theorem convex_distiller_expensive {p_0 β Q target : ℝ}
(_hp : 0 < p_0) (_hβ : 0 < β) (_hQ : 0 < Q)
(hlarge : 2 * target / (p_0 * β) < Q) :
target < convexAvgCost p_0 β Q := by
unfold convexAvgCost
have hpβ : 0 < p_0 * β := mul_pos _hp _hβ
rw [div_lt_iff₀ hpβ] at hlarge; nlinarithPaper 10: The Knowledge Commons Paradox: