------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between lattice-like structures
------------------------------------------------------------------------
 
-- See Data.Nat.Binary.Properties for examples of how this and similar
-- modules can be used to easily translate properties between types.
 
{-# OPTIONS --cubical-compatible --safe #-}
 
open import Algebra
open import Algebra.Lattice
open import Algebra.Lattice.Morphism.Structures
import Algebra.Consequences.Setoid as Consequences
import Algebra.Morphism.MagmaMonomorphism as MagmaMonomorphisms
import Algebra.Lattice.Properties.Lattice as LatticeProperties
open import Data.Product.Base using (_,_; map)
open import Relation.Binary.Bundles using (Setoid)
import Relation.Binary.Morphism.RelMonomorphism as RelMonomorphisms
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
module Algebra.Lattice.Morphism.LatticeMonomorphism
  {a b ℓ₁ ℓ₂} {L₁ : RawLattice a ℓ₁} {L₂ : RawLattice b ℓ₂} {⟦_⟧}
  (isLatticeMonomorphism : IsLatticeMonomorphism L₁ L₂ ⟦_⟧)
  where
 
open IsLatticeMonomorphism isLatticeMonomorphism
open RawLattice L₁ renaming (_≈_ to _≈₁_; _∨_ to _∨_; _∧_ to _∧_)
open RawLattice L₂ renaming (_≈_ to _≈₂_; _∨_ to _⊔_; _∧_ to _⊓_)
 
------------------------------------------------------------------------
-- Re-export all properties of magma monomorphisms
 
open MagmaMonomorphisms ∨-isMagmaMonomorphism public
  using () renaming
  ( cong    to ∨-cong
  ; assoc   to ∨-assoc
  ; comm    to ∨-comm
  ; idem    to ∨-idem
  ; sel     to ∨-sel
  ; cancelˡ to ∨-cancelˡ
  ; cancelʳ to ∨-cancelʳ
  ; cancel  to ∨-cancel
  )
 
open MagmaMonomorphisms ∧-isMagmaMonomorphism public
  using () renaming
  ( cong    to ∧-cong
  ; assoc   to ∧-assoc
  ; comm    to ∧-comm
  ; idem    to ∧-idem
  ; sel     to ∧-sel
  ; cancelˡ to ∧-cancelˡ
  ; cancelʳ to ∧-cancelʳ
  ; cancel  to ∧-cancel
  )
 
------------------------------------------------------------------------
-- Lattice-specific properties
 
module _ (⊔-⊓-isLattice : IsLattice _≈₂_ _⊔_ _⊓_) where
 
  open IsLattice ⊔-⊓-isLattice using (isEquivalence) renaming
    ( ∨-congˡ     to ⊔-congˡ
    ; ∨-congʳ     to ⊔-congʳ
    ; ∧-cong      to ⊓-cong
    ; ∧-congˡ     to ⊓-congˡ
    ; ∨-absorbs-∧ to ⊔-absorbs-⊓
    ; ∧-absorbs-∨ to ⊓-absorbs-⊔
    )
 
  setoid : Setoid b ℓ₂
  setoid = record { isEquivalence = isEquivalence }
 
  open ≈-Reasoning setoid
 
  ∨-absorbs-∧ : _Absorbs_ _≈₁_ _∨_ _∧_
  ∨-absorbs-∧ x y = injective (begin
    ⟦ x ∨ x ∧ y ⟧                        ≈⟨ ∨-homo x (x ∧ y) ⟩
    ⟦ x ⟧ ⊔ ⟦ x ∧ y ⟧                    ≈⟨ ⊔-congˡ (∧-homo x y) ⟩
    ⟦ x ⟧ ⊔ ⟦ x ⟧ ⊓ ⟦ y ⟧                ≈⟨ ⊔-absorbs-⊓ ⟦ x ⟧ ⟦ y ⟧ ⟩
    ⟦ x ⟧                                ∎)
 
  ∧-absorbs-∨ : _Absorbs_ _≈₁_ _∧_ _∨_
  ∧-absorbs-∨ x y = injective (begin
    ⟦ x ∧ (x ∨ y) ⟧                      ≈⟨ ∧-homo x (x ∨ y) ⟩
    ⟦ x ⟧ ⊓ ⟦ x ∨ y ⟧                    ≈⟨ ⊓-congˡ (∨-homo x y) ⟩
    ⟦ x ⟧ ⊓ (⟦ x ⟧ ⊔ ⟦ y ⟧)              ≈⟨ ⊓-absorbs-⊔ ⟦ x ⟧ ⟦ y ⟧ ⟩
    ⟦ x ⟧                                ∎)
 
  absorptive : Absorptive _≈₁_ _∨_ _∧_
  absorptive = ∨-absorbs-∧ , ∧-absorbs-∨
 
  distribʳ : _DistributesOverʳ_ _≈₂_ _⊔_ _⊓_ → _DistributesOverʳ_ _≈₁_ _∨_ _∧_
  distribʳ distribʳ x y z = injective (begin
    ⟦ y ∧ z ∨ x ⟧                        ≈⟨ ∨-homo (y ∧ z) x ⟩
    ⟦ y ∧ z ⟧ ⊔ ⟦ x ⟧                    ≈⟨ ⊔-congʳ (∧-homo y z) ⟩
    ⟦ y ⟧ ⊓ ⟦ z ⟧ ⊔ ⟦ x ⟧                ≈⟨ distribʳ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
    (⟦ y ⟧ ⊔ ⟦ x ⟧) ⊓ (⟦ z ⟧ ⊔ ⟦ x ⟧)    ≈⟨ ⊓-cong (∨-homo y x) (∨-homo z x) ⟨
    ⟦ y ∨ x ⟧ ⊓ ⟦ z ∨ x ⟧                ≈⟨ ∧-homo (y ∨ x) (z ∨ x) ⟨
    ⟦ (y ∨ x) ∧ (z ∨ x) ⟧                ∎)
 
isLattice : IsLattice _≈₂_ _⊔_ _⊓_ → IsLattice _≈₁_ _∨_ _∧_
isLattice isLattice = record
  { isEquivalence = RelMonomorphisms.isEquivalence isRelMonomorphism L.isEquivalence
  ; ∨-comm        = ∨-comm  LP.∨-isMagma L.∨-comm
  ; ∨-assoc       = ∨-assoc LP.∨-isMagma L.∨-assoc
  ; ∨-cong        = ∨-cong  LP.∨-isMagma
  ; ∧-comm        = ∧-comm  LP.∧-isMagma L.∧-comm
  ; ∧-assoc       = ∧-assoc LP.∧-isMagma L.∧-assoc
  ; ∧-cong        = ∧-cong  LP.∧-isMagma
  ; absorptive    = absorptive isLattice
  } where
    module L  = IsLattice isLattice
    module LP = LatticeProperties (record { isLattice = isLattice })
 
isDistributiveLattice : IsDistributiveLattice _≈₂_ _⊔_ _⊓_ →
                        IsDistributiveLattice _≈₁_ _∨_ _∧_
isDistributiveLattice isDL = isDistributiveLatticeʳʲᵐ (record
  { isLattice     = isLattice L.isLattice
  ; ∨-distribʳ-∧  = distribʳ  L.isLattice L.∨-distribʳ-∧
  }) where module L = IsDistributiveLattice isDL