{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Data.List.Properties where open import Agda.Builtin.List open import Cubical.Core.Everything open import Cubical.Foundations.GroupoidLaws open import Cubical.Foundations.HLevels open import Cubical.Foundations.Prelude open import Cubical.Data.Empty as ⊥ open import Cubical.Data.Nat open import Cubical.Data.Sigma open import Cubical.Data.Unit open import Cubical.Relation.Nullary open import Cubical.Data.List.Base module _ {ℓ} {A : Type ℓ} where ++-unit-r : (xs : List A) → xs ++ [] ≡ xs ++-unit-r [] = refl ++-unit-r (x ∷ xs) = cong (_∷_ x) (++-unit-r xs) ++-assoc : (xs ys zs : List A) → (xs ++ ys) ++ zs ≡ xs ++ ys ++ zs ++-assoc [] ys zs = refl ++-assoc (x ∷ xs) ys zs = cong (_∷_ x) (++-assoc xs ys zs) rev-snoc : (xs : List A) (y : A) → rev (xs ++ [ y ]) ≡ y ∷ rev xs rev-snoc [] y = refl rev-snoc (x ∷ xs) y = cong (_++ [ x ]) (rev-snoc xs y) rev-++ : (xs ys : List A) → rev (xs ++ ys) ≡ rev ys ++ rev xs rev-++ [] ys = sym (++-unit-r (rev ys)) rev-++ (x ∷ xs) ys = cong (λ zs → zs ++ [ x ]) (rev-++ xs ys) ∙ ++-assoc (rev ys) (rev xs) [ x ] rev-rev : (xs : List A) → rev (rev xs) ≡ xs rev-rev [] = refl rev-rev (x ∷ xs) = rev-snoc (rev xs) x ∙ cong (_∷_ x) (rev-rev xs) rev-rev-snoc : (xs : List A) (y : A) → Square (rev-rev (xs ++ [ y ])) (cong (_++ [ y ]) (rev-rev xs)) (cong rev (rev-snoc xs y)) refl rev-rev-snoc [] y = sym (lUnit refl) rev-rev-snoc (x ∷ xs) y i j = hcomp (λ k → λ { (i = i1) → compPath-filler (rev-snoc (rev xs) x) (cong (x ∷_) (rev-rev xs)) k j ++ [ y ] ; (j = i0) → rev (rev-snoc xs y i ++ [ x ]) ; (j = i1) → x ∷ rev-rev-snoc xs y i k }) (rev-snoc (rev-snoc xs y i) x j) data SnocView : List A → Type ℓ where nil : SnocView [] snoc : (x : A) → (xs : List A) → (sx : SnocView xs) → SnocView (xs ∷ʳ x) snocView : (xs : List A) → SnocView xs snocView xs = helper nil xs where helper : {l : List A} -> SnocView l -> (r : List A) -> SnocView (l ++ r) helper {l} sl [] = subst SnocView (sym (++-unit-r l)) sl helper {l} sl (x ∷ r) = subst SnocView (++-assoc l (x ∷ []) r) (helper (snoc x l sl) r) -- Path space of list type module ListPath {ℓ} {A : Type ℓ} where Cover : List A → List A → Type ℓ Cover [] [] = Lift Unit Cover [] (_ ∷ _) = Lift ⊥ Cover (_ ∷ _) [] = Lift ⊥ Cover (x ∷ xs) (y ∷ ys) = (x ≡ y) × Cover xs ys reflCode : ∀ xs → Cover xs xs reflCode [] = lift tt reflCode (_ ∷ xs) = refl , reflCode xs encode : ∀ xs ys → (p : xs ≡ ys) → Cover xs ys encode xs _ = J (λ ys _ → Cover xs ys) (reflCode xs) encodeRefl : ∀ xs → encode xs xs refl ≡ reflCode xs encodeRefl xs = JRefl (λ ys _ → Cover xs ys) (reflCode xs) decode : ∀ xs ys → Cover xs ys → xs ≡ ys decode [] [] _ = refl decode [] (_ ∷ _) (lift ()) decode (x ∷ xs) [] (lift ()) decode (x ∷ xs) (y ∷ ys) (p , c) = cong₂ _∷_ p (decode xs ys c) decodeRefl : ∀ xs → decode xs xs (reflCode xs) ≡ refl decodeRefl [] = refl decodeRefl (x ∷ xs) = cong (cong₂ _∷_ refl) (decodeRefl xs) decodeEncode : ∀ xs ys → (p : xs ≡ ys) → decode xs ys (encode xs ys p) ≡ p decodeEncode xs _ = J (λ ys p → decode xs ys (encode xs ys p) ≡ p) (cong (decode xs xs) (encodeRefl xs) ∙ decodeRefl xs) isOfHLevelCover : (n : HLevel) (p : isOfHLevel (suc (suc n)) A) (xs ys : List A) → isOfHLevel (suc n) (Cover xs ys) isOfHLevelCover n p [] [] = isOfHLevelLift (suc n) (isProp→isOfHLevelSuc n isPropUnit) isOfHLevelCover n p [] (y ∷ ys) = isOfHLevelLift (suc n) (isProp→isOfHLevelSuc n isProp⊥) isOfHLevelCover n p (x ∷ xs) [] = isOfHLevelLift (suc n) (isProp→isOfHLevelSuc n isProp⊥) isOfHLevelCover n p (x ∷ xs) (y ∷ ys) = isOfHLevelΣ (suc n) (p x y) (\ _ → isOfHLevelCover n p xs ys) isOfHLevelList : ∀ {ℓ} (n : HLevel) {A : Type ℓ} → isOfHLevel (suc (suc n)) A → isOfHLevel (suc (suc n)) (List A) isOfHLevelList n ofLevel xs ys = isOfHLevelRetract (suc n) (ListPath.encode xs ys) (ListPath.decode xs ys) (ListPath.decodeEncode xs ys) (ListPath.isOfHLevelCover n ofLevel xs ys) private variable ℓ : Level A : Type ℓ caseList : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (n c : B) → List A → B caseList n _ [] = n caseList _ c (_ ∷ _) = c safe-head : A → List A → A safe-head x [] = x safe-head _ (x ∷ _) = x safe-tail : List A → List A safe-tail [] = [] safe-tail (_ ∷ xs) = xs cons-inj₁ : ∀ {x y : A} {xs ys} → x ∷ xs ≡ y ∷ ys → x ≡ y cons-inj₁ {x = x} p = cong (safe-head x) p cons-inj₂ : ∀ {x y : A} {xs ys} → x ∷ xs ≡ y ∷ ys → xs ≡ ys cons-inj₂ = cong safe-tail ¬cons≡nil : ∀ {x : A} {xs} → ¬ (x ∷ xs ≡ []) ¬cons≡nil {A = A} p = lower (subst (caseList (Lift ⊥) (List A)) p []) ¬nil≡cons : ∀ {x : A} {xs} → ¬ ([] ≡ x ∷ xs) ¬nil≡cons {A = A} p = lower (subst (caseList (List A) (Lift ⊥)) p []) ¬snoc≡nil : ∀ {x : A} {xs} → ¬ (xs ∷ʳ x ≡ []) ¬snoc≡nil {xs = []} contra = ¬cons≡nil contra ¬snoc≡nil {xs = x ∷ xs} contra = ¬cons≡nil contra ¬nil≡snoc : ∀ {x : A} {xs} → ¬ ([] ≡ xs ∷ʳ x) ¬nil≡snoc contra = ¬snoc≡nil (sym contra) cons≡rev-snoc : (x : A) → (xs : List A) → x ∷ rev xs ≡ rev (xs ∷ʳ x) cons≡rev-snoc _ [] = refl cons≡rev-snoc x (y ∷ ys) = λ i → cons≡rev-snoc x ys i ++ y ∷ [] isContr[]≡[] : isContr (Path (List A) [] []) isContr[]≡[] = refl , ListPath.decodeEncode [] [] isPropXs≡[] : {xs : List A} → isProp (xs ≡ []) isPropXs≡[] {xs = []} = isOfHLevelSuc 0 isContr[]≡[] isPropXs≡[] {xs = x ∷ xs} = λ p _ → ⊥.rec (¬cons≡nil p) discreteList : Discrete A → Discrete (List A) discreteList eqA [] [] = yes refl discreteList eqA [] (y ∷ ys) = no ¬nil≡cons discreteList eqA (x ∷ xs) [] = no ¬cons≡nil discreteList eqA (x ∷ xs) (y ∷ ys) with eqA x y | discreteList eqA xs ys ... | yes p | yes q = yes (λ i → p i ∷ q i) ... | yes _ | no ¬q = no (λ p → ¬q (cons-inj₂ p)) ... | no ¬p | _ = no (λ q → ¬p (cons-inj₁ q)) foldrCons : (xs : List A) → foldr _∷_ [] xs ≡ xs foldrCons [] = refl foldrCons (x ∷ xs) = cong (x ∷_) (foldrCons xs)

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