import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finite.Basic
import Mathlib.Tactic

/-!
# USA Mathematical Olympiad 2002, Problem 1

Let S be a set with 2002 elements, and let N be an integer with

  0 ≤ N ≤ 2^2002.

Prove that it is possible to color every subset of S either blue or
red so that the following conditions hold:

 a) the union of any two red subsets is red;
 b) the union of any two blue subsets is blue;
 c) there are exactly N red subsets.
-/

namespace Usa2002P1

inductive Color : Type where
| red : Color
| blue : Color
deriving DecidableEq, Fintype

theorem usa2002_p1
    {α : Type} [DecidableEq α] [Fintype α] (hs : Fintype.card α = 2002)
    (N : ℕ) (hN : N ≤ 2 ^ 2002) :
    ∃ f : Finset α → Color,
      ((∀ s1 s2 : Finset α, f s1 = f s2 → f (s1 ∪ s2) = f s1) ∧
       (Fintype.card
           { a : Finset α // f a = Color.red } = N)) := sorry

This problem has a complete solution written by David Renshaw.