name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.MorphismProperty.Under.pushoutComp._proof_10 | Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | ∀ {T : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} T] {P Q : CategoryTheory.MorphismProperty T} {X Y Z : T}
(f : X ⟶ Y) (g : Y ⟶ Z) [P.IsStableUnderCobaseChangeAlong f] [P.HasPushoutsAlong f] [P.HasPushoutsAlong g]
(X_1 : P.Under Q X), CategoryTheory.Limits.HasPushout X_1.hom (CategoryTheory.CategoryStruct... | null | false |
AlgebraicGeometry.surjective_isZariskiLocalAtTarget | Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | AlgebraicGeometry.IsZariskiLocalAtTarget @AlgebraicGeometry.Surjective | null | true |
PiTensorProduct.instSemiring._proof_9 | Mathlib.RingTheory.PiTensorProduct | ∀ {ι : Type u_1} {R : Type u_2} {A : ι → Type u_3} [inst : CommSemiring R] [inst_1 : (i : ι) → Semiring (A i)]
[inst_2 : (i : ι) → Algebra R (A i)] (a : PiTensorProduct R fun i => A i), a * 0 = 0 | null | false |
IsAddFoelner.comp_tendsto | Mathlib.MeasureTheory.Group.FoelnerFilter | ∀ {G : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} [inst_1 : AddGroup G]
[inst_2 : AddAction G X] {ι : Type u_3} {l : Filter ι} {F : ι → Set X} {ι' : Type u_4} {l' : Filter ι'} {φ : ι' → ι},
IsAddFoelner G μ l F → Filter.Tendsto φ l' l → IsAddFoelner G μ l' (F ∘ φ) | null | true |
CategoryTheory.MonoidalCategory.DayConvolution.associator_hom_unit_unit_assoc | Mathlib.CategoryTheory.Monoidal.DayConvolution | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {V : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} V]
[inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : CategoryTheory.MonoidalCategory V]
(F G : CategoryTheory.Functor C V) [inst_4 : CategoryTheory.MonoidalCategory.DayConvolution F G]
(H : Ca... | Characterizing the forward direction of the associator isomorphism
with respect to the unit transformations. | true |
ArithmeticFunction.instSemiring._proof_5 | Mathlib.NumberTheory.ArithmeticFunction.Defs | ∀ {R : Type u_1} [inst : Semiring R] (a b c : ArithmeticFunction R), (a + b) * c = a * c + b * c | null | false |
_private.Init.Data.String.Lemmas.Pattern.String.ForwardSearcher.0.String.Slice.Pattern.Model.ForwardSliceSearcher.Invariants.of_prefixFunction_eq | Init.Data.String.Lemmas.Pattern.String.ForwardSearcher | ∀ {pat s : String.Slice} {stackPos needlePos : String.Pos.Raw},
String.Slice.Pattern.Model.ForwardSliceSearcher.Invariants✝ pat s needlePos stackPos →
needlePos ≠ 0 →
∀ {k : ℕ} {hki : needlePos.byteIdx - 1 < pat.copy.toByteArray.size},
String.Slice.Pattern.Model.ForwardSliceSearcher.prefixFunction✝ ... | null | true |
Urysohns.CU.mk.injEq | Mathlib.Topology.UrysohnsLemma | ∀ {X : Type u_2} [inst : TopologicalSpace X] {P : Set X → Set X → Prop} (C U : Set X) (P_C_U : P C U)
(closed_C : IsClosed C) (open_U : IsOpen U) (subset : C ⊆ U)
(hP :
∀ {c u : Set X},
IsClosed c → P c u → IsOpen u → c ⊆ u → ∃ v, IsOpen v ∧ c ⊆ v ∧ closure v ⊆ u ∧ P c v ∧ P (closure v) u)
(C_1 U_1 : Se... | null | true |
Std.DHashMap.Internal.Raw₀.filter_key_equiv_self_iff | Std.Data.DHashMap.Internal.RawLemmas | ∀ {α : Type u} {β : α → Type v} (m : Std.DHashMap.Internal.Raw₀ α β) [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α]
[LawfulHashable α] {f : α → Bool},
(↑m).WF →
((↑(Std.DHashMap.Internal.Raw₀.filter (fun k x => f k) m)).Equiv ↑m ↔
∀ (a : α) (h : m.contains a = true), f (m.getKey a h) = true) | null | true |
Action.FunctorCategoryEquivalence.counitIso._proof_4 | Mathlib.CategoryTheory.Action.Basic | ∀ {V : Type u_3} [inst : CategoryTheory.Category.{u_2, u_3} V] {G : Type u_1} [inst_1 : Monoid G]
{X Y : CategoryTheory.Functor (CategoryTheory.SingleObj G) V} (f : X ⟶ Y),
CategoryTheory.CategoryStruct.comp
((Action.FunctorCategoryEquivalence.inverse.comp Action.FunctorCategoryEquivalence.functor).map f)
... | null | false |
ExceptCpsT.run_pure | Init.Control.ExceptCps | ∀ {m : Type u_1 → Type u_2} {ε α : Type u_1} {x : α} [inst : Monad m], (pure x).run = pure (Except.ok x) | null | true |
Matrix.transpose_nonsing_inv | Mathlib.LinearAlgebra.Matrix.NonsingularInverse | ∀ {n : Type u'} {α : Type v} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing α] (A : Matrix n n α),
A⁻¹.transpose = A.transpose⁻¹ | null | true |
CategoryTheory.Subfunctor.equivalenceMonoOver | Mathlib.CategoryTheory.Subfunctor.Subobject | {C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
(F : CategoryTheory.Functor C (Type w)) → CategoryTheory.Subfunctor F ≌ CategoryTheory.MonoOver F | The equivalence of categories `Subfunctor F ≌ MonoOver F`. | true |
MeasureTheory.Lp.simpleFunc._proof_5 | Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | ∀ {α : Type u_1} (E : Type u_2) [inst : MeasurableSpace α] [inst_1 : NormedAddCommGroup E] (p : ENNReal)
(μ : MeasureTheory.Measure α), ∃ s, MeasureTheory.AEEqFun.mk ⇑s ⋯ = ↑0 | null | false |
_private.Mathlib.Topology.UniformSpace.Equiv.0.UniformEquiv.isUniformInducing._simp_1_1 | Mathlib.Topology.UniformSpace.Equiv | ∀ {α : Type u} [inst : UniformSpace α], IsUniformInducing id = True | null | false |
IsPrimitiveRoot.adjoinEquivRingOfIntegersOfPrimePow.congr_simp | Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | ∀ {p p_1 : ℕ} (e_p : p = p_1) {k k_1 : ℕ} (e_k : k = k_1) {K : Type u} [inst : Field K] {ζ : K}
[hp : Fact (Nat.Prime p)] [inst_1 : CharZero K] [inst_2 : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ (p ^ k)), hζ.adjoinEquivRingOfIntegersOfPrimePow = ⋯.adjoinEquivRingOfIntegersOfPrimePow | null | true |
Nat.sub_lt_sub_iff_right | Init.Data.Nat.Lemmas | ∀ {a b c : ℕ}, c ≤ a → (a - c < b - c ↔ a < b) | null | true |
Computation.destruct_pure | Mathlib.Data.Seq.Computation | ∀ {α : Type u} (a : α), (Computation.pure a).destruct = Sum.inl a | null | true |
Bicategory.Opposite.unop2_comp | Mathlib.CategoryTheory.Bicategory.Opposites | ∀ {B : Type u} [inst : CategoryTheory.Bicategory B] {a b : Bᵒᵖ} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h),
(CategoryTheory.CategoryStruct.comp η θ).unop2 = CategoryTheory.CategoryStruct.comp η.unop2 θ.unop2 | null | true |
eq_of_fin_equiv | Mathlib.LinearAlgebra.InvariantBasisNumber | ∀ (R : Type u) [inst : Semiring R] [InvariantBasisNumber R] {n m : ℕ} (a : (Fin n → R) ≃ₗ[R] Fin m → R), n = m | null | true |
ContinuousAlgEquiv.continuous_toFun | Mathlib.Topology.Algebra.Algebra.Equiv | ∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : TopologicalSpace A] [inst_3 : Semiring B] [inst_4 : TopologicalSpace B] [inst_5 : Algebra R A]
[inst_6 : Algebra R B] (self : A ≃A[R] B), Continuous self.toFun | The forward map of a homeomorphism is a continuous function. | true |
trivialVectorBundleCore._proof_2 | Mathlib.Topology.VectorBundle.Basic | ∀ (F : Type u_1) [inst : NormedAddCommGroup F], IsTopologicalAddGroup F | null | false |
AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_assoc | Mathlib.Geometry.RingedSpace.LocallyRingedSpace | ∀ {X Y : AlgebraicGeometry.LocallyRingedSpace} (f g : X ⟶ Y) (hfg : f = g) (x x' : ↑X.toTopCat) (hxx' : x = x')
{Z : CommRingCat} (h : X.presheaf.stalk x' ⟶ Z),
CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x)
(CategoryTheory.CategoryStruct.comp (X.presheaf.stalkSpeci... | null | true |
IsArtinianRing.isSemisimpleRing_iff_jacobson | Mathlib.RingTheory.Artinian.Module | ∀ {R : Type u_1} [inst : Ring R] [IsArtinianRing R], IsSemisimpleRing R ↔ Ring.jacobson R = ⊥ | null | true |
_private.Mathlib.Analysis.Normed.Operator.Fredholm.Open.0.ContinuousLinearMap.FredholmPackage.eventually_nhds_isFredholm_and_index_eq._proof_1_1 | Mathlib.Analysis.Normed.Operator.Fredholm.Open | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜], RingHomCompTriple (RingHom.id 𝕜) (RingHom.id 𝕜) (RingHom.id 𝕜) | null | false |
TopologicalSpace.IsCompletelyMetrizableSpace.mk._flat_ctor | Mathlib.Topology.Metrizable.CompletelyMetrizable | ∀ {X : Type u_3} [t : TopologicalSpace X],
(∃ m, PseudoMetricSpace.toUniformSpace.toTopologicalSpace = t ∧ CompleteSpace X) →
TopologicalSpace.IsCompletelyMetrizableSpace X | null | false |
MeasurableSet.const | Mathlib.MeasureTheory.MeasurableSpace.Defs | ∀ {α : Type u_1} {m : MeasurableSpace α} (p : Prop), MeasurableSet {_a | p} | null | true |
CategoryTheory.Injective.enoughInjectives_of_enoughProjectives_op | Mathlib.CategoryTheory.Preadditive.Injective.Basic | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.EnoughProjectives Cᵒᵖ],
CategoryTheory.EnoughInjectives C | null | true |
WithAbs.algebraLeft._proof_3 | Mathlib.Analysis.Normed.Ring.WithAbs | ∀ {S : Type u_3} [inst : Semiring S] [inst_1 : PartialOrder S] {R : Type u_2} (T : Type u_1) [inst_2 : CommSemiring R]
[inst_3 : Semiring T] [inst_4 : Algebra R T] (v : AbsoluteValue R S),
(algebraMap R T) ((WithAbs.equiv v).toRingHom 1) = 1 | null | false |
AlgebraicGeometry.Spec.mapMulEquiv._proof_4 | Mathlib.AlgebraicGeometry.Group.Affine | ∀ {R S T : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Bialgebra R S]
[inst_4 : Algebra R T]
(f :
(AlgebraicGeometry.Spec ↧T).asOver (AlgebraicGeometry.Spec ↧R) ⟶
(AlgebraicGeometry.Spec ↧S).asOver (AlgebraicGeometry.Spec ↧R)),
(fun f =>
AlgebraicGeometry.... | null | false |
AddMonoidAlgebra.support_coeff_one_subset | Mathlib.Algebra.MonoidAlgebra.Support | ∀ {k : Type u₁} {G : Type u₂} [inst : Semiring k] [inst_1 : Zero G], (AddMonoidAlgebra.coeff 1).support ⊆ 0 | null | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.SimplexCategoryGenRel.simplicialInsert.match_1.eq_1 | Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | ∀ (motive : List ℕ → Sort u_1) (h_1 : Unit → motive []) (h_2 : (b : ℕ) → (l : List ℕ) → motive (b :: l)),
(match [] with
| [] => h_1 ()
| b :: l => h_2 b l) =
h_1 () | null | true |
Std.Tactic.BVDecide.BVExpr.bitblast.goCache | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Expr | {w : ℕ} →
(aig : Std.Sat.AIG Std.Tactic.BVDecide.BVBit) →
Std.Tactic.BVDecide.BVExpr w → Std.Tactic.BVDecide.BVExpr.Cache aig → Std.Tactic.BVDecide.BVExpr.Return aig w | null | true |
PSum.inhabitedLeft | Init.Core | {α : Sort u_1} → {β : Sort u_2} → [Inhabited α] → Inhabited (α ⊕' β) | If the left type in a sum is inhabited then the sum is inhabited.
This is not an instance to avoid non-canonical instances when both the left and right types are
inhabited.
| true |
Std.DTreeMap.Internal.Impl.Const.mem_alter!_self | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {β : Type v} {t : Std.DTreeMap.Internal.Impl α fun x => β} [Std.TransOrd α],
t.WF →
∀ {k : α} {f : Option β → Option β},
k ∈ Std.DTreeMap.Internal.Impl.Const.alter! k f t ↔ (f (Std.DTreeMap.Internal.Impl.Const.get? t k)).isSome = true | null | true |
WithCStarModule.instNormedAddCommGroupForall._proof_7 | Mathlib.Analysis.CStarAlgebra.Module.Constructions | ∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] {ι : Type u_2} {E : ι → Type u_3}
[inst_2 : Fintype ι] [inst_3 : (i : ι) → NormedAddCommGroup (E i)] [inst_4 : (i : ι) → Module ℂ (E i)]
[inst_5 : (i : ι) → SMul A (E i)] [inst_6 : (i : ι) → CStarModule A (E i)] [inst_7 : StarOrderedRing A]... | null | false |
Topology.IsGeneratedBy.instPUnit | Mathlib.Topology.Convenient.GeneratedBy | ∀ {ι : Type t} {X : ι → Type u} [inst : (i : ι) → TopologicalSpace (X i)], Topology.IsGeneratedBy X PUnit.{v + 1} | null | true |
ArchimedeanClass.instCommRingFiniteElement._proof_27 | Mathlib.Algebra.Order.Ring.StandardPart | ∀ (K : Type u_1) [inst : LinearOrder K] [inst_1 : Field K] [inst_2 : IsOrderedRing K]
(a b c : ArchimedeanClass.FiniteElement K), (a + b) * c = a * c + b * c | null | false |
_private.Std.Time.Zoned.Database.Basic.0.Std.Time.TimeZone.convertTZifV1.match_4 | Std.Time.Zoned.Database.Basic | (motive : Option Std.Time.TimeZone.Transition → Sort u_1) →
(x : Option Std.Time.TimeZone.Transition) →
((result : Std.Time.TimeZone.Transition) → motive (some result)) →
((x : Option Std.Time.TimeZone.Transition) → motive x) → motive x | null | false |
Nat.toList_roo_eq_append | Init.Data.Range.Polymorphic.NatLemmas | ∀ {m n : ℕ}, m + 1 < n → (m<...n).toList = (m<...n - 1).toList ++ [n - 1] | null | true |
AlgebraicGeometry.Etale.etale_comp | Mathlib.AlgebraicGeometry.Morphisms.Etale | ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) {Z : AlgebraicGeometry.Scheme} (g : Y ⟶ Z) [AlgebraicGeometry.Etale f]
[AlgebraicGeometry.Etale g], AlgebraicGeometry.Etale (CategoryTheory.CategoryStruct.comp f g) | The composition of étale morphisms is étale. | true |
CategoryTheory.FreeMonoidalCategory.Hom.comp.sizeOf_spec | Mathlib.CategoryTheory.Monoidal.Free.Basic | ∀ {C : Type u} [inst : SizeOf C] {X Y Z : CategoryTheory.FreeMonoidalCategory C} (f : X.Hom Y) (g : Y.Hom Z),
sizeOf (f.comp g) = 1 + sizeOf X + sizeOf Y + sizeOf Z + sizeOf f + sizeOf g | null | true |
AlgEquiv.ofInjective_apply | Mathlib.Algebra.Algebra.Subalgebra.Basic | ∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
[inst_3 : Algebra R A] [inst_4 : Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (x : A),
↑((AlgEquiv.ofInjective f hf) x) = f x | null | true |
CategoryTheory.Iso.groupIsoToMulEquiv._proof_1 | Mathlib.Algebra.Category.Grp.Basic | ∀ {X Y : GrpCat} (i : X ≅ Y), (GrpCat.Hom.hom i.inv).comp (GrpCat.Hom.hom i.hom) = MonoidHom.id ↑X | null | false |
_private.Std.Sat.AIG.CNF.0.Std.Sat.AIG.toCNF.projectRightAssign_property | Std.Sat.AIG.CNF | ∀ {assign : ℕ → Bool} {x : ℕ}, Std.Sat.AIG.toCNF.projectRightAssign✝ assign x = assign x | null | true |
Set.inter_singleton_nonempty | Mathlib.Data.Set.Insert | ∀ {α : Type u_1} {s : Set α} {a : α}, (s ∩ {a}).Nonempty ↔ a ∈ s | null | true |
Std.DHashMap.Raw.mem_of_mem_insert | Std.Data.DHashMap.RawLemmas | ∀ {α : Type u} {β : α → Type v} {m : Std.DHashMap.Raw α β} [inst : BEq α] [inst_1 : Hashable α] [EquivBEq α]
[LawfulHashable α], m.WF → ∀ {k a : α} {v : β k}, a ∈ m.insert k v → (k == a) = false → a ∈ m | null | true |
_private.Mathlib.Data.Finset.Card.0.Finset.two_lt_card._simp_1_2 | Mathlib.Data.Finset.Card | ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, b ∧ p x) = (b ∧ ∃ x, p x) | null | false |
Finset.prod_range_div' | Mathlib.Algebra.BigOperators.Group.Finset.Basic | ∀ {G : Type u_3} [inst : CommGroup G] (f : ℕ → G) (n : ℕ), ∏ i ∈ Finset.range n, f i / f (i + 1) = f 0 / f n | A reversed telescoping product along `{0, ..., n - 1}` of a commutative-group-valued function
reduces to the ratio of the first and last factors. | true |
Nat.add_descFactorial_eq_ascFactorial._f | Mathlib.Data.Nat.Factorial.Basic | ∀ (n x : ℕ) (f : Nat.below x), (n + x).descFactorial x = (n + 1).ascFactorial x | null | false |
FirstOrder.Language.exists_cg_is_age_of | Mathlib.ModelTheory.Fraisse | ∀ {L : FirstOrder.Language} {K : Set (CategoryTheory.Bundled L.Structure)},
K.Nonempty →
(Quotient.mk' '' K).Countable →
(∀ M ∈ K, FirstOrder.Language.Structure.FG L ↑M) →
FirstOrder.Language.Hereditary K →
FirstOrder.Language.JointEmbedding K → ∃ M, FirstOrder.Language.Structure.CG L ↑M ∧... | Sufficient conditions for a class to be the age of a countably-generated structure. | true |
_private.Lean.Meta.MkIffOfInductiveProp.0.Lean.Meta.List.init.match_1 | Lean.Meta.MkIffOfInductiveProp | {α : Type u_1} →
(motive : List α → Sort u_2) →
(x : List α) →
(Unit → motive []) → ((head : α) → motive [head]) → ((a : α) → (l : List α) → motive (a :: l)) → motive x | null | false |
SuccOrder.limitRecOn.eq_1 | Mathlib.Order.SuccPred.Limit | ∀ {α : Type u_1} (b : α) {motive : α → Sort u_2} [inst : PartialOrder α] [inst_1 : SuccOrder α]
[inst_2 : WellFoundedLT α] (isMin : (a : α) → IsMin a → motive a)
(succ : (a : α) → ¬IsMax a → motive a → motive (Order.succ a))
(isSuccLimit : (a : α) → Order.IsSuccLimit a → ((b : α) → b < a → motive b) → motive a),
... | null | true |
Array.extract.eq_1 | Init.Data.Array.Lemmas | ∀ {α : Type u_1} (as : Array α) (start stop : ℕ),
as.extract start stop =
Array.extract.loop as ((min stop as.size).sub start) start (Array.emptyWithCapacity ((min stop as.size).sub start)) | null | true |
MvPolynomial.IsPrime.vanishingIdeal_zeroLocus | Mathlib.RingTheory.Nullstellensatz | ∀ {k : Type u_1} {K : Type u_2} [inst : Field k] [inst_1 : Field K] [inst_2 : Algebra k K] {σ : Type u_3}
[IsAlgClosed K] [Finite σ] (P : Ideal (MvPolynomial σ k)) [h : P.IsPrime],
MvPolynomial.vanishingIdeal k (MvPolynomial.zeroLocus K P) = P | null | true |
FreeAddMonoid.instMembership | Mathlib.Algebra.FreeMonoid.Basic | {α : Type u_1} → Membership α (FreeAddMonoid α) | null | true |
Lean.Elab.WF.EqnInfo.value | Lean.Elab.PreDefinition.WF.Eqns | Lean.Elab.WF.EqnInfo → Lean.Expr | null | true |
_private.Mathlib.Algebra.Star.Unitary.0.Pi.unitary_eq._simp_1_1 | Mathlib.Algebra.Star.Unitary | ∀ {ι : Type u_3} {R : ι → Type u_2} [inst : (i : ι) → Monoid (R i)] [inst_1 : (i : ι) → StarMul (R i)]
{p : (i : ι) → R i}, (p ∈ unitary ((i : ι) → R i)) = ∀ (i : ι), p i ∈ unitary (R i) | null | false |
instDecidableEqOrderedFinpartition.decEq._proof_10 | Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | ∀ {n : ℕ} (a : ℕ) (a_1 : Fin a → ℕ) (a_2 : ∀ (m : Fin a), 0 < a_1 m) (a_3 : (m : Fin a) → Fin (a_1 m) → Fin n)
(a_4 : ∀ (m : Fin a), StrictMono (a_3 m)) (a_5 : StrictMono fun m => a_3 m ⟨a_1 m - 1, ⋯⟩)
(a_6 : Set.univ.PairwiseDisjoint fun m => Set.range (a_3 m)) (a_7 : ∀ (x : Fin n), ∃ m, x ∈ Set.range (a_3 m)),
... | null | false |
Lists'.recOfList | Mathlib.SetTheory.Lists | {α : Type u_1} →
{motive : Lists' α true → Sort u_2} →
((l : List (Lists α)) → motive (Lists'.ofList l)) → (l : Lists' α true) → motive l | Recursion/induction principle for `Lists'.ofList`. | true |
isSeqCompact_iff_isSeqCompact_univ | Mathlib.Topology.Compactness.CountablyCompact | ∀ {E : Type u_2} [inst : TopologicalSpace E] {A : Set E}, IsSeqCompact A ↔ IsSeqCompact Set.univ | null | true |
Condensed.fintypeCatAsCofan | Mathlib.Condensed.Discrete.Colimit | (X : Profinite) → CategoryTheory.Limits.Cofan fun x => ↧PUnit.{u + 1} | A finite set as a coproduct cocone in `Profinite` over itself. | true |
Aesop.IndexMatchLocation.hyp.sizeOf_spec | Aesop.Index.Basic | ∀ (ldecl : Lean.LocalDecl), sizeOf (Aesop.IndexMatchLocation.hyp ldecl) = 1 + sizeOf ldecl | null | true |
Bipointed.hasForget._proof_6 | Mathlib.CategoryTheory.Category.Bipointed | ∀ {X Y : Bipointed} (f : X.HomSubtype Y), ↑f X.toProd.1 = Y.toProd.1 | null | false |
Ordinal.sInf_compl_lt_lift_ord_succ | Mathlib.SetTheory.Ordinal.Family | ∀ {ι : Type u} (f : ι → Ordinal.{max u v}), sInf (Set.range f)ᶜ < Ordinal.lift.{v, u} (Order.succ (Cardinal.mk ι)).ord | null | true |
HurwitzKernelBounds.isBigO_exp_neg_mul_of_le | Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | ∀ {c d : ℝ}, c ≤ d → (fun t => Real.exp (-d * t)) =O[Filter.atTop] fun t => Real.exp (-c * t) | null | true |
Multiset.ext | Mathlib.Data.Multiset.Count | ∀ {α : Type u_1} [inst : DecidableEq α] {s t : Multiset α}, s = t ↔ ∀ (a : α), Multiset.count a s = Multiset.count a t | null | true |
_private.Init.Data.Range.Polymorphic.SInt.0.BitVec.Signed.rotate.eq_1 | Init.Data.Range.Polymorphic.SInt | ∀ {n : ℕ} (x : BitVec n), BitVec.Signed.rotate✝ x = x + BitVec.Signed.intMinSealed✝ n | null | true |
divisibleByIntOfCharZero._proof_2 | Mathlib.GroupTheory.Divisible | ∀ {𝕜 : Type u_1} [inst : DivisionRing 𝕜] [CharZero 𝕜] {n : ℤ} (q : 𝕜), n ≠ 0 → n • (q / ↑n) = q | null | false |
ValuationSubring.eq_iff_unitGroup | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] {A B : ValuationSubring K}, A = B ↔ A.unitGroup = B.unitGroup | null | true |
String.Slice.RevByteIterator.instIteratorUInt8OfPure | Init.Data.String.Iterate | {m : Type → Type u_1} → [Pure m] → Std.Iterator String.Slice.RevByteIterator m UInt8 | null | true |
_private.Mathlib.Tactic.Abel.0.Mathlib.Tactic.Abel.instEvalExprConfig.evalExpr | Mathlib.Tactic.Abel | Lean.Expr → Lean.MetaM Mathlib.Tactic.AtomM.Recurse.Config | null | true |
CardinalInterFilter | Mathlib.Order.Filter.CardinalInter | {α : Type u} → Filter α → Cardinal.{u} → Prop | A filter `l` has the cardinal `c` intersection property if for any collection
of less than `c` sets `s ∈ l`, their intersection belongs to `l` as well. | true |
Submodule.lipschitzWith_orthogonalProjection | Mathlib.Analysis.InnerProductSpace.Projection.Basic | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E) [inst_3 : K.HasOrthogonalProjection], LipschitzWith 1 ⇑K.orthogonalProjectionOnto | **Alias** of `Submodule.lipschitzWith_orthogonalProjectionOnto`.
---
The orthogonal projection onto a closed subspace is a `1`-Lipschitz map. | true |
String.Slice.RevByteIterator.recOn | Init.Data.String.Iterate | {motive : String.Slice.RevByteIterator → Sort u} →
(t : String.Slice.RevByteIterator) →
((s : String.Slice) →
(offset : String.Pos.Raw) → (hinv : offset ≤ s.rawEndPos) → motive { s := s, offset := offset, hinv := hinv }) →
motive t | null | false |
_private.Mathlib.LinearAlgebra.RootSystem.Base.0.RootPairing.Base.sub_notMem_range_root._simp_1_1 | Mathlib.LinearAlgebra.RootSystem.Base | ∀ {α : Type u} {ι : Sort u_1} {f : ι → α} {x : α}, (x ∈ Set.range f) = ∃ y, f y = x | null | false |
_private.Mathlib.Combinatorics.SimpleGraph.CycleGraph.0.SimpleGraph.cycleGraph_neighborSet._simp_1_2 | Mathlib.Combinatorics.SimpleGraph.CycleGraph | ∀ {α : Type u_1} {x a : α} {s : Set α}, (x ∈ insert a s) = (x = a ∨ x ∈ s) | null | false |
Algebra.charZero_of_charZero | Mathlib.Algebra.Algebra.Basic | ∀ (R : Type u_1) (A : Type u_2) [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] [FaithfulSMul R A]
[CharZero R], CharZero A | null | true |
MvPolynomial.HomogeneousSubmodule.gradedMonoid | Mathlib.RingTheory.MvPolynomial.Homogeneous | ∀ {σ : Type u_1} {R : Type u_3} [inst : CommSemiring R], SetLike.GradedMonoid (MvPolynomial.homogeneousSubmodule σ R) | The homogeneous submodules form a graded ring.
This instance is used by `DirectSum.commSemiring` and `DirectSum.algebra`. | true |
Std.TreeMap.Raw.compare_maxKey?_modify_eq | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap.Raw α β cmp} [inst : Std.TransCmp cmp] (h : t.WF)
{k : α} {f : β → β} {km kmm : α} (hkm : t.maxKey? = some km),
(t.modify k f).maxKey?.get ⋯ = kmm → cmp kmm km = Ordering.eq | null | true |
IsUnital.toNonAssocRing._proof_1 | Mathlib.Algebra.Ring.Defs | ∀ {A : Type u_1} [inst : NonUnitalNonAssocRing A] [inst_1 : IsUnital A], Nat.unaryCast 0 = 0 | null | false |
CategoryTheory.uliftYoneda_map_app | Mathlib.CategoryTheory.Yoneda | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : X ⟶ Y) (X_1 : Cᵒᵖ),
(CategoryTheory.uliftYoneda.{w, v₁, u₁}.map f).app X_1 =
TypeCat.ofHom fun x => { down := CategoryTheory.CategoryStruct.comp x.down f } | null | true |
_private.Init.Data.Range.Polymorphic.Internal.SignedBitVec.0.BitVec.Signed.ofNat_eq_rotate_ofInt_sub | Init.Data.Range.Polymorphic.Internal.SignedBitVec | ∀ {n k : ℕ}, BitVec.ofNat n k = BitVec.Signed.rotate✝ (BitVec.ofInt n (↑k - ↑(BitVec.Signed.intMinSealed✝ n).toNat)) | null | true |
Algebra.Extension.instAddCommGroupH1Cotangent._proof_11 | Mathlib.RingTheory.Extension.Cotangent.Basic | ∀ {R : Type u_2} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
{P : Algebra.Extension R S},
autoParam (∀ (n : ℕ) (x : P.H1Cotangent), (n + 1) • x = n • x + x) AddMonoid.nsmul_succ._autoParam | null | false |
_private.Init.Data.Array.Lemmas.0.Array.erase.match_1.eq_2 | Init.Data.Array.Lemmas | ∀ {α : Type u_1} (as : Array α) (motive : Option (Fin as.size) → Sort u_2) (i : Fin as.size) (h_1 : Unit → motive none)
(h_2 : (i : Fin as.size) → motive (some i)),
(match some i with
| none => h_1 ()
| some i => h_2 i) =
h_2 i | null | true |
Aesop.GlobalRuleSetMember.name | Aesop.RuleSet.Member | Aesop.GlobalRuleSetMember → Aesop.RuleName | null | true |
PiTensorProduct.instAlgebra._proof_4 | Mathlib.RingTheory.PiTensorProduct | ∀ {ι : Type u_1} {R' : Type u_4} {R : Type u_2} {A : ι → Type u_3} [inst : CommSemiring R'] [inst_1 : CommSemiring R]
[inst_2 : (i : ι) → Semiring (A i)] [inst_3 : Algebra R' R] [inst_4 : (i : ι) → Algebra R (A i)] (x y : R'),
(x + y) • 1 = x • 1 + y • 1 | null | false |
_private.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs.0.SimpleGraph.binomialRandom_singleton._simp_1_2 | Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | ∀ {V : Type u} (G : SimpleGraph V), (G.edgeSet ⊆ Sym2.diagSetᶜ) = True | null | false |
AlgebraicGeometry.Scheme.isGrothendieckAbelian_sheaf_smallEtaleTopology | Mathlib.AlgebraicGeometry.Sites.AffineEtale | ∀ (S : AlgebraicGeometry.Scheme) (A : Type u') [inst : CategoryTheory.Category.{u, u'} A] {FA : A → A → Type u_1}
{CD : A → Type u} [inst_1 : (X Y : A) → FunLike (FA X Y) (CD X) (CD Y)]
[inst_2 : CategoryTheory.ConcreteCategory A FA]
[inst_3 : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)]
[in... | null | true |
Submodule.reflection | Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {𝕜 : Type u_1} →
{E : Type u_2} →
[inst : RCLike 𝕜] →
[inst_1 : NormedAddCommGroup E] →
[inst_2 : InnerProductSpace 𝕜 E] → (K : Submodule 𝕜 E) → [K.HasOrthogonalProjection] → E ≃ₗᵢ[𝕜] E | Reflection in a complete subspace of an inner product space. The word "reflection" is
sometimes understood to mean specifically reflection in a codimension-one subspace, and sometimes
more generally to cover operations such as reflection in a point. The definition here, of
reflection in a subspace, is a more general ... | true |
HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv | Mathlib.Algebra.Homology.Opposite | ∀ {ι : Type u_1} {V : Type u_2} [inst : CategoryTheory.Category.{v_1, u_2} V] {c : ComplexShape ι}
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] (K : HomologicalComplex V c) (i : ι) [inst_2 : K.HasHomology i]
(j : ι), CategoryTheory.CategoryStruct.comp (K.fromOpcycles i j).op (K.cyclesOpIso i).inv = K.op.toCy... | null | true |
Nat.Internal.Linear.Expr.rec | Init.Data.Nat.Internal.Linear | {motive : Nat.Internal.Linear.Expr → Sort u} →
((v : ℕ) → motive (Nat.Internal.Linear.Expr.num v)) →
((i : Nat.Internal.Linear.Var) → motive (Nat.Internal.Linear.Expr.var i)) →
((a b : Nat.Internal.Linear.Expr) → motive a → motive b → motive (a.add b)) →
((k : ℕ) → (a : Nat.Internal.Linear.Expr) → m... | null | false |
UniformConvergenceCLM.smulCommClass | Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM | ∀ {𝕜₁ : Type u_1} {𝕜₂ : Type u_2} [inst : NormedField 𝕜₁] [inst_1 : NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E : Type u_3}
(F : Type u_4) [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜₁ E] [inst_4 : TopologicalSpace E]
[inst_5 : AddCommGroup F] [inst_6 : Module 𝕜₂ F] {S : Type u_6} {T : Type u_7} [inst_7 : Topologica... | null | true |
Std.Internal.Parsec.ParseResult | Std.Internal.Parsec.Basic | Type → Type → Type | The result of parsing some string.
| true |
Mathlib.Tactic.Ring.Common.RingCompute.casesOn | Mathlib.Tactic.Ring.Common | {u : Lean.Level} →
{α : Q(Type u)} →
{BaseType : Q(«$α») → Type} →
{sα : Q(CommSemiring «$α»)} →
{motive : Mathlib.Tactic.Ring.Common.RingCompute BaseType sα → Sort u} →
(t : Mathlib.Tactic.Ring.Common.RingCompute BaseType sα) →
((toRingCompare : Mathlib.Tactic.Ring.Common.Ring... | null | false |
MeasureTheory.VectorMeasure.rec | Mathlib.MeasureTheory.VectorMeasure.Defs | {α : Type u_2} →
[inst : MeasurableSpace α] →
{M : Type u_3} →
[inst_1 : AddCommMonoid M] →
[inst_2 : TopologicalSpace M] →
{motive : MeasureTheory.VectorMeasure α M → Sort u} →
((measureOf' : Set α → M) →
(empty' : measureOf' ∅ = 0) →
(not_mea... | null | false |
Tactic.ComputeAsymptotics.Seq.FriendlyOperationClass.comp | Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion | ∀ {α : Type u_1} {γ : Type u_3} {γ' : Type u_4} (F : γ → Stream'.Seq α → Stream'.Seq α) (g : γ' → γ)
[h : Tactic.ComputeAsymptotics.Seq.FriendlyOperationClass F],
Tactic.ComputeAsymptotics.Seq.FriendlyOperationClass fun c => F (g c) | null | true |
_private.Mathlib.Algebra.Colimit.Ring.0.Ring.DirectLimit.Polynomial.exists_of.match_1_3 | Mathlib.Algebra.Colimit.Ring | ∀ {ι : Type u_1} [inst : Preorder ι] (i₁ i₂ : ι) (motive : (∃ c, i₁ ≤ c ∧ i₂ ≤ c) → Prop) (x : ∃ c, i₁ ≤ c ∧ i₂ ≤ c),
(∀ (i : ι) (h1 : i₁ ≤ i) (h2 : i₂ ≤ i), motive ⋯) → motive x | null | false |
_private.Lean.Compiler.IR.Basic.0.Lean.IR.Expr.alphaEqv._sparseCasesOn_13 | Lean.Compiler.IR.Basic | {motive : Lean.IR.Expr → Sort u} →
(t : Lean.IR.Expr) →
((x : Lean.IR.VarId) → motive (Lean.IR.Expr.isShared x)) → (Nat.hasNotBit 4096 t.ctorIdx → motive t) → motive t | null | false |
CategoryTheory.ShortComplex.homology_π_ι | Mathlib.Algebra.Homology.ShortComplex.Homology | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : S.HasHomology],
CategoryTheory.CategoryStruct.comp S.homologyπ S.homologyι = CategoryTheory.CategoryStruct.comp S.iCycles S.pOpcycles | null | true |
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