theorem knockoutPunishment_pos {ρ : ℝ}
(hρ0 : 0 < ρ) (_hρ1 : ρ < 1)
{c : ℝ} (hc : 0 < c) (hJ : 2 ≤ J) :
0 < knockoutPunishment J ρ c := by
unfold knockoutPunishment
apply mul_pos hc
have hR := knockout_partial_retained hρ0
(by omega : 0 < J) (by omega : 0 < 1)
(by omega : 1 < J)
linarith [hR.2]Multi-Agent CES Game Theory (Gap #14)