```
import Batteries.Data.List.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
/-!
USA Mathematical Olympiad 1999, Problem 1
Some checkers placed on an n × n checkerboard satisfy the following conditions:
a. every square that does not contain a checker shares a side with one that does;
b. given any pair of squares that contain checkers, there is a sequence of squares
containing checkers, starting and ending with the given squares, such that
every two consecutive squares of the sequence share a side.
Prove that at least (n²-2)/3 checkers have been placed on the board.
-/
namespace Usa1999P1
def checkerboard (n : ℕ) := Fin n × Fin n
def adjacent {n : ℕ} : checkerboard n → checkerboard n → Prop
| ⟨a1, a2⟩, ⟨b1, b2⟩ =>
(a1.val = b1.val ∧ a2.val = b2.val + 1) ∨
(a1.val = b1.val ∧ a2.val + 1 = b2.val) ∨
(a2.val = b2.val ∧ a1.val = b1.val + 1) ∨
(a2.val = b2.val ∧ a1.val + 1 = b1.val )
theorem usa1999_p1 (n : ℕ) (c : Finset (checkerboard n))
(ha : ∀ x : checkerboard n, x ∈ c ∨ (∃ y ∈ c, adjacent x y))
(hb : ∀ x ∈ c, ∀ y ∈ c,
∃ p : List (checkerboard n),
List.Chain' adjacent p ∧
List.head? p = x ∧
List.getLast? p = y) :
n^2 ≤ c.card * 3 + 2 := sorry
end Usa1999P1
```

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