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

open import Algebra

module Algebra.Properties.BooleanAlgebra
         {b₁ b₂} (B : BooleanAlgebra b₁ b₂)
         where

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

------------------------------------------------------------------------
-- Some simple generalisations

∨-complement : Inverse  ¬_ _∨_
∨-complement = ∨-complementˡ , ∨-complementʳ
  where
  ∨-complementˡ : LeftInverse  ¬_ _∨_
  ∨-complementˡ x = begin
    ¬ x  x    ≈⟨ ∨-comm _ _ 
    x    ¬ x  ≈⟨ ∨-complementʳ _ 
              

∧-complement : Inverse  ¬_ _∧_
∧-complement = ∧-complementˡ , ∧-complementʳ
  where
  ∧-complementˡ : LeftInverse  ¬_ _∧_
  ∧-complementˡ x = begin
    ¬ x  x    ≈⟨ ∧-comm _ _ 
    x    ¬ x  ≈⟨ ∧-complementʳ _ 
              

------------------------------------------------------------------------
-- The dual construction is also a boolean algebra

∧-∨-isBooleanAlgebra : IsBooleanAlgebra _≈_ _∧_ _∨_ ¬_  
∧-∨-isBooleanAlgebra = record
  { isDistributiveLattice = ∧-∨-isDistributiveLattice
  ; ∨-complementʳ         = proj₂ ∧-complement
  ; ∧-complementʳ         = proj₂ ∨-complement
  ; ¬-cong                = ¬-cong
  }

∧-∨-booleanAlgebra : BooleanAlgebra _ _
∧-∨-booleanAlgebra = record
  { _∧_              = _∨_
  ; _∨_              = _∧_
  ;                 = 
  ;                 = 
  ; isBooleanAlgebra = ∧-∨-isBooleanAlgebra
  }

------------------------------------------------------------------------
-- (∨, ∧, ⊥, ⊤) is a commutative semiring

private

  ∧-identity : Identity  _∧_
  ∧-identity =  _  ∧-comm _ _  trans  x∧⊤=x _) , x∧⊤=x
    where
    x∧⊤=x :  x  x    x
    x∧⊤=x x = begin
      x            ≈⟨ refl  ∧-cong  sym (proj₂ ∨-complement _) 
      x  (x  ¬ x)  ≈⟨ proj₂ absorptive _ _ 
      x              

  ∨-identity : Identity  _∨_
  ∨-identity =  _  ∨-comm _ _  trans  x∨⊥=x _) , x∨⊥=x
    where
    x∨⊥=x :  x  x    x
    x∨⊥=x x = begin
      x            ≈⟨ refl  ∨-cong  sym (proj₂ ∧-complement _) 
      x  x  ¬ x    ≈⟨ proj₁ absorptive _ _ 
      x              

  ∧-zero : Zero  _∧_
  ∧-zero =  _  ∧-comm _ _  trans  x∧⊥=⊥ _) , x∧⊥=⊥
    where
    x∧⊥=⊥ :  x  x    
    x∧⊥=⊥ x = begin
      x            ≈⟨ refl  ∧-cong  sym (proj₂ ∧-complement _) 
      x   x   ¬ x  ≈⟨ sym $ ∧-assoc _ _ _ 
      (x  x)  ¬ x  ≈⟨ ∧-idempotent _  ∧-cong  refl 
      x        ¬ x  ≈⟨ proj₂ ∧-complement _ 
                    

∨-∧-isCommutativeSemiring : IsCommutativeSemiring _≈_ _∨_ _∧_  
∨-∧-isCommutativeSemiring = record
  { +-isCommutativeMonoid = record
    { isSemigroup = record
      { isEquivalence = isEquivalence
      ; assoc         = ∨-assoc
      ; ∙-cong        = ∨-cong
      }
    ; identityˡ = proj₁ ∨-identity
    ; comm      = ∨-comm
    }
  ; *-isCommutativeMonoid = record
    { isSemigroup = record
      { isEquivalence = isEquivalence
      ; assoc         = ∧-assoc
      ; ∙-cong        = ∧-cong
      }
    ; identityˡ = proj₁ ∧-identity
    ; comm      = ∧-comm
    }
  ; distribʳ = proj₂ ∧-∨-distrib
  ; zeroˡ    = proj₁ ∧-zero
  }

∨-∧-commutativeSemiring : CommutativeSemiring _ _
∨-∧-commutativeSemiring = record
  { _+_                   = _∨_
  ; _*_                   = _∧_
  ; 0#                    = 
  ; 1#                    = 
  ; isCommutativeSemiring = ∨-∧-isCommutativeSemiring
  }

------------------------------------------------------------------------
-- (∧, ∨, ⊤, ⊥) is a commutative semiring

private

  ∨-zero : Zero  _∨_
  ∨-zero =  _  ∨-comm _ _  trans  x∨⊤=⊤ _) , x∨⊤=⊤
    where
    x∨⊤=⊤ :  x  x    
    x∨⊤=⊤ x = begin
      x            ≈⟨ refl  ∨-cong  sym (proj₂ ∨-complement _) 
      x   x   ¬ x  ≈⟨ sym $ ∨-assoc _ _ _ 
      (x  x)  ¬ x  ≈⟨ ∨-idempotent _  ∨-cong  refl 
      x        ¬ x  ≈⟨ proj₂ ∨-complement _ 
                    

∧-∨-isCommutativeSemiring : IsCommutativeSemiring _≈_ _∧_ _∨_  
∧-∨-isCommutativeSemiring = record
  { +-isCommutativeMonoid = record
    { isSemigroup = record
      { isEquivalence = isEquivalence
      ; assoc         = ∧-assoc
      ; ∙-cong        = ∧-cong
      }
    ; identityˡ = proj₁ ∧-identity
    ; comm      = ∧-comm
    }
  ; *-isCommutativeMonoid = record
    { isSemigroup = record
      { isEquivalence = isEquivalence
      ; assoc         = ∨-assoc
      ; ∙-cong        = ∨-cong
      }
    ; identityˡ = proj₁ ∨-identity
    ; comm      = ∨-comm
    }
  ; distribʳ = proj₂ ∨-∧-distrib
  ; zeroˡ    = proj₁ ∨-zero
  }

∧-∨-commutativeSemiring : CommutativeSemiring _ _
∧-∨-commutativeSemiring =
  record { isCommutativeSemiring = ∧-∨-isCommutativeSemiring }

------------------------------------------------------------------------
-- Some other properties

-- I took the statement of this lemma (called Uniqueness of
-- Complements) from some course notes, "Boolean Algebra", written
-- by Gert Smolka.

private
  lemma :  x y  x  y    x  y    ¬ x  y
  lemma x y x∧y=⊥ x∨y=⊤ = begin
    ¬ x                ≈⟨ sym $ proj₂ ∧-identity _ 
    ¬ x              ≈⟨ refl  ∧-cong  sym x∨y=⊤ 
    ¬ x  (x  y)      ≈⟨ proj₁ ∧-∨-distrib _ _ _ 
    ¬ x  x  ¬ x  y  ≈⟨ proj₁ ∧-complement _  ∨-cong  refl 
      ¬ x  y        ≈⟨ sym x∧y=⊥  ∨-cong  refl 
    x  y  ¬ x  y    ≈⟨ sym $ proj₂ ∧-∨-distrib _ _ _ 
    (x  ¬ x)  y      ≈⟨ proj₂ ∨-complement _  ∧-cong  refl 
      y              ≈⟨ proj₁ ∧-identity _ 
    y                  

¬⊥=⊤ : ¬   
¬⊥=⊤ = lemma   (proj₂ ∧-identity _) (proj₂ ∨-zero _)

¬⊤=⊥ : ¬   
¬⊤=⊥ = lemma   (proj₂ ∧-zero _) (proj₂ ∨-identity _)

¬-involutive : Involutive ¬_
¬-involutive x = lemma (¬ x) x (proj₁ ∧-complement _) (proj₁ ∨-complement _)

deMorgan₁ :  x y  ¬ (x  y)  ¬ x  ¬ y
deMorgan₁ x y = lemma (x  y) (¬ x  ¬ y) lem₁ lem₂
  where
  lem₁ = begin
    (x  y)  (¬ x  ¬ y)          ≈⟨ proj₁ ∧-∨-distrib _ _ _ 
    (x  y)  ¬ x  (x  y)  ¬ y  ≈⟨ (∧-comm _ _  ∧-cong  refl)  ∨-cong  refl 
    (y  x)  ¬ x  (x  y)  ¬ y  ≈⟨ ∧-assoc _ _ _  ∨-cong  ∧-assoc _ _ _ 
    y  (x  ¬ x)  x  (y  ¬ y)  ≈⟨ (refl  ∧-cong  proj₂ ∧-complement _)  ∨-cong 
                                      (refl  ∧-cong  proj₂ ∧-complement _) 
    (y  )  (x  )              ≈⟨ proj₂ ∧-zero _  ∨-cong  proj₂ ∧-zero _ 
                                ≈⟨ proj₂ ∨-identity _ 
                                  

  lem₃ = begin
    (x  y)  ¬ x          ≈⟨ proj₂ ∨-∧-distrib _ _ _ 
    (x  ¬ x)  (y  ¬ x)  ≈⟨ proj₂ ∨-complement _  ∧-cong  refl 
      (y  ¬ x)          ≈⟨ proj₁ ∧-identity _ 
    y  ¬ x                ≈⟨ ∨-comm _ _ 
    ¬ x  y                

  lem₂ = begin
    (x  y)  (¬ x  ¬ y)  ≈⟨ sym $ ∨-assoc _ _ _ 
    ((x  y)  ¬ x)  ¬ y  ≈⟨ lem₃  ∨-cong  refl 
    (¬ x  y)  ¬ y        ≈⟨ ∨-assoc _ _ _ 
    ¬ x  (y  ¬ y)        ≈⟨ refl  ∨-cong  proj₂ ∨-complement _ 
    ¬ x                  ≈⟨ proj₂ ∨-zero _ 
                          

deMorgan₂ :  x y  ¬ (x  y)  ¬ x  ¬ y
deMorgan₂ x y = begin
  ¬ (x  y)          ≈⟨ ¬-cong $ sym (¬-involutive _)  ∨-cong 
                                 sym (¬-involutive _) 
  ¬ (¬ ¬ x  ¬ ¬ y)  ≈⟨ ¬-cong $ sym $ deMorgan₁ _ _ 
  ¬ ¬ (¬ x  ¬ y)    ≈⟨ ¬-involutive _ 
  ¬ x  ¬ y          

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

replace-equality :
  {_≈′_ : Rel Carrier b₂} 
  (∀ {x y}  x  y  x ≈′ y)  BooleanAlgebra _ _
replace-equality {_≈′_} ≈⇔≈′ = record
  { _≈_              = _≈′_
  ; _∨_              = _∨_
  ; _∧_              = _∧_
  ; ¬_               = ¬_
  ;                 = 
  ;                 = 
  ; isBooleanAlgebra =  record
    { isDistributiveLattice = DistributiveLattice.isDistributiveLattice
                                (DL.replace-equality ≈⇔≈′)
    ; ∨-complementʳ         = λ x  to ⟨$⟩ ∨-complementʳ x
    ; ∧-complementʳ         = λ x  to ⟨$⟩ ∧-complementʳ x
    ; ¬-cong                = λ i≈j  to ⟨$⟩ ¬-cong (from ⟨$⟩ i≈j)
    }
  } where open module E {x y} = Equivalence (≈⇔≈′ {x} {y})

------------------------------------------------------------------------
-- (⊕, ∧, id, ⊥, ⊤) is a commutative ring

-- This construction is parameterised over the definition of xor.

module XorRing
  (xor : Op₂ Carrier)
  (⊕-def :  x y  xor x y  (x  y)  ¬ (x  y))
  where

  private
    infixl 6 _⊕_

    _⊕_ : Op₂ Carrier
    _⊕_ = xor

  private
    helper :  {x y u v}  x  y  u  v  x  ¬ u  y  ¬ v
    helper x≈y u≈v = x≈y  ∧-cong  ¬-cong u≈v

  ⊕-¬-distribˡ :  x y  ¬ (x  y)  ¬ x  y
  ⊕-¬-distribˡ x y = begin
    ¬ (x  y)                              ≈⟨ ¬-cong $ ⊕-def _ _ 
    ¬ ((x  y)  (¬ (x  y)))              ≈⟨ ¬-cong (proj₂ ∧-∨-distrib _ _ _) 
    ¬ ((x  ¬ (x  y))  (y  ¬ (x  y)))  ≈⟨ ¬-cong $
                                                refl  ∨-cong 
                                                  (refl  ∧-cong 
                                                     ¬-cong (∧-comm _ _)) 
    ¬ ((x  ¬ (x  y))  (y  ¬ (y  x)))  ≈⟨ ¬-cong $ lem _ _  ∨-cong  lem _ _ 
    ¬ ((x  ¬ y)  (y  ¬ x))              ≈⟨ deMorgan₂ _ _ 
    ¬ (x  ¬ y)  ¬ (y  ¬ x)              ≈⟨ deMorgan₁ _ _  ∧-cong  refl 
    (¬ x  (¬ ¬ y))  ¬ (y  ¬ x)          ≈⟨ helper (refl  ∨-cong  ¬-involutive _)
                                                     (∧-comm _ _) 
    (¬ x  y)  ¬ (¬ x  y)                ≈⟨ sym $ ⊕-def _ _ 
    ¬ x  y                                
    where
    lem :  x y  x  ¬ (x  y)  x  ¬ y
    lem x y = begin
      x  ¬ (x  y)          ≈⟨ refl  ∧-cong  deMorgan₁ _ _ 
      x  (¬ x  ¬ y)        ≈⟨ proj₁ ∧-∨-distrib _ _ _ 
      (x  ¬ x)  (x  ¬ y)  ≈⟨ proj₂ ∧-complement _  ∨-cong  refl 
        (x  ¬ y)          ≈⟨ proj₁ ∨-identity _ 
      x  ¬ y                

  private
    ⊕-comm : Commutative _⊕_
    ⊕-comm x y = begin
      x  y                ≈⟨ ⊕-def _ _ 
      (x  y)  ¬ (x  y)  ≈⟨ helper (∨-comm _ _) (∧-comm _ _) 
      (y  x)  ¬ (y  x)  ≈⟨ sym $ ⊕-def _ _ 
      y  x                

  ⊕-¬-distribʳ :  x y  ¬ (x  y)  x  ¬ y
  ⊕-¬-distribʳ x y = begin
    ¬ (x  y)  ≈⟨ ¬-cong $ ⊕-comm _ _ 
    ¬ (y  x)  ≈⟨ ⊕-¬-distribˡ _ _ 
    ¬ y  x    ≈⟨ ⊕-comm _ _ 
    x  ¬ y    

  ⊕-annihilates-¬ :  x y  x  y  ¬ x  ¬ y
  ⊕-annihilates-¬ x y = begin
    x  y        ≈⟨ sym $ ¬-involutive _ 
    ¬ ¬ (x  y)  ≈⟨ ¬-cong $ ⊕-¬-distribˡ _ _ 
    ¬ (¬ x  y)  ≈⟨ ⊕-¬-distribʳ _ _ 
    ¬ x  ¬ y    

  private
    ⊕-cong : _⊕_ Preserves₂ _≈_  _≈_  _≈_
    ⊕-cong {x} {y} {u} {v} x≈y u≈v = begin
      x  u                ≈⟨ ⊕-def _ _ 
      (x  u)  ¬ (x  u)  ≈⟨ helper (x≈y  ∨-cong  u≈v)
                                     (x≈y  ∧-cong  u≈v) 
      (y  v)  ¬ (y  v)  ≈⟨ sym $ ⊕-def _ _ 
      y  v                

    ⊕-identity : Identity  _⊕_
    ⊕-identity = ⊥⊕x=x ,  _  ⊕-comm _ _  trans  ⊥⊕x=x _)
      where
      ⊥⊕x=x :  x    x  x
      ⊥⊕x=x x = begin
          x                ≈⟨ ⊕-def _ _ 
        (  x)  ¬ (  x)  ≈⟨ helper (proj₁ ∨-identity _)
                                       (proj₁ ∧-zero _) 
        x  ¬               ≈⟨ refl  ∧-cong  ¬⊥=⊤ 
        x                  ≈⟨ proj₂ ∧-identity _ 
        x                    

    ⊕-inverse : Inverse  id _⊕_
    ⊕-inverse = x⊕x=⊥ ,  _  ⊕-comm _ _  trans  x⊕x=⊥ _)
      where
      x⊕x=⊥ :  x  x  x  
      x⊕x=⊥ x = begin
        x  x                ≈⟨ ⊕-def _ _ 
        (x  x)  ¬ (x  x)  ≈⟨ helper (∨-idempotent _)
                                       (∧-idempotent _) 
        x  ¬ x              ≈⟨ proj₂ ∧-complement _ 
                            

    distrib-∧-⊕ : _∧_ DistributesOver _⊕_
    distrib-∧-⊕ = distˡ , distʳ
      where
      distˡ : _∧_ DistributesOverˡ _⊕_
      distˡ x y z = begin
        x  (y  z)                ≈⟨ refl  ∧-cong  ⊕-def _ _ 
        x  ((y  z)  ¬ (y  z))  ≈⟨ sym $ ∧-assoc _ _ _ 
        (x  (y  z))  ¬ (y  z)  ≈⟨ refl  ∧-cong  deMorgan₁ _ _ 
        (x  (y  z)) 
        (¬ y  ¬ z)                ≈⟨ sym $ proj₁ ∨-identity _ 
         
        ((x  (y  z)) 
         (¬ y  ¬ z))              ≈⟨ lem₃  ∨-cong  refl 
        ((x  (y  z))  ¬ x) 
        ((x  (y  z)) 
         (¬ y  ¬ z))              ≈⟨ sym $ proj₁ ∧-∨-distrib _ _ _ 
        (x  (y  z)) 
        (¬ x  (¬ y  ¬ z))        ≈⟨  refl  ∧-cong 
                                      (refl  ∨-cong  sym (deMorgan₁ _ _)) 
        (x  (y  z)) 
        (¬ x  ¬ (y  z))          ≈⟨ refl  ∧-cong  sym (deMorgan₁ _ _) 
          (x  (y  z)) 
        ¬ (x  (y  z))            ≈⟨ helper refl lem₁ 
          (x  (y  z)) 
        ¬ ((x  y)  (x  z))      ≈⟨ proj₁ ∧-∨-distrib _ _ _  ∧-cong 
                                      refl 
          ((x  y)  (x  z)) 
        ¬ ((x  y)  (x  z))      ≈⟨ sym $ ⊕-def _ _ 
        (x  y)  (x  z)          
        where
        lem₂ = begin
          x  (y  z)  ≈⟨ sym $ ∧-assoc _ _ _ 
          (x  y)  z  ≈⟨ ∧-comm _ _  ∧-cong  refl 
          (y  x)  z  ≈⟨ ∧-assoc _ _ _ 
          y  (x  z)  

        lem₁ = begin
          x  (y  z)        ≈⟨ sym (∧-idempotent _)  ∧-cong  refl 
          (x  x)  (y  z)  ≈⟨ ∧-assoc _ _ _ 
          x  (x  (y  z))  ≈⟨ refl  ∧-cong  lem₂ 
          x  (y  (x  z))  ≈⟨ sym $ ∧-assoc _ _ _ 
          (x  y)  (x  z)  

        lem₃ = begin
                                ≈⟨ sym $ proj₂ ∧-zero _ 
          (y  z)              ≈⟨ refl  ∧-cong  sym (proj₂ ∧-complement _) 
          (y  z)  (x  ¬ x)    ≈⟨ sym $ ∧-assoc _ _ _ 
          ((y  z)  x)  ¬ x    ≈⟨ ∧-comm _ _  ∧-cong  refl  
          (x  (y  z))  ¬ x    

      distʳ : _∧_ DistributesOverʳ _⊕_
      distʳ x y z = begin
        (y  z)  x        ≈⟨ ∧-comm _ _ 
        x  (y  z)        ≈⟨ distˡ _ _ _ 
        (x  y)  (x  z)  ≈⟨ ∧-comm _ _  ⊕-cong  ∧-comm _ _ 
        (y  x)  (z  x)  

    lemma₂ :  x y u v 
             (x  y)  (u  v) 
             ((x  u)  (y  u)) 
             ((x  v)  (y  v))
    lemma₂ x y u v = begin
        (x  y)  (u  v)              ≈⟨ proj₁ ∨-∧-distrib _ _ _ 
        ((x  y)  u)  ((x  y)  v)  ≈⟨ proj₂ ∨-∧-distrib _ _ _
                                             ∧-cong 
                                          proj₂ ∨-∧-distrib _ _ _ 
        ((x  u)  (y  u)) 
        ((x  v)  (y  v))            

    ⊕-assoc : Associative _⊕_
    ⊕-assoc x y z = sym $ begin
      x  (y  z)                                ≈⟨ refl  ⊕-cong  ⊕-def _ _ 
      x  ((y  z)  ¬ (y  z))                  ≈⟨ ⊕-def _ _ 
        (x  ((y  z)  ¬ (y  z))) 
      ¬ (x  ((y  z)  ¬ (y  z)))              ≈⟨ lem₃  ∧-cong  lem₄ 
      (((x  y)  z)  ((x  ¬ y)  ¬ z)) 
      (((¬ x  ¬ y)  z)  ((¬ x  y)  ¬ z))    ≈⟨ ∧-assoc _ _ _ 
      ((x  y)  z) 
      (((x  ¬ y)  ¬ z) 
       (((¬ x  ¬ y)  z)  ((¬ x  y)  ¬ z)))  ≈⟨ refl  ∧-cong  lem₅ 
      ((x  y)  z) 
      (((¬ x  ¬ y)  z) 
       (((x  ¬ y)  ¬ z)  ((¬ x  y)  ¬ z)))  ≈⟨ sym $ ∧-assoc _ _ _ 
      (((x  y)  z)  ((¬ x  ¬ y)  z)) 
      (((x  ¬ y)  ¬ z)  ((¬ x  y)  ¬ z))    ≈⟨ lem₁  ∧-cong  lem₂ 
        (((x  y)  ¬ (x  y))  z) 
      ¬ (((x  y)  ¬ (x  y))  z)              ≈⟨ sym $ ⊕-def _ _ 
      ((x  y)  ¬ (x  y))  z                  ≈⟨ sym $ ⊕-def _ _  ⊕-cong  refl 
      (x  y)  z                                
      where
      lem₁ = begin
        ((x  y)  z)  ((¬ x  ¬ y)  z)  ≈⟨ sym $ proj₂ ∨-∧-distrib _ _ _ 
        ((x  y)  (¬ x  ¬ y))  z        ≈⟨ (refl  ∧-cong  sym (deMorgan₁ _ _))
                                                     ∨-cong  refl 
        ((x  y)  ¬ (x  y))  z          

      lem₂' = begin
        (x  ¬ y)  (¬ x  y)              ≈⟨ sym $
                                                proj₁ ∧-identity _
                                                   ∧-cong 
                                                proj₂ ∧-identity _ 
        (  (x  ¬ y))  ((¬ x  y)  )  ≈⟨ sym $
                                                (proj₁ ∨-complement _  ∧-cong  ∨-comm _ _)
                                                   ∧-cong 
                                                (refl  ∧-cong  proj₁ ∨-complement _) 
        ((¬ x  x)  (¬ y  x)) 
        ((¬ x  y)  (¬ y  y))            ≈⟨ sym $ lemma₂ _ _ _ _ 
        (¬ x  ¬ y)  (x  y)              ≈⟨ sym $ deMorgan₂ _ _  ∨-cong  ¬-involutive _ 
        ¬ (x  y)  ¬ ¬ (x  y)            ≈⟨ sym (deMorgan₁ _ _) 
        ¬ ((x  y)  ¬ (x  y))            

      lem₂ = begin
        ((x  ¬ y)  ¬ z)  ((¬ x  y)  ¬ z)  ≈⟨ sym $ proj₂ ∨-∧-distrib _ _ _ 
        ((x  ¬ y)  (¬ x  y))  ¬ z          ≈⟨ lem₂'  ∨-cong  refl 
        ¬ ((x  y)  ¬ (x  y))  ¬ z          ≈⟨ sym $ deMorgan₁ _ _ 
        ¬ (((x  y)  ¬ (x  y))  z)          

      lem₃ = begin
        x  ((y  z)  ¬ (y  z))          ≈⟨ refl  ∨-cong 
                                                (refl  ∧-cong  deMorgan₁ _ _) 
        x  ((y  z)  (¬ y  ¬ z))        ≈⟨ proj₁ ∨-∧-distrib _ _ _ 
        (x  (y  z))  (x  (¬ y  ¬ z))  ≈⟨ sym (∨-assoc _ _ _)  ∧-cong 
                                              sym (∨-assoc _ _ _) 
        ((x  y)  z)  ((x  ¬ y)  ¬ z)  

      lem₄' = begin
        ¬ ((y  z)  ¬ (y  z))    ≈⟨ deMorgan₁ _ _ 
        ¬ (y  z)  ¬ ¬ (y  z)    ≈⟨ deMorgan₂ _ _  ∨-cong  ¬-involutive _ 
        (¬ y  ¬ z)  (y  z)      ≈⟨ lemma₂ _ _ _ _ 
        ((¬ y  y)  (¬ z  y)) 
        ((¬ y  z)  (¬ z  z))    ≈⟨ (proj₁ ∨-complement _  ∧-cong  ∨-comm _ _)
                                         ∧-cong 
                                      (refl  ∧-cong  proj₁ ∨-complement _) 
        (  (y  ¬ z)) 
        ((¬ y  z)  )            ≈⟨ proj₁ ∧-identity _  ∧-cong 
                                      proj₂ ∧-identity _ 
        (y  ¬ z)  (¬ y  z)      

      lem₄ = begin
        ¬ (x  ((y  z)  ¬ (y  z)))  ≈⟨ deMorgan₁ _ _ 
        ¬ x  ¬ ((y  z)  ¬ (y  z))  ≈⟨ refl  ∨-cong  lem₄' 
        ¬ x  ((y  ¬ z)  (¬ y  z))  ≈⟨ proj₁ ∨-∧-distrib _ _ _ 
        (¬ x  (y      ¬ z)) 
        (¬ x  (¬ y  z))              ≈⟨ sym (∨-assoc _ _ _)  ∧-cong 
                                          sym (∨-assoc _ _ _) 
        ((¬ x  y)      ¬ z) 
        ((¬ x  ¬ y)  z)              ≈⟨ ∧-comm _ _ 
        ((¬ x  ¬ y)  z) 
        ((¬ x  y)      ¬ z)          

      lem₅ = begin
        ((x  ¬ y)  ¬ z) 
        (((¬ x  ¬ y)  z)  ((¬ x  y)  ¬ z))    ≈⟨ sym $ ∧-assoc _ _ _ 
        (((x  ¬ y)  ¬ z)  ((¬ x  ¬ y)  z)) 
        ((¬ x  y)  ¬ z)                          ≈⟨ ∧-comm _ _  ∧-cong  refl 
        (((¬ x  ¬ y)  z)  ((x  ¬ y)  ¬ z)) 
        ((¬ x  y)  ¬ z)                          ≈⟨ ∧-assoc _ _ _ 
        ((¬ x  ¬ y)  z) 
        (((x  ¬ y)  ¬ z)  ((¬ x  y)  ¬ z))    

  isCommutativeRing : IsCommutativeRing _≈_ _⊕_ _∧_ id  
  isCommutativeRing = record
    { isRing = record
      { +-isAbelianGroup = record
        { isGroup = record
          { isMonoid = record
            { isSemigroup = record
              { isEquivalence = isEquivalence
              ; assoc         = ⊕-assoc
              ; ∙-cong        = ⊕-cong
              }
            ; identity = ⊕-identity
            }
          ; inverse   = ⊕-inverse
          ; ⁻¹-cong   = id
          }
        ; comm = ⊕-comm
        }
      ; *-isMonoid = record
        { isSemigroup = record
          { isEquivalence = isEquivalence
          ; assoc         = ∧-assoc
          ; ∙-cong        = ∧-cong
          }
        ; identity = ∧-identity
        }
      ; distrib = distrib-∧-⊕
      }
    ; *-comm = ∧-comm
    }

  commutativeRing : CommutativeRing _ _
  commutativeRing = record
    { _+_               = _⊕_
    ; _*_               = _∧_
    ; -_                = id
    ; 0#                = 
    ; 1#                = 
    ; isCommutativeRing = isCommutativeRing
    }

infixl 6 _⊕_

_⊕_ : Op₂ Carrier
x  y = (x  y)  ¬ (x  y)

module DefaultXorRing = XorRing _⊕_  _ _  refl)