GPT4 ChurchList conversion

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 →