Compatibility of algebraic operations with metric space structures

In this file we define mixin typeclasses LipschitzMul, LipschitzAdd, IsBoundedSMul expressing compatibility of multiplication, addition and scalar-multiplication operations with an underlying metric space structure. The intended use case is to abstract certain properties shared by normed groups and by R≥0.

Implementation notes

We deduce a ContinuousMul instance from LipschitzMul, etc. In principle there should be an intermediate typeclass for uniform spaces, but the algebraic hierarchy there (see IsUniformGroup) is structured differently.

class LipschitzAdd (β : Type u_2) [PseudoMetricSpace β] [AddMonoid β] : Prop

Class LipschitzAdd M says that the addition (+) : X × X → X is Lipschitz jointly in the two arguments.

• lipschitz_add : ∃ (C : NNReal), LipschitzWith C fun (p : β × β) => p.1 + p.2

class LipschitzMul (β : Type u_2) [PseudoMetricSpace β] [Monoid β] : Prop

Class LipschitzMul M says that the multiplication (*) : X × X → X is Lipschitz jointly in the two arguments.

• lipschitz_mul : ∃ (C : NNReal), LipschitzWith C fun (p : β × β) => p.1 * p.2

def LipschitzAdd.C (β : Type u_2) [PseudoMetricSpace β] [AddMonoid β] [_i : LipschitzAdd β] : NNReal

The Lipschitz constant of an AddMonoid β satisfying LipschitzAdd

Equations

• LipschitzAdd.C β = Classical.choose ⋯

def LipschitzMul.C (β : Type u_2) [PseudoMetricSpace β] [Monoid β] [_i : LipschitzMul β] : NNReal

The Lipschitz constant of a monoid β satisfying LipschitzMul

Equations

• LipschitzMul.C β = Classical.choose ⋯

theorem lipschitzWith_lipschitz_const_mul_edist {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [_i : LipschitzMul β] : LipschitzWith (LipschitzMul.C β) fun (p : β × β) => p.1 * p.2

theorem lipschitzWith_lipschitz_const_add_edist {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [_i : LipschitzAdd β] : LipschitzWith (LipschitzAdd.C β) fun (p : β × β) => p.1 + p.2

theorem lipschitz_with_lipschitz_const_mul {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [LipschitzMul β] (p q : β × β) : dist (p.1 * p.2) (q.1 * q.2) ≤ ↑(LipschitzMul.C β) * dist p q

theorem lipschitz_with_lipschitz_const_add {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] (p q : β × β) : dist (p.1 + p.2) (q.1 + q.2) ≤ ↑(LipschitzAdd.C β) * dist p q

@[instance 100]

instance LipschitzMul.continuousMul {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [LipschitzMul β] : ContinuousMul β

@[instance 100]

instance LipschitzAdd.continuousAdd {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] : ContinuousAdd β

instance Submonoid.lipschitzMul {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [LipschitzMul β] (s : Submonoid β) : LipschitzMul ↥s

instance AddSubmonoid.lipschitzAdd {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] (s : AddSubmonoid β) : LipschitzAdd ↥s

instance MulOpposite.lipschitzMul {β : Type u_2} [PseudoMetricSpace β] [Monoid β] [LipschitzMul β] : LipschitzMul βᵐᵒᵖ

instance AddOpposite.lipschitzAdd {β : Type u_2} [PseudoMetricSpace β] [AddMonoid β] [LipschitzAdd β] : LipschitzAdd βᵃᵒᵖ

instance Real.hasLipschitzAdd : LipschitzAdd ℝ

instance NNReal.hasLipschitzAdd : LipschitzAdd NNReal

class IsBoundedSMul (α : Type u_1) (β : Type u_2) [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] : Prop

Mixin typeclass on a scalar action of a metric space α on a metric space β both with distinguished points 0, requiring compatibility of the action in the sense that dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ and dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0.

If [NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β] are assumed, then prefer writing [NormSMulClass α β] instead of using [IsBoundedSMul α β], since while equivalent, typeclass search can only infer the latter from the former and not vice versa.

• dist_smul_pair' (x : α) (y₁ y₂ : β) : dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
• dist_pair_smul' (x₁ x₂ : α) (y : β) : dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0

@[deprecated IsBoundedSMul (since := "2025-03-10")]

def BoundedSMul (α : Type u_1) (β : Type u_2) [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] : Prop

Alias of IsBoundedSMul.

---

Mixin typeclass on a scalar action of a metric space α on a metric space β both with distinguished points 0, requiring compatibility of the action in the sense that dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ and dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0.

If [NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β] are assumed, then prefer writing [NormSMulClass α β] instead of using [IsBoundedSMul α β], since while equivalent, typeclass search can only infer the latter from the former and not vice versa.

Equations

• BoundedSMul = IsBoundedSMul

theorem dist_smul_pair {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] (x : α) (y₁ y₂ : β) : dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂

theorem dist_pair_smul {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] (x₁ x₂ : α) (y : β) : dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0

@[instance 100]

instance IsBoundedSMul.continuousSMul {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] : ContinuousSMul α β

The typeclass IsBoundedSMul on a metric-space scalar action implies continuity of the action.

@[instance 100]

instance IsBoundedSMul.toUniformContinuousConstSMul {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] : UniformContinuousConstSMul α β

instance Real.isBoundedSMul : IsBoundedSMul ℝ ℝ

instance NNReal.isBoundedSMul : IsBoundedSMul NNReal NNReal

instance IsBoundedSMul.op {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] [SMul αᵐᵒᵖ β] [IsCentralScalar α β] : IsBoundedSMul αᵐᵒᵖ β

If a scalar is central, then its right action is bounded when its left action is.

instance instLipschitzAddAdditiveOfLipschitzMul (α : Type u_1) [PseudoMetricSpace α] [Monoid α] [LipschitzMul α] : LipschitzAdd (Additive α)

instance instLipschitzMulMultiplicativeOfLipschitzAdd (α : Type u_1) [PseudoMetricSpace α] [AddMonoid α] [LipschitzAdd α] : LipschitzMul (Multiplicative α)

instance instLipschitzMulOrderDual (α : Type u_1) [PseudoMetricSpace α] [Monoid α] [LipschitzMul α] : LipschitzMul αᵒᵈ

instance instLipschitzAddOrderDual (α : Type u_1) [PseudoMetricSpace α] [AddMonoid α] [LipschitzAdd α] : LipschitzAdd αᵒᵈ

instance Pi.instIsBoundedSMul {ι : Type u_3} [Fintype ι] {α : Type u_4} {β : ι → Type u_5} [PseudoMetricSpace α] [(i : ι) → PseudoMetricSpace (β i)] [Zero α] [(i : ι) → Zero (β i)] [(i : ι) → SMul α (β i)] [∀ (i : ι), IsBoundedSMul α (β i)] : IsBoundedSMul α ((i : ι) → β i)

instance Pi.instIsBoundedSMul' {ι : Type u_3} [Fintype ι] {α : ι → Type u_4} {β : ι → Type u_5} [(i : ι) → PseudoMetricSpace (α i)] [(i : ι) → PseudoMetricSpace (β i)] [(i : ι) → Zero (α i)] [(i : ι) → Zero (β i)] [(i : ι) → SMul (α i) (β i)] [∀ (i : ι), IsBoundedSMul (α i) (β i)] : IsBoundedSMul ((i : ι) → α i) ((i : ι) → β i)

instance Prod.instIsBoundedSMul {α : Type u_4} {β : Type u_5} {γ : Type u_6} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] [Zero α] [Zero β] [Zero γ] [SMul α β] [SMul α γ] [IsBoundedSMul α β] [IsBoundedSMul α γ] : IsBoundedSMul α (β × γ)

instance instIsBoundedSMulSeparationQuotient {α : Type u_4} {β : Type u_5} [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [IsBoundedSMul α β] : IsBoundedSMul α (SeparationQuotient β)