------------------------------------------------------------------------
-- 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.Lattice
  using (RawLattice; IsLattice; IsDistributiveLattice; isDistributiveLatticeʳʲᵐ)
open import Algebra.Lattice.Morphism.Structures using (IsLatticeMonomorphism)

module Algebra.Lattice.Morphism.LatticeMonomorphism
  {a b ℓ₁ ℓ₂} {L₁ : RawLattice a ℓ₁} {L₂ : RawLattice b ℓ₂} {⟦_⟧}
  (isLatticeMonomorphism : IsLatticeMonomorphism L₁ L₂ ⟦_⟧)
  where

open import Algebra using (Absorptive; _Absorbs_; _DistributesOverʳ_)
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

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
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2654}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{2656}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2658}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{2660}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2627}{\htmlId{2662}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$                        ≈⟨ ∨-homo x (x ∧ y) ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2717}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2725}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{2727}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2627}{\htmlId{2729}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$                    ≈⟨ ⊔-congˡ (∧-homo x y) ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2784}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2792}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊓ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2627}{\htmlId{2800}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$                ≈⟨ ⊔-absorbs-⊓ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2836}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2627}{\htmlId{2842}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2625}{\htmlId{2854}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                                ∎)

  ∧-absorbs-∨ : _Absorbs_ _≈₁_ _∧_ _∨_
  ∧-absorbs-∨ x y = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{2975}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{2977}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\htmlId{2979}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{2980}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{2982}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2948}{\htmlId{2984}{\htmlClass{Bound}{\text{y}}}}\,\,\htmlId{2985}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$                      ≈⟨ ∧-homo x (x ∨ y) ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3038}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊓ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3046}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3048}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2948}{\htmlId{3050}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$                    ≈⟨ ⊓-congˡ (∨-homo x y) ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3105}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊓ ($\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3114}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2948}{\htmlId{3122}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$)              ≈⟨ ⊓-absorbs-⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3157}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2948}{\htmlId{3163}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#2946}{\htmlId{3175}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                                ∎)

  absorptive : Absorptive _≈₁_ _∨_ _∧_
  absorptive = ∨-absorbs-∧ , ∧-absorbs-∨

  distribʳ : _DistributesOverʳ_ _≈₂_ _⊔_ _⊓_ → _DistributesOverʳ_ _≈₁_ _∨_ _∧_
  distribʳ distribʳ x y z = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3425}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{3427}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3429}{\htmlClass{Bound}{\text{z}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3431}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3433}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                        ≈⟨ ∨-homo (y ∧ z) x ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3488}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{3490}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3492}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3500}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                    ≈⟨ ⊔-congʳ (∧-homo y z) ⟩
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3555}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊓ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3563}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3571}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                ≈⟨ distribʳ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3604}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3610}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3616}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⟩
    ($\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3629}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3637}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$) ⊓ ($\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3647}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⊔ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3655}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$)    ≈⟨ ⊓-cong (∨-homo y x) (∨-homo z x) ⟨
    $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3707}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3709}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3711}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊓ $\begin{pmatrix} \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3719}{\htmlClass{Bound}{\text{z}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3721}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3723}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$                ≈⟨ ∧-homo (y ∨ x) (z ∨ x) ⟨
    $\begin{pmatrix} \,\htmlId{3776}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3396}{\htmlId{3777}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3779}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3781}{\htmlClass{Bound}{\text{x}}}}\,\,\htmlId{3782}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1336}{\htmlId{3784}{\htmlClass{Function Operator}{\text{∧}}}}\, \,\htmlId{3786}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3398}{\htmlId{3787}{\htmlClass{Bound}{\text{z}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#1324}{\htmlId{3789}{\htmlClass{Function Operator}{\text{∨}}}}\, \,\href{Algebra.Lattice.Morphism.LatticeMonomorphism.html#3394}{\htmlId{3791}{\htmlClass{Bound}{\text{x}}}}\,\,\htmlId{3792}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$                ∎)

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