```------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of unique lists (setoid equality)
------------------------------------------------------------------------

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

module Data.List.Relation.Unary.Unique.Setoid.Properties where

open import Data.List.Base
open import Data.List.Membership.Setoid.Properties
open import Data.List.Relation.Binary.Disjoint.Setoid
open import Data.List.Relation.Binary.Disjoint.Setoid.Properties
open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
open import Data.List.Relation.Unary.All.Properties using (All¬⇒¬Any)
open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)
open import Data.List.Relation.Unary.Unique.Setoid
open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
import Data.List.Relation.Unary.AllPairs.Properties as AllPairs
open import Data.Fin.Base using (Fin)
open import Data.Nat.Base using (_<_)
open import Function.Base using (_∘_; id)
open import Level using (Level)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.PropositionalEquality.Core using (_≡_)
open import Relation.Unary using (Pred; Decidable)
open import Relation.Nullary.Negation using (¬_)
open import Relation.Nullary.Negation using (contraposition)

private
variable
a b c p ℓ ℓ₁ ℓ₂ ℓ₃ : Level

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

module _ (S : Setoid a ℓ₁) (R : Setoid b ℓ₂) where

open Setoid S renaming (_≈_ to _≈₁_)
open Setoid R renaming (_≈_ to _≈₂_)

map⁺ : ∀ {f} → (∀ {x y} → f x ≈₂ f y → x ≈₁ y) →
∀ {xs} → Unique S xs → Unique R (map f xs)
map⁺ inj xs! = AllPairs.map⁺ (AllPairs.map (contraposition inj) xs!)

------------------------------------------------------------------------
-- ++

module _ (S : Setoid a ℓ) where

++⁺ : ∀ {xs ys} → Unique S xs → Unique S ys → Disjoint S xs ys → Unique S (xs ++ ys)
++⁺ xs! ys! xs#ys = AllPairs.++⁺ xs! ys! (Disjoint⇒AllAll S xs#ys)

------------------------------------------------------------------------
-- concat

module _ (S : Setoid a ℓ) where

concat⁺ : ∀ {xss} → All (Unique S) xss → AllPairs (Disjoint S) xss → Unique S (concat xss)
concat⁺ xss! xss# = AllPairs.concat⁺ xss! (AllPairs.map (Disjoint⇒AllAll S) xss#)

------------------------------------------------------------------------
-- cartesianProductWith

module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) (U : Setoid c ℓ₃) where

open Setoid S using () renaming (_≈_ to _≈₁_)
open Setoid T using () renaming (_≈_ to _≈₂_)
open Setoid U using () renaming (_≈_ to _≈₃_; sym to sym₃; trans to trans₃)

cartesianProductWith⁺ : ∀ {xs ys} f → (∀ {w x y z} → f w y ≈₃ f x z → w ≈₁ x × y ≈₂ z) →
Unique S xs → Unique T ys →
Unique U (cartesianProductWith f xs ys)
cartesianProductWith⁺ {_}      {_}  f f-inj  []          ys! = [] {S = U}
cartesianProductWith⁺ {x ∷ xs} {ys} f f-inj (x∉xs ∷ xs!) ys! = ++⁺ U
(map⁺ T U (proj₂ ∘ f-inj) ys!)
(cartesianProductWith⁺ f f-inj xs! ys!)
map#cartesianProductWith
where
map#cartesianProductWith : Disjoint U (map (f x) ys) (cartesianProductWith f xs ys)
map#cartesianProductWith (v∈map , v∈com) with
∈-map⁻ T U v∈map | ∈-cartesianProductWith⁻ S T U f xs ys v∈com
... | (c , _ , v≈fxc) | (a , b , a∈xs , _ , v≈fab) =
All¬⇒¬Any x∉xs (∈-resp-≈ S (proj₁ (f-inj (trans₃ (sym₃ v≈fab) v≈fxc))) a∈xs)

------------------------------------------------------------------------
-- cartesianProduct

module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) {xs ys} where

cartesianProduct⁺ : Unique S xs → Unique T ys →
Unique (S ×ₛ T) (cartesianProduct xs ys)
cartesianProduct⁺ = cartesianProductWith⁺ S T (S ×ₛ T) _,_ id

------------------------------------------------------------------------
-- take & drop

module _ (S : Setoid a ℓ) where

drop⁺ : ∀ {xs} n → Unique S xs → Unique S (drop n xs)
drop⁺ = AllPairs.drop⁺

take⁺ : ∀ {xs} n → Unique S xs → Unique S (take n xs)
take⁺ = AllPairs.take⁺

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

module _ (S : Setoid a ℓ) where

open Setoid S

applyUpTo⁺₁ : ∀ f n → (∀ {i j} → i < j → j < n → f i ≉ f j) →
Unique S (applyUpTo f n)
applyUpTo⁺₁ = AllPairs.applyUpTo⁺₁

applyUpTo⁺₂ : ∀ f n → (∀ i j → f i ≉ f j) →
Unique S (applyUpTo f n)
applyUpTo⁺₂ = AllPairs.applyUpTo⁺₂

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

module _ (S : Setoid a ℓ) where

open Setoid S

applyDownFrom⁺₁ : ∀ f n → (∀ {i j} → j < i → i < n → f i ≉ f j) →
Unique S (applyDownFrom f n)
applyDownFrom⁺₁ = AllPairs.applyDownFrom⁺₁

applyDownFrom⁺₂ : ∀ f n → (∀ i j → f i ≉ f j) →
Unique S (applyDownFrom f n)
applyDownFrom⁺₂ = AllPairs.applyDownFrom⁺₂

------------------------------------------------------------------------
-- tabulate

module _ (S : Setoid a ℓ) where

open Setoid S renaming (Carrier to A)

tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≈ f j → i ≡ j) →
Unique S (tabulate f)
tabulate⁺ f-inj = AllPairs.tabulate⁺ (_∘ f-inj)

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

module _ (S : Setoid a ℓ) {P : Pred _ p} (P? : Decidable P) where

filter⁺ : ∀ {xs} → Unique S xs → Unique S (filter P? xs)
filter⁺ = AllPairs.filter⁺ P?
```