complexDemand

formal source

 164        demand(L, Z) = barrierPeriod · J(L) · (1 + |Z| · k_R)
 165
 166    Higher Z requires more recognition events per barrier cycle. -/
 167noncomputable def complexDemand (L : ℝ) (Z : ℤ) : ℝ :=
 168  maintenanceDemand L * (1 + |Z| * k_R)
 169
 170/-- Complex demand ≥ simple demand for any Z. -/
 171theorem complexDemand_ge {L : ℝ} (hL : 0 < L) (Z : ℤ) :
 172    maintenanceDemand L ≤ complexDemand L Z := by
 173  unfold complexDemand
 174  have hd := maintenanceDemand_nonneg hL
 175  have hfac : 1 ≤ 1 + ↑|Z| * k_R := by
 176    have : 0 ≤ ↑|Z| * k_R := mul_nonneg (by exact_mod_cast abs_nonneg Z) (le_of_lt k_R_pos)
 177    linarith
 178  calc maintenanceDemand L
 179      = maintenanceDemand L * 1 := (mul_one _).symm
 180    _ ≤ maintenanceDemand L * (1 + ↑|Z| * k_R) := by
 181        apply mul_le_mul_of_nonneg_left hfac hd
 182
 183/-- Higher Z-complexity strictly increases demand (when J > 0). -/
 184theorem higher_Z_more_demand {L : ℝ} (hL : 0 < L) (hL1 : L ≠ 1)
 185    {Z₁ Z₂ : ℤ} (hZ : |Z₁| < |Z₂|) :
 186    complexDemand L Z₁ < complexDemand L Z₂ := by
 187  unfold complexDemand
 188  have hd : 0 < maintenanceDemand L := by
 189    unfold maintenanceDemand
 190    apply mul_pos barrierPeriod_pos
 191    have : Cost.Jcost L ≠ 0 := by
 192      intro h
 193      exact hL1 ((Cost.Jcost_eq_zero_iff L hL).mp h)
 194    exact lt_of_le_of_ne (Cost.Jcost_nonneg hL) (Ne.symm this)
 195  apply mul_lt_mul_of_pos_left _ hd
 196  have : (↑|Z₁| : ℝ) < ↑|Z₂| := Int.cast_lt.mpr hZ
 197  linarith [mul_lt_mul_of_pos_right this k_R_pos]