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------------------------------------------------------------------------
-- The Agda standard library
--
-- Sorted lists
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.List.Relation.Unary.Sorted.TotalOrder.Properties where

open import Data.List.Base
open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.AllPairs using (AllPairs)
open import Data.List.Relation.Unary.Linked as Linked
  using (Linked; []; [-]; _∷_; _∷′_; head′; tail)
import Data.List.Relation.Unary.Linked.Properties as Linked
open import Data.List.Relation.Unary.Sorted.TotalOrder hiding (head)
open import Data.Maybe.Base using (just; nothing)
open import Data.Maybe.Relation.Binary.Connected using (Connected; just)
open import Data.Nat.Base using (; zero; suc; _<_; z≤n; s≤s)
open import Level using (Level)
open import Relation.Binary hiding (Decidable)
import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
open import Relation.Unary using (Pred; Decidable)
open import Relation.Nullary using (yes; no)
private
  variable
    a b p ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level

------------------------------------------------------------------------
-- Relationship to other predicates
------------------------------------------------------------------------

module _ (O : TotalOrder a ℓ₁ ℓ₂) where
  open TotalOrder O

  AllPairs⇒Sorted :  {xs}  AllPairs _≤_ xs  Sorted O xs
  AllPairs⇒Sorted = Linked.AllPairs⇒Linked

  Sorted⇒AllPairs :  {xs}  Sorted O xs  AllPairs _≤_ xs
  Sorted⇒AllPairs = Linked.Linked⇒AllPairs trans

------------------------------------------------------------------------
-- Introduction (⁺) and elimination (⁻) rules for list operations
------------------------------------------------------------------------
-- map

module _ (O₁ : TotalOrder a ℓ₁ ℓ₂) (O₂ : TotalOrder a ℓ₁ ℓ₂) where
  private
    module O₁ = TotalOrder O₁
    module O₂ = TotalOrder O₂

  map⁺ :  {f xs}  f Preserves O₁._≤_  O₂._≤_ 
         Sorted O₁ xs  Sorted O₂ (map f xs)
  map⁺ pres xs↗ = Linked.map⁺ (Linked.map pres xs↗)

  map⁻ :  {f xs}  (∀ {x y}  f x O₂.≤ f y  x O₁.≤ y) 
         Sorted O₂ (map f xs)  Sorted O₁ xs
  map⁻ pres fxs↗ = Linked.map pres (Linked.map⁻ fxs↗)

------------------------------------------------------------------------
-- _++_

module _ (O : TotalOrder a ℓ₁ ℓ₂) where
  open TotalOrder O

  ++⁺ :  {xs ys}  Sorted O xs  Connected _≤_ (last xs) (head ys) 
        Sorted O ys  Sorted O (xs ++ ys)
  ++⁺ = Linked.++⁺

------------------------------------------------------------------------
-- applyUpTo

module _ (O : TotalOrder a ℓ₁ ℓ₂) where
  open TotalOrder O

  applyUpTo⁺₁ :  f n  (∀ {i}  suc i < n  f i  f (suc i)) 
                Sorted O (applyUpTo f n)
  applyUpTo⁺₁ = Linked.applyUpTo⁺₁

  applyUpTo⁺₂ :  f n  (∀ i  f i  f (suc i)) 
                Sorted O (applyUpTo f n)
  applyUpTo⁺₂ = Linked.applyUpTo⁺₂

------------------------------------------------------------------------
-- applyDownFrom

module _ (O : TotalOrder a ℓ₁ ℓ₂) where
  open TotalOrder O

  applyDownFrom⁺₁ :  f n  (∀ {i}  suc i < n  f (suc i)  f i) 
                    Sorted O (applyDownFrom f n)
  applyDownFrom⁺₁ = Linked.applyDownFrom⁺₁

  applyDownFrom⁺₂ :  f n  (∀ i  f (suc i)  f i) 
                    Sorted O (applyDownFrom f n)
  applyDownFrom⁺₂ = Linked.applyDownFrom⁺₂


------------------------------------------------------------------------
-- merge

module _ (DO : DecTotalOrder a ℓ₁ ℓ₂) where
  open DecTotalOrder DO renaming (totalOrder to O)
  open TotalOrderProperties O using (≰⇒≥)

  private
    merge-con :  {v xs ys} 
                Connected _≤_ (just v) (head xs) 
                Connected _≤_ (just v) (head ys) 
                Connected _≤_ (just v) (head (merge _≤?_ xs ys))
    merge-con {xs = []}     {[]}     cxs cys = cys
    merge-con {xs = []}     {y  ys} cxs cys = cys
    merge-con {xs = x  xs} {[]}     cxs cys = cxs
    merge-con {xs = x  xs} {y  ys} cxs cys with x ≤? y
    ... | yes x≤y = cxs
    ... | no  x≰y = cys

  merge⁺ :  {xs ys}  Sorted O xs  Sorted O ys  Sorted O (merge _≤?_ xs ys)
  merge⁺ {[]}     {[]}     rxs rys = []
  merge⁺ {[]}     {x  ys} rxs rys = rys
  merge⁺ {x  xs} {[]}     rxs rys = rxs
  merge⁺ {x  xs} {y  ys} rxs rys with x ≤? y |
    merge⁺ (Linked.tail rxs) rys | merge⁺ rxs (Linked.tail rys)
  ... | yes x≤y | rec | _   = merge-con (head′ rxs)      (just x≤y)  ∷′ rec
  ... | no  x≰y | _   | rec = merge-con (just (≰⇒≥ x≰y)) (head′ rys) ∷′ rec

------------------------------------------------------------------------
-- filter

module _ (O : TotalOrder a ℓ₁ ℓ₂) {P : Pred _ p} (P? : Decidable P) where
  open TotalOrder O

  filter⁺ :  {xs}  Sorted O xs  Sorted O (filter P? xs)
  filter⁺ = Linked.filter⁺ P? trans