-- The Agda standard library
-- Properties of the generalised view of appending two lists

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

module Data.List.Relation.Ternary.Appending.Properties where

open import Data.List.Base using (List; [])
open import Data.List.Relation.Ternary.Appending
open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)
open import Level using (Level)
open import Relation.Binary using (REL; Rel; Trans)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)

    a a′ b b′ c c′ l r : Level
    A : Set a
    A′ : Set a′
    B : Set b
    B′ : Set b′
    C : Set c
    C′ : Set c′
    L : REL A C l
    R : REL B C r
    as : List A
    bs : List B
    cs : List C

module _  {e} {E : REL C C′ e} {L′ : REL A C′ l} {R′ : REL B C′ r}
          (LEL′ : Trans L E L′) (RER′ : Trans R E R′)

  respʳ-≋ :  {cs′}  Appending L R as bs cs  Pointwise E cs cs′  Appending L′ R′ as bs cs′
  respʳ-≋ ([]++ rs) es       = []++ Pw.transitive RER′ rs es
  respʳ-≋ (l  lrs) (e  es) = LEL′ l e  respʳ-≋ lrs es

module _  {eᴬ eᴮ} {Eᴬ : REL A′ A eᴬ} {Eᴮ : REL B′ B eᴮ}
          {L′ : REL A′ C l} (ELL′ : Trans Eᴬ L L′)
          {R′ : REL B′ C r} (ERR′ : Trans Eᴮ R R′)

  respˡ-≋ :  {as′ bs′}  Pointwise Eᴬ as′ as  Pointwise Eᴮ bs′ bs 
            Appending L R as bs cs  Appending L′ R′ as′ bs′ cs
  respˡ-≋ []         esʳ ([]++ rs) = []++ Pw.transitive ERR′ esʳ rs
  respˡ-≋ (  esˡ) esʳ (l  lrs) = ELL′  l  respˡ-≋ esˡ esʳ lrs

conicalˡ : Appending L R as bs []  as  []
conicalˡ ([]++ rs) = refl

conicalʳ : Appending L R as bs []  bs  []
conicalʳ ([]++ []) = refl