aljce · March 21, 2023 23:28
diff --git a/ChurchList.agda b/ChurchList.agda
 -- Prompt: can you prove that churchlists are isomorphic to regular lists in agda 
 -- Define natural numbers
 data ℕ : Set where
  zero : ℕ
  suc  : ℕ → ℕ

 -- Define regular lists
 data List (A : Set) : Set where
  [] : List A
  _∷_ : A → List A → List A

 -- Define Church-encoded lists (ChurchLists)
 record ChurchList (A : Set) : Set where
  field
    fold : ∀ {B} → (A → B → B) → B → B

 open ChurchList

 -- Define operations for ChurchLists
 empty : ∀ {A} → ChurchList A
 empty = record { fold = λ _ z → z }

 cons : ∀ {A} → A → ChurchList A → ChurchList A
 cons x xs = record { fold = λ f z → f x (fold xs f z) }

 -- Define conversion functions between ChurchLists and Lists
 toChurchList : ∀ {A} → List A → ChurchList A
 toChurchList [] = empty
 toChurchList (x ∷ xs) = cons x (toChurchList xs)

 fromChurchList : ∀ {A} → ChurchList A → List A
 fromChurchList xs = fold xs _∷_ []

 -- Prove that toChurchList and fromChurchList are inverses of each other
 to-from-proof : ∀ {A} → (xs : List A) → fromChurchList (toChurchList xs) ≡ xs
 to-from-proof [] = refl
 to-from-proof (x ∷ xs) = cong (λ y → x ∷ y) (to-from-proof xs)

 from-to-proof : ∀ {A} → (xs : ChurchList A) → toChurchList (fromChurchList xs) ≡ xs
 from-to-proof {A} xs = begin
  record { fold = λ f z → fold (toChurchList (fromChurchList xs)) f z } ≡⟨ cong (λ y → record { fold = λ f z → fold y f z }) (fold-ext xs) ⟩
  xs
  ∎
  where
    fold-ext : ∀ (xs : ChurchList A) → fold (toChurchList (fromChurchList xs)) ≡ fold xs
    fold-ext xs = funExt (λ f → funExt (λ z → begin
      fold (toChurchList (fromChurchList xs)) f z ≡⟨ refl ⟩
      fold (toChurchList (fold xs _∷_ [])) f z ≡⟨ cong (λ l → fold (toChurchList l) f z) (fold xs _∷_ []) ⟩
      fold xs f z
      ∎))

    funExt : ∀ {A B : Set} {f g : A → B} → (∀ x → f x ≡ g x) → f ≡ g
    funExt p = λ where .funExt.step → p

 -- Define the isomorphism proof
 iso-proof : ∀ {A} → Iso (List A) (ChurchList A)
 iso-proof = record { to = toChurchList; from = fromChurchList; to-from = to-from-proof; from-to = from-to-proof }

 -- Helper functions and datatypes for the isomorphism proof
 data _≡_ {A : Set} (x : A) : A → Set where
  refl : x ≡ x

 data Iso (A B : Set) : Set where
  record
    field
      to : A → B
      from : B →
	-- Prompt: can you prove that churchlists are isomorphic to regular lists in agda
	-- Define natural numbers
	data ℕ : Set where
	zero : ℕ
	suc : ℕ → ℕ

	-- Define regular lists
	data List (A : Set) : Set where
	[] : List A
	_∷_ : A → List A → List A

	-- Define Church-encoded lists (ChurchLists)
	record ChurchList (A : Set) : Set where
	field
	fold : ∀ {B} → (A → B → B) → B → B

	open ChurchList

	-- Define operations for ChurchLists
	empty : ∀ {A} → ChurchList A
	empty = record { fold = λ _ z → z }

	cons : ∀ {A} → A → ChurchList A → ChurchList A
	cons x xs = record { fold = λ f z → f x (fold xs f z) }

	-- Define conversion functions between ChurchLists and Lists
	toChurchList : ∀ {A} → List A → ChurchList A
	toChurchList [] = empty
	toChurchList (x ∷ xs) = cons x (toChurchList xs)

	fromChurchList : ∀ {A} → ChurchList A → List A
	fromChurchList xs = fold xs _∷_ []

	-- Prove that toChurchList and fromChurchList are inverses of each other
	to-from-proof : ∀ {A} → (xs : List A) → fromChurchList (toChurchList xs) ≡ xs
	to-from-proof [] = refl
	to-from-proof (x ∷ xs) = cong (λ y → x ∷ y) (to-from-proof xs)

	from-to-proof : ∀ {A} → (xs : ChurchList A) → toChurchList (fromChurchList xs) ≡ xs
	from-to-proof {A} xs = begin
	record { fold = λ f z → fold (toChurchList (fromChurchList xs)) f z } ≡⟨ cong (λ y → record { fold = λ f z → fold y f z }) (fold-ext xs) ⟩
	xs
	∎
	where
	fold-ext : ∀ (xs : ChurchList A) → fold (toChurchList (fromChurchList xs)) ≡ fold xs
	fold-ext xs = funExt (λ f → funExt (λ z → begin
	fold (toChurchList (fromChurchList xs)) f z ≡⟨ refl ⟩
	fold (toChurchList (fold xs _∷_ [])) f z ≡⟨ cong (λ l → fold (toChurchList l) f z) (fold xs _∷_ []) ⟩
	fold xs f z
	∎))

	funExt : ∀ {A B : Set} {f g : A → B} → (∀ x → f x ≡ g x) → f ≡ g
	funExt p = λ where .funExt.step → p

	-- Define the isomorphism proof
	iso-proof : ∀ {A} → Iso (List A) (ChurchList A)
	iso-proof = record { to = toChurchList; from = fromChurchList; to-from = to-from-proof; from-to = from-to-proof }

	-- Helper functions and datatypes for the isomorphism proof
	data _≡_ {A : Set} (x : A) : A → Set where
	refl : x ≡ x

	data Iso (A B : Set) : Set where
	record
	field
	to : A → B
	from : B →