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

module Relation.Unary where

open import Data.Empty
open import Function
open import Data.Unit.Base using ()
open import Data.Product
open import Data.Sum
open import Level
open import Relation.Nullary
open import Relation.Binary.Core using (_≡_)

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

Pred :  {a}  Set a  ( : Level)  Set (a  suc )
Pred A  = A  Set 

------------------------------------------------------------------------
-- 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 4 _∈_ _∉_

  _∈_ :  {}  A  Pred A   Set _
  x  P = P x

  _∉_ :  {}  A  Pred A   Set _
  x  P = ¬ x  P

  -- The empty set.

   : Pred A zero
   = λ _  

  -- The property of being empty.

  Empty :  {}  Pred A   Set _
  Empty P =  x  x  P

  ∅-Empty : Empty 
  ∅-Empty x ()

  -- The singleton set.
  {_} : A  Pred A a
   x  = _≡_ x

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

  U : Pred A zero
  U = λ _  

  -- The property of being universal.

  Universal :  {}  Pred A   Set _
  Universal P =  x  x  P

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

  -- Set complement.

   :  {}  Pred A   Pred A 
   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 4 _⊆_ _⊇_ _⊆′_ _⊇′_

  _⊆_ :  {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

  ∅-⊆ :  {}  (P : Pred A )    P
  ∅-⊆ P ()

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

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

  infix 4 _≬_

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

  -- Set union.

  infixr 6 _∪_

  _∪_ :  {ℓ₁ ℓ₂}  Pred A ℓ₁  Pred A ℓ₂  Pred A _
  P  Q = λ x  x  P  x  Q

  -- Set intersection.

  infixr 7 _∩_

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

  -- Implication.

  infixl 8 _⇒_

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

  -- Infinitary union and intersection.

  infix 9 ⋃ ⋂

   :  { i} (I : Set i)  (I  Pred A )  Pred A _
   I P = λ x  Σ[ i  I ] P i x

  syntax ⋃ I  i  P) = ⋃[ i ∶ I ] P

   :  { i} (I : Set i)  (I  Pred A )  Pred A _
   I P = λ x  (i : I)  P i x

  syntax ⋂ I  i  P) = ⋂[ i ∶ I ] P

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

infixr  2 _⟨×⟩_
infixr  2 _⟨⊙⟩_
infixr  1 _⟨⊎⟩_
infixr  0 _⟨→⟩_
infixl  9 _⟨·⟩_
infix  10 _~
infixr  9 _⟨∘⟩_
infixr  2 _//_ _\\_

_⟨×⟩_ :  {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 } {A : Set a} {B : Set b} 
        Pred A   Pred B   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 } {A : Set a} {B : Set b} 
     Pred (A × B)   Pred (B × A) 
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 } {A : Set a} (P : Pred A )  Set _
Decidable P =  x  Dec (P x)