```------------------------------------------------------------------------
-- The Agda standard library
--
-- Some derivable properties
------------------------------------------------------------------------

open import Algebra

module Algebra.Props.DistributiveLattice
{dl₁ dl₂} (DL : DistributiveLattice dl₁ dl₂)
where

open DistributiveLattice DL
import Algebra.Props.Lattice
private
open module L = Algebra.Props.Lattice lattice public
hiding (replace-equality)
open import Algebra.Structures
import Algebra.FunctionProperties as P; open P _≈_
open import Relation.Binary
import Relation.Binary.EqReasoning as EqR; open EqR setoid
open import Function
open import Function.Equality using (_⟨\$⟩_)
open import Function.Equivalence using (_⇔_; module Equivalence)
open import Data.Product

∨-∧-distrib : _∨_ DistributesOver _∧_
∨-∧-distrib = ∨-∧-distribˡ , ∨-∧-distribʳ
where
∨-∧-distribˡ : _∨_ DistributesOverˡ _∧_
∨-∧-distribˡ x y z = begin
x ∨ y ∧ z          ≈⟨ ∨-comm _ _ ⟩
y ∧ z ∨ x          ≈⟨ ∨-∧-distribʳ _ _ _ ⟩
(y ∨ x) ∧ (z ∨ x)  ≈⟨ ∨-comm _ _ ⟨ ∧-cong ⟩ ∨-comm _ _ ⟩
(x ∨ y) ∧ (x ∨ z)  ∎

∧-∨-distrib : _∧_ DistributesOver _∨_
∧-∨-distrib = ∧-∨-distribˡ , ∧-∨-distribʳ
where
∧-∨-distribˡ : _∧_ DistributesOverˡ _∨_
∧-∨-distribˡ x y z = begin
x ∧ (y ∨ z)                ≈⟨ sym (proj₂ absorptive _ _) ⟨ ∧-cong ⟩ refl ⟩
(x ∧ (x ∨ y)) ∧ (y ∨ z)    ≈⟨ (refl ⟨ ∧-cong ⟩ ∨-comm _ _) ⟨ ∧-cong ⟩ refl ⟩
(x ∧ (y ∨ x)) ∧ (y ∨ z)    ≈⟨ ∧-assoc _ _ _ ⟩
x ∧ ((y ∨ x) ∧ (y ∨ z))    ≈⟨ refl ⟨ ∧-cong ⟩ sym (proj₁ ∨-∧-distrib _ _ _) ⟩
x ∧ (y ∨ x ∧ z)            ≈⟨ sym (proj₁ absorptive _ _) ⟨ ∧-cong ⟩ refl ⟩
(x ∨ x ∧ z) ∧ (y ∨ x ∧ z)  ≈⟨ sym \$ proj₂ ∨-∧-distrib _ _ _ ⟩
x ∧ y ∨ x ∧ z              ∎

∧-∨-distribʳ : _∧_ DistributesOverʳ _∨_
∧-∨-distribʳ x y z = begin
(y ∨ z) ∧ x    ≈⟨ ∧-comm _ _ ⟩
x ∧ (y ∨ z)    ≈⟨ ∧-∨-distribˡ _ _ _ ⟩
x ∧ y ∨ x ∧ z  ≈⟨ ∧-comm _ _ ⟨ ∨-cong ⟩ ∧-comm _ _ ⟩
y ∧ x ∨ z ∧ x  ∎

-- The dual construction is also a distributive lattice.

∧-∨-isDistributiveLattice : IsDistributiveLattice _≈_ _∧_ _∨_
∧-∨-isDistributiveLattice = record
{ isLattice    = ∧-∨-isLattice
; ∨-∧-distribʳ = proj₂ ∧-∨-distrib
}

∧-∨-distributiveLattice : DistributiveLattice _ _
∧-∨-distributiveLattice = record
{ _∧_                   = _∨_
; _∨_                   = _∧_
; isDistributiveLattice = ∧-∨-isDistributiveLattice
}

-- One can replace the underlying equality with an equivalent one.

replace-equality :
{_≈′_ : Rel Carrier dl₂} →
(∀ {x y} → x ≈ y ⇔ x ≈′ y) → DistributiveLattice _ _
replace-equality {_≈′_} ≈⇔≈′ = record
{ _≈_                   = _≈′_
; _∧_                   = _∧_
; _∨_                   = _∨_
; isDistributiveLattice = record
{ isLattice    = Lattice.isLattice (L.replace-equality ≈⇔≈′)
; ∨-∧-distribʳ = λ x y z → to ⟨\$⟩ ∨-∧-distribʳ x y z
}
} where open module E {x y} = Equivalence (≈⇔≈′ {x} {y})
```