{-# OPTIONS --cubical-compatible --safe #-}
open import Algebra.Core using (Op₂)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Definitions using (Transitive; Symmetric; Reflexive; _Respects_)
open import Relation.Binary.Structures using (IsEquivalence)
module Data.List.Relation.Binary.Equality.Setoid {a ℓ} (S : Setoid a ℓ) where
open import Data.Fin.Base using (Fin)
open import Data.List.Base
open import Data.List.Relation.Binary.Pointwise as PW using (Pointwise)
open import Data.List.Relation.Unary.Unique.Setoid S using (Unique)
open import Function.Base using (_∘_)
open import Level
open import Relation.Binary.Core renaming (Rel to Rel₂)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
open import Relation.Unary as U using (Pred)
open Setoid S renaming (Carrier to A)
private
variable
p q : Level
infix 4 _≋_
_≋_ : Rel₂ (List A) (a ⊔ ℓ)
_≋_ = Pointwise _≈_
open PW public
using ([]; _∷_)
≋-refl : Reflexive _≋_
≋-refl = PW.refl refl
≋-reflexive : _≡_ ⇒ _≋_
≋-reflexive P.refl = ≋-refl
≋-sym : Symmetric _≋_
≋-sym = PW.symmetric sym
≋-trans : Transitive _≋_
≋-trans = PW.transitive trans
≋-isEquivalence : IsEquivalence _≋_
≋-isEquivalence = PW.isEquivalence isEquivalence
≋-setoid : Setoid _ _
≋-setoid = PW.setoid S
open PW public
using () renaming
( Any-resp-Pointwise to Any-resp-≋
; All-resp-Pointwise to All-resp-≋
; AllPairs-resp-Pointwise to AllPairs-resp-≋
)
Unique-resp-≋ : Unique Respects _≋_
Unique-resp-≋ = AllPairs-resp-≋ ≉-resp₂
≋-length : ∀ {xs ys} → xs ≋ ys → length xs ≡ length ys
≋-length = PW.Pointwise-length
module _ {b ℓ₂} (T : Setoid b ℓ₂) where
open Setoid T using () renaming (_≈_ to _≈′_)
private
_≋′_ = Pointwise _≈′_
map⁺ : ∀ {f} → f Preserves _≈_ ⟶ _≈′_ →
∀ {xs ys} → xs ≋ ys → map f xs ≋′ map f ys
map⁺ {f} pres xs≋ys = PW.map⁺ f f (PW.map pres xs≋ys)
foldr⁺ : ∀ {_•_ : Op₂ A} {_◦_ : Op₂ A} →
(∀ {w x y z} → w ≈ x → y ≈ z → (w • y) ≈ (x ◦ z)) →
∀ {xs ys e f} → e ≈ f → xs ≋ ys →
foldr _•_ e xs ≈ foldr _◦_ f ys
foldr⁺ ∙⇔◦ e≈f xs≋ys = PW.foldr⁺ ∙⇔◦ e≈f xs≋ys
++⁺ : ∀ {ws xs ys zs} → ws ≋ xs → ys ≋ zs → ws ++ ys ≋ xs ++ zs
++⁺ = PW.++⁺
++-cancelˡ : ∀ xs {ys zs} → xs ++ ys ≋ xs ++ zs → ys ≋ zs
++-cancelˡ xs = PW.++-cancelˡ xs
++-cancelʳ : ∀ {xs} ys zs → ys ++ xs ≋ zs ++ xs → ys ≋ zs
++-cancelʳ = PW.++-cancelʳ
concat⁺ : ∀ {xss yss} → Pointwise _≋_ xss yss → concat xss ≋ concat yss
concat⁺ = PW.concat⁺
module _ {n} {f g : Fin n → A}
where
tabulate⁺ : (∀ i → f i ≈ g i) → tabulate f ≋ tabulate g
tabulate⁺ = PW.tabulate⁺
tabulate⁻ : tabulate f ≋ tabulate g → (∀ i → f i ≈ g i)
tabulate⁻ = PW.tabulate⁻
module _ {P : Pred A p} (P? : U.Decidable P) (resp : P Respects _≈_)
where
filter⁺ : ∀ {xs ys} → xs ≋ ys → filter P? xs ≋ filter P? ys
filter⁺ xs≋ys = PW.filter⁺ P? P? resp (resp ∘ sym) xs≋ys
ʳ++⁺ : ∀{xs xs′ ys ys′} → xs ≋ xs′ → ys ≋ ys′ → xs ʳ++ ys ≋ xs′ ʳ++ ys′
ʳ++⁺ = PW.ʳ++⁺
reverse⁺ : ∀ {xs ys} → xs ≋ ys → reverse xs ≋ reverse ys
reverse⁺ = PW.reverse⁺