name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
FiniteField.frobeniusAlgEquivOfAlgebraic | Mathlib.FieldTheory.Finite.Basic | (K : Type u_1) →
[inst : Field K] →
[Fintype K] → (L : Type u_3) → [inst_2 : Field L] → [inst_3 : Algebra K L] → [Algebra.IsAlgebraic K L] → Gal(L/K) | If `L/K` is an algebraic extension of a finite field, the Frobenius `K`-algebra endomorphism
of `L` is an automorphism. | true |
List.rtakeWhile_concat | Mathlib.Data.List.DropRight | ∀ {α : Type u_1} (p : α → Bool) (l : List α) (x : α),
List.rtakeWhile p (l ++ [x]) = if p x = true then List.rtakeWhile p l ++ [x] else [] | null | true |
Module.Basis.ofIsCoprimeDifferentIdeal | Mathlib.RingTheory.DedekindDomain.LinearDisjoint | (A : Type u_1) →
(B : Type u_2) →
{K : Type u_3} →
{L : Type u_4} →
[inst : CommRing A] →
[inst_1 : Field K] →
[inst_2 : Algebra A K] →
[IsFractionRing A K] →
[inst_4 : CommRing B] →
[inst_5 : Field L] →
[inst_... | Let `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two
subrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint
over `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal
and `Frac R` denotes the frac... | true |
CategoryTheory.MonoidalCategory.DayFunctor.equiv._proof_1 | Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor | ∀ (C : Type u_3) [inst : CategoryTheory.Category.{u_1, u_3} C] (V : Type u_4)
[inst_1 : CategoryTheory.Category.{u_2, u_4} V] [inst_2 : CategoryTheory.MonoidalCategory C]
[inst_3 : CategoryTheory.MonoidalCategory V] (X : CategoryTheory.MonoidalCategory.DayFunctor C V),
(CategoryTheory.CategoryStruct.id X).natTran... | null | false |
OreLocalization.instMonoidWithZero._proof_2 | Mathlib.RingTheory.OreLocalization.Basic | ∀ {R : Type u_1} [inst : MonoidWithZero R] {S : Submonoid R} [inst_1 : OreLocalization.OreSet S]
(x : OreLocalization S R), x * 0 = 0 | null | false |
Lean.Elab.HeaderProcessedSnapshot.mk.sizeOf_spec | Lean.Elab.DefView | ∀ (toSnapshot : Lean.Language.Snapshot) (view : Lean.Elab.DefViewElabHeaderData) (state : Lean.Elab.Term.SavedState)
(tacStx? : Option Lean.Syntax) (tacSnap? : Option (Lean.Language.SnapshotTask Lean.Elab.Tactic.TacticParsedSnapshot))
(bodyStx : Lean.Syntax) (bodySnap : Lean.Language.SnapshotTask (Option Lean.Elab.... | null | true |
String.Slice.Pattern.Model.NoSuffixPatternModel | Init.Data.String.Lemmas.Pattern.Basic | {ρ : Type} → (pat : ρ) → [String.Slice.Pattern.Model.PatternModel pat] → Prop | Predicate stating that a match for a given pattern is never a proper suffix of another match.
This implies that the notion of reverse match and longest reverse match coincide.
| true |
CliffordAlgebra.foldr'Aux._proof_1 | Mathlib.LinearAlgebra.CliffordAlgebra.Fold | ∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
(Q : QuadraticForm R M), SMulCommClass R R (CliffordAlgebra Q) | null | false |
_private.Init.Data.Dyadic.Basic.0.Rat.toDyadic.match_1.splitter | Init.Data.Dyadic.Basic | (motive : ℤ → Sort u_1) →
(prec : ℤ) → ((n : ℕ) → motive (Int.ofNat n)) → ((n : ℕ) → motive (Int.negSucc n)) → motive prec | null | true |
BitVec.reduceDiv | Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec | Lean.Meta.Simp.DSimproc | Simplification procedure for division of `BitVec`s. | true |
Meromorphic.const_smul | Mathlib.Analysis.Meromorphic.Basic | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {R : Type u_4} [inst_3 : NormedRing R] [inst_4 : Module R E] [IsBoundedSMul R E]
{f : 𝕜 → E} [SMulCommClass 𝕜 R E], Meromorphic f → ∀ (c : R), Meromorphic (c • f) | null | true |
Std.Iter.toArray.eq_1 | Init.Data.Iterators.Lemmas.Combinators.FlatMap | ∀ {α β : Type w} [inst : Std.Iterator α Id β] (it : Std.Iter β), it.toArray = it.toIterM.toArray.run | null | true |
Nat.minFac_eq | Mathlib.Data.Nat.Prime.Defs | ∀ (n : ℕ), n.minFac = if 2 ∣ n then 2 else n.minFacAux 3 | null | true |
_private.Init.Data.BitVec.Lemmas.0.BitVec.toInt_ushiftRight_of_lt._proof_1_5 | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {x : BitVec w} {n : ℕ}, 0 < n → n ≤ w → ¬(w - n).succ ≤ w → False | null | false |
Real.iteratedDerivWithin_cos_Icc | Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | ∀ (n : ℕ) {a b : ℝ},
a < b → ∀ {x : ℝ}, x ∈ Set.Icc a b → iteratedDerivWithin n Real.cos (Set.Icc a b) x = iteratedDeriv n Real.cos x | null | true |
Algebra.Generators.Hom.mk.inj | Mathlib.RingTheory.Extension.Generators | ∀ {R : Type u} {S : Type v} {ι : Type w} {inst : CommRing R} {inst_1 : CommRing S} {inst_2 : Algebra R S}
{P : Algebra.Generators R S ι} {R' : Type u_1} {S' : Type u_2} {ι' : Type u_3} {inst_3 : CommRing R'}
{inst_4 : CommRing S'} {inst_5 : Algebra R' S'} {P' : Algebra.Generators R' S' ι'} {inst_6 : Algebra S S'}
... | null | true |
Lean.PrettyPrinter.Parenthesizer.categoryParser.parenthesizer.match_1 | Lean.PrettyPrinter.Parenthesizer | (motive : List Lean.PrettyPrinter.CategoryParenthesizer → Sort u_1) →
(x : List Lean.PrettyPrinter.CategoryParenthesizer) →
((p : Lean.PrettyPrinter.CategoryParenthesizer) →
(tail : List Lean.PrettyPrinter.CategoryParenthesizer) → motive (p :: tail)) →
((x : List Lean.PrettyPrinter.CategoryParenthes... | null | false |
CategoryTheory.Bicategory.whiskerLeft_hom_inv_whiskerRight_assoc | Mathlib.CategoryTheory.Bicategory.Basic | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) (k : c ⟶ d)
{Z : a ⟶ d} (h_1 : CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g k) ⟶ Z),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory... | null | true |
IsTensorProduct.equiv_apply | Mathlib.RingTheory.IsTensorProduct | ∀ {R : Type u_1} [inst : CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [inst_1 : AddCommMonoid M₁]
[inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M] [inst_4 : Module R M₁] [inst_5 : Module R M₂]
[inst_6 : Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) (x : TensorProduct R M₁ M₂)... | null | true |
Lean.Elab.Term.Do.ToTerm.Context.mk.injEq | Lean.Elab.Do.Legacy | ∀ (m returnType : Lean.Syntax) (uvars : Array Lean.Elab.Term.Do.Var) (kind : Lean.Elab.Term.Do.ToTerm.Kind)
(m_1 returnType_1 : Lean.Syntax) (uvars_1 : Array Lean.Elab.Term.Do.Var) (kind_1 : Lean.Elab.Term.Do.ToTerm.Kind),
({ m := m, returnType := returnType, uvars := uvars, kind := kind } =
{ m := m_1, retur... | null | true |
_private.Mathlib.InformationTheory.KullbackLeibler.Basic.0.InformationTheory.toReal_klDiv_smul_right._simp_1_4 | Mathlib.InformationTheory.KullbackLeibler.Basic | ∀ {α : Type u_2} [inst : Zero α] [inst_1 : OfNat α 4] [NeZero 4], (4 = 0) = False | null | false |
SemiNormedGrp.instCoeSortType | Mathlib.Analysis.Normed.Group.SemiNormedGrp | CoeSort SemiNormedGrp (Type u_1) | null | true |
_private.Mathlib.Basic.Denumerable.0.Encodable.decodeSum.match_1.eq_2 | Mathlib.Basic.Denumerable | ∀ (motive : Bool → ℕ → Sort u_1) (x : Bool) (m : ℕ) (h_1 : (m : ℕ) → motive false m)
(h_2 : (x : Bool) → (m : ℕ) → motive x m),
(x = false → False) →
(match x, m with
| false, m => h_1 m
| x, m => h_2 x m) =
h_2 x m | null | true |
_private.Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct.0.SSet.PtSimplex.MulStruct.assocAux.α_of_lt._proof_1_3 | Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | ∀ {n : ℕ} {i : Fin n}, ∀ j < i.castSucc.castSucc.castSucc, ¬j = i.succ.succ.succ | null | false |
Mathlib.Meta.NormNum.isInt_add | Mathlib.Tactic.NormNum.Basic | ∀ {α : Type u_1} [inst : Ring α] {f : α → α → α} {a b : α} {a' b' c : ℤ},
f = HAdd.hAdd →
Mathlib.Meta.NormNum.IsInt a a' →
Mathlib.Meta.NormNum.IsInt b b' → a'.add b' = c → Mathlib.Meta.NormNum.IsInt (f a b) c | null | true |
_private.Mathlib.Topology.LocallyFinite.0.LocallyFinite.comp_injOn.match_1_1 | Mathlib.Topology.LocallyFinite | ∀ {ι : Type u_2} {X : Type u_1} [inst : TopologicalSpace X] {f : ι → Set X} (x : X)
(motive : (∃ t ∈ nhds x, {i | (f i ∩ t).Nonempty}.Finite) → Prop)
(x_1 : ∃ t ∈ nhds x, {i | (f i ∩ t).Nonempty}.Finite),
(∀ (t : Set X) (htx : t ∈ nhds x) (htf : {i | (f i ∩ t).Nonempty}.Finite), motive ⋯) → motive x_1 | null | false |
_private.Mathlib.Order.Cover.0.Pi.covBy_iff_exists_left_eq._simp_1_2 | Mathlib.Order.Cover | ∀ {α : Sort u_1} {a b : α}, (a = b) = (b = a) | null | false |
ZNum.pred | Mathlib.Data.Num.Basic | ZNum → ZNum | The predecessor of a `ZNum`. | true |
MeasureTheory.Measure.addHaarMeasure_eq_iff | Mathlib.MeasureTheory.Measure.Haar.Basic | ∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : TopologicalSpace G] [inst_2 : IsTopologicalAddGroup G]
[inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [SecondCountableTopology G]
(K₀ : TopologicalSpace.PositiveCompacts G) (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ]
[μ.IsAddLeftInvariant], Meas... | Let `μ` be a σ-finite left invariant measure on `G`. Then `μ` is equal to the
additive Haar measure defined by `K₀` iff `μ K₀ = 1`. | true |
Composition.ones | Mathlib.Combinatorics.Enumerative.Composition | (n : ℕ) → Composition n | The composition made of blocks all of size `1`. | true |
TopologicalSpace.Closeds.complOrderIso | Mathlib.Topology.Sets.Closeds | (α : Type u_2) → [inst : TopologicalSpace α] → TopologicalSpace.Closeds α ≃o (TopologicalSpace.Opens α)ᵒᵈ | `TopologicalSpace.Closeds.compl` as an `OrderIso` to the order dual of
`TopologicalSpace.Opens α`. | true |
AddSubmonoid.gciMapComap.eq_1 | Mathlib.Algebra.Group.Submonoid.Operations | ∀ {M : Type u_1} {N : Type u_2} [inst : AddZeroClass M] [inst_1 : AddZeroClass N] {F : Type u_4}
[inst_2 : FunLike F M N] [mc : AddMonoidHomClass F M N] {f : F} (hf : Function.Injective ⇑f),
AddSubmonoid.gciMapComap hf = ⋯.toGaloisCoinsertion ⋯ | null | true |
_private.Mathlib.Data.Num.Lemmas.0.PosNum.cmp.match_1.eq_6 | Mathlib.Data.Num.Lemmas | ∀ (motive : PosNum → PosNum → Sort u_1) (a b : PosNum) (h_1 : Unit → motive PosNum.one PosNum.one)
(h_2 : (x : PosNum) → motive x PosNum.one) (h_3 : (x : PosNum) → motive PosNum.one x)
(h_4 : (a b : PosNum) → motive a.bit0 b.bit0) (h_5 : (a b : PosNum) → motive a.bit0 b.bit1)
(h_6 : (a b : PosNum) → motive a.bit1... | null | true |
CategoryTheory.Coreflective.mk.noConfusion | Mathlib.CategoryTheory.Adjunction.Reflective | {C : Type u₁} →
{D : Type u₂} →
{inst : CategoryTheory.Category.{v₁, u₁} C} →
{inst_1 : CategoryTheory.Category.{v₂, u₂} D} →
{L : CategoryTheory.Functor C D} →
{P : Sort u} →
{toFull : L.Full} →
{toFaithful : L.Faithful} →
{R : CategoryTheory.Func... | null | false |
GenLoop.fromLoop_coe | Mathlib.Topology.Homotopy.HomotopyGroup | ∀ {N : Type u_1} {X : Type u_2} [inst : TopologicalSpace X] {x : X} [inst_1 : DecidableEq N] (i : N)
(p : LoopSpace (↑(GenLoop { j // j ≠ i } X x)) GenLoop.const),
↑(GenLoop.fromLoop i p) =
({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp p.toContinuousMap).uncurry.comp ↑(Cube.splitAt i) | null | true |
Set.ne_univ_iff_exists_notMem | Mathlib.Data.Set.Basic | ∀ {α : Type u_1} (s : Set α), s ≠ Set.univ ↔ ∃ a, a ∉ s | null | true |
_private.Mathlib.SetTheory.Cardinal.Basic.0.Cardinal.add_lt_aleph0_iff.match_1_1 | Mathlib.SetTheory.Cardinal.Basic | ∀ {a b : Cardinal.{u_1}} (motive : a < Cardinal.aleph0 ∧ b < Cardinal.aleph0 → Prop)
(x : a < Cardinal.aleph0 ∧ b < Cardinal.aleph0),
(∀ (h1 : a < Cardinal.aleph0) (h2 : b < Cardinal.aleph0), motive ⋯) → motive x | null | false |
Algebra.PreSubmersivePresentation.baseChange | Mathlib.RingTheory.Extension.Presentation.Submersive | {R : Type u} →
{S : Type v} →
{ι : Type w} →
{σ : Type t} →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
(T : Type u_1) →
[inst_3 : CommRing T] →
[inst_4 : Algebra R T] →
Algebra.PreSu... | If `P` is a pre-submersive presentation of `S` over `R` and `T` is an `R`-algebra, we
obtain a natural pre-submersive presentation of `T ⊗[R] S` over `T`. | true |
RBTree.RBNode.IsCut.gt_trans | Batteries.Recycling.RBTree.Lemmas | ∀ {α : Type u_1} {cmp : α → α → Ordering} {cut : α → Ordering} {x y : α} [RBTree.RBNode.IsCut cmp cut]
[Std.TransCmp cmp], cmp x y = Ordering.lt → cut y = Ordering.gt → cut x = Ordering.gt | null | true |
GrpCat.hasLimitsOfShape | Mathlib.Algebra.Category.Grp.Limits | ∀ {J : Type v} [inst : CategoryTheory.Category.{w, v} J] [Small.{u, v} J],
CategoryTheory.Limits.HasLimitsOfShape J GrpCat | If `J` is `u`-small, `GrpCat.{u}` has limits of shape `J`. | true |
ShrinkingLemma.PartialRefinement.noConfusionType | Mathlib.Topology.ShrinkingLemma | Sort u →
{ι : Type u_1} →
{X : Type u_2} →
[inst : TopologicalSpace X] →
{u : ι → Set X} →
{s : Set X} →
{p : Set X → Prop} →
ShrinkingLemma.PartialRefinement u s p →
{ι' : Type u_1} →
{X' : Type u_2} →
[inst' ... | null | false |
ContinuousLinearMap.toBilinForm_inj | Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | ∀ {𝕜 : Type u_2} {E : Type u_5} [inst : NormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] (L₁ L₂ : E →L[𝕜] E →L[𝕜] 𝕜), L₁.toBilinForm = L₂.toBilinForm ↔ L₁ = L₂ | null | true |
CategoryTheory.CostructuredArrow.faithful_map₂ | Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{T : D} {S : CategoryTheory.Functor C D} {A : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} A] {B : Type u₄}
[inst_3 : CategoryTheory.Category.{v₄, u₄} B] {U : CategoryTheory.Functor A B} {V... | null | true |
CategoryTheory.Pretriangulated.Triangle.epi₁ | Mathlib.CategoryTheory.Triangulated.Pretriangulated | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C]
[inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C]
[inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C],
∀ T ∈ CategoryTheory.Pr... | null | true |
TopRep.d._proof_3 | Mathlib.RepresentationTheory.Homological.ContCohomology.Basic | ∀ {k : Type u_3} {G : Type u_2} [inst : Ring k] [inst_1 : Group G] [inst_2 : TopologicalSpace k]
[inst_3 : TopologicalSpace G] [inst_4 : IsTopologicalGroup G] (X : TopRep k G),
IsTopologicalAddGroup ↑(X.resolutionX 0) | null | false |
IsPrimitiveRoot.ne_one | Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | ∀ {M : Type u_1} [inst : CommMonoid M] {k : ℕ} {ζ : M}, IsPrimitiveRoot ζ k → 1 < k → ζ ≠ 1 | null | true |
WeierstrassCurve.Jacobian.baseChange_addXYZ | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | ∀ {R : Type r} {S : Type s} {A : Type u} {B : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing A]
[inst_3 : CommRing B] {W' : WeierstrassCurve.Jacobian R} [inst_4 : Algebra R S] [inst_5 : Algebra R A]
[inst_6 : Algebra S A] [IsScalarTower R S A] [inst_8 : Algebra R B] [inst_9 : Algebra S B] [IsS... | null | true |
Lean.Elab.Term.registerMVarErrorInfo | Lean.Elab.Term.TermElabM | Lean.Elab.Term.MVarErrorInfo → Lean.Elab.TermElabM Unit | null | true |
Cardinal.cantorFunctionAux_true | Mathlib.Analysis.Real.Cardinality | ∀ {c : ℝ} {f : ℕ → Bool} {n : ℕ}, f n = true → Cardinal.cantorFunctionAux c f n = c ^ n | null | true |
_private.Mathlib.LinearAlgebra.Pi.0.LinearMap.disjoint_single_single._simp_1_3 | Mathlib.LinearAlgebra.Pi | ∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Sort u_4}
(p : ι → Submodule R M) {x : M}, (x ∈ ⨅ i, p i) = ∀ (i : ι), x ∈ p i | null | false |
Metric.edistLtTopSetoid._proof_2 | Mathlib.Topology.EMetricSpace.Defs | ∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : WeakPseudoEMetricSpace α] {x y : α},
edist x y < ⊤ → edist y x < ⊤ | null | false |
lp.evalCLM._proof_2 | Mathlib.Analysis.Normed.Lp.lpSpace | ∀ (𝕜 : Type u_3) {α : Type u_2} (E : α → Type u_1) [inst : (i : α) → NormedAddCommGroup (E i)] [inst_1 : NormedRing 𝕜]
[inst_2 : (i : α) → Module 𝕜 (E i)] [inst_3 : ∀ (i : α), IsBoundedSMul 𝕜 (E i)] (p : ENNReal) [inst_4 : Fact (1 ≤ p)]
(i : α) (x : ↥(lp E p)), ‖(lp.evalₗ E p i) x‖ ≤ 1 * ‖x‖ | null | false |
List.mem_pi_toList | Mathlib.Data.FinEnum | ∀ {α : Type u_1} [inst : FinEnum α] {β : α → Type u_2} [inst_1 : (a : α) → FinEnum (β a)] (xs : List α)
(f : (a : α) → a ∈ xs → β a), f ∈ xs.pi fun x => FinEnum.toList (β x) | null | true |
IsAlgClosed.degree_eq_one_of_irreducible | Mathlib.FieldTheory.IsAlgClosed.Basic | ∀ (k : Type u) [inst : Field k] [IsAlgClosed k] {p : Polynomial k}, Irreducible p → p.degree = 1 | If `k` is algebraically closed, then every irreducible polynomial over `k` is linear.
| true |
_private.Mathlib.Order.Antisymmetrization.0.wellFoundedLT_antisymmetrization_iff._simp_1_1 | Mathlib.Order.Antisymmetrization | ∀ {α : Type u_1} [inst : Preorder α], WellFounded LT.lt = WellFounded LT.lt | null | false |
Applicative.seqRight._default | Init.Prelude | {f : Type u → Type v} →
({α β : Type u} → (α → β) → f α → f β) →
({α β : Type u} → α → f β → f α) →
({α β : Type u} → f (α → β) → (Unit → f α) → f β) → {α β : Type u} → f α → (Unit → f β) → f β | null | false |
groupCohomology.exists_mul_galRestrict_of_norm_eq_one | Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | ∀ {K L : Type} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] [FiniteDimensional K L]
[inst_4 : IsGalois K L] [IsCyclic Gal(L/K)] {g : Gal(L/K)} {A : Type u_1} {B : Type u_2} [inst_6 : CommRing A]
[inst_7 : CommRing B] [inst_8 : Algebra A B] [inst_9 : Algebra A L] [inst_10 : Algebra A K] [inst_11 : Alge... | The integral version of the classical formulation of Hilbert's theorem 90: in the `ABKL`
setting, suppose that `L/K` is a finite Galois extension such that the Galois group is cyclic
generated by `g` and let `η : B` be an element of norm `1` (when viewed as an element of `L`).
Then there exists `ε : B` such that `ε ≠ 0... | true |
Nat.Primrec.below.comp | Mathlib.Computability.Primrec.Basic | ∀ {motive : (a : ℕ → ℕ) → Nat.Primrec a → Prop} {f g : ℕ → ℕ} (a : Nat.Primrec f) (a_1 : Nat.Primrec g),
Nat.Primrec.below a → motive f a → Nat.Primrec.below a_1 → motive g a_1 → Nat.Primrec.below ⋯ | null | true |
_private.Mathlib.Tactic.ClickSuggestions.TryPremises.0.Mathlib.Tactic.ClickSuggestions.getCandidates | Mathlib.Tactic.ClickSuggestions.TryPremises | Mathlib.Tactic.ClickSuggestions.RwInfo →
Array Mathlib.Tactic.ClickSuggestions.GrwPos →
Mathlib.Tactic.ClickSuggestions.PreDiscrTrees →
Mathlib.Tactic.ClickSuggestions.ClickSuggestionsM (Array Mathlib.Tactic.ClickSuggestions.Candidates✝) | Get the candidate theorems from `pres`.
Used for current file declarations and local hypotheses | true |
_private.Init.Data.UInt.Lemmas.0.USize.lt_of_lt_of_le._simp_1_1 | Init.Data.UInt.Lemmas | ∀ {a b : USize}, (a ≤ b) = (a.toNat ≤ b.toNat) | null | false |
RestrictedProduct.instNSMul.eq_1 | Mathlib.Topology.Algebra.RestrictedProduct.Basic | ∀ {ι : Type u_1} (R : ι → Type u_2) {𝓕 : Filter ι} {S : ι → Type u_3} [inst : (i : ι) → SetLike (S i) (R i)]
{B : (i : ι) → S i} [inst_1 : (i : ι) → AddMonoid (R i)] [inst_2 : ∀ (i : ι), AddSubmonoidClass (S i) (R i)],
RestrictedProduct.instNSMul R = { smul := fun n x => ⟨fun i => n • x i, ⋯⟩ } | null | true |
CategoryTheory.Pretopology.toGrothendieck | Mathlib.CategoryTheory.Sites.Pretopology | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.Limits.HasPullbacks C] →
CategoryTheory.Pretopology C → CategoryTheory.GrothendieckTopology C | A pretopology `K` can be completed to a Grothendieck topology `J` by declaring a sieve to be
`J`-covering if it contains a family in `K`.
See also [MM92] Chapter III, Section 2, Equation (2).
| true |
CategoryTheory.Profunctor.whiskerLeft._proof_4 | Mathlib.CategoryTheory.Profunctor.Comp | ∀ {C D E : Type u_1} [inst : CategoryTheory.Category.{u_4, u_1} C] [inst_1 : CategoryTheory.Category.{u_5, u_1} D]
[inst_2 : CategoryTheory.Category.{u_3, u_1} E] (P : CategoryTheory.Profunctor.{max u_1 u_2, u_4, u_5, u_1, u_1} C D)
{Q R : CategoryTheory.Profunctor.{max u_1 u_2, u_5, u_3, u_1, u_1} D E} (f : Q ⟶ R)... | null | false |
_private.Lean.Meta.IndPredBelow.0.Lean.Meta.IndPredBelow.Context._sizeOf_1 | Lean.Meta.IndPredBelow | Lean.Meta.IndPredBelow.Context✝ → ℕ | null | false |
Lean.Omega.Fin.lt_or_gt_of_ne | Init.Omega.Int | ∀ {n : ℕ} {i j : Fin n}, i ≠ j → i < j ∨ i > j | null | true |
Field.lift_rank_mul_lift_sepDegree_of_isSeparable | Mathlib.FieldTheory.PurelyInseparable.Tower | ∀ (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] [inst_5 : Algebra E K] [IsScalarTower F E K] [Algebra.IsSeparable F E],
Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Field.sepDegree E K) =
Cardinal.lif... | If `K / E / F` is a field extension tower, such that `E / F` is separable,
then $[E:F] [K:E]_s = [K:F]_s$.
It is a special case of `Field.lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic`, and is an
intermediate result used to prove it. | true |
Matrix.blockDiagonalAddMonoidHom | Mathlib.Data.Matrix.Block | (m : Type u_2) →
(n : Type u_3) →
(o : Type u_4) →
(α : Type u_11) → [DecidableEq o] → [inst : AddZeroClass α] → (o → Matrix m n α) →+ Matrix (m × o) (n × o) α | `Matrix.blockDiagonal` as an `AddMonoidHom`. | true |
_private.Mathlib.CategoryTheory.Triangulated.TriangleShift.0.CategoryTheory.Pretriangulated.Triangle.shiftFunctor._simp_1 | Mathlib.CategoryTheory.Triangulated.TriangleShift | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(self : CategoryTheory.Functor C D) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp (self.map f) (self.map g) = self.map (CategoryTheory.CategoryStruct.comp f g) | null | false |
_private.Mathlib.Tactic.ClickSuggestions.Util.0.Mathlib.Tactic.ClickSuggestions.Premise.toString.match_1 | Mathlib.Tactic.ClickSuggestions.Util | (motive : Mathlib.Tactic.ClickSuggestions.Premise → Sort u_1) →
(x : Mathlib.Tactic.ClickSuggestions.Premise) →
((name : Lean.Name) → motive (Mathlib.Tactic.ClickSuggestions.Premise.const name)) →
((name : Lean.Name) → motive (Mathlib.Tactic.ClickSuggestions.Premise.fvar { name := name })) → motive x | null | false |
QuaternionAlgebra.instStarRing._proof_3 | Mathlib.Algebra.Quaternion | ∀ {R : Type u_1} {c₁ c₂ c₃ : R} [inst : CommRing R] (a b : QuaternionAlgebra R c₁ c₂ c₃), star (a + b) = star a + star b | null | false |
_private.Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char.0._regBuiltin.Char.reduceLT.declare_76._@.Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char.1447824035._hygCtx._hyg.27 | Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char | IO Unit | null | false |
Std.Http.URI.EncodedString.new | Std.Http.Data.URI.Encoding | {r : UInt8 → Bool} →
(ba : ByteArray) →
Std.Http.URI.IsAllowedEncodedChars r ba →
Std.Http.URI.isValidPercentEncoding ba = true → Std.Http.URI.EncodedString r | Creates an `EncodedString` from a `ByteArray` with compile-time proofs.
Use this when you have proofs that the byte array is valid.
| true |
Lean.Lsp.DidChangeWatchedFilesParams.mk.injEq | Lean.Data.Lsp.Workspace | ∀ (changes changes_1 : Array Lean.Lsp.FileEvent),
({ changes := changes } = { changes := changes_1 }) = (changes = changes_1) | null | true |
_private.Init.Data.List.Basic.0.List.lengthTRAux.match_1.splitter | Init.Data.List.Basic | {α : Type u_1} →
(motive : List α → ℕ → Sort u_2) →
(x : List α) →
(x_1 : ℕ) →
((n : ℕ) → motive [] n) → ((head : α) → (as : List α) → (n : ℕ) → motive (head :: as) n) → motive x x_1 | null | true |
_private.Lean.Meta.IndPredBelow.0.Lean.Meta.IndPredBelow.NewDecl.below.noConfusion | Lean.Meta.IndPredBelow | {P : Sort u} →
{decl : Lean.LocalDecl} →
{indName : Lean.Name} →
{vars : Array Lean.FVarId} →
{decl' : Lean.LocalDecl} →
{indName' : Lean.Name} →
{vars' : Array Lean.FVarId} →
Lean.Meta.IndPredBelow.NewDecl.below✝ decl indName vars =
Lean.Meta.In... | null | false |
Action.instConcreteCategoryHomSubtypeV._proof_7 | Mathlib.CategoryTheory.Action.Basic | ∀ (V : Type u_2) [inst : CategoryTheory.Category.{u_1, u_2} V] (G : Type u_3) [inst_1 : Monoid G]
{FV : V → V → Type u_5} {CV : V → Type u_4} [inst_2 : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)]
[inst_3 : CategoryTheory.ConcreteCategory V FV] {X Y Z : Action V G} (x : X ⟶ Y) (x_1 : Y ⟶ Z) (x_2 : CV X.V),
(Catego... | null | false |
SemiNormedGrp._sizeOf_1 | Mathlib.Analysis.Normed.Group.SemiNormedGrp | SemiNormedGrp → ℕ | null | false |
Matrix.SpecialLinearGroup.coe_int_neg | Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup | ∀ {n : Type u} [inst : DecidableEq n] [inst_1 : Fintype n] {R : Type v} [inst_2 : CommRing R]
[inst_3 : Fact (Even (Fintype.card n))] (g : Matrix.SpecialLinearGroup n ℤ),
(Matrix.SpecialLinearGroup.map (Int.castRingHom R)) (-g) = -(Matrix.SpecialLinearGroup.map (Int.castRingHom R)) g | null | true |
_private.Std.Time.Format.Basic.0.Std.Time.GenericFormat.builderParser.go._f | Std.Time.Format.Basic | {α : Type} →
Std.Time.FormatConfig →
(format : Std.Time.FormatString) →
List.below (motive := fun format => Std.Time.FormatType (Option α) format → Std.Internal.Parsec.String.Parser α)
format →
Std.Time.FormatType (Option α) format → Std.Internal.Parsec.String.Parser α | null | false |
List.find?_some._f | Init.Data.List.Find | ∀ {α : Type u_1} {p : α → Bool} {a : α} (x : List α)
(f : List.below (motive := fun x => List.find? p x = some a → p a = true) x), List.find? p x = some a → p a = true | null | false |
UpperSet.erase._proof_1 | Mathlib.Order.UpperLower.Closure | ∀ {α : Type u_1} [inst : Preorder α] (s : UpperSet α) (a : α), IsUpperSet (↑s \ ↑(LowerSet.Iic a)) | null | false |
HVertexOperator.coeff._proof_5 | Mathlib.Algebra.Vertex.HVertexOperator | ∀ {Γ : Type u_2} [inst : PartialOrder Γ] {R : Type u_1} {W : Type u_3} [inst_1 : CommRing R] [inst_2 : AddCommGroup W]
[inst_3 : Module R W], SMulCommClass R R (HahnModule Γ R W) | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Const.maxKey!_modify._simp_1_3 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : Ord α} {t : Std.DTreeMap.Internal.Impl α β} {k : α},
(k ∈ t) = (Std.DTreeMap.Internal.Impl.contains k t = true) | null | false |
CategoryTheory.GrothendieckTopology.W_of_preservesSheafification | Mathlib.CategoryTheory.Sites.PreservesSheafification | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {A : Type u_1}
{B : Type u_2} [inst_1 : CategoryTheory.Category.{v_1, u_1} A] [inst_2 : CategoryTheory.Category.{v_2, u_2} B]
(F : CategoryTheory.Functor A B) [J.PreservesSheafification F] {P₁ P₂ : CategoryTheory.Fu... | null | true |
DFinsupp.instDecidableEq.match_1 | Mathlib.Data.DFinsupp.Defs | ∀ {ι : Type u_1} {β : ι → Type u_2} [inst : DecidableEq ι] [inst_1 : (i : ι) → Zero (β i)]
[inst_2 : (i : ι) → DecidableEq (β i)] (f g : Π₀ (i : ι), β i)
(motive : (f.support = g.support ∧ ∀ i ∈ f.support, f i = g i) → Prop)
(x : f.support = g.support ∧ ∀ i ∈ f.support, f i = g i),
(∀ (h₁ : f.support = g.suppor... | null | false |
_private.Mathlib.CategoryTheory.ComposableArrows.Basic.0.CategoryTheory.ComposableArrows.isoMk₅._proof_5 | Mathlib.CategoryTheory.ComposableArrows.Basic | ¬2 + 1 ≤ 5 → False | null | false |
Std.DHashMap.Const.get_alter_self | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {x : BEq α} {x_1 : Hashable α} {β : Type v} {m : Std.DHashMap α fun x => β} [inst : EquivBEq α]
[inst_1 : LawfulHashable α] {k : α} {f : Option β → Option β} {h : k ∈ Std.DHashMap.Const.alter m k f},
Std.DHashMap.Const.get (Std.DHashMap.Const.alter m k f) k h = (f (Std.DHashMap.Const.get? m k)).get ⋯ | null | true |
_private.Mathlib.Data.Nat.Choose.Sum.0.Nat.sum_range_choose_halfway._proof_1_1 | Mathlib.Data.Nat.Choose.Sum | ∀ (m : ℕ), m + 1 ≤ 2 * m + 1 + 1 | null | false |
IsUnital.toCommCStarAlgebra._proof_6 | Mathlib.Analysis.CStarAlgebra.Classes | ∀ {A : Type u_1} [inst : NonUnitalCommCStarAlgebra A] [inst_1 : IsUnital A] (a : A), 0 * a = 0 | null | false |
Complex.sin.eq_1 | Mathlib.Analysis.Complex.Trigonometric | ∀ (z : ℂ), Complex.sin z = (Complex.exp (-z * Complex.I) - Complex.exp (z * Complex.I)) * Complex.I / 2 | null | true |
CategoryTheory.SimplicialObject.Augmented.w₀_assoc | Mathlib.AlgebraicTopology.SimplicialObject.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : CategoryTheory.SimplicialObject.Augmented C} (f : X ⟶ Y)
{Z : C} (h : Y.right ⟶ Z),
CategoryTheory.CategoryStruct.comp (f.left.app (Opposite.op { len := 0 }))
(CategoryTheory.CategoryStruct.comp (Y.hom.app (Opposite.op { len := 0 })) h) =
Cat... | The compatibility of a morphism with the augmentation, on 0-simplices | true |
InvolutiveNeg.recOn | Mathlib.Algebra.Group.DivInvMonoid | {A : Type u_2} →
{motive : InvolutiveNeg A → Sort u} →
(t : InvolutiveNeg A) →
([toNeg : Neg A] → (neg_neg : ∀ (x : A), - -x = x) → motive { toNeg := toNeg, neg_neg := neg_neg }) → motive t | null | false |
Polynomial.Splits.X._simp_1 | Mathlib.Algebra.Polynomial.Splits | ∀ {R : Type u_1} [inst : Semiring R], Polynomial.X.Splits = True | null | false |
contDiffOn_congr | Mathlib.Analysis.Calculus.ContDiff.Defs | ∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
{f f₁ : E → F} {n : WithTop ℕ∞}, (∀ x ∈ s, f₁ x = f x) → (ContDiffOn 𝕜 n f₁ s ↔ ContDiffOn 𝕜 n f s) | null | true |
translate_add' | Mathlib.Algebra.Group.Translate | ∀ {α : Type u_2} {G : Type u_5} [inst : AddCommGroup G] (a b : G) (f : G → α),
translate (a + b) f = translate b (translate a f) | See `translate_add` | true |
compHausToTop.createsLimits | Mathlib.Topology.Category.CompHaus.Basic | CategoryTheory.CreatesLimits compHausToTop | null | true |
_private.Mathlib.CategoryTheory.GradedObject.Unitor.0.CategoryTheory.GradedObject.mapBifunctor_triangle._simp_1_1 | Mathlib.CategoryTheory.GradedObject.Unitor | ∀ {obj : Type u} [self : CategoryTheory.Category.{v, u} obj] {W X Y Z : obj} (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h | null | false |
HomologicalComplex.HomologySequence.snakeInput._proof_27 | Mathlib.Algebra.Homology.HomologySequence | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Abelian C],
CategoryTheory.ShortComplex.π₃.PreservesZeroMorphisms | null | false |
Lean.Elab.Tactic.Conv.evalNestedTactic | Lean.Elab.Tactic.Conv.Basic | Lean.Elab.Tactic.Tactic | null | true |
Lean.Elab.Term.Do.Alt.noConfusionType | Lean.Elab.Do.Legacy | Sort u → {σ : Type} → Lean.Elab.Term.Do.Alt σ → {σ' : Type} → Lean.Elab.Term.Do.Alt σ' → Sort u | null | false |
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