------------------------------------------------------------------------
-- The Agda standard library
--
-- The core definition of a sorting algorithm
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
open import Relation.Binary.Bundles using (TotalOrder)
module Data.List.Sort.Base
{a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂)
where
open import Data.List.Base using (List)
open import Data.List.Relation.Binary.Permutation.Propositional
using (_↭_; ↭⇒↭ₛ)
import Data.List.Relation.Binary.Permutation.Homogeneous as Homo
open import Level using (_⊔_)
open TotalOrder totalOrder renaming (Carrier to A)
open import Data.List.Relation.Unary.Sorted.TotalOrder totalOrder
import Data.List.Relation.Binary.Permutation.Setoid Eq.setoid as S
------------------------------------------------------------------------
-- Definition of a sorting algorithm
record SortingAlgorithm : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
sort : List A → List A
-- The output of `sort` is a propositional permutation of the input.
-- Note that the choice of using a propositional equality here
-- is very deliberate. A sorting algorithm should only be capable
-- of altering the order of the elements of the list, and should not
-- be capable of altering the elements themselves in any way.
sort-↭ : ∀ xs → sort xs ↭ xs
-- The output of `sort` is sorted.
sort-↗ : ∀ xs → Sorted (sort xs)
-- Lists are also permutations under the setoid versions of
-- permutation.
sort-↭ₛ : ∀ xs → sort xs S.↭ xs
sort-↭ₛ xs = Homo.map Eq.reflexive (↭⇒↭ₛ (sort-↭ xs))
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