------------------------------------------------------------------------
-- The Agda standard library
--
-- Operations on and properties of decidable relations
------------------------------------------------------------------------

module Relation.Nullary.Decidable where

open import Data.Bool
open import Data.Empty
open import Data.Product hiding (map)
open import Data.Unit
open import Function
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence
  using (_⇔_; equivalence; module Equivalence)
open import Level using (Lift)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary

⌊_⌋ :  {p} {P : Set p}  Dec P  Bool
 yes _  = true
 no  _  = false

True :  {p} {P : Set p}  Dec P  Set
True Q = T  Q 

False :  {p} {P : Set p}  Dec P  Set
False Q = T (not  Q )

-- Gives a witness to the "truth".

toWitness :  {p} {P : Set p} {Q : Dec P}  True Q  P
toWitness {Q = yes p} _  = p
toWitness {Q = no  _} ()

-- Establishes a "truth", given a witness.

fromWitness :  {p} {P : Set p} {Q : Dec P}  P  True Q
fromWitness {Q = yes p} = const _
fromWitness {Q = no ¬p} = ¬p

-- Variants for False.

toWitnessFalse :  {p} {P : Set p} {Q : Dec P}  False Q  ¬ P
toWitnessFalse {Q = yes _}  ()
toWitnessFalse {Q = no  ¬p} _  = ¬p

fromWitnessFalse :  {p} {P : Set p} {Q : Dec P}  ¬ P  False Q
fromWitnessFalse {Q = yes p} = flip _$_ p
fromWitnessFalse {Q = no ¬p} = const _

map :  {p q} {P : Set p} {Q : Set q}  P  Q  Dec P  Dec Q
map P⇔Q (yes p) = yes (Equivalence.to P⇔Q ⟨$⟩ p)
map P⇔Q (no ¬p) = no (¬p  _⟨$⟩_ (Equivalence.from P⇔Q))

map′ :  {p q} {P : Set p} {Q : Set q} 
       (P  Q)  (Q  P)  Dec P  Dec Q
map′ P→Q Q→P = map (equivalence P→Q Q→P)

-- If a decision procedure returns "yes", then we can extract the
-- proof using from-yes.

From-yes :  {p} {P : Set p}  Dec P  Set p
From-yes {P = P} (yes _) = P
From-yes         (no  _) = Lift 

from-yes :  {p} {P : Set p} (p : Dec P)  From-yes p
from-yes (yes p) = p
from-yes (no  _) = _

-- If a decision procedure returns "no", then we can extract the proof
-- using from-no.

From-no :  {p} {P : Set p}  Dec P  Set p
From-no {P = P} (no  _) = ¬ P
From-no         (yes _) = Lift 

from-no :  {p} {P : Set p} (p : Dec P)  From-no p
from-no (no ¬p) = ¬p
from-no (yes _) = _