module peano where

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

_+_ : ℕ → ℕ → ℕ
zero + zero = zero
zero + n = n
(suc n) + n′ = suc (n + n′)

_*_ : ℕ → ℕ → ℕ
zero * n = zero
(suc zero) * n = n
(suc n) * n′ = n′ + (n * n′)

data _even : ℕ → Set where
  ZERO : zero even
  STEP : ∀ x → x even → suc (suc x) even
  -- alternative:
  --   : (x : ℕ) → x even → suc (suc x) even

proof₁ : suc (suc (suc (suc zero))) even
proof₁ = STEP _ (STEP _ ZERO)

data _∧_ (P : Set) (Q : Set) : Set where
  ∧-intro : P → Q → (P ∧ Q)

proof₃ : {P Q : Set} → (P ∧ Q) → P
proof₃ (∧-intro p q) = p

_⇔_ : (P : Set) → (Q : Set) → Set
a ⇔ b = (a → b) ∧ (b → a)

∧-comm′ : {P Q : Set} → (P ∧ Q) → (Q ∧ P)
∧-comm′ (∧-intro p q) = ∧-intro q p

--∧-comm : {P Q : Set} → (P ∧ Q) ⇔ (Q ∧ P)
--∧-comm = ∧-intro (∧-comm′ {P} {Q}) (∧-comm′ {Q} {P})

∧-assoc₁ : {P Q R : Set} → ((P ∧ Q) ∧ R) → (P ∧ (Q ∧ R))
∧-assoc₁ (∧-intro (∧-intro p q) r) = ∧-intro p (∧-intro q r)

∧-assoc₂ : {P Q R : Set} → (P ∧ (Q ∧ R)) → ((P ∧ Q) ∧ R)
∧-assoc₂ (∧-intro p (∧-intro q r)) = ∧-intro (∧-intro p q) r

∧-assoc : {P Q R : Set} → ((P ∧ Q) ∧ R) ⇔  (P ∧ (Q ∧ R))
∧-assoc = ∧-intro ∧-assoc₁ ∧-assoc₂

data _∨_ (P Q : Set) : Set where
  ∨-intro₁ : P → P ∨ Q
  ∨-intro₂ : Q → P ∨ Q

∨-elim : {A B C : Set} → (A → C) → (B → C) → (A ∨ B) → C
∨-elim ac bc (∨-intro₁ a) = ac a
∨-elim ac bc (∨-intro₂ b) = bc b

∨-comm′ : {P Q : Set} → (P ∨ Q) → (Q ∨ P)
∨-comm′ (∨-intro₁ p) = ∨-intro₂ p
∨-comm′ (∨-intro₂ q) = ∨-intro₁ q

∨-comm : {P Q : Set} → (P ∨ Q) ⇔ (Q ∨ P)
∨-comm = ∧-intro ∨-comm′ ∨-comm′


-- this is my first own proof! The associativity of disjunction

∨-assoc₁ : {P Q R : Set} → ((P ∨ Q) ∨ R) → (P ∨ (Q ∨ R))
∨-assoc₁ (∨-intro₁ (∨-intro₁ p)) = ∨-intro₁ p
∨-assoc₁ (∨-intro₁ (∨-intro₂ q)) = ∨-intro₂ (∨-intro₁ q)
∨-assoc₁ (∨-intro₂ r) = ∨-intro₂ (∨-intro₂ r)

∨-assoc₂ : {P Q R : Set} → (P ∨ (Q ∨ R)) → ((P ∨ Q) ∨ R)
∨-assoc₂ (∨-intro₁ p) = ∨-intro₁ (∨-intro₁ p)
∨-assoc₂ (∨-intro₂ (∨-intro₁ q)) = ∨-intro₁ (∨-intro₂ q)
∨-assoc₂ (∨-intro₂ (∨-intro₂ r)) = ∨-intro₂ r

∨-assoc : {P Q R : Set} → ((P ∨ Q) ∨ R) ⇔  (P ∨ (Q ∨ R))
∨-assoc = ∧-intro ∨-assoc₁ ∨-assoc₂

--

data ⊥ : Set where -- nothing

¬ : Set → Set
¬ A = A → ⊥