------------------------------------------------------------------------
-- The Agda standard library
--
-- The core definition of a sorting algorithm
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
open import Data.List.Base using (List)
open import Data.List.Relation.Binary.Permutation.Propositional
open import Level using (_⊔_)
open import Relation.Binary.Bundles using (TotalOrder)
 
module Data.List.Sort.Base
  {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂)
  where
 
open TotalOrder totalOrder renaming (Carrier to A)
open import Data.List.Relation.Unary.Sorted.TotalOrder totalOrder
 
------------------------------------------------------------------------
-- Definition of a sorting algorithm
 
record SortingAlgorithm : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
  field
    sort   : List A → List A
 
    -- The output of `sort` is a permutation of the input
    sort-↭ : ∀ xs → sort xs ↭ xs
 
    -- The output of `sort` is sorted.
    sort-↗ : ∀ xs → Sorted (sort xs)