-- The Agda standard library
-- A definition for the permutation relation using setoid equality

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

module Data.List.Relation.Binary.Permutation.Homogeneous where

open import Data.List.Base using (List; _∷_)
open import Data.List.Relation.Binary.Pointwise as Pointwise
  using (Pointwise)
open import Level using (Level; _⊔_)
open import Relation.Binary

    a r s : Level
    A : Set a

data Permutation {A : Set a} (R : Rel A r) : Rel (List A) (a  r) where
  refl  :  {xs ys}  Pointwise R xs ys  Permutation R xs ys
  prep  :  {xs ys x y} (eq : R x y)  Permutation R xs ys  Permutation R (x  xs) (y  ys)
  swap  :  {xs ys x y x′ y′} (eq₁ : R x x′) (eq₂ : R y y′)  Permutation R xs ys  Permutation R (x  y  xs) (y′  x′  ys)
  trans :  {xs ys zs}  Permutation R xs ys  Permutation R ys zs  Permutation R xs zs

-- The Permutation relation is an equivalence

module _ {R : Rel A r}  where

  sym : Symmetric R  Symmetric (Permutation R)
  sym R-sym (refl xs∼ys)           = refl (Pointwise.symmetric R-sym xs∼ys)
  sym R-sym (prep x∼x′ xs↭ys)      = prep (R-sym x∼x′) (sym R-sym xs↭ys)
  sym R-sym (swap x∼x′ y∼y′ xs↭ys) = swap (R-sym y∼y′) (R-sym x∼x′) (sym R-sym xs↭ys)
  sym R-sym (trans xs↭ys ys↭zs)    = trans (sym R-sym ys↭zs) (sym R-sym xs↭ys)

  isEquivalence : Reflexive R  Symmetric R  IsEquivalence (Permutation R)
  isEquivalence R-refl R-sym = record
    { refl  = refl (Pointwise.refl R-refl)
    ; sym   = sym R-sym
    ; trans = trans

  setoid : Reflexive R  Symmetric R  Setoid _ _
  setoid R-refl R-sym = record
    { isEquivalence = isEquivalence R-refl R-sym

map :  {R : Rel A r} {S : Rel A s} 
      (R  S)  (Permutation R  Permutation S)
map R⇒S (refl xs∼ys)         = refl (Pointwise.map R⇒S xs∼ys)
map R⇒S (prep e xs∼ys)       = prep (R⇒S e) (map R⇒S xs∼ys)
map R⇒S (swap e₁ e₂ xs∼ys)   = swap (R⇒S e₁) (R⇒S e₂) (map R⇒S xs∼ys)
map R⇒S (trans xs∼ys ys∼zs)  = trans (map R⇒S xs∼ys) (map R⇒S ys∼zs)