module distrib where

data _∨_ (A B : Set) : Set where
  inl : A → A ∨ B
  inr : B → A ∨ B

data _∧_ (A B : Set) : Set where
  _and_ : A → B → A ∧ B

and_dist_over_or : {a b c : Set} → a ∧ (b ∨ c) → (a ∧ b) ∨ (a ∧ c)
and_dist_over_or (a₁ and (inl x)) = inl (a₁ and x)
and_dist_over_or (a₁ and (inr x)) = inr (a₁ and x)

and_dist_over_or′ : {a b c : Set} → (a ∧ b) ∨ (a ∧ c) → a ∧ (b ∨ c)
and_dist_over_or′ (inl (x and x′)) = x and (inl x′)
and_dist_over_or′ (inr (x and x′)) = x and (inr x′)

or_dist_over_and : {a b c : Set} → (a ∨ b) ∧ (a ∨ c) -> a ∨ (b ∧ c)
or_dist_over_and ((inl x) and (inl y)) = inl x
or_dist_over_and ((inl x) and (inr y)) = inl x
or_dist_over_and ((inr x) and (inl y)) = inl y
or_dist_over_and ((inr x) and (inr y)) = inr (x and y)

or_dist_over_and′ : {a b c : Set} → a ∨ (b ∧ c) → (a ∨ b) ∧ (a ∨ c)
or_dist_over_and′ (inl x) = (inl x) and (inl x)
or_dist_over_and′ (inr (x and y)) = (inr x) and (inr y)