493 lines
14 KiB
Plaintext
493 lines
14 KiB
Plaintext
import Bookshelf.Enderton.Set.OrderedPair
|
||
|
||
/-! # Enderton.Set.Relation
|
||
|
||
A representation of a relation, i.e. a set of ordered pairs. Like `Set`, it is
|
||
assumed a relation is homogeneous.
|
||
-/
|
||
|
||
namespace Set
|
||
|
||
/--
|
||
A relation type as defined by Enderton.
|
||
|
||
We choose to use tuples to represent our ordered pairs, as opposed to
|
||
Kuratowski's definition of a set.
|
||
|
||
Not to be confused with the Lean-provided `Rel`.
|
||
-/
|
||
abbrev HRelation (α β : Type ) := Set (α × β)
|
||
|
||
/--
|
||
A homogeneous variant of the `HRelation` type.
|
||
-/
|
||
abbrev Relation (α : Type _) := HRelation α α
|
||
|
||
namespace Relation
|
||
|
||
/-! ## Domain and Range -/
|
||
|
||
/--
|
||
The domain of a `Relation`.
|
||
-/
|
||
def dom (R : HRelation α β) : Set α := Prod.fst '' R
|
||
|
||
/--
|
||
The first component of any pair in a `Relation` must be a member of the
|
||
`Relation`'s domain.
|
||
-/
|
||
theorem mem_pair_imp_fst_mem_dom {R : HRelation α β} (h : (x, y) ∈ R)
|
||
: x ∈ dom R := by
|
||
unfold dom Prod.fst
|
||
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right]
|
||
exact ⟨y, h⟩
|
||
|
||
/--
|
||
If `x ∈ dom R`, there exists some `y` such that `⟨x, y⟩ ∈ R`.
|
||
-/
|
||
theorem dom_exists {R : HRelation α β} (hx : x ∈ dom R)
|
||
: ∃ y : β, (x, y) ∈ R := by
|
||
unfold dom at hx
|
||
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] at hx
|
||
exact hx
|
||
|
||
/--
|
||
The range of a `Relation`.
|
||
-/
|
||
def ran (R : HRelation α β) : Set β := Prod.snd '' R
|
||
|
||
theorem mem_pair_imp_snd_mem_ran {R : HRelation α β} (h : (x, y) ∈ R)
|
||
: y ∈ ran R := by
|
||
unfold ran Prod.snd
|
||
simp only [mem_image, Prod.exists, exists_eq_right]
|
||
exact ⟨x, h⟩
|
||
|
||
/--
|
||
If `x ∈ ran R`, there exists some `t` such that `⟨t, x⟩ ∈ R`.
|
||
-/
|
||
theorem ran_exists {R : HRelation α β} (hx : x ∈ ran R)
|
||
: ∃ t : α, (t, x) ∈ R := by
|
||
unfold ran at hx
|
||
simp only [mem_image, Prod.exists, exists_eq_right] at hx
|
||
exact hx
|
||
|
||
/--
|
||
The field of a `Relation`.
|
||
-/
|
||
def fld (R : Relation α) : Set α := dom R ∪ ran R
|
||
|
||
/--
|
||
The inverse of a `Relation`.
|
||
-/
|
||
def inv (R : HRelation α β) : HRelation β α := { (p.2, p.1) | p ∈ R }
|
||
|
||
/--
|
||
`(x, y)` is a member of relation `R` **iff** `(y, x)` is a member of `R⁻¹`.
|
||
-/
|
||
@[simp]
|
||
theorem mem_self_comm_mem_inv {R : HRelation α β}
|
||
: (y, x) ∈ inv R ↔ (x, y) ∈ R := by
|
||
unfold inv
|
||
simp only [Prod.exists, mem_setOf_eq, Prod.mk.injEq]
|
||
apply Iff.intro
|
||
· intro ⟨x', y', hxy⟩
|
||
rw [← hxy.right.left, ← hxy.right.right]
|
||
exact hxy.left
|
||
· intro hxy
|
||
exact ⟨x, y, hxy, rfl, rfl⟩
|
||
|
||
/--
|
||
The inverse of the inverse of a `Relation` is the `Relation`.
|
||
-/
|
||
@[simp]
|
||
theorem inv_inv_eq_self (R : HRelation α β)
|
||
: inv (inv R) = R := by
|
||
unfold inv
|
||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
|
||
ext x
|
||
apply Iff.intro
|
||
· intro hx
|
||
have ⟨a₁, b₁, ⟨⟨a₂, b₂, h₁⟩, h₂⟩⟩ := hx
|
||
rw [← h₂, ← h₁.right.right, ← h₁.right.left]
|
||
exact h₁.left
|
||
· intro hx
|
||
have (p, q) := x
|
||
refine ⟨q, p, ⟨?_, ?_⟩⟩
|
||
· exact ⟨p, q, hx, rfl, rfl⟩
|
||
· rfl
|
||
|
||
/--
|
||
For a set `F`, `dom F⁻¹ = ran F`.
|
||
-/
|
||
@[simp]
|
||
theorem dom_inv_eq_ran_self {F : HRelation α β}
|
||
: dom (inv F) = ran F := by
|
||
ext x
|
||
unfold dom ran inv
|
||
simp only [
|
||
Prod.exists,
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.mk.injEq,
|
||
exists_and_right,
|
||
exists_eq_right
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨y, a, _, h⟩
|
||
rw [← h.right.left]
|
||
exact ⟨a, h.left⟩
|
||
· intro ⟨y, hy⟩
|
||
exact ⟨y, y, x, hy, rfl, rfl⟩
|
||
|
||
/--
|
||
For a set `F`, `ran F⁻¹ = dom F`.
|
||
-/
|
||
@[simp]
|
||
theorem ran_inv_eq_dom_self {F : HRelation α β}
|
||
: ran (inv F) = dom F := by
|
||
ext x
|
||
unfold dom ran inv
|
||
simp only [
|
||
Prod.exists,
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.mk.injEq,
|
||
exists_eq_right,
|
||
exists_and_right
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨a, y, b, h⟩
|
||
rw [← h.right.right]
|
||
exact ⟨b, h.left⟩
|
||
· intro ⟨y, hy⟩
|
||
exact ⟨y, x, y, hy, rfl, rfl⟩
|
||
|
||
/-! ## Restriction -/
|
||
|
||
/--
|
||
The restriction of a `Relation` to a `Set`.
|
||
-/
|
||
def restriction (R : HRelation α β) (A : Set α) : HRelation α β :=
|
||
{ p ∈ R | p.1 ∈ A }
|
||
|
||
/-! ## Image -/
|
||
|
||
/--
|
||
The image of a `Relation` under a `Set`.
|
||
-/
|
||
def image (R : HRelation α β) (A : Set α) : Set β :=
|
||
{ y | ∃ u ∈ A, (u, y) ∈ R }
|
||
|
||
/-! ## Single-Rooted and Single-Valued -/
|
||
|
||
/--
|
||
A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, there
|
||
exists exactly one `x` such that `⟨x, y⟩ ∈ R`.
|
||
-/
|
||
def isSingleRooted (R : HRelation α β) : Prop :=
|
||
∀ y ∈ ran R, ∃! x, x ∈ dom R ∧ (x, y) ∈ R
|
||
|
||
/--
|
||
A single-rooted `Relation` should map the same output to the same input.
|
||
-/
|
||
theorem single_rooted_eq_unique {R : HRelation α β} {x₁ x₂ : α} {y : β}
|
||
(hR : isSingleRooted R)
|
||
: (x₁, y) ∈ R → (x₂, y) ∈ R → x₁ = x₂ := by
|
||
intro hx₁ hx₂
|
||
unfold isSingleRooted at hR
|
||
have := hR y (mem_pair_imp_snd_mem_ran hx₁)
|
||
have ⟨y₁, hy₁⟩ := this
|
||
simp only [and_imp] at hy₁
|
||
have h₁ := hy₁.right x₁ (mem_pair_imp_fst_mem_dom hx₁) hx₁
|
||
have h₂ := hy₁.right x₂ (mem_pair_imp_fst_mem_dom hx₂) hx₂
|
||
rw [h₁, h₂]
|
||
|
||
/--
|
||
A `Relation` `R` is said to be single-valued **iff** for all `x ∈ dom R`, there
|
||
exists exactly one `y` such that `⟨x, y⟩ ∈ R`.
|
||
|
||
Notice, a `Relation` that is single-valued is a function.
|
||
-/
|
||
def isSingleValued (R : HRelation α β) : Prop :=
|
||
∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R
|
||
|
||
/--
|
||
A single-valued `Relation` should map the same input to the same output.
|
||
-/
|
||
theorem single_valued_eq_unique {R : HRelation α β} {x : α} {y₁ y₂ : β}
|
||
(hR : isSingleValued R)
|
||
: (x, y₁) ∈ R → (x, y₂) ∈ R → y₁ = y₂ := by
|
||
intro hy₁ hy₂
|
||
unfold isSingleValued at hR
|
||
have := hR x (mem_pair_imp_fst_mem_dom hy₁)
|
||
have ⟨x₁, hx₁⟩ := this
|
||
simp only [and_imp] at hx₁
|
||
have h₁ := hx₁.right y₁ (mem_pair_imp_snd_mem_ran hy₁) hy₁
|
||
have h₂ := hx₁.right y₂ (mem_pair_imp_snd_mem_ran hy₂) hy₂
|
||
rw [h₁, h₂]
|
||
|
||
/--
|
||
For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted.
|
||
-/
|
||
theorem single_valued_inv_iff_single_rooted_self {F : HRelation α β}
|
||
: isSingleValued (inv F) ↔ isSingleRooted F := by
|
||
apply Iff.intro
|
||
· intro hF
|
||
unfold isSingleValued at hF
|
||
simp only [
|
||
dom_inv_eq_ran_self,
|
||
ran_inv_eq_dom_self,
|
||
mem_self_comm_mem_inv
|
||
] at hF
|
||
suffices ∀ x ∈ ran F, ∃! y, (y, x) ∈ F from hF
|
||
intro x hx
|
||
have ⟨y, hy⟩ := hF x hx
|
||
simp only [
|
||
ran_inv_eq_dom_self,
|
||
mem_self_comm_mem_inv,
|
||
and_imp
|
||
] at hy
|
||
refine ⟨y, hy.left.right, ?_⟩
|
||
intro y₁ hy₁
|
||
exact hy.right y₁ (mem_pair_imp_fst_mem_dom hy₁) hy₁
|
||
· intro hF
|
||
unfold isSingleRooted at hF
|
||
unfold isSingleValued
|
||
simp only [
|
||
dom_inv_eq_ran_self,
|
||
ran_inv_eq_dom_self,
|
||
mem_self_comm_mem_inv
|
||
]
|
||
exact hF
|
||
|
||
/--
|
||
For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted.
|
||
-/
|
||
theorem single_valued_self_iff_single_rooted_inv {F : HRelation α β}
|
||
: isSingleValued F ↔ isSingleRooted (inv F) := by
|
||
conv => lhs; rw [← inv_inv_eq_self F]
|
||
rw [single_valued_inv_iff_single_rooted_self]
|
||
|
||
/--
|
||
The subset of a function must also be a function.
|
||
-/
|
||
theorem single_valued_subset {F G : HRelation α β}
|
||
(hG : isSingleValued G) (h : F ⊆ G)
|
||
: isSingleValued F := by
|
||
unfold isSingleValued
|
||
intro x hx
|
||
have ⟨y, hy⟩ := dom_exists hx
|
||
unfold ExistsUnique
|
||
simp only
|
||
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hy, hy⟩, ?_⟩
|
||
intro y₁ hy₁
|
||
exact single_valued_eq_unique hG (h hy₁.right) (h hy)
|
||
|
||
/-! ## Injections -/
|
||
|
||
/--
|
||
A `Relation` `R` is one-to-one if it is a single-rooted function.
|
||
-/
|
||
def isOneToOne (R : HRelation α β) : Prop :=
|
||
isSingleValued R ∧ isSingleRooted R
|
||
|
||
/--
|
||
A `Relation` is one-to-one **iff** it's inverse is.
|
||
-/
|
||
theorem one_to_one_self_iff_one_to_one_inv {R : HRelation α β}
|
||
: isOneToOne R ↔ isOneToOne (inv R) := by
|
||
unfold isOneToOne isSingleValued isSingleRooted
|
||
conv => rhs; simp only [
|
||
dom_inv_eq_ran_self,
|
||
ran_inv_eq_dom_self,
|
||
mem_self_comm_mem_inv,
|
||
eq_iff_iff
|
||
]
|
||
apply Iff.intro <;>
|
||
· intro ⟨hx, hy⟩
|
||
exact ⟨hy, hx⟩
|
||
|
||
/-! ## Surjections -/
|
||
|
||
/--
|
||
Indicates `Relation` `F` is a function from `A` to `B`.
|
||
|
||
This is usually denoted as `F : A → B`.
|
||
-/
|
||
def mapsInto (F : HRelation α β) (A : Set α) (B : Set β) :=
|
||
isSingleValued F ∧ dom F = A ∧ ran F ⊆ B
|
||
|
||
/--
|
||
Indicates `Relation` `F` is a function from `A` to `ran F = B`.
|
||
-/
|
||
def mapsOnto (F : HRelation α β) (A : Set α) (B : Set β) :=
|
||
isSingleValued F ∧ dom F = A ∧ ran F = B
|
||
|
||
/-! ## Composition -/
|
||
|
||
/--
|
||
The composition of two `Relation`s.
|
||
-/
|
||
def comp (F : HRelation β γ) (G : HRelation α β) : HRelation α γ :=
|
||
{ p | ∃ t : β, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
|
||
|
||
/--
|
||
If `x ∈ dom (F ∘ G)`, then `x ∈ dom G`.
|
||
-/
|
||
theorem dom_comp_imp_dom_self {F : HRelation β γ} {G : HRelation α β}
|
||
: x ∈ dom (comp F G) → x ∈ dom G := by
|
||
unfold dom comp
|
||
simp only [
|
||
mem_image,
|
||
mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right,
|
||
forall_exists_index
|
||
]
|
||
intro y t ht
|
||
exact ⟨t, ht.left⟩
|
||
|
||
/--
|
||
If `y ∈ ran (F ∘ G)`, then `y ∈ ran F`.
|
||
-/
|
||
theorem ran_comp_imp_ran_self {F : HRelation β γ} {G : HRelation α β}
|
||
: y ∈ ran (comp F G) → y ∈ ran F := by
|
||
unfold ran comp
|
||
simp only [
|
||
mem_image,
|
||
mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_eq_right,
|
||
forall_exists_index
|
||
]
|
||
intro x t ht
|
||
exact ⟨t, ht.right⟩
|
||
|
||
/--
|
||
Composition of functions is associative.
|
||
-/
|
||
theorem comp_assoc {R : HRelation γ δ} {S : HRelation β γ} {T : HRelation α β}
|
||
: comp (comp R S) T = comp R (comp S T) := by
|
||
calc comp (comp R S) T
|
||
_ = { p | ∃ t, (p.1, t) ∈ T ∧ (t, p.2) ∈ comp R S} := rfl
|
||
_ = { p | ∃ t, (p.1, t) ∈ T ∧ (∃ a, (t, a) ∈ S ∧ (a, p.2) ∈ R) } := rfl
|
||
_ = { p | ∃ t, ∃ a, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
|
||
ext p
|
||
simp only [mem_setOf_eq]
|
||
apply Iff.intro
|
||
· intro ⟨t, ht, a, ha⟩
|
||
exact ⟨t, a, ⟨ht, ha.left⟩, ha.right⟩
|
||
· intro ⟨t, a, h₁, h₂⟩
|
||
exact ⟨t, h₁.left, a, h₁.right, h₂⟩
|
||
_ = { p | ∃ a, ∃ t, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
|
||
ext p
|
||
simp only [mem_setOf_eq]
|
||
apply Iff.intro
|
||
· intro ⟨t, a, h⟩
|
||
exact ⟨a, t, h⟩
|
||
· intro ⟨a, t, h⟩
|
||
exact ⟨t, a, h⟩
|
||
_ = { p | ∃ a, (∃ t, (p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
|
||
ext p
|
||
simp only [mem_setOf_eq]
|
||
apply Iff.intro
|
||
· intro ⟨a, t, h⟩
|
||
exact ⟨a, ⟨t, h.left⟩, h.right⟩
|
||
· intro ⟨a, ⟨t, ht⟩, ha⟩
|
||
exact ⟨a, t, ht, ha⟩
|
||
_ = { p | ∃ a, (p.1, a) ∈ comp S T ∧ (a, p.2) ∈ R } := rfl
|
||
_ = comp R (comp S T) := rfl
|
||
|
||
/--
|
||
The composition of two functions is itself a function.
|
||
-/
|
||
theorem single_valued_comp_is_single_valued
|
||
{F : HRelation β γ} {G : HRelation α β}
|
||
(hF : isSingleValued F) (hG : isSingleValued G)
|
||
: isSingleValued (comp F G) := by
|
||
unfold isSingleValued
|
||
intro x hx
|
||
have ⟨y, hxy⟩ := dom_exists hx
|
||
have hy := mem_pair_imp_snd_mem_ran hxy
|
||
refine ⟨y, ⟨hy, hxy⟩, ?_⟩
|
||
simp only [and_imp]
|
||
|
||
intro y₁ _ hxy₁
|
||
unfold comp at hxy hxy₁
|
||
simp only [mem_setOf_eq] at hxy hxy₁
|
||
have ⟨t₁, ht₁⟩ := hxy
|
||
have ⟨t₂, ht₂⟩ := hxy₁
|
||
|
||
-- First show `t₁ = t₂` and then show `y = y₁`.
|
||
have t_eq : t₁ = t₂ := by
|
||
unfold isSingleValued at hG
|
||
have ⟨t', ht'⟩ := hG x (mem_pair_imp_fst_mem_dom ht₁.left)
|
||
simp only [and_imp] at ht'
|
||
have ht₁' := ht'.right t₁ (mem_pair_imp_snd_mem_ran ht₁.left) ht₁.left
|
||
have ht₂' := ht'.right t₂ (mem_pair_imp_snd_mem_ran ht₂.left) ht₂.left
|
||
rw [ht₁', ht₂']
|
||
|
||
unfold isSingleValued at hF
|
||
rw [t_eq] at ht₁
|
||
have ⟨y', hy'⟩ := hF t₂ (mem_pair_imp_fst_mem_dom ht₁.right)
|
||
simp only [and_imp] at hy'
|
||
have hk₁ := hy'.right y (mem_pair_imp_snd_mem_ran ht₁.right) ht₁.right
|
||
have hk₂ := hy'.right y₁ (mem_pair_imp_snd_mem_ran ht₂.right) ht₂.right
|
||
rw [hk₁, hk₂]
|
||
|
||
/--
|
||
For `Relation`s `F` and `G`, `(F ∘ G)⁻¹ = G⁻¹ ∘ F⁻¹`.
|
||
-/
|
||
theorem comp_inv_eq_inv_comp_inv {F : HRelation β γ} {G : HRelation α β}
|
||
: inv (comp F G) = comp (inv G) (inv F) := by
|
||
calc inv (comp F G)
|
||
_ = {p | ∃ t, (p.2, t) ∈ G ∧ (t, p.1) ∈ F} := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· unfold inv comp
|
||
intro t ht
|
||
simp only [mem_setOf_eq, Prod.exists] at ht
|
||
have ⟨a, b, ⟨⟨p, hp⟩, hab⟩⟩ := ht
|
||
rw [← hab]
|
||
exact ⟨p, hp⟩
|
||
· unfold inv comp
|
||
intro (a, b) ⟨p, hp⟩
|
||
simp only [mem_setOf_eq, Prod.exists, Prod.mk.injEq]
|
||
exact ⟨b, a, ⟨p, hp⟩, rfl, rfl⟩
|
||
_ = {p | ∃ t, (t, p.1) ∈ F ∧ (p.2, t) ∈ G} := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· intro (a, b) ht
|
||
simp only [mem_setOf_eq] at *
|
||
have ⟨t, p, q⟩ := ht
|
||
exact ⟨t, q, p⟩
|
||
· intro (a, b) ht
|
||
simp only [mem_setOf_eq] at *
|
||
have ⟨t, p, q⟩ := ht
|
||
exact ⟨t, q, p⟩
|
||
_ = {p | ∃ t, (p.1, t) ∈ inv F ∧ (t, p.2) ∈ inv G } := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· intro (a, b) ht
|
||
simp only [mem_setOf_eq] at *
|
||
have ⟨t, p, q⟩ := ht
|
||
refine ⟨t, ?_, ?_⟩ <;> rwa [mem_self_comm_mem_inv]
|
||
· intro (a, b) ht
|
||
simp only [mem_setOf_eq] at *
|
||
have ⟨t, p, q⟩ := ht
|
||
refine ⟨t, ?_, ?_⟩ <;> rwa [← mem_self_comm_mem_inv]
|
||
_ = comp (inv G) (inv F) := rfl
|
||
|
||
/-! ## Ordered Pairs -/
|
||
|
||
/--
|
||
Convert a `Relation` into an equivalent representation using `OrderedPair`s.
|
||
-/
|
||
def toOrderedPairs (R : Relation α) : Set (Set (Set α)) :=
|
||
-- Notice here we are using `Set.image` and *not* `Set.Relation.image`.
|
||
Set.image (fun (x, y) => OrderedPair x y) R
|
||
|
||
end Relation
|
||
|
||
end Set |