Locally bounded maps

This file defines locally bounded maps between bornologies.

We use the DFunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms

• LocallyBoundedMap: Locally bounded maps. Maps which preserve boundedness.

Typeclasses

• LocallyBoundedMapClass

structure LocallyBoundedMap (α : Type u_6) (β : Type u_7) [Bornology α] [Bornology β] : Type (max u_6 u_7)

The type of bounded maps from α to β, the maps which send a bounded set to a bounded set.

• toFun : α → β

  The function underlying a locally bounded map

• comap_cobounded_le' : Filter.comap self.toFun (Bornology.cobounded β) ≤ Bornology.cobounded α

  The pullback of the Bornology.cobounded filter under the function is contained in the cobounded filter. Equivalently, the function maps bounded sets to bounded sets.

theorem LocallyBoundedMap.comap_cobounded_le' {α : Type u_6} {β : Type u_7} [Bornology α] [Bornology β] (self : LocallyBoundedMap α β) : Filter.comap self.toFun (Bornology.cobounded β) ≤ Bornology.cobounded α

The pullback of the Bornology.cobounded filter under the function is contained in the cobounded filter. Equivalently, the function maps bounded sets to bounded sets.

class LocallyBoundedMapClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [Bornology α] [Bornology β] [FunLike F α β] : Prop

LocallyBoundedMapClass F α β states that F is a type of bounded maps.

You should extend this class when you extend LocallyBoundedMap.

• comap_cobounded_le : ∀ (f : F), Filter.comap (⇑f) (Bornology.cobounded β) ≤ Bornology.cobounded α

  The pullback of the Bornology.cobounded filter under the function is contained in the cobounded filter. Equivalently, the function maps bounded sets to bounded sets.

theorem LocallyBoundedMapClass.comap_cobounded_le {F : Type u_6} {α : outParam (Type u_7)} {β : outParam (Type u_8)} : ∀ {inst : Bornology α} {inst_1 : Bornology β} {inst_2 : FunLike F α β} [self : LocallyBoundedMapClass F α β] (f : F), Filter.comap (⇑f) (Bornology.cobounded β) ≤ Bornology.cobounded α

The pullback of the Bornology.cobounded filter under the function is contained in the cobounded filter. Equivalently, the function maps bounded sets to bounded sets.

theorem Bornology.IsBounded.image {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F) {s : Set α} (hs : Bornology.IsBounded s) : Bornology.IsBounded (⇑f '' s)

def LocallyBoundedMapClass.toLocallyBoundedMap {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F) : LocallyBoundedMap α β

Turn an element of a type F satisfying LocallyBoundedMapClass F α β into an actual LocallyBoundedMap. This is declared as the default coercion from F to LocallyBoundedMap α β.

Equations

• ↑f = { toFun := ⇑f, comap_cobounded_le' := ⋯ }

instance instCoeTCLocallyBoundedMapOfLocallyBoundedMapClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] : CoeTC F (LocallyBoundedMap α β)

Equations

• instCoeTCLocallyBoundedMapOfLocallyBoundedMapClass = { coe := fun (f : F) => { toFun := ⇑f, comap_cobounded_le' := ⋯ } }

instance LocallyBoundedMap.instFunLike {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] : FunLike (LocallyBoundedMap α β) α β

Equations

• LocallyBoundedMap.instFunLike = { coe := fun (f : LocallyBoundedMap α β) => f.toFun, coe_injective' := ⋯ }

instance LocallyBoundedMap.instLocallyBoundedMapClass {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] : LocallyBoundedMapClass (LocallyBoundedMap α β) α β

Equations

• ⋯ = ⋯

theorem LocallyBoundedMap.ext_iff {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] {f : LocallyBoundedMap α β} {g : LocallyBoundedMap α β} : f = g ↔ ∀ (a : α), f a = g a

theorem LocallyBoundedMap.ext {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] {f : LocallyBoundedMap α β} {g : LocallyBoundedMap α β} (h : ∀ (a : α), f a = g a) : f = g

def LocallyBoundedMap.copy {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = ⇑f) : LocallyBoundedMap α β

Copy of a LocallyBoundedMap with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toFun := f', comap_cobounded_le' := ⋯ }

@[simp]

theorem LocallyBoundedMap.coe_copy {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem LocallyBoundedMap.copy_eq {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = ⇑f) : f.copy f' h = f

def LocallyBoundedMap.ofMapBounded {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : α → β) (h : ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)) : LocallyBoundedMap α β

Construct a LocallyBoundedMap from the fact that the function maps bounded sets to bounded sets.

Equations

• LocallyBoundedMap.ofMapBounded f h = { toFun := f, comap_cobounded_le' := ⋯ }

@[simp]

theorem LocallyBoundedMap.coe_ofMapBounded {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : α → β) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)} : ⇑(LocallyBoundedMap.ofMapBounded f h) = f

@[simp]

theorem LocallyBoundedMap.ofMapBounded_apply {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : α → β) {h : ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)} (a : α) : (LocallyBoundedMap.ofMapBounded f h) a = f a

def LocallyBoundedMap.id (α : Type u_2) [Bornology α] : LocallyBoundedMap α α

id as a LocallyBoundedMap.

Equations

• LocallyBoundedMap.id α = { toFun := id, comap_cobounded_le' := ⋯ }

instance LocallyBoundedMap.instInhabited (α : Type u_2) [Bornology α] : Inhabited (LocallyBoundedMap α α)

Equations

• LocallyBoundedMap.instInhabited α = { default := LocallyBoundedMap.id α }

@[simp]

theorem LocallyBoundedMap.coe_id (α : Type u_2) [Bornology α] : ⇑(LocallyBoundedMap.id α) = id

@[simp]

theorem LocallyBoundedMap.id_apply {α : Type u_2} [Bornology α] (a : α) : (LocallyBoundedMap.id α) a = a

def LocallyBoundedMap.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) : LocallyBoundedMap α γ

Composition of LocallyBoundedMaps as a LocallyBoundedMap.

Equations

• f.comp g = { toFun := ⇑f ∘ ⇑g, comap_cobounded_le' := ⋯ }

@[simp]

theorem LocallyBoundedMap.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) : ⇑(f.comp g) = ⇑f ∘ ⇑g

@[simp]

theorem LocallyBoundedMap.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α) : (f.comp g) a = f (g a)

@[simp]

theorem LocallyBoundedMap.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Bornology α] [Bornology β] [Bornology γ] [Bornology δ] (f : LocallyBoundedMap γ δ) (g : LocallyBoundedMap β γ) (h : LocallyBoundedMap α β) : (f.comp g).comp h = f.comp (g.comp h)

@[simp]

theorem LocallyBoundedMap.comp_id {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) : f.comp (LocallyBoundedMap.id α) = f

@[simp]

theorem LocallyBoundedMap.id_comp {α : Type u_2} {β : Type u_3} [Bornology α] [Bornology β] (f : LocallyBoundedMap α β) : (LocallyBoundedMap.id β).comp f = f

@[simp]

theorem LocallyBoundedMap.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] {g₁ : LocallyBoundedMap β γ} {g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α β} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

@[simp]

theorem LocallyBoundedMap.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Bornology α] [Bornology β] [Bornology γ] {g : LocallyBoundedMap β γ} {f₁ : LocallyBoundedMap α β} {f₂ : LocallyBoundedMap α β} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂