{-# OPTIONS --cubical #-}

open import Cubical.Foundations.Prelude

data ℕ : Set where
  zero : ℕ
  suc : ℕ → ℕ

add : ℕ → ℕ → ℕ
add zero n = n
add (suc m) n = add m (suc n)

data ℤ₀ : Set where
  pos : ℕ → ℤ₀
  negsuc : ℕ → ℤ₀

data ℤ : Set where
  z : ℕ → ℕ → ℤ
  p : (m n : ℕ) → z m n ≡ z (suc m) (suc n)

ℤ→ℤ₀ : ℤ → ℤ₀
ℤ→ℤ₀ (z zero zero) = pos zero
ℤ→ℤ₀ (z zero (suc x₁)) = negsuc x₁
ℤ→ℤ₀ (z (suc x) zero) = pos (suc x)
ℤ→ℤ₀ (z (suc x) (suc x₁)) = ℤ→ℤ₀ (z x x₁)
ℤ→ℤ₀ (p m n i) = ℤ→ℤ₀ {!   !}

ℤ₀→ℤ : ℤ₀ → ℤ
ℤ₀→ℤ (pos x) = z x zero
ℤ₀→ℤ (negsuc x) = z zero (suc x)

ap : ∀ {A B x y} → (f : A → B) → (x ≡ y) → (f x ≡ f y)
ap f x i = f (x i)

add-ℤ : ℤ → ℤ → ℤ 
add-ℤ (z x x₁) (z x₂ x₃) = z (add x x₂) (add x₁ x₃)
add-ℤ (z x x₁) (p m n i) = z (add x {!   !}) {!   !}
add-ℤ (p m n i) (z x x₁) = {!   !}    
add-ℤ (p m n i) (p m₁ n₁ i₁) = {!   !}