dlnw_pos

formal source

  52    · repeat apply mul_nonneg <;> (try exact le_of_lt hB) <;> (try exact le_of_lt hXinv_pow)
  53
  54/-- Lemma: In ILG baseline, the kernel log-derivative dlnw/dlna is positive. -/
  55theorem dlnw_pos (P : KernelParams) (k a : ℝ) (ha : 0 < a) (hk : 0 < k)
  56    (halpha : 0 < P.alpha) (hC : 0 < P.C) :
  57    0 < (P.alpha * P.C * (a / (k * P.tau0)) ^ P.alpha) / (1 + P.C * (a / (k * P.tau0)) ^ P.alpha) := by
  58  set Xinv := a / (k * P.tau0)
  59  have hXinv : 0 < Xinv := div_pos ha (mul_pos hk P.tau0_pos)
  60  have hXinv_pow : 0 < Xinv ^ P.alpha := rpow_pos_of_pos hXinv _
  61  apply div_pos
  62  · repeat apply mul_pos <;> assumption
  63  · apply add_pos_of_pos_of_nonneg
  64    · exact one_pos
  65    · repeat apply mul_nonneg <;> (try exact le_of_lt hC) <;> (try exact le_of_lt hXinv_pow)
  66
  67/-- Main Theorem (Target E): The ISW driver B(a,k) is strictly positive in ILG baseline. -/
  68theorem isw_driver_positive (P : KernelParams) (k a : ℝ) (ha : 0 < a) (hk : 0 < k)
  69    (halpha : 0 < P.alpha) (hC : 0 < P.C) :
  70    0 < isw_driver P k a := by
  71  unfold isw_driver
  72  have hf := f_growth_gt_one P k a ha hk halpha hC
  73  have hd := dlnw_pos P k a ha hk halpha hC
  74  linarith
  75
  76end ILG
  77end IndisputableMonolith