------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of binary relations
------------------------------------------------------------------------

-- The contents of this module should be accessed via `Relation.Binary`.

{-# OPTIONS --without-K --safe #-}

module Relation.Binary.Core where

open import Function.Base using (_on_)
open import Level using (Level; _⊔_; suc)

private
variable
a b c  ℓ₁ ℓ₂ ℓ₃ : Level
A : Set a
B : Set b
C : Set c

------------------------------------------------------------------------
-- Definitions
------------------------------------------------------------------------

-- Heterogeneous binary relations

REL : Set a  Set b  ( : Level)  Set (a  b  suc )
REL A B  = A  B  Set

-- Homogeneous binary relations

Rel : Set a  ( : Level)  Set (a  suc )
Rel A  = REL A A

------------------------------------------------------------------------
-- Relationships between relations
------------------------------------------------------------------------

infixr 4 _⇒_ _=[_]⇒_

-- Implication/containment - could also be written _⊆_.

_⇒_ : REL A B ℓ₁  REL A B ℓ₂  Set _
P  Q =  {x y}  P x y  Q x y

-- Generalised implication - if P ≡ Q it can be read as "f preserves P".

_=[_]⇒_ : Rel A ℓ₁  (A  B)  Rel B ℓ₂  Set _
P =[ f ]⇒ Q = P  (Q on f)

-- A synonym for _=[_]⇒_.

_Preserves_⟶_ : (A  B)  Rel A ℓ₁  Rel B ℓ₂  Set _
f Preserves P  Q = P =[ f ]⇒ Q

-- A binary variant of _Preserves_⟶_.

_Preserves₂_⟶_⟶_ : (A  B  C)  Rel A ℓ₁  Rel B ℓ₂  Rel C ℓ₃  Set _
_∙_ Preserves₂ P  Q  R =  {x y u v}  P x y  Q u v  R (x  u) (y  v)