name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
UInt64.ofFin_shiftLeft | Init.Data.UInt.Bitwise | ∀ (a b : Fin UInt64.size), ↑b < 64 → UInt64.ofFin (a <<< b) = UInt64.ofFin a <<< UInt64.ofFin b | null | true |
_private.Mathlib.Combinatorics.Hall.Finite.0.HallMarriageTheorem.hall_cond_of_compl._simp_1_9 | Mathlib.Combinatorics.Hall.Finite | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) | null | false |
instCommMonoidUniformOnFun._proof_1 | Mathlib.Topology.Algebra.UniformConvergence | ∀ {α : Type u_1} {β : Type u_2} {𝔖 : Set (Set α)} [inst : CommMonoid β] (a b : UniformOnFun α β 𝔖), a * b = b * a | null | false |
_private.Mathlib.Data.Vector.Basic.0.List.Vector.mOfFn.match_1.eq_1 | Mathlib.Data.Vector.Basic | ∀ {m : Type u_3 → Type u_2} {α : Type u_3} (motive : (x : ℕ) → (Fin x → m α) → Sort u_1) (x : Fin 0 → m α)
(h_1 : (x : Fin 0 → m α) → motive 0 x) (h_2 : (n : ℕ) → (f : Fin (n + 1) → m α) → motive n.succ f),
(match 0, x with
| 0, x => h_1 x
| n.succ, f => h_2 n f) =
h_1 x | null | true |
CategoryTheory.eId | Mathlib.CategoryTheory.Enriched.Basic | (V : Type v) →
[inst : CategoryTheory.Category.{w, v} V] →
[inst_1 : CategoryTheory.MonoidalCategory V] →
{C : Type u₁} →
[inst_2 : CategoryTheory.EnrichedCategory V C] →
(X : C) → CategoryTheory.MonoidalCategoryStruct.tensorUnit V ⟶ X ⟶[V] X | The `𝟙_ V`-shaped generalized element giving the identity in a `V`-enriched category.
| true |
ENNReal.ofReal_lt_ofReal_iff._simp_1 | Mathlib.Basic.ENNReal.Real | ∀ {p q : ℝ}, 0 < q → (ENNReal.ofReal p < ENNReal.ofReal q) = (p < q) | null | false |
CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem_assoc | Mathlib.CategoryTheory.Idempotents.HomologicalComplex | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C] {ι : Type u_2}
{c : ComplexShape ι} (P : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) (n : ι) {Z : C}
(h : P.X.X n ⟶ Z),
CategoryTheory.CategoryStruct.comp (P.p.f n) (CategoryTheory.CategoryStruc... | null | true |
LinearMap.instDistribMulAction | Mathlib.Algebra.Module.LinearMap.Defs | {R : Type u_1} →
{R₂ : Type u_3} →
{S : Type u_5} →
{M : Type u_8} →
{M₂ : Type u_10} →
[inst : Semiring R] →
[inst_1 : Semiring R₂] →
[inst_2 : AddCommMonoid M] →
[inst_3 : AddCommMonoid M₂] →
[inst_4 : Module R M] →
... | null | true |
TwoSidedIdeal.orderIsoIdeal._proof_5 | Mathlib.RingTheory.TwoSidedIdeal.Operations | ∀ {R : Type u_1} [inst : CommRing R] {a b : TwoSidedIdeal R},
{ toFun := ⇑TwoSidedIdeal.asIdeal, invFun := ⇑TwoSidedIdeal.fromIdeal, left_inv := ⋯, right_inv := ⋯ } a ≤
{ toFun := ⇑TwoSidedIdeal.asIdeal, invFun := ⇑TwoSidedIdeal.fromIdeal, left_inv := ⋯, right_inv := ⋯ } b ↔
a ≤ b | null | false |
Rep.coinvariantsTensorMk._proof_2 | Mathlib.RepresentationTheory.Coinvariants | ∀ {k : Type u_1} {G : Type u_2} [inst : CommRing k] [inst_1 : Monoid G] (A B : Rep.{u_1, u_1, u_2} k G),
SMulCommClass k k (TensorProduct k ↑A ↑B) | null | false |
Int.Internal.Linear.Poly.combine'.match_3 | Init.Data.Int.Linear | (motive : ℕ → Sort u_1) → (fuel : ℕ) → (Unit → motive 0) → ((fuel : ℕ) → motive fuel.succ) → motive fuel | null | false |
Submodule.LinearDisjoint.mk._flat_ctor | Mathlib.LinearAlgebra.LinearDisjoint | ∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S] {M N : Submodule R S},
Function.Injective ⇑(M.mulMap N) → M.LinearDisjoint N | null | false |
Rat.sqrt_eq | Mathlib.Data.Rat.Sqrt | ∀ (q : ℚ), Rat.sqrt (q * q) = |q| | null | true |
SSet.stdSimplex.finSuccAboveOrderIsoFinset | Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex | {n : ℕ} → (i : Fin (n + 2)) → Fin (n + 1) ≃o ↥{i}ᶜ | If `i : Fin (n + 2)`, this is the order isomorphism between `Fin (n +1)`
and the complement of `{i}` as a finset. | true |
Finmap.mk.sizeOf_spec | Mathlib.Data.Finmap | ∀ {α : Type u} {β : α → Type v} [inst : SizeOf α] [inst_1 : (a : α) → SizeOf (β a)] (entries : Multiset (Sigma β))
(nodupKeys : entries.NodupKeys),
sizeOf { entries := entries, nodupKeys := nodupKeys } = 1 + sizeOf entries + sizeOf nodupKeys | null | true |
_private.Mathlib.CategoryTheory.Monoidal.Functor.0._auto_48 | Mathlib.CategoryTheory.Monoidal.Functor | Lean.Syntax | null | false |
_private.Mathlib.GroupTheory.GroupAction.Hom.0.MulActionHom.End.equivMulOpposite._simp_3 | Mathlib.GroupTheory.GroupAction.Hom | ∀ {F : Type u_8} {M : Type u_9} {X : Type u_10} {Y : Type u_11} [inst : SMul M X] [inst_1 : SMul M Y]
[inst_2 : FunLike F X Y] [MulActionHomClass F M X Y] (f : F) (c : M) (x : X), c • f x = f (c • x) | null | false |
Mathlib.Tactic.BicategoryLike.StructuralAtom.coherenceHom.elim | Mathlib.Tactic.CategoryTheory.Coherence.Datatypes | {motive : Mathlib.Tactic.BicategoryLike.StructuralAtom → Sort u} →
(t : Mathlib.Tactic.BicategoryLike.StructuralAtom) →
t.ctorIdx = 4 →
((α : Mathlib.Tactic.BicategoryLike.CoherenceHom) →
motive (Mathlib.Tactic.BicategoryLike.StructuralAtom.coherenceHom α)) →
motive t | null | false |
CategoryTheory.Limits.Bicone.toBinaryBiconeIsColimit._proof_1 | Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{X Y : C} (b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y))
(j : CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair),
CategoryTheory.CategoryStruct.comp (b.toBinaryBi... | null | false |
Std.IterM.DefaultConsumers.forIn'._proof_4 | Init.Data.Iterators.Consumers.Monadic.Loop | ∀ {m : Type u_1 → Type u_2} {α β : Type u_1} [inst : Std.Iterator α m β] (γ : Type u_3)
(PlausibleForInStep : β → γ → ForInStep γ → Prop) (P : β → Prop) (it : Std.IterM m β) (acc : γ)
(hP : ∀ (b : β), it.IsPlausibleIndirectOutput b → P b) (it' : Std.IterM m β) (out : β)
(h : it.IsPlausibleStep (Std.IterStep.yield... | null | false |
Std.DHashMap.Const.get!_eq_get! | 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 : LawfulBEq α]
[inst_1 : Inhabited β] {a : α}, Std.DHashMap.Const.get! m a = m.get! a | null | true |
CategoryTheory.SemiadditiveOfBinaryBiproducts.addCommMonoidHomOfHasBinaryBiproducts._proof_4 | Mathlib.CategoryTheory.Preadditive.OfBiproducts | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] (X Y : C) (a : X ⟶ Y),
CategoryTheory.SemiadditiveOfBinaryBiproducts.rightAdd X Y 0 a = a | null | false |
Set.IsPWO.mono | Mathlib.Order.WellFoundedSet | ∀ {α : Type u_2} [inst : Preorder α] {s t : Set α}, t.IsPWO → s ⊆ t → s.IsPWO | null | true |
_private.Lean.Meta.Sym.InstantiateS.0.Lean.Meta.Sym.instantiateRevBetaS'.visit._unsafe_rec | Lean.Meta.Sym.InstantiateS | Array Lean.Expr → Lean.Expr → ℕ → Lean.Meta.Sym.M✝ Lean.Expr | null | false |
AlgebraicGeometry.IsReduced.mk._flat_ctor | Mathlib.AlgebraicGeometry.Properties | ∀ {X : AlgebraicGeometry.Scheme},
autoParam (∀ (U : X.Opens), IsReduced ↑(X.presheaf.obj (Opposite.op U)))
AlgebraicGeometry.IsReduced.component_reduced._autoParam →
AlgebraicGeometry.IsReduced X | null | false |
Ideal.mem_toTwoSided | Mathlib.RingTheory.TwoSidedIdeal.Operations | ∀ {R : Type u_1} [inst : Ring R] {I : Ideal R} [inst_1 : I.IsTwoSided] {x : R}, x ∈ I.toTwoSided ↔ x ∈ I | null | true |
AddGroup.residuallyFinite_iff_forall_finiteIndex | Mathlib.GroupTheory.ResiduallyFinite | ∀ {G : Type u_1} [inst : AddGroup G],
AddGroup.ResiduallyFinite G ↔ ∀ (g : G), (∀ (H : AddSubgroup G) [H.FiniteIndex], g ∈ H) → g = 0 | null | true |
_private.Mathlib.Order.Filter.AtTopBot.Field.0.Filter.tendsto_mul_const_atBot_iff._simp_1_1 | Mathlib.Order.Filter.AtTopBot.Field | ∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] {l : Filter β}
{f : β → α} {r : α} [l.NeBot],
Filter.Tendsto (fun x => r * f x) l Filter.atBot =
(0 < r ∧ Filter.Tendsto f l Filter.atBot ∨ r < 0 ∧ Filter.Tendsto f l Filter.atTop) | null | false |
_private.Mathlib.RingTheory.DedekindDomain.SelmerGroup.0.«_aux_Mathlib_RingTheory_DedekindDomain_SelmerGroup___macroRules__private_Mathlib_RingTheory_DedekindDomain_SelmerGroup_0_term_/__1_1» | Mathlib.RingTheory.DedekindDomain.SelmerGroup | Lean.Macro | null | false |
CategoryTheory.SmallObject.SuccStruct.extendToSuccRestrictionLEIso_hom_app | Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] {J : Type u} [inst_1 : LinearOrder J]
[inst_2 : SuccOrder J] {j : J} (hj : ¬IsMax j) (F : CategoryTheory.Functor (↑(Set.Iic j)) C) {X : C}
(τ : F.obj ⟨j, ⋯⟩ ⟶ X) (X_1 : ↑(Set.Iic j)),
(CategoryTheory.SmallObject.SuccStruct.extendToSuccRestrictionLEIso... | null | true |
SimpleGraph.cliqueFree_completeMultipartiteGraph | Mathlib.Combinatorics.SimpleGraph.Clique | ∀ {n : ℕ} {ι : Type u_3} [inst : Fintype ι] (V : ι → Type u_4),
Fintype.card ι < n → (SimpleGraph.completeMultipartiteGraph V).CliqueFree n | A complete `r`-partite graph has no `n`-cliques for `r < n`. | true |
_private.Mathlib.Geometry.Convex.Cone.Face.Basic.0.PointedCone.IsFaceOf.sInf._simp_1_1 | Mathlib.Geometry.Convex.Cone.Face.Basic | ∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{p q : Submodule R M} {x : M}, (x ∈ p ⊓ q) = (x ∈ p ∧ x ∈ q) | null | false |
MeasureTheory.AEEqFun.pow_toGerm | Mathlib.MeasureTheory.Function.AEEqFun | ∀ {α : Type u_1} {γ : Type u_3} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace γ]
[inst_2 : Monoid γ] [inst_3 : ContinuousMul γ] (f : α →ₘ[μ] γ) (n : ℕ), (f ^ n).toGerm = f.toGerm ^ n | null | true |
_private.Mathlib.NumberTheory.NumberField.Cyclotomic.Galois.0.IsCyclotomicExtension.Rat.mem_subgroupGalEquivSubgroupChar_symm_iff._simp_1_3 | Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | ∀ {M : Type u_1} {R : Type u_2} [inst : CommMonoid M] [inst_1 : CommRing R] [inst_2 : Finite M]
[inst_3 : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)] {X : Subgroup (MulChar M R)} {m : Mˣ},
(m ∈ (MulChar.subgroupOrderIsoSubgroupMulChar M R).symm (OrderDual.toDual X)) = ∀ χ ∈ X, χ ↑m = 1 | null | false |
MulRingNorm.mulRingNormEquivAbsoluteValue_symm_apply | Mathlib.Analysis.Normed.Unbundled.RingSeminorm | ∀ {R : Type u_2} [inst : Ring R] [inst_1 : Nontrivial R] (v : AbsoluteValue R ℝ),
MulRingNorm.mulRingNormEquivAbsoluteValue.symm v = MulRingNorm.ofAbsoluteValue v | null | true |
_private.Mathlib.Tactic.NormNum.Ordinal.0.Mathlib.Meta.NormNum.evalOrdinalMod._proof_1 | Mathlib.Tactic.NormNum.Ordinal | ∀ (an bn rn : Q(ℕ)), («$an» % «$bn») =Q «$rn» | null | false |
AlgebraicGeometry.prodComparison_algSpec_left | Mathlib.AlgebraicGeometry.Group.Affine | ∀ {R : CommRingCat} (X Y : (CommAlgCat ↑R)ᵒᵖ),
CategoryTheory.Over.Hom.left
(CategoryTheory.CartesianMonoidalCategory.prodComparison (AlgebraicGeometry.algSpec R) X Y) =
(AlgebraicGeometry.pullbackSpecIso ↑R ↑(Opposite.unop X) ↑(Opposite.unop Y)).inv | null | true |
Associates.irreducible_iff_prime_iff | Mathlib.Algebra.GroupWithZero.Associated | ∀ {M : Type u_1} [inst : CommMonoidWithZero M],
(∀ (a : M), Irreducible a ↔ Prime a) ↔ ∀ (a : Associates M), Irreducible a ↔ Prime a | null | true |
OrdinalApprox.lfpApprox_zero | Mathlib.SetTheory.Ordinal.FixedPointApproximants | ∀ {α : Type u} [inst : CompleteLattice α] (f : α →o α) {x : α}, OrdinalApprox.lfpApprox f x 0 = x | null | true |
hfdifferential._proof_3 | Mathlib.Geometry.Manifold.DerivationBundle | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_6} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_7} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_5}
[inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_3} [inst_6 : NormedAddComm... | null | false |
Std.Internal.List.mem_iff_getKey?_eq_some_and_getValue?_eq_some | Std.Data.Internal.List.Associative | ∀ {α : Type u} [inst : BEq α] [EquivBEq α] {β : Type v} {k : α} {v : β} {l : List ((_ : α) × β)},
Std.Internal.List.DistinctKeys l →
(⟨k, v⟩ ∈ l ↔ Std.Internal.List.getKey? k l = some k ∧ Std.Internal.List.getValue? k l = some v) | null | true |
Composition.recOnAppendSingle._proof_1 | Mathlib.Combinatorics.Enumerative.Composition | ∀ {motive : (n : ℕ) → Composition n → Sort u_1} (k n : ℕ) (c : Composition n),
motive (k + 1 + n) ((Composition.single (k + 1) ⋯).append c).reverse =
motive (n + (k + 1)) (c.reverse.append (Composition.single (k + 1) ⋯)) | null | false |
Equiv.Perm.congr_arg | Mathlib.Logic.Equiv.Defs | ∀ {α : Sort u} {f : Equiv.Perm α} {x x' : α}, x = x' → f x = f x' | null | true |
FinVec.prod.eq_def | Mathlib.Data.Fin.Tuple.Reflection | ∀ {α : Type u_1} [inst : Mul α] [inst_1 : One α] (x : ℕ) (x_1 : Fin x → α),
FinVec.prod x_1 =
match x, x_1 with
| 0, x => 1
| 1, v => v 0
| n.succ.succ, v => (FinVec.prod fun i => v i.castSucc) * v (Fin.last (n + 1)) | null | true |
_private.Mathlib.FieldTheory.SeparableClosure.0.IntermediateField.normalClosure.isSeparable._simp_1 | Mathlib.FieldTheory.SeparableClosure | ∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] (L : IntermediateField F E),
Algebra.IsSeparable F ↥L = (L ≤ separableClosure F E) | null | false |
one_lt_mul_self_iff._simp_2 | Mathlib.Algebra.Order.Monoid.Defs | ∀ {α : Type u_1} [inst : CommMonoid α] [inst_1 : LinearOrder α] [IsOrderedMonoid α] {a : α}, (1 < a * a) = (1 < a) | null | false |
Bot.ctorIdx | Mathlib.Order.Notation | {α : Type u_1} → Bot α → ℕ | null | false |
Quiver.Path.reverse | Mathlib.Combinatorics.Quiver.Symmetric | {V : Type u_2} → [inst : Quiver V] → [Quiver.HasReverse V] → {a b : V} → Quiver.Path a b → Quiver.Path b a | Reverse the direction of a path. | true |
Homeomorph.addRight.eq_1 | Mathlib.Topology.Algebra.Group.Basic | ∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : SeparatelyContinuousAdd G] (a : G),
Homeomorph.addRight a = { toEquiv := Equiv.addRight a, continuous_toFun := ⋯, continuous_invFun := ⋯ } | null | true |
_private.Init.Data.Nat.MinMax.0.Nat.add_min_add_right._simp_1_1 | Init.Data.Nat.MinMax | ∀ {m k n : ℕ}, (m + n ≤ k + n) = (m ≤ k) | null | false |
Quiver.Arborescence.ctorIdx | Mathlib.Combinatorics.Quiver.Arborescence | {V : Type u} → {inst : Quiver V} → Quiver.Arborescence V → ℕ | null | false |
_private.Lean.Compiler.LCNF.ExplicitRC.0.Lean.Compiler.LCNF.Context.mk._flat_ctor | Lean.Compiler.LCNF.ExplicitRC | Lean.FVarIdHashSet →
Lean.Compiler.LCNF.DerivedValMap✝ →
Lean.Compiler.LCNF.VarMap✝ → Lean.Compiler.LCNF.JPLiveVarMap✝ → ℕ → Lean.Compiler.LCNF.Context✝ | null | false |
Lean.Elab.Do.withDoBlockResultType | Lean.Elab.Do.Basic | {α : Type} → Lean.Expr → Lean.Elab.Do.DoElabM α → Lean.Elab.Do.DoElabM α | Set the new `do` block result type for the scope of the continuation `k`. | true |
Option.isSome.eq_1 | Init.Data.Option.Lemmas | ∀ {α : Type u_1} (val : α), (some val).isSome = true | null | true |
_private.Init.Data.BitVec.Lemmas.0.BitVec.cons_append_append._proof_1_2 | Init.Data.BitVec.Lemmas | ∀ {w₁ w₂ w₃ : ℕ}, ∀ i < w₁ + 1 + w₂ + w₃, ¬i < w₁ + w₂ + w₃ → i - w₃ - w₂ < w₁ → False | null | false |
Mathlib.Tactic.Algebra.RingCompute.cast | Mathlib.Tactic.Algebra.Basic | {u v : Lean.Level} →
{R : Q(Type u)} →
{A : Q(Type v)} →
{sR : Q(CommSemiring «$R»)} →
{sA : Q(CommSemiring «$A»)} →
(sAlg : Q(Algebra «$R» «$A»)) →
Mathlib.Tactic.Algebra.Cache sR →
(u' : Lean.Level) →
(R' : Q(Type u')) →
Q(CommS... | Take an expression `r'` in a ring `R'` such that `R` is an `R'`-algebra and cast `r'` to `R`
using `algebraMap R' R`, so that the scalar multiplication action on `A` is preserved. | true |
Lean.Lsp.instFromJsonPosition | Lean.Data.Lsp.BasicAux | Lean.FromJson Lean.Lsp.Position | null | true |
CategoryTheory.SimplicialThickening.Path.mk.inj | Mathlib.AlgebraicTopology.SimplicialNerve | ∀ {J : Type u_1} {inst : LinearOrder J} {i j : J} {I : Set J}
{left : autoParam (i ∈ I) CategoryTheory.SimplicialThickening.Path.left._autoParam}
{right : autoParam (j ∈ I) CategoryTheory.SimplicialThickening.Path.right._autoParam}
{left_le : autoParam (∀ k ∈ I, i ≤ k) CategoryTheory.SimplicialThickening.Path.lef... | null | true |
instRingUniversalEnvelopingAlgebra._proof_39 | Mathlib.Algebra.Lie.UniversalEnveloping | ∀ (R : Type u_1) (L : Type u_2) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L],
autoParam (∀ (a : UniversalEnvelopingAlgebra R L), 0 • a = 0) SubNegMonoid.zsmul_zero'._autoParam | null | false |
AddSubgroup.instTop.eq_1 | Mathlib.Algebra.Group.Subgroup.Lattice | ∀ {G : Type u_1} [inst : AddGroup G],
AddSubgroup.instTop =
{
top :=
let __src := ⊤;
{ toAddSubmonoid := __src, neg_mem' := ⋯ } } | null | true |
Lean.Parser.Attr._aux_Mathlib_Tactic_Simps_Basic___macroRules_Lean_Parser_Attr_attrSimps!?__1 | Mathlib.Tactic.Simps.Basic | Lean.Macro | null | false |
IntermediateField.copy | Mathlib.FieldTheory.IntermediateField.Basic | {K : Type u_1} →
{L : Type u_2} →
[inst : Field K] →
[inst_1 : Field L] →
[inst_2 : Algebra K L] → (S : IntermediateField K L) → (s : Set L) → s = ↑S → IntermediateField K L | Copy of an intermediate field with a new `carrier` equal to the old one. Useful to fix
definitional equalities. | true |
Sublattice.comap | Mathlib.Order.Sublattice | {α : Type u_2} →
{β : Type u_3} → [inst : Lattice α] → [inst_1 : Lattice β] → LatticeHom α β → Sublattice β → Sublattice α | The preimage of a sublattice along a lattice homomorphism. | true |
Finsupp.smul_single | Mathlib.Data.Finsupp.SMulWithZero | ∀ {α : Type u_1} {M : Type u_2} {R : Type u_4} [inst : Zero M] [inst_1 : SMulZeroClass R M] (c : R) (a : α) (b : M),
(c • fun₀ | a => b) = fun₀ | a => c • b | null | true |
CategoryTheory.GrothendieckTopology.isoToPlus.congr_simp | Mathlib.CategoryTheory.Sites.Plus | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {D : Type w}
[inst_1 : CategoryTheory.Category.{w', w} D]
[inst_2 :
∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X), CategoryTheory.Limits.HasMultiequalizer (S.index P)]
(P : CategoryTheory.Functo... | null | true |
PFunctor.M.bisim' | Mathlib.Data.PFunctor.Univariate.M | ∀ {P : PFunctor.{uA, uB}} {α : Type u_3} (Q : α → Prop) (u v : α → P.M),
(∀ (x : α),
Q x →
∃ a f f',
(u x).dest = PFunctor.Obj.mk a f ∧
(v x).dest = PFunctor.Obj.mk a f' ∧ ∀ (i : P.B a), ∃ x', Q x' ∧ f i = u x' ∧ f' i = v x') →
∀ (x : α), Q x → u x = v x | null | true |
Asymptotics.isBigO_top._simp_1 | Mathlib.Analysis.Asymptotics.Lemmas | ∀ {α : Type u_1} {E : Type u_3} {F : Type u_4} [inst : Norm E] [inst_1 : Norm F] {f : α → E} {g : α → F},
f =O[⊤] g = ∃ C, ∀ (x : α), ‖f x‖ ≤ C * ‖g x‖ | null | false |
CategoryTheory.OrthogonalReflection.D₁.ιLeft_comp_t_assoc | Mathlib.CategoryTheory.Presentable.OrthogonalReflection | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {W : CategoryTheory.MorphismProperty C} {Z : C}
[inst_1 : CategoryTheory.Limits.HasCoproduct CategoryTheory.OrthogonalReflection.D₁.obj₁]
[inst_2 : CategoryTheory.Limits.HasCoproduct CategoryTheory.OrthogonalReflection.D₁.obj₂] {X Y : C} (f : X ⟶ Y)
(hf : W... | null | true |
CategoryTheory.Adjunction.comp_unit_app_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]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] {F : CategoryTheory.Functor C D}
{G : CategoryTheory.Functor D C} {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj... | null | false |
CategoryTheory.Dial.recOn | Mathlib.CategoryTheory.Dialectica.Basic | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.Limits.HasFiniteProducts C] →
{motive : CategoryTheory.Dial C → Sort u_1} →
(t : CategoryTheory.Dial C) →
((src tgt : C) →
(rel : CategoryTheory.Subobject (src ⨯ tgt)) → motive { src := src, t... | null | false |
List.length_destutter_le_length_destutter_cons | Mathlib.Data.List.Destutter | ∀ {α : Type u_1} {R : α → α → Prop} [inst : DecidableRel R] {a : α} [IsEquiv α Rᶜ] {l : List α},
(List.destutter R l).length ≤ (List.destutter R (a :: l)).length | `List.destutter` on a relation like ≠, whose negation is an equivalence, has length
monotone under List.cons | true |
StarSubsemiring.center | Mathlib.Algebra.Star.Subsemiring | (R : Type u_1) → [inst : NonAssocSemiring R] → [inst_1 : StarRing R] → StarSubsemiring R | The center of a semiring `R` is the set of elements that commute and associate with everything
in `R` | true |
Subtype.coe_bot | Mathlib.Order.BoundedOrder.Basic | ∀ {α : Type u} {p : α → Prop} [inst : PartialOrder α] [inst_1 : OrderBot α] [inst_2 : OrderBot (Subtype p)],
p ⊥ → ↑⊥ = ⊥ | null | true |
Std.Internal.instPureForInIterOfLawfulMonadOfFiniteOfLawfulIteratorLoopId | Std.Internal.ForIn.Lemmas | ∀ {α γ : Type w} {m : Type w → Type v} [inst : Monad m] [LawfulMonad m] [inst_2 : Std.Iterator α Id γ]
[Std.Iterators.Finite α Id] [inst_4 : Std.IteratorLoop α Id m] [Std.LawfulIteratorLoop α Id m]
[inst_6 : Std.IteratorLoop α Id Id] [Std.LawfulIteratorLoop α Id Id], Std.Internal.PureForIn m (Std.Iter γ) γ | null | true |
CategoryTheory.MorphismProperty.IsStableUnderCobaseChangeAlong.mk._flat_ctor | Mathlib.CategoryTheory.MorphismProperty.Limits | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} {X Y : C} {f : X ⟶ Y},
(∀ {Z W : C} {f' : Z ⟶ W} {g' : Y ⟶ W} {g : X ⟶ Z}, CategoryTheory.IsPushout f g g' f' → P g → P g') →
P.IsStableUnderCobaseChangeAlong f | null | false |
Rep.RepToAction_obj_V_carrier | Mathlib.RepresentationTheory.Rep.Basic | ∀ (k : Type u) (G : Type v) [inst : Ring k] [inst_1 : Monoid G] (X : Rep.{w, u, v} k G),
↑((Rep.RepToAction k G).obj X).V = ↑X | null | true |
AddAction.ext | Mathlib.Algebra.Group.Action.Defs | ∀ {G : Type u_9} {P : Type u_10} {inst : AddMonoid G} {x y : AddAction G P}, VAdd.vadd = VAdd.vadd → x = y | null | true |
Lean.getPPAnalyzeExplicitHoles | Lean.PrettyPrinter.Delaborator.TopDownAnalyze | Lean.Options → Bool | null | true |
TopologicalSpace.Closeds.iInf_def | Mathlib.Topology.Sets.Closeds | ∀ {α : Type u_2} [inst : TopologicalSpace α] {ι : Sort u_4} (s : ι → TopologicalSpace.Closeds α),
⨅ i, s i = { carrier := ⋂ i, ↑(s i), isClosed' := ⋯ } | null | true |
CategoryTheory.Arrow.LeftHomotopy.ofEq._proof_2 | Mathlib.CategoryTheory.Preadditive.FreydCategory.Homotopy | ∀ {V : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} V] [inst_1 : CategoryTheory.Preadditive V]
{u v : CategoryTheory.Arrow V} {f g : u ⟶ v},
f = g → CategoryTheory.Arrow.Hom.left f - CategoryTheory.Arrow.Hom.left g = CategoryTheory.CategoryStruct.comp u.hom 0 | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.getKeyD_diff_of_contains_eq_false_left._simp_1_1 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {x : Ord α} {x_1 : BEq α} [Std.LawfulBEqOrd α] {a b : α}, (compare a b = Ordering.eq) = ((a == b) = true) | null | false |
SimpleGraph.nonempty_hom_of_forall_finite_subgraph_hom | Mathlib.Combinatorics.SimpleGraph.Finsubgraph | ∀ {V : Type u} {W : Type v} {G : SimpleGraph V} {F : SimpleGraph W} [Finite W]
(h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F), Nonempty (G →g F) | If every finite subgraph of a graph `G` has a homomorphism to a finite graph `F`, then there is
a homomorphism from the whole of `G` to `F`. | true |
Ideal.Quotient.divisionRing._proof_9 | Mathlib.RingTheory.Ideal.Quotient.Basic | ∀ {R : Type u_1} [inst : Ring R] (I : Ideal R) [inst_1 : I.IsTwoSided] [inst_2 : I.IsMaximal],
autoParam (∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den) DivisionRing.nnratCast_def._autoParam | null | false |
WellFoundedRelation.isWellFounded | Mathlib.Order.RelClasses | ∀ {α : Type u} [h : WellFoundedRelation α], WellFounded WellFoundedRelation.rel | null | true |
Float32.recOn | Init.Data.Float.Float32 | {motive : Float32 → Sort u} → (t : Float32) → ((toModel : Float32.Model) → motive { toModel := toModel }) → motive t | null | false |
_private.Mathlib.Topology.Semicontinuity.Hemicontinuity.0.upperHemicontinuousWithinAt_singleton_iff._simp_1_2 | Mathlib.Topology.Semicontinuity.Hemicontinuity | ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l₁ : Filter α} {l₂ : Filter β},
Filter.Tendsto f l₁ l₂ = ∀ s ∈ l₂, ∀ᶠ (x : α) in l₁, f x ∈ s | null | false |
Graph.IsBond.subset | Mathlib.Combinatorics.Graph.Connected.EdgeCut | ∀ {α : Type u_1} {β : Type u_2} {G : Graph α β} {B : Set β}, G.IsBond B → B ⊆ G.edgeSet | null | true |
CategoryTheory.shrinkCoyonedaEquiv_shrinkCoyoneda_map | Mathlib.CategoryTheory.ShrinkYoneda | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.LocallySmall.{w, v, u} C] {X Y : Cᵒᵖ}
(f : X ⟶ Y),
CategoryTheory.shrinkCoyonedaEquiv (CategoryTheory.shrinkCoyoneda.{w, v, u}.map f) =
CategoryTheory.shrinkCoyonedaObjObjEquiv.symm f.unop | null | true |
LinearIsometryEquiv.symm_apply_apply | Mathlib.Analysis.Normed.Operator.LinearIsometry | ∀ {R : Type u_1} {R₂ : Type u_2} {E : Type u_4} {E₂ : Type u_5} [inst : Semiring R] [inst_1 : Semiring R₂]
{σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [inst_2 : RingHomInvPair σ₁₂ σ₂₁] [inst_3 : RingHomInvPair σ₂₁ σ₁₂]
[inst_4 : SeminormedAddCommGroup E] [inst_5 : SeminormedAddCommGroup E₂] [inst_6 : Module R E] [inst_7 : Mo... | null | true |
_private.Lean.Parser.Term.Basic.0.Lean.Parser.Term.implicitBinder._regBuiltin.Lean.Parser.Term.implicitBinder.docString_1 | Lean.Parser.Term.Basic | IO Unit | null | false |
Std.Iterators.Types.Append.snd | Init.Data.Iterators.Combinators.Monadic.Append | {α₁ α₂ : Type w} → {m : Type w → Type w'} → {β : Type w} → Std.IterM m β → Std.Iterators.Types.Append α₁ α₂ m β | null | true |
Int.dvd_emod_sub_self | Init.Data.Int.DivMod.Lemmas | ∀ {x m : ℤ}, m ∣ x % m - x | null | true |
_private.Mathlib.Algebra.Homology.ShortComplex.ExactFunctor.0.CategoryTheory.Functor.preservesFiniteLimits_tfae.match_1_1 | Mathlib.Algebra.Homology.ShortComplex.ExactFunctor | ∀ {C : Type u_2} {D : Type u_4} [inst : CategoryTheory.Category.{u_1, u_2} C]
[inst_1 : CategoryTheory.Category.{u_3, u_4} D] [inst_2 : CategoryTheory.Abelian C]
[inst_3 : CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [inst_4 : F.Additive]
(motive :
(∀ (S : CategoryTheory.ShortComplex C), S.Short... | null | false |
isAddCyclic_of_card_nsmul_eq_zero_le | Mathlib.GroupTheory.SpecificGroups.Cyclic | ∀ {α : Type u_1} [inst : AddGroup α] [inst_1 : DecidableEq α] [inst_2 : Fintype α],
(∀ (n : ℕ), 0 < n → {a | n • a = 0}.card ≤ n) → IsAddCyclic α | null | true |
InfTopHom.dual._proof_1 | Mathlib.Order.Hom.BoundedLattice | ∀ {α : Type u_1} {β : Type u_2} [inst : Min α] [inst_1 : Top α] [inst_2 : Min β] [inst_3 : Top β],
Function.LeftInverse (fun f => { toInfHom := InfHom.dual.symm f.toSupHom, map_top' := ⋯ }) fun f =>
{ toSupHom := InfHom.dual f.toInfHom, map_bot' := ⋯ } | null | false |
Set.bounded_ge_inter_ge | Mathlib.Order.Bounded | ∀ {α : Type u_1} {s : Set α} [inst : LinearOrder α] (a : α),
Set.Bounded (fun x1 x2 => x1 ≥ x2) (s ∩ {b | b ≤ a}) ↔ Set.Bounded (fun x1 x2 => x1 ≥ x2) s | null | true |
DirectSum.IsInternal.collectedBasis_orthonormal | Mathlib.Analysis.InnerProductSpace.Subspace | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{ι : Type u_4} [inst_3 : DecidableEq ι] {V : ι → Submodule 𝕜 E},
(OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) →
∀ (hV_sum : DirectSum.IsInternal fun i => V i) {α : ι → Type ... | null | true |
Function.Injective.torsor | Mathlib.Algebra.Torsor.Basic | {G : Type u_1} →
{P : Type u_2} →
{Q : Type u_3} →
[inst : Group G] →
[inst_1 : Torsor G P] →
[inst_2 : SMul G Q] →
[inst_3 : SDiv G Q] →
[Nonempty Q] →
(f : Q → P) →
Function.Injective f →
(∀ (c : G) (x : Q), ... | Pullback of a torsor along an injective map. | true |
MaxTropical.instLinearOrder._proof_2 | Mathlib.Algebra.Tropical.Basic | ∀ {R : Type u_1} [inst : LinearOrder R] (a b : MaxTropical R),
MaxTropical.untrop (a + b) = MaxTropical.untrop (if a ≤ b then b else a) | null | false |
CategoryTheory.CommShift₂Setup.hε | Mathlib.CategoryTheory.Shift.CommShiftTwo | ∀ {D : Type u_5} [inst : CategoryTheory.Category.{v_5, u_5} D] {M : Type u_6} [inst_1 : AddCommMonoid M]
[inst_2 : CategoryTheory.HasShift D M] (self : CategoryTheory.CommShift₂Setup D M) (m n : M),
self.ε m n = (self.z (0, n) (m, 0))⁻¹ * self.z (m, 0) (0, n) | null | true |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.