇑f = g),
f.base = TopCat.ofHom { toFun := g, continuous_toFun := ⋯ } | null | false |
Std.HashMap.Raw.getKeyD_insert_self | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} {m : Std.HashMap.Raw α β} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α]
[LawfulHashable α], m.WF → ∀ {k fallback : α} {v : β}, (m.insert k v).getKeyD k fallback = k | null | true |
basisOfPiSpaceOfLinearIndependent | Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {K : Type u} →
[inst : DivisionRing K] →
{ι : Type u_1} →
[Fintype ι] → [Decidable (Nonempty ι)] → {b : ι → ι → K} → LinearIndependent K b → Module.Basis ι K (ι → K) | In a vector space `ι → K`, a linear independent family indexed by `ι` is a basis. | true |
HomotopicalAlgebra.Cylinder.ofFactorizationData_i₀ | Mathlib.AlgebraicTopology.ModelCategory.Cylinder | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : HomotopicalAlgebra.ModelCategory C] {A : C}
(h :
(HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C)
(CategoryTheory.Limits.codiag A)),
(HomotopicalAlgebra.Cylinder.ofFactorizationData h).i₀ =
... | null | true |
_private.Init.Data.Range.Polymorphic.Lemmas.0.Std.Rci.succ_mem_succ_succ_iff._simp_1_1 | Init.Data.Range.Polymorphic.Lemmas | ∀ {α : Type u} [inst : LE α] [inst_1 : Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α]
[Std.PRange.LawfulUpwardEnumerableLE α] [inst_4 : Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxi.IsAlwaysFinite α]
[Std.PRange.LawfulUpwardEnumerable α] {lo a : α},
(a ∈ (Std.PRange.succ lo)...*) = ∃ a... | null | false |
_private.Mathlib.Algebra.Module.Injective.0.Module.Baer.extension_property_addMonoidHom.match_1_1 | Mathlib.Algebra.Module.Injective | ∀ {Q : Type u_1} [inst : AddCommGroup Q] {M : Type u_3} {N : Type u_2} [inst_1 : AddCommGroup M]
[inst_2 : AddCommGroup N] (f : M →+ N) (g : M →+ Q) (motive : (∃ h, h ∘ₗ f.toIntLinearMap = g.toIntLinearMap) → Prop)
(x : ∃ h, h ∘ₗ f.toIntLinearMap = g.toIntLinearMap),
(∀ (g' : N →ₗ[ℤ] Q) (hg' : g' ∘ₗ f.toIntLinear... | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.0._regBuiltin.UInt64.reduceOfNat.declare_305._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt.4002762760._hygCtx._hyg.336 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | IO Unit | null | false |
CommRingCat.prodFanIsLimit._proof_3 | Mathlib.Algebra.Category.Ring.Constructions | ∀ (A B : CommRingCat) (s : CategoryTheory.Limits.Cone (CategoryTheory.Limits.pair A B)) (m : s.pt ⟶ (A.prodFan B).pt),
(∀ (j : CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair),
CategoryTheory.CategoryStruct.comp m ((A.prodFan B).π.app j) = s.π.app j) →
m =
CommRingCat.ofHom
((CommRin... | null | false |
Nat.Primes.coe_pnat_inj | Mathlib.Data.PNat.Prime | ∀ (p q : Nat.Primes), ↑p = ↑q ↔ p = q | null | true |
_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope.0.MonotoneOn.intervalIntegral_slope_le._simp_1_24 | Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [CharZero R] (n : ℕ), (↑n + 1 = 0) = False | null | false |
BitVec.sdivOverflow.eq_1 | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} (x y : BitVec w),
x.sdivOverflow y = (decide (2 ^ (w - 1) ≤ x.toInt / y.toInt) || decide (x.toInt / y.toInt < -2 ^ (w - 1))) | null | true |
Lean.Meta.Grind.Arith.CommRing.State.ncSemirings | Lean.Meta.Tactic.Grind.Arith.CommRing.Types | Lean.Meta.Grind.Arith.CommRing.State → Array Lean.Meta.Grind.Arith.CommRing.Semiring | Non commutative semirings.
| true |
Lists.Equiv.symm | Mathlib.SetTheory.Lists | ∀ {α : Type u_1} {l₁ l₂ : Lists α}, l₁.Equiv l₂ → l₂.Equiv l₁ | null | true |
Submodule.colon._proof_1 | Mathlib.RingTheory.Ideal.Colon | ∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(N : Submodule R M), ↑N + ↑N = ↑N | null | false |
Batteries.TransCmp.toOrientedCmp | Batteries.Classes.Deprecated | ∀ {α : Sort u_1} {cmp : α → α → Ordering} [self : Batteries.TransCmp cmp], Batteries.OrientedCmp cmp | null | true |
fderivWithin_fun_sub | Mathlib.Analysis.Calculus.FDeriv.Add | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f g : E → F}
{x : E} {s : Set E},
UniqueDiffWithinAt 𝕜 s x →
DifferentiableWithinAt 𝕜 f s x →
Dif... | null | true |
ContinuousMulEquiv.coe_trans | Mathlib.Topology.Algebra.ContinuousMonoidHom | ∀ {M : Type u_1} {N : Type u_2} [inst : TopologicalSpace M] [inst_1 : TopologicalSpace N] [inst_2 : Mul M]
[inst_3 : Mul N] {L : Type u_3} [inst_4 : Mul L] [inst_5 : TopologicalSpace L] (e₁ : M ≃ₜ* N) (e₂ : N ≃ₜ* L),
⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁ | null | true |
_private.Batteries.Data.Array.Pairwise.0.Array.pairwise_iff_getElem._simp_1_1 | Batteries.Data.Array.Pairwise | ∀ {α : Type u_1} {R : α → α → Prop} {l : List α},
List.Pairwise R l = ∀ (i j : ℕ) (_hi : i < l.length) (_hj : j < l.length), i < j → R l[i] l[j] | null | false |
SetLike.GradeZero.instSemiring._aux_5 | Mathlib.Algebra.DirectSum.Internal | {ι : Type u_3} →
{σ : Type u_2} →
{R : Type u_1} →
[inst : Semiring R] →
[inst_1 : AddMonoid ι] →
[inst_2 : SetLike σ R] → [AddSubmonoidClass σ R] → (A : ι → σ) → [SetLike.GradedMonoid A] → ℕ → ↥(A 0) | null | false |
_private.Init.Data.ToString.Name.0.Lean.Name.toStringWithToken.maybePseudoSyntax.match_1 | Init.Data.ToString.Name | (motive : Lean.Name → Sort u_1) →
(x : Lean.Name) → ((pre : Lean.Name) → (s : String) → motive (pre.str s)) → ((x : Lean.Name) → motive x) → motive x | null | false |
PiTensorProduct.lift_reindex | Mathlib.LinearAlgebra.PiTensorProduct | ∀ {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [inst : CommSemiring R] {s : ι → Type u_7}
[inst_1 : (i : ι) → AddCommMonoid (s i)] [inst_2 : (i : ι) → Module R (s i)] {E : Type u_9} [inst_3 : AddCommMonoid E]
[inst_4 : Module R E] (e : ι ≃ ι₂) (φ : MultilinearMap R (fun i => s (e.symm i)) E)
(x : PiTensorProduct... | null | true |
_private.Mathlib.RingTheory.Polynomial.Nilpotent.0.Polynomial.not_isUnit_of_natDegree_pos_of_isReduced._simp_1_1 | Mathlib.RingTheory.Polynomial.Nilpotent | ∀ {R : Type u_3} {x : R} [inst : MonoidWithZero R] [IsReduced R], IsNilpotent x = (x = 0) | null | false |
Aesop.RuleState.enqueueRawHyp | Aesop.Forward.State | Aesop.RawHyp → Aesop.PremiseIndex → Aesop.RuleState → Aesop.RuleState | Add a hypothesis or pattern substitution to the rule state. The
hypothesis's type does not necessarily need to match the given premise. If it
does not, this is detected by `update` and the hyp is not added. | true |
Lean.Compiler.LCNF.CodeDecl.casesOn | Lean.Compiler.LCNF.Basic | {pu : Lean.Compiler.LCNF.Purity} →
{motive : Lean.Compiler.LCNF.CodeDecl pu → Sort u} →
(t : Lean.Compiler.LCNF.CodeDecl pu) →
((decl : Lean.Compiler.LCNF.LetDecl pu) → motive (Lean.Compiler.LCNF.CodeDecl.let decl)) →
((decl : Lean.Compiler.LCNF.FunDecl pu) →
(h : pu = Lean.Compiler.LCNF... | null | false |
Lean.Parser.ParserState.keepNewError | Lean.Parser.Basic | Lean.Parser.ParserState → ℕ → Lean.Parser.ParserState | null | true |
Lean.Meta.Grind.Filter.and | Lean.Meta.Tactic.Grind.Filter | Lean.Meta.Grind.Filter → Lean.Meta.Grind.Filter → Lean.Meta.Grind.Filter | null | true |
Tactic.ComputeAsymptotics.WellFormedBasis.tendsto_atTop | Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis | ∀ {basis : Tactic.ComputeAsymptotics.Basis},
Tactic.ComputeAsymptotics.WellFormedBasis basis →
∀ {f : ℝ → ℝ}, f ∈ basis → Filter.Tendsto f Filter.atTop Filter.atTop | All functions from a well-formed basis tend to `atTop`. | true |
SSet.Subcomplex.Pairing.instIsRegularOp | Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op | ∀ {X : SSet} {A : X.Subcomplex} (P : A.Pairing) [P.IsRegular], P.op.IsRegular | null | true |
CategoryTheory.Functor.representableByUliftFunctorEquiv._proof_3 | Mathlib.CategoryTheory.Yoneda | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_3, u_1} C] {F : CategoryTheory.Functor Cᵒᵖ (Type u_2)} {X : C},
Function.RightInverse (fun R => { homEquiv := fun {Y} => R.homEquiv.trans Equiv.ulift.symm, homEquiv_comp := ⋯ })
fun R => { homEquiv := fun {Y} => R.homEquiv.trans Equiv.ulift, homEquiv_comp := ⋯ } | null | false |
Lean.Grind.Semiring.pow_add_congr | Init.Grind.Ring.Basic | ∀ {α : Type u} [inst : Lean.Grind.Semiring α] (a r : α) (k k₁ k₂ : ℕ), k = k₁ + k₂ → a ^ k₁ * a ^ k₂ = r → a ^ k = r | null | true |
_private.Mathlib.Data.Nat.Nth.0.Nat.le_nth_of_monotoneOn_of_surjOn._proof_1_4 | Mathlib.Data.Nat.Nth | ∀ {p : ℕ → Prop} (f : ℕ → ℕ) (n : ℕ),
((∀ (hf : (setOf p).Finite), n < hf.toFinset.card) → f n ≤ Nat.nth p n) →
(∀ (hf : (setOf p).Finite), n + 1 < hf.toFinset.card) → ∀ (m : ℕ), (∀ k < n + 1, Nat.nth p k < f m) → f n < f m | null | false |
Std.Internal.List.insertListIfNew._unsafe_rec | Std.Data.Internal.List.Associative | {α : Type u} → {β : α → Type v} → [BEq α] → List ((a : α) × β a) → List ((a : α) × β a) → List ((a : α) × β a) | null | false |
_private.Mathlib.MeasureTheory.Measure.LevyProkhorovMetric.0.MeasureTheory.levyProkhorovEDist_le_of_forall._simp_1_2 | Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + (b + c) = a + b + c | null | false |
Std.DTreeMap.Internal.Unit.RocSliceData.ctorIdx | Std.Data.DTreeMap.Internal.Zipper | {α : Type u} → {inst : Ord α} → Std.DTreeMap.Internal.Unit.RocSliceData α → ℕ | null | false |
Bornology.isVonNBounded_add_self | Mathlib.Analysis.LocallyConvex.Bounded | ∀ {𝕜 : Type u_1} {E : Type u_3} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] [ContinuousSMul 𝕜 E] [ContinuousAdd E] {s : Set E},
Bornology.IsVonNBounded 𝕜 (s + s) ↔ Bornology.IsVonNBounded 𝕜 s | null | true |
CategoryTheory.Presieve.BindStruct.mk.sizeOf_spec | Mathlib.CategoryTheory.Sites.Sieves | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X : C} {S : CategoryTheory.Presieve X}
{R : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → S f → CategoryTheory.Presieve Y} {Z : C} {h : Z ⟶ X} [inst_1 : SizeOf C]
[inst_2 : ⦃Y : C⦄ → (a : Y ⟶ X) → SizeOf (S a)]
[inst_3 : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → (a : S f) → ⦃Y_1 : C⦄ → (a_1 :... | null | true |
HomologicalComplex.Hom.isoOfComponents | Mathlib.Algebra.Homology.HomologicalComplex | {ι : Type u_1} →
{V : Type u} →
[inst : CategoryTheory.Category.{v, u} V] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] →
{c : ComplexShape ι} →
{C₁ C₂ : HomologicalComplex V c} →
(f : (i : ι) → C₁.X i ≅ C₂.X i) →
autoParam
(∀ (i j : ι),
... | Construct an isomorphism of chain complexes from isomorphism of the objects
which commute with the differentials. | true |
Lean.Elab.Do.ControlLifter | Lean.Elab.Do.Control | Type | null | true |
FirstOrder.Language.BoundedFormula.IsQF.below | Mathlib.ModelTheory.Complexity | {L : FirstOrder.Language} →
{α : Type u'} →
{n : ℕ} → {motive : (a : L.BoundedFormula α n) → a.IsQF → Prop} → {a : L.BoundedFormula α n} → a.IsQF → Prop | null | true |
_private.Mathlib.RingTheory.Valuation.ValuativeRel.Basic.0.ValuativeRel.ValueGroupWithZero.embed._simp_14 | Mathlib.RingTheory.Valuation.ValuativeRel.Basic | ∀ {M₀ : Type u_1} [inst : MonoidWithZero M₀] {a : M₀} [IsReduced M₀] (n : ℕ), a ≠ 0 → (a ^ n = 0) = False | null | false |
Lean.Elab.Term.ToDepElimPattern.Context.userName._default | Lean.Elab.Match | Lean.Name | null | false |
LE.opposite_def | Init.Data.Order.Opposite | ∀ {α : Type u_1} {le : LE α}, le.opposite = { le := fun a b => b ≤ a } | null | true |
Matrix.transpose_hasOrthogonalCols_iff_hasOrthogonalRows._simp_1 | Mathlib.LinearAlgebra.Matrix.Orthogonal | ∀ {α : Type u_1} {n : Type u_2} {m : Type u_3} [inst : Mul α] [inst_1 : AddCommMonoid α] (A : Matrix m n α)
[inst_2 : Fintype n], A.transpose.HasOrthogonalCols = A.HasOrthogonalRows | null | false |
_private.Mathlib.RingTheory.Extension.Cotangent.BaseChange.0.Algebra.Extension.tensorCotangentOfFlat._proof_2 | Mathlib.RingTheory.Extension.Cotangent.BaseChange | ∀ {R : Type u_3} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
(P : Algebra.Extension R S) (T : Type u_2) [inst_3 : CommRing T] [inst_4 : Algebra R T],
LinearMap.CompatibleSMul (Ideal.map Algebra.TensorProduct.includeRight.toRingHom P.ker).Cotangent
P.baseChange.ker.Cotangent T... | null | false |
AddMonoidAlgebra.prod_single | Mathlib.Algebra.MonoidAlgebra.Defs | ∀ {R : Type u_1} {M : Type u_4} {ι : Type u_7} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] (s : Finset ι)
(m : ι → M) (r : ι → R),
∏ i ∈ s, AddMonoidAlgebra.single (m i) (r i) = AddMonoidAlgebra.single (∑ i ∈ s, m i) (∏ i ∈ s, r i) | null | true |
DirectSum.lie_of_same | Mathlib.Algebra.Lie.DirectSum | ∀ {ι : Type v} (L : ι → Type w) [inst : (i : ι) → LieRing (L i)] [inst_1 : DecidableEq ι] {i : ι} (x y : L i),
⁅(DirectSum.of L i) x, (DirectSum.of L i) y⁆ = (DirectSum.of L i) ⁅x, y⁆ | null | true |
Polynomial.IsUnitTrinomial.irreducible_of_coprime' | Mathlib.Analysis.Complex.Polynomial.UnitTrinomial | ∀ {p : Polynomial ℤ},
p.IsUnitTrinomial → (∀ (z : ℂ), ¬((Polynomial.aeval z) p = 0 ∧ (Polynomial.aeval z) p.mirror = 0)) → Irreducible p | A unit trinomial is irreducible if it has no complex roots in common with its mirror. | true |
WithVal.instField | Mathlib.Topology.Algebra.Valued.WithVal | {R : Type u_1} →
{Γ₀ : Type u_2} →
[inst : LinearOrderedCommGroupWithZero Γ₀] → [inst_1 : Field R] → (v : Valuation R Γ₀) → Field (WithVal v) | null | true |
CategoryTheory.Arrow.mapCechNerve._proof_1 | Mathlib.AlgebraicTopology.CechNerve | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {f g : CategoryTheory.Arrow C}
[inst_1 : ∀ (n : ℕ), CategoryTheory.Limits.HasWidePullback f.right (fun x => f.left) fun x => f.hom] (F : f ⟶ g)
(n : SimplexCategoryᵒᵖ) (j : Fin ((Opposite.unop n).len + 1)),
CategoryTheory.CategoryStruct.comp
(Ca... | null | false |
ContinuousGeneratedByCat.coe_of | Mathlib.Topology.Convenient.Category | ∀ {ι : Type t} {X : ι → Type u} [inst : (i : ι) → TopologicalSpace (X i)] (Y : Type v) [inst_1 : TopologicalSpace Y],
↑{ carrier := Y, str := inst_1 } = Y | null | true |
CategoryTheory.MorphismProperty.cons_mem_paths_iff._simp_1 | Mathlib.CategoryTheory.PathCategory.MorphismProperty | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y Z : C}
{p : Quiver.Path X Y} {f : Y ⟶ Z}, W.paths (p.cons f) = (W.paths p ∧ W f) | null | false |
Submodule.sndEquiv_apply | Mathlib.LinearAlgebra.Prod | ∀ (R : Type u) (M : Type v) (M₂ : Type w) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]
[inst_3 : Module R M] [inst_4 : Module R M₂] (x : ↥(Submodule.snd R M M₂)), (Submodule.sndEquiv R M M₂) x = (↑x).2 | null | true |
ProbabilityTheory.centralMoment_two_eq_variance | Mathlib.Probability.Moments.Basic | ∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω},
AEMeasurable X μ → ProbabilityTheory.centralMoment X 2 μ = ProbabilityTheory.variance X μ | null | true |
MeasureTheory.IntegrableOn.congr_set_ae | Mathlib.MeasureTheory.Integral.IntegrableOn | ∀ {α : Type u_1} {ε : Type u_3} {mα : MeasurableSpace α} {f : α → ε} {s t : Set α} {μ : MeasureTheory.Measure α}
[inst : TopologicalSpace ε] [inst_1 : ContinuousENorm ε],
MeasureTheory.IntegrableOn f t μ → s =ᵐ[μ] t → MeasureTheory.IntegrableOn f s μ | null | true |
SSet.Truncated.instCategoryHomotopyCategory._proof_8 | Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | ∀ (V : SSet.Truncated 2),
autoParam
(∀ {X Y : V.HomotopyCategory} (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id Y) = f)
CategoryTheory.Category.comp_id._autoParam | null | false |
_private.Mathlib.CategoryTheory.ComposableArrows.Basic.0._auto_282 | Mathlib.CategoryTheory.ComposableArrows.Basic | Lean.Syntax | null | false |
_private.Mathlib.Algebra.Order.Floor.Ring.0.Int.image_fract._simp_1_2 | Mathlib.Algebra.Order.Floor.Ring | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) | null | false |
AddRightCancelMonoid.mk._flat_ctor | Mathlib.Algebra.Group.Defs | {M : Type u} →
(add : M → M → M) →
(∀ (a b c : M), a + b + c = a + (b + c)) →
(zero : M) →
(∀ (a : M), 0 + a = a) →
(∀ (a : M), a + 0 = a) →
(nsmul : ℕ → M → M) →
autoParam (∀ (x : M), nsmul 0 x = 0) AddMonoid.nsmul_zero._autoParam →
autoParam (∀ (... | null | false |
HomologicalComplex.instHasHomotopyFiberOfHasBinaryBiproducts | Mathlib.Algebra.Homology.HomotopyFiber | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {α : Type u_2}
{c : ComplexShape α} {F G : HomologicalComplex C c} (φ : F ⟶ G) [CategoryTheory.Limits.HasBinaryBiproducts C],
HomologicalComplex.HasHomotopyFiber φ | null | true |
_private.Mathlib.Order.CountableSupClosed.0.countableInfClosure._proof_4 | Mathlib.Order.CountableSupClosed | ∀ {α : Type u_1} (A : Set α) (i : α) (i_1 : i ∈ A), ↑⟨i, ⋯⟩ ∈ A | null | false |
ONote.repr_ofNat | Mathlib.SetTheory.Ordinal.Notation | ∀ (n : ℕ), (↑n).repr = ↑n | null | true |
LinearMap.convIntrinsicStarRing._proof_4 | Mathlib.Algebra.Star.LinearMap | ∀ {R : Type u_3} {A : Type u_1} {C : Type u_2} [inst : CommSemiring R] [inst_1 : StarRing R]
[inst_2 : NonUnitalNonAssocSemiring A] [inst_3 : Module R A] [inst_4 : StarRing A] [inst_5 : StarModule R A]
[inst_6 : AddCommMonoid C] [inst_7 : Module R C] [inst_8 : StarAddMonoid C] [inst_9 : StarModule R C]
(r s : Wit... | null | false |
_private.Batteries.Data.Fin.Lemmas.0.Fin.findSome?_eq_none_iff._simp_1_3 | Batteries.Data.Fin.Lemmas | ∀ {n : ℕ} {P : Fin (n + 1) → Prop}, (∀ (i : Fin (n + 1)), P i) = (P 0 ∧ ∀ (i : Fin n), P i.succ) | null | false |
LinearMap.iterateKer._proof_1 | Mathlib.Algebra.Module.Submodule.Ker | ∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (f : M →ₗ[R] M)
(n m : ℕ), n ≤ m → ∀ x ∈ (fun n => (f ^ n).ker) n, x ∈ (fun n => (f ^ n).ker) m | null | false |
_private.Mathlib.Tactic.ClickSuggestions.FindPremises.0.Mathlib.Tactic.ClickSuggestions.blacklist.match_1 | Mathlib.Tactic.ClickSuggestions.FindPremises | (motive : Lean.Name → Sort u_1) →
(declName : Lean.Name) →
((pre : Lean.Name) → (s : String) → motive (pre.str s)) → ((x : Lean.Name) → motive x) → motive declName | null | false |
orthonormal_vecCons_iff | Mathlib.Analysis.InnerProductSpace.Orthonormal | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{n : ℕ} {v : E} {vs : Fin n → E},
Orthonormal 𝕜 (Matrix.vecCons v vs) ↔ ‖v‖ = 1 ∧ (∀ (i : Fin n), inner 𝕜 v (vs i) = 0) ∧ Orthonormal 𝕜 vs | null | true |
_private.Lean.Linter.MissingDocs.0.Lean.Linter.MissingDocs.docOptStatus | Lean.Linter.MissingDocs | Lean.Syntax → Lean.Syntax → optParam Bool false → Lean.Elab.Command.CommandElabM (Option Bool) | null | true |
Subsemiring.sumSq._proof_4 | Mathlib.Algebra.Ring.SumsOfSquares | ∀ (T : Type u_1) [inst : CommSemiring T], 0 ∈ (NonUnitalSubsemiring.sumSq T).carrier | null | false |
Prod.isRegular_mk._simp_2 | Mathlib.Algebra.Regular.Prod | ∀ {R : Type u_2} {S : Type u_3} [inst : Mul R] [inst_1 : Mul S] {a : R} {b : S},
IsRegular (a, b) = (IsRegular a ∧ IsRegular b) | null | false |
SimpleGraph.Walk.IsHamiltonianCycle.mk | Mathlib.Combinatorics.SimpleGraph.Hamiltonian | ∀ {α : Type u_1} [inst : DecidableEq α] {G : SimpleGraph α} {a : α} {p : G.Walk a a},
p.IsCycle → p.tail.IsHamiltonian → p.IsHamiltonianCycle | null | true |
_private.Mathlib.MeasureTheory.Constructions.BorelSpace.Order.0.borel_eq_generateFrom_Ico._simp_1_1 | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b) | null | false |
List.maxIdxOn_lt_length | Init.Data.List.MinMaxIdx | ∀ {β : Type u_1} {α : Type u_2} [inst : LE β] [inst_1 : DecidableLE β] {f : α → β} {xs : List α} (h : xs ≠ []),
List.maxIdxOn f xs h < xs.length | null | true |
Mathlib.Tactic.GCongr.GCongrHyp._sizeOf_1 | Mathlib.Tactic.GCongr.Core | Mathlib.Tactic.GCongr.GCongrHyp → ℕ | null | false |
_private.Mathlib.Tactic.GRewrite.Core.0.Mathlib.Tactic.GRewrite.Progress.ctorIdx | Mathlib.Tactic.GRewrite.Core | Mathlib.Tactic.GRewrite.Progress✝ → ℕ | null | false |
Int.negOfNat_eq | Init.Data.Int.Lemmas | ∀ {n : ℕ}, Int.negOfNat n = -Int.ofNat n | null | true |
CategoryTheory.Functor.CoreMonoidal.ofOplaxMonoidal._proof_6 | Mathlib.CategoryTheory.Monoidal.Functor | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] [inst_1 : CategoryTheory.MonoidalCategory C]
{D : Type u_2} [inst_2 : CategoryTheory.Category.{u_1, u_2} D] [inst_3 : CategoryTheory.MonoidalCategory D]
(F : CategoryTheory.Functor C D) [inst_4 : F.OplaxMonoidal]
[inst_5 : CategoryTheory.IsIso (Catego... | null | false |
_private.Lean.Meta.Tactic.Cases.0.Lean.Meta.Cases.unifyEqs?.match_1 | Lean.Meta.Tactic.Cases | (motive : Option Lean.Meta.UnifyEqResult → Sort u_1) →
(__do_lift : Option Lean.Meta.UnifyEqResult) →
((mvarId : Lean.MVarId) →
(subst : Lean.Meta.FVarSubst) →
(numNewEqs : ℕ) → motive (some { mvarId := mvarId, subst := subst, numNewEqs := numNewEqs })) →
((x : Option Lean.Meta.UnifyEqResu... | null | false |
Prod.edist_eq | Mathlib.Topology.EMetricSpace.Defs | ∀ {α : Type u} {β : Type v} [inst : PseudoEMetricSpace α] [inst_1 : PseudoEMetricSpace β] (x y : α × β),
edist x y = max (edist x.1 y.1) (edist x.2 y.2) | null | true |
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