------------------------------------------------------------------------
-- The Agda standard library
--
-- Intersection of two binary relations
------------------------------------------------------------------------

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

module Relation.Binary.Construct.Intersection where

open import Data.Product
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
open import Function.Base using (_∘_)
open import Level using (Level; _⊔_)
open import Relation.Binary
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Product using (_×-dec_)

private
  variable
    a b ℓ₁ ℓ₂ ℓ₃ : Level
    A B : Set a
     L R : Rel A ℓ₁

------------------------------------------------------------------------
-- Definition

_∩_ : REL A B ℓ₁  REL A B ℓ₂  REL A B (ℓ₁  ℓ₂)
L  R = λ i j  L i j × R i j

------------------------------------------------------------------------
-- Properties

module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) where

  reflexive : Reflexive L  Reflexive R  Reflexive (L  R)
  reflexive L-refl R-refl = L-refl , R-refl

  symmetric : Symmetric L  Symmetric R  Symmetric (L  R)
  symmetric L-sym R-sym = map L-sym R-sym

  transitive : Transitive L  Transitive R  Transitive (L  R)
  transitive L-trans R-trans = zip L-trans R-trans

  respects :  {p} (P : A  Set p) 
             P Respects L  P Respects R  P Respects (L  R)
  respects P resp (Lxy , Rxy) = [  x  x Lxy) ,  x  x Rxy) ] resp

  min :  {}  Minimum L   Minimum R   Minimum (L  R) 
  min L-min R-min = < L-min , R-min >

  max :  {}  Maximum L   Maximum R   Maximum (L  R) 
  max L-max R-max = < L-max , R-max >

module _ ( : REL A B ℓ₁) {L : REL A B ℓ₂} {R : REL A B ℓ₃} where

  implies : (  L)  (  R)    (L  R)
  implies ≈⇒L ≈⇒R = < ≈⇒L , ≈⇒R >

module _ ( : REL A B ℓ₁) (L : REL A B ℓ₂) (R : REL A B ℓ₃) where

  irreflexive : Irreflexive  L  Irreflexive  R  Irreflexive  (L  R)
  irreflexive irrefl x≈y (Lxy , Rxy) = [  x  x x≈y Lxy) ,  x  x x≈y Rxy) ] irrefl

module _ ( : Rel A ℓ₁) (L : Rel A ℓ₂) (R : Rel A ℓ₃) where

  respectsˡ : L Respectsˡ   R Respectsˡ   (L  R) Respectsˡ 
  respectsˡ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)

  respectsʳ : L Respectsʳ   R Respectsʳ   (L  R) Respectsʳ 
  respectsʳ L-resp R-resp x≈y = map (L-resp x≈y) (R-resp x≈y)

  respects₂ : L Respects₂   R Respects₂   (L  R) Respects₂ 
  respects₂ ( , ) ( , ) = respectsʳ   , respectsˡ  

  antisymmetric : Antisymmetric  L  Antisymmetric  R  Antisymmetric  (L  R)
  antisymmetric (inj₁ L-antisym) (Lxy , _) (Lyx , _) = L-antisym Lxy Lyx
  antisymmetric (inj₂ R-antisym) (_ , Rxy) (_ , Ryx) = R-antisym Rxy Ryx

module _ {L : REL A B ℓ₁} {R : REL A B ℓ₂} where

  decidable : Decidable L  Decidable R  Decidable (L  R)
  decidable L? R? x y = L? x y ×-dec R? x y

------------------------------------------------------------------------
-- Structures

isEquivalence : IsEquivalence L  IsEquivalence R  IsEquivalence (L  R)
isEquivalence {L = L} {R = R} eqₗ eqᵣ = record
  { refl  = reflexive  L R L.refl  R.refl
  ; sym   = symmetric  L R L.sym   R.sym
  ; trans = transitive L R L.trans R.trans
  } where module L = IsEquivalence eqₗ; module R = IsEquivalence eqᵣ

isDecEquivalence : IsDecEquivalence L  IsDecEquivalence R  IsDecEquivalence (L  R)
isDecEquivalence eqₗ eqᵣ = record
  { isEquivalence = isEquivalence L.isEquivalence R.isEquivalence
  ; _≟_           = decidable L._≟_ R._≟_
  } where module L = IsDecEquivalence eqₗ; module R = IsDecEquivalence eqᵣ

isPreorder : IsPreorder  L  IsPreorder  R  IsPreorder  (L  R)
isPreorder { = } {L = L} {R = R} Oₗ Oᵣ = record
  { isEquivalence = Oₗ.isEquivalence
  ; reflexive     = implies  Oₗ.reflexive Oᵣ.reflexive
  ; trans         = transitive L R Oₗ.trans Oᵣ.trans
  }
  where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPreorder Oᵣ

isPartialOrderˡ : IsPartialOrder  L  IsPreorder  R  IsPartialOrder  (L  R)
isPartialOrderˡ { = } {L = L} {R = R} Oₗ Oᵣ = record
  { isPreorder = isPreorder Oₗ.isPreorder Oᵣ
  ; antisym    = antisymmetric  L R (inj₁ Oₗ.antisym)
  } where module Oₗ = IsPartialOrder Oₗ; module Oᵣ = IsPreorder Oᵣ

isPartialOrderʳ : IsPreorder  L  IsPartialOrder  R  IsPartialOrder  (L  R)
isPartialOrderʳ { = } {L = L} {R = R} Oₗ Oᵣ = record
  { isPreorder = isPreorder Oₗ Oᵣ.isPreorder
  ; antisym    = antisymmetric  L R (inj₂ Oᵣ.antisym)
  } where module Oₗ = IsPreorder Oₗ; module Oᵣ = IsPartialOrder Oᵣ

isStrictPartialOrderˡ : IsStrictPartialOrder  L 
                        Transitive R  R Respects₂  
                        IsStrictPartialOrder  (L  R)
isStrictPartialOrderˡ { = } {L = L} {R = R}  Oₗ transᵣ respᵣ = record
  { isEquivalence = Oₗ.isEquivalence
  ; irrefl        = irreflexive  L R (inj₁ Oₗ.irrefl)
  ; trans         = transitive L R Oₗ.trans transᵣ
  ; <-resp-≈      = respects₂  L R Oₗ.<-resp-≈ respᵣ
  } where module Oₗ = IsStrictPartialOrder Oₗ

isStrictPartialOrderʳ : Transitive L  L Respects₂  
                        IsStrictPartialOrder  R 
                        IsStrictPartialOrder  (L  R)
isStrictPartialOrderʳ {L = L} { = } {R = R} transₗ respₗ Oᵣ = record
  { isEquivalence = Oᵣ.isEquivalence
  ; irrefl        = irreflexive  L R (inj₂ Oᵣ.irrefl)
  ; trans         = transitive L R transₗ Oᵣ.trans
  ; <-resp-≈      = respects₂  L R respₗ Oᵣ.<-resp-≈
  } where module Oᵣ = IsStrictPartialOrder Oᵣ