------------------------------------------------------------------------
-- The Agda standard library
--
-- Unary relations
------------------------------------------------------------------------

module Relation.Unary where

open import Data.Empty
open import Function
open import Data.Unit hiding (setoid)
open import Data.Product
open import Data.Sum
open import Level
open import Relation.Nullary
open import Relation.Binary using (Setoid; IsEquivalence)

------------------------------------------------------------------------
-- Unary relations

Pred : ∀ {a} → Set a → (ℓ : Level) → Set (a ⊔ suc ℓ)
Pred A ℓ = A → Set ℓ

------------------------------------------------------------------------
-- Unary relations can be seen as sets

-- I.e., they can be seen as subsets of the universe of discourse.

module _ {a} {A : Set a} -- The universe of discourse.
         where

  -- Set membership.

  infix  _∈_ _∉_

  _∈_ : ∀ {ℓ} → A → Pred A ℓ → Set _
  x ∈ P = P x

  _∉_ : ∀ {ℓ} → A → Pred A ℓ → Set _
  x ∉ P = ¬ x ∈ P

  -- The empty set.

  ∅ : Pred A zero
  ∅ = λ _ → ⊥

  -- The property of being empty.

  Empty : ∀ {ℓ} → Pred A ℓ → Set _
  Empty P = ∀ x → x ∉ P

  ∅-Empty : Empty ∅
  ∅-Empty x ()

  -- The universe, i.e. the subset containing all elements in A.

  U : Pred A zero
  U = λ _ → ⊤

  -- The property of being universal.

  Universal : ∀ {ℓ} → Pred A ℓ → Set _
  Universal P = ∀ x → x ∈ P

  U-Universal : Universal U
  U-Universal = λ _ → _

  -- Set complement.

  ∁ : ∀ {ℓ} → Pred A ℓ → Pred A ℓ
  ∁ P = λ x → x ∉ P

  ∁∅-Universal : Universal (∁ ∅)
  ∁∅-Universal = λ x x∈∅ → x∈∅

  ∁U-Empty : Empty (∁ U)
  ∁U-Empty = λ x x∈∁U → x∈∁U _

  -- P ⊆ Q means that P is a subset of Q. _⊆′_ is a variant of _⊆_.

  infix  _⊆_ _⊇_ _⊆′_ _⊇′_

  _⊆_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Set _
  P ⊆ Q = ∀ {x} → x ∈ P → x ∈ Q

  _⊆′_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Set _
  P ⊆′ Q = ∀ x → x ∈ P → x ∈ Q

  _⊇_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Set _
  Q ⊇ P = P ⊆ Q

  _⊇′_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Set _
  Q ⊇′ P = P ⊆′ Q

  ∅-⊆ : ∀ {ℓ} → (P : Pred A ℓ) → ∅ ⊆ P
  ∅-⊆ P ()

  ⊆-U : ∀ {ℓ} → (P : Pred A ℓ) → P ⊆ U
  ⊆-U P _ = _

  -- Positive version of non-disjointness, dual to inclusion.

  infix  _≬_

  _≬_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Set _
  P ≬ Q = ∃ λ x → x ∈ P × x ∈ Q

  -- Set union.

  infixr  _∪_

  _∪_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Pred A _
  P ∪ Q = λ x → x ∈ P ⊎ x ∈ Q

  -- Set intersection.

  infixr  _∩_

  _∩_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Pred A _
  P ∩ Q = λ x → x ∈ P × x ∈ Q

  -- Implication.

  infixl  _⇒_

  _⇒_ : ∀ {ℓ₁ ℓ₂} → Pred A ℓ₁ → Pred A ℓ₂ → Pred A _
  P ⇒ Q = λ x → x ∈ P → x ∈ Q

  -- Infinitary union and intersection.

  infix  ⋃ ⋂

  ⋃ : ∀ {ℓ i} (I : Set i) → (I → Pred A ℓ) → Pred A _
  ⋃ I P = λ x → Σ[ i ∈ I ] P i x

  syntax ⋃ I (λ i → P) = ⋃[ i ∶ I ] P

  ⋂ : ∀ {ℓ i} (I : Set i) → (I → Pred A ℓ) → Pred A _
  ⋂ I P = λ x → (i : I) → P i x

  syntax ⋂ I (λ i → P) = ⋂[ i ∶ I ] P

------------------------------------------------------------------------
-- Unary relation combinators

infixr  _⟨×⟩_
infixr  _⟨⊙⟩_
infixr  _⟨⊎⟩_
infixr  _⟨→⟩_
infixl  _⟨·⟩_
infixr  _⟨∘⟩_
infixr  _//_ _\\_

_⟨×⟩_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b} →
        Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _
(P ⟨×⟩ Q) (x , y) = x ∈ P × y ∈ Q

_⟨⊙⟩_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b} →
        Pred A ℓ₁ → Pred B ℓ₂ → Pred (A × B) _
(P ⟨⊙⟩ Q) (x , y) = x ∈ P ⊎ y ∈ Q

_⟨⊎⟩_ : ∀ {a b ℓ} {A : Set a} {B : Set b} →
        Pred A ℓ → Pred B ℓ → Pred (A ⊎ B) _
P ⟨⊎⟩ Q = [ P , Q ]

_⟨→⟩_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b} →
        Pred A ℓ₁ → Pred B ℓ₂ → Pred (A → B) _
(P ⟨→⟩ Q) f = P ⊆ Q ∘ f

_⟨·⟩_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
        (P : Pred A ℓ₁) (Q : Pred B ℓ₂) →
        (P ⟨×⟩ (P ⟨→⟩ Q)) ⊆ Q ∘ uncurry (flip _$_)
(P ⟨·⟩ Q) (p , f) = f p

-- Converse.

_~ : ∀ {a b ℓ} {A : Set a} {B : Set b} →
     Pred (A × B) ℓ → Pred (B × A) ℓ
P ~ = P ∘ swap

-- Composition.

_⟨∘⟩_ : ∀ {a b c ℓ₁ ℓ₂} {A : Set a} {B : Set b} {C : Set c} →
        Pred (A × B) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × C) _
(P ⟨∘⟩ Q) (x , z) = ∃ λ y → (x , y) ∈ P × (y , z) ∈ Q

-- Post and pre-division.

_//_ : ∀ {a b c ℓ₁ ℓ₂} {A : Set a} {B : Set b} {C : Set c} →
       Pred (A × C) ℓ₁ → Pred (B × C) ℓ₂ → Pred (A × B) _
(P // Q) (x , y) = Q ∘ _,_ y ⊆ P ∘ _,_ x

_\\_ : ∀ {a b c ℓ₁ ℓ₂} {A : Set a} {B : Set b} {C : Set c} →
       Pred (A × C) ℓ₁ → Pred (A × B) ℓ₂ → Pred (B × C) _
P \\ Q = (P ~ // Q ~) ~

------------------------------------------------------------------------
-- Properties of unary relations

Decidable : ∀ {a ℓ} {A : Set a} (P : Pred A ℓ) → Set _
Decidable P = ∀ x → Dec (P x)