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January 2, 2022 00:30
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{-# OPTIONS --cubical #-} | |
open import Cubical.Foundations.Prelude | |
open import Cubical.Foundations.Isomorphism | |
open import Cubical.Data.List | |
private | |
variable | |
ℓ : Level | |
A : Type ℓ | |
data Free (A : Type ℓ) : Type ℓ where | |
∅ : Free A | |
⟨_⟩ : A → Free A | |
_⋆_ : Free A → Free A → Free A | |
⋆-idl : ∀ x → ∅ ⋆ x ≡ x | |
⋆-idr : ∀ x → x ⋆ ∅ ≡ x | |
⋆-assoc : ∀ x y z → (x ⋆ y) ⋆ z ≡ x ⋆ (y ⋆ z) | |
infixr 5 _⋆_ | |
List→Free : List A → Free A | |
List→Free = foldr (λ x f → ⟨ x ⟩ ⋆ f) ∅ | |
Free→List : Free A → List A | |
Free→List ∅ = [] | |
Free→List ⟨ x ⟩ = [ x ] | |
Free→List (xs ⋆ ys) = Free→List xs ++ Free→List ys | |
Free→List (⋆-idl xs i) = Free→List xs | |
Free→List (⋆-idr xs i) = ++-unit-r (Free→List xs) i | |
Free→List (⋆-assoc xs ys zs i) = ++-assoc (Free→List xs) (Free→List ys) (Free→List zs) i | |
example-List→Free : List→Free (1 ∷ 2 ∷ []) ≡ ⟨ 1 ⟩ ⋆ ⟨ 2 ⟩ ⋆ ∅ | |
example-List→Free = refl | |
example-Free→List : Free→List (∅ ⋆ ∅ ⋆ ⟨ 1 ⟩ ⋆ ∅ ⋆ ∅ ⋆ ⟨ 2 ⟩ ⋆ ∅) ≡ 1 ∷ 2 ∷ [] | |
example-Free→List = refl | |
List→Free→List : (l : List A) → Free→List (List→Free l) ≡ l | |
List→Free→List [] = refl | |
List→Free→List (x ∷ xs) = | |
Free→List (List→Free (x ∷ xs)) | |
≡⟨ refl ⟩ | |
x ∷ Free→List (List→Free xs) | |
≡⟨ cong (x ∷_) (List→Free→List xs) ⟩ | |
x ∷ xs ∎ | |
List→Free-++ : (xs ys : List A) → List→Free (xs ++ ys) ≡ List→Free xs ⋆ List→Free ys | |
List→Free-++ [] ys = sym (⋆-idl (List→Free ys)) | |
List→Free-++ (x ∷ xs) ys = | |
⟨ x ⟩ ⋆ List→Free (xs ++ ys) | |
≡⟨ cong (⟨ x ⟩ ⋆_) (List→Free-++ xs ys) ⟩ | |
⟨ x ⟩ ⋆ List→Free xs ⋆ List→Free ys | |
≡⟨ sym (⋆-assoc _ _ _) ⟩ | |
(⟨ x ⟩ ⋆ List→Free xs) ⋆ List→Free ys ∎ | |
Free→List→Free : (f : Free A) → List→Free (Free→List f) ≡ f | |
Free→List→Free ∅ = refl | |
Free→List→Free ⟨ x ⟩ = ⋆-idr ⟨ x ⟩ | |
Free→List→Free (xs ⋆ ys) = | |
List→Free (Free→List xs ++ Free→List ys) | |
≡⟨ List→Free-++ (Free→List xs) (Free→List ys) ⟩ | |
List→Free (Free→List xs) ⋆ List→Free (Free→List ys) | |
≡⟨ cong₂ _⋆_ (Free→List→Free xs) (Free→List→Free ys) ⟩ | |
xs ⋆ ys ∎ | |
Free→List→Free (⋆-idl x i) = {! !} | |
Free→List→Free (⋆-idr x i) = {! !} | |
Free→List→Free (⋆-assoc a b c i) = {! !} | |
List≡Free : List A ≡ Free A | |
List≡Free = isoToPath (iso List→Free Free→List Free→List→Free List→Free→List) |
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