bookshelf/Bookshelf/Enderton/Set/Relation.lean

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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