module _ where

open import nat
open import eq
open import sum
open import bool
open import bool-thms
open import nat-thms

{- Be aware that you can define helper functions (aka lemmas).  Often,
   it is easier to prove some specific lemma on its own and then use
   it in a larger proof. This is very handy even when the main proof
   is only two or three lines, in fact. -}



{-

We did right distributive in class; state and prove that multiplication
also distributes on the right, eg, for any three numbers, x, y, and z:

  z * (x + y) ≡ z * x + z * y

-}


-- This predicate determines if a number is a multiple of three or not
m3 : ℕ -> 𝔹
m3 zero = tt
m3 (suc zero) = ff
m3 (suc (suc zero)) = ff
m3 (suc (suc (suc n))) = m3 n

{- prove that the sum of two multiples of three is also a multiple of three -}
summ3 : ∀ n m -> m3 n ≡ tt -> m3 m ≡ tt -> m3 (n + m) ≡ tt
summ3 = {!!}

{- prove that the product of a multiple of three and any other number
   is also a multiple of three -}
multm3 : ∀ n m -> m3 n ≡ tt -> m3 (n * m) ≡ tt
multm3 = {!!}

{- consider this alternative definition of less than: -}

_≥2_ : nat -> nat -> bool
zero ≥2  zero = tt
zero ≥2 (suc m) = ff
(suc n) ≥2 zero = tt
(suc n) ≥2 (suc m) = n ≥2 m

{- prove that it is the same as the definition in the ial,
   i.e.  prove these two results: -}

≥2->≥ : ∀ n m -> n ≥2 m ≡ tt -> n ≥ m ≡ tt
≥2->≥ = {!!}

≥->≥2 : ∀ n m -> n ≥ m ≡ tt -> n ≥2 m ≡ tt
≥->≥2 = {!!}