------------------------------------------------------------------------
-- The Agda standard library
--
-- Operations on nullary relations (like negation and decidability)
------------------------------------------------------------------------

-- Some operations on/properties of nullary relations, i.e. sets.

{-# OPTIONS --without-K #-}

module Relation.Nullary where

open import Data.Empty hiding (⊥-elim)
open import Data.Empty.Irrelevant
open import Level

-- Negation.

infix 3 ¬_

¬_ :  {}  Set   Set 
¬ P = P  

-- Decidable relations.

data Dec {p} (P : Set p) : Set p where
  yes : ( p :   P)  Dec P
  no  : (¬p : ¬ P)  Dec P

-- Given an irrelevant proof of a decidable type, a proof can
-- be recomputed and subsequently used in relevant contexts.
recompute :  {a} {A : Set a}  Dec A  .A  A
recompute (yes x) _ = x
recompute (no ¬p) x = ⊥-elim (¬p x)