------------------------------------------------------------------------
-- The Agda standard library
--
-- Recomputable types and their algebra as Harrop formulas
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Relation.Nullary.Recomputable where
 
open import Data.Empty using (⊥)
open import Data.Irrelevant using (Irrelevant; irrelevant; [_])
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
open import Level using (Level)
open import Relation.Nullary.Negation.Core using (¬_)
 
private
  variable
    a b : Level
    A : Set a
    B : Set b
 
------------------------------------------------------------------------
-- Re-export
 
open import Relation.Nullary.Recomputable.Core public
 
 
------------------------------------------------------------------------
-- Constructions
 
-- Irrelevant types are Recomputable
 
irrelevant-recompute : Recomputable (Irrelevant A)
irrelevant (irrelevant-recompute [ a ]) = a
 
-- Corollary: so too is ⊥
 
⊥-recompute : Recomputable ⊥
⊥-recompute = irrelevant-recompute
 
_×-recompute_ : Recomputable A → Recomputable B → Recomputable (A × B)
(rA ×-recompute rB) p = rA (p .proj₁) , rB (p .proj₂)
 
_→-recompute_ : (A : Set a) → Recomputable B → Recomputable (A → B)
(A →-recompute rB) f a = rB (f a)
 
Π-recompute : (B : A → Set b) → (∀ x → Recomputable (B x)) → Recomputable (∀ x → B x)
Π-recompute B rB f a = rB a (f a)
 
∀-recompute : (B : A → Set b) → (∀ {x} → Recomputable (B x)) → Recomputable (∀ {x} → B x)
∀-recompute B rB f = rB f
 
-- Corollary: negations are Recomputable
 
¬-recompute : Recomputable (¬ A)
¬-recompute {A = A} = A →-recompute ⊥-recompute