module _ where

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

{- prove that if the maximum and the minimum of two numbers are the
   same, the numbers themselves must also be the same 

    minmax≡ : ∀ n m -> min n m ≡ max n m -> n ≡ m

   HINT: consider all four cases of `n` and `m` as either
   `zero` or `(suc n)`. Three will either be impossible
   or trivial. For the last one, look in nat-thms.agda
   to see if you find some useful rewrites. You may need
   to do a case split on n < m (which is interesting, as you
   are trying to prove that it never happens)
-}

minmax≡ : ∀ n m -> min n m ≡ max n m -> n ≡ m
minmax≡ = {!!} 

{- 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 other definition, i.e.
   prove these two results: 

≥2->≥ : ∀ n m -> n ≥2 m ≡ tt -> n ≥ m ≡ tt

≥->≥2 : ∀ n m -> n ≥ m ≡ tt -> n ≥2 m ≡ tt

-}


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

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