Relations

This file defines bundled relations. A relation between α and β is a function α → β → Prop. Relations are also known as set-valued functions, or partial multifunctions.

Main declarations

• Rel α β: Relation between α and β.
• Rel.inv: r.inv is the Rel β α obtained by swapping the arguments of r.
• Rel.dom: Domain of a relation. x ∈ r.dom iff there exists y such that r x y.
• Rel.codom: Codomain, aka range, of a relation. y ∈ r.codom iff there exists x such that r x y.
• Rel.comp: Relation composition. Note that the arguments order follows the CategoryTheory/ one, so r.comp s x z ↔ ∃ y, r x y ∧ s y z.
• Rel.image: Image of a set under a relation. r.image s is the set of f x over all x ∈ s.
• Rel.preimage: Preimage of a set under a relation. Note that r.preimage = r.inv.image.
• Rel.core: Core of a set. For s : Set β, r.core s is the set of x : α such that all y related to x are in s.
• Rel.restrict_domain: Domain-restriction of a relation to a subtype.
• Function.graph: Graph of a function as a relation.

def Rel (α : Type u_4) (β : Type u_5) : Type (max u_4 u_5)

A relation on α and β, aka a set-valued function, aka a partial multifunction

Equations

• Rel α β = (α → β → Prop)

instance instCompleteLatticeRel {α : Type u_1} {β : Type u_2} : CompleteLattice (Rel α β)

Equations

• instCompleteLatticeRel = let_fun this := inferInstance; this

instance instInhabitedRel {α : Type u_1} {β : Type u_2} : Inhabited (Rel α β)

Equations

• instInhabitedRel = let_fun this := inferInstance; this

theorem Rel.ext {α : Type u_1} {β : Type u_2} {r : Rel α β} {s : Rel α β} : (∀ (a : α), r a = s a) → r = s

def Rel.inv {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel β α

The inverse relation : r.inv x y ↔ r y x. Note that this is not a groupoid inverse.

Equations

• Rel.inv r = flip r

theorem Rel.inv_def {α : Type u_1} {β : Type u_2} (r : Rel α β) (x : α) (y : β) : Rel.inv r y x ↔ r x y

theorem Rel.inv_inv {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.inv (Rel.inv r) = r

def Rel.dom {α : Type u_1} {β : Type u_2} (r : Rel α β) : Set α

Domain of a relation

Equations

• Rel.dom r = {x : α | ∃ (y : β), r x y}

theorem Rel.dom_mono {α : Type u_1} {β : Type u_2} {r : Rel α β} {s : Rel α β} (h : r ≤ s) : Rel.dom r ⊆ Rel.dom s

def Rel.codom {α : Type u_1} {β : Type u_2} (r : Rel α β) : Set β

Codomain aka range of a relation

Equations

• Rel.codom r = {y : β | ∃ (x : α), r x y}

theorem Rel.codom_inv {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.codom (Rel.inv r) = Rel.dom r

theorem Rel.dom_inv {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.dom (Rel.inv r) = Rel.codom r

def Rel.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) (s : Rel β γ) : Rel α γ

Composition of relation; note that it follows the CategoryTheory/ order of arguments.

Equations

• Rel.comp r s x z = ∃ (y : β), r x y ∧ s y z

theorem Rel.comp_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (r : Rel α β) (s : Rel β γ) (t : Rel γ δ) : Rel.comp (Rel.comp r s) t = Rel.comp r (Rel.comp s t)

@[simp]

theorem Rel.comp_right_id {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.comp r Eq = r

@[simp]

theorem Rel.comp_left_id {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.comp Eq r = r

@[simp]

theorem Rel.comp_right_bot {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) : Rel.comp r ⊥ = ⊥

@[simp]

theorem Rel.comp_left_bot {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) : Rel.comp ⊥ r = ⊥

@[simp]

theorem Rel.comp_right_top {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) : Rel.comp r ⊤ = fun (x : α) (x_1 : γ) => x ∈ Rel.dom r

@[simp]

theorem Rel.comp_left_top {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) : Rel.comp ⊤ r = fun (x : γ) (y : β) => y ∈ Rel.codom r

theorem Rel.inv_id {α : Type u_1} : Rel.inv Eq = Eq

theorem Rel.inv_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) (s : Rel β γ) : Rel.inv (Rel.comp r s) = Rel.comp (Rel.inv s) (Rel.inv r)

@[simp]

theorem Rel.inv_bot {α : Type u_1} {β : Type u_2} : Rel.inv ⊥ = ⊥

@[simp]

theorem Rel.inv_top {α : Type u_1} {β : Type u_2} : Rel.inv ⊤ = ⊤

def Rel.image {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) : Set β

Image of a set under a relation

Equations

• Rel.image r s = {y : β | ∃ x ∈ s, r x y}

theorem Rel.mem_image {α : Type u_1} {β : Type u_2} (r : Rel α β) (y : β) (s : Set α) : y ∈ Rel.image r s ↔ ∃ x ∈ s, r x y

theorem Rel.image_subset {α : Type u_1} {β : Type u_2} (r : Rel α β) : ((fun (x x_1 : Set α) => x ⊆ x_1) ⇒ fun (x x_1 : Set β) => x ⊆ x_1) (Rel.image r) (Rel.image r)

theorem Rel.image_mono {α : Type u_1} {β : Type u_2} (r : Rel α β) : Monotone (Rel.image r)

theorem Rel.image_inter {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) (t : Set α) : Rel.image r (s ∩ t) ⊆ Rel.image r s ∩ Rel.image r t

theorem Rel.image_union {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) (t : Set α) : Rel.image r (s ∪ t) = Rel.image r s ∪ Rel.image r t

@[simp]

theorem Rel.image_id {α : Type u_1} (s : Set α) : Rel.image Eq s = s

theorem Rel.image_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) (s : Rel β γ) (t : Set α) : Rel.image (Rel.comp r s) t = Rel.image s (Rel.image r t)

theorem Rel.image_univ {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.image r Set.univ = Rel.codom r

@[simp]

theorem Rel.image_empty {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.image r ∅ = ∅

@[simp]

theorem Rel.image_bot {α : Type u_1} {β : Type u_2} (s : Set α) : Rel.image ⊥ s = ∅

@[simp]

theorem Rel.image_top {α : Type u_1} {β : Type u_2} {s : Set α} (h : Set.Nonempty s) : Rel.image ⊤ s = Set.univ

def Rel.preimage {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) : Set α

Preimage of a set under a relation r. Same as the image of s under r.inv

Equations

• Rel.preimage r s = Rel.image (Rel.inv r) s

theorem Rel.mem_preimage {α : Type u_1} {β : Type u_2} (r : Rel α β) (x : α) (s : Set β) : x ∈ Rel.preimage r s ↔ ∃ y ∈ s, r x y

theorem Rel.preimage_def {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) : Rel.preimage r s = {x : α | ∃ y ∈ s, r x y}

theorem Rel.preimage_mono {α : Type u_1} {β : Type u_2} (r : Rel α β) {s : Set β} {t : Set β} (h : s ⊆ t) : Rel.preimage r s ⊆ Rel.preimage r t

theorem Rel.preimage_inter {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) (t : Set β) : Rel.preimage r (s ∩ t) ⊆ Rel.preimage r s ∩ Rel.preimage r t

theorem Rel.preimage_union {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) (t : Set β) : Rel.preimage r (s ∪ t) = Rel.preimage r s ∪ Rel.preimage r t

theorem Rel.preimage_id {α : Type u_1} (s : Set α) : Rel.preimage Eq s = s

theorem Rel.preimage_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) (s : Rel β γ) (t : Set γ) : Rel.preimage (Rel.comp r s) t = Rel.preimage r (Rel.preimage s t)

theorem Rel.preimage_univ {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.preimage r Set.univ = Rel.dom r

@[simp]

theorem Rel.preimage_empty {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.preimage r ∅ = ∅

@[simp]

theorem Rel.preimage_inv {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) : Rel.preimage (Rel.inv r) s = Rel.image r s

@[simp]

theorem Rel.preimage_bot {α : Type u_1} {β : Type u_2} (s : Set β) : Rel.preimage ⊥ s = ∅

@[simp]

theorem Rel.preimage_top {α : Type u_1} {β : Type u_2} {s : Set β} (h : Set.Nonempty s) : Rel.preimage ⊤ s = Set.univ

theorem Rel.image_eq_dom_of_codomain_subset {α : Type u_1} {β : Type u_2} (r : Rel α β) {s : Set β} (h : Rel.codom r ⊆ s) : Rel.preimage r s = Rel.dom r

theorem Rel.preimage_eq_codom_of_domain_subset {α : Type u_1} {β : Type u_2} (r : Rel α β) {s : Set α} (h : Rel.dom r ⊆ s) : Rel.image r s = Rel.codom r

theorem Rel.image_inter_dom_eq {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) : Rel.image r (s ∩ Rel.dom r) = Rel.image r s

@[simp]

theorem Rel.preimage_inter_codom_eq {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) : Rel.preimage r (s ∩ Rel.codom r) = Rel.preimage r s

theorem Rel.inter_dom_subset_preimage_image {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) : s ∩ Rel.dom r ⊆ Rel.preimage r (Rel.image r s)

theorem Rel.image_preimage_subset_inter_codom {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) : s ∩ Rel.codom r ⊆ Rel.image r (Rel.preimage r s)

def Rel.core {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) : Set α

Core of a set s : Set β w.r.t r : Rel α β is the set of x : α that are related only to elements of s. Other generalization of Function.preimage.

Equations

• Rel.core r s = {x : α | ∀ (y : β), r x y → y ∈ s}

theorem Rel.mem_core {α : Type u_1} {β : Type u_2} (r : Rel α β) (x : α) (s : Set β) : x ∈ Rel.core r s ↔ ∀ (y : β), r x y → y ∈ s

theorem Rel.core_subset {α : Type u_1} {β : Type u_2} (r : Rel α β) : ((fun (x x_1 : Set β) => x ⊆ x_1) ⇒ fun (x x_1 : Set α) => x ⊆ x_1) (Rel.core r) (Rel.core r)

theorem Rel.core_mono {α : Type u_1} {β : Type u_2} (r : Rel α β) : Monotone (Rel.core r)

theorem Rel.core_inter {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) (t : Set β) : Rel.core r (s ∩ t) = Rel.core r s ∩ Rel.core r t

theorem Rel.core_union {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set β) (t : Set β) : Rel.core r s ∪ Rel.core r t ⊆ Rel.core r (s ∪ t)

@[simp]

theorem Rel.core_univ {α : Type u_1} {β : Type u_2} (r : Rel α β) : Rel.core r Set.univ = Set.univ

theorem Rel.core_id {α : Type u_1} (s : Set α) : Rel.core Eq s = s

theorem Rel.core_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} (r : Rel α β) (s : Rel β γ) (t : Set γ) : Rel.core (Rel.comp r s) t = Rel.core r (Rel.core s t)

def Rel.restrictDomain {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) : Rel { x : α // x ∈ s } β

Restrict the domain of a relation to a subtype.

Equations

• Rel.restrictDomain r s x y = r (↑x) y

theorem Rel.image_subset_iff {α : Type u_1} {β : Type u_2} (r : Rel α β) (s : Set α) (t : Set β) : Rel.image r s ⊆ t ↔ s ⊆ Rel.core r t

theorem Rel.image_core_gc {α : Type u_1} {β : Type u_2} (r : Rel α β) : GaloisConnection (Rel.image r) (Rel.core r)

def Function.graph {α : Type u_1} {β : Type u_2} (f : α → β) : Rel α β

The graph of a function as a relation.

Equations

• Function.graph f x y = (f x = y)

@[simp]

theorem Function.graph_def {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) (y : β) : Function.graph f x y ↔ f x = y

theorem Function.graph_id {α : Type u_1} : Function.graph id = Eq

theorem Function.graph_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : β → γ} {g : α → β} : Function.graph (f ∘ g) = Rel.comp (Function.graph g) (Function.graph f)

theorem Equiv.graph_inv {α : Type u_1} {β : Type u_2} (f : α ≃ β) : Function.graph ⇑f.symm = Rel.inv (Function.graph ⇑f)

theorem Relation.is_graph_iff {α : Type u_1} {β : Type u_2} (r : Rel α β) : (∃! f : α → β, Function.graph f = r) ↔ ∀ (x : α), ∃! y : β, r x y

theorem Set.image_eq {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) : f '' s = Rel.image (Function.graph f) s

theorem Set.preimage_eq {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set β) : f ⁻¹' s = Rel.preimage (Function.graph f) s

theorem Set.preimage_eq_core {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set β) : f ⁻¹' s = Rel.core (Function.graph f) s