capabilityCost

formal source

  40
  41/-- J-cost on the relative-capability ratio
  42    (rising_power_capabilities / incumbent_power_capabilities). -/
  43def capabilityCost (rising incumbent : ℝ) : ℝ :=
  44  Jcost (rising / incumbent)
  45
  46theorem capabilityCost_at_parity (c : ℝ) (h : c ≠ 0) :
  47    capabilityCost c c = 0 := by
  48  unfold capabilityCost; rw [div_self h]; exact Jcost_unit0
  49
  50theorem capabilityCost_nonneg (r i : ℝ) (hr : 0 < r) (hi : 0 < i) :
  51    0 ≤ capabilityCost r i := by
  52  unfold capabilityCost; exact Jcost_nonneg (div_pos hr hi)
  53
  54/-- War window: capability ratio in (1/φ, φ). -/
  55def warWindowLow : ℝ := phi⁻¹
  56def warWindowHigh : ℝ := phi
  57
  58theorem warWindowLow_pos : 0 < warWindowLow :=
  59  inv_pos.mpr phi_pos
  60
  61theorem warWindowHigh_gt_one : 1 < warWindowHigh := one_lt_phi
  62
  63theorem warWindow_ordered : warWindowLow < warWindowHigh := by
  64  unfold warWindowLow warWindowHigh
  65  -- phi⁻¹ < phi since phi > 1 implies phi * phi > 1 > phi * phi⁻¹ = 1... direct:
  66  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  67  rw [inv_eq_one_div]
  68  rw [div_lt_iff₀ phi_pos]
  69  nlinarith [phi_gt_onePointFive]
  70
  71/-- At the φ-boundary, J-cost equals the canonical recognition quantum. -/
  72theorem capabilityCost_at_phi_boundary :
  73    capabilityCost phi 1 = phi - 3 / 2 := by