{-# OPTIONS --safe #-}
module Cubical.Data.Bool.Base where

open import Cubical.Foundations.Prelude

open import Cubical.Data.Empty
open import Cubical.Data.Sum.Base
open import Cubical.Data.Unit.Base

open import Cubical.Relation.Nullary.Base

-- Obtain the booleans
open import Agda.Builtin.Bool public

private
  variable
     : Level
    A : Type 

infixr 6 _and_
infixr 5 _or_
infix  0 if_then_else_

not : Bool  Bool
not true = false
not false = true

_or_ : Bool  Bool  Bool
false or x = x
true  or _ = true

_and_ : Bool  Bool  Bool
false and _ = false
true  and x = x

-- xor / mod-2 addition
_⊕_ : Bool  Bool  Bool
false  x = x
true   x = not x

if_then_else_ : Bool  A  A  A
if true  then x else y = x
if false then x else y = y

_≟_ : Discrete Bool
false  false = yes refl
false  true  = no λ p  subst  b  if b then  else Bool) p true
true   false = no λ p  subst  b  if b then Bool else ) p true
true   true  = yes refl

Dec→Bool : Dec A  Bool
Dec→Bool (yes p) = true
Dec→Bool (no ¬p) = false

-- Helpers for automatic proof
Bool→Type : Bool  Type₀
Bool→Type true = Unit
Bool→Type false = 

Bool→Type* : Bool  Type 
Bool→Type* true = Unit*
Bool→Type* false = ⊥*

True : Dec A  Type₀
True Q = Bool→Type (Dec→Bool Q)

False : Dec A  Type₀
False Q = Bool→Type (not (Dec→Bool Q))

toWitness : {Q : Dec A}  True Q  A
toWitness {Q = yes p} _ = p

toWitnessFalse : {Q : Dec A}  False Q  ¬ A
toWitnessFalse {Q = no ¬p} _ = ¬p

dichotomyBool : (x : Bool)  (x  true)  (x  false)
dichotomyBool true  = inl refl
dichotomyBool false = inr refl

dichotomyBoolSym : (x : Bool)  (x  false)  (x  true)
dichotomyBoolSym false = inl refl
dichotomyBoolSym true = inr refl

-- TODO: this should be uncommented and implemented using instance arguments
-- _==_ : {dA : Discrete A} → A → A → Bool
-- _==_ {dA = dA} x y = Dec→Bool (dA x y)

-- Universe lifted booleans
Bool* :  {}  Type 
Bool* = Lift Bool

true* :  {}  Bool* {}
true* = lift true

false* :  {}  Bool* {}
false* = lift false

-- Pointed version
Bool*∙ :  {}  Σ[ X  Type  ] X
fst Bool*∙ = Bool*
snd Bool*∙ = true*