kernel_at_ratio_one_alpha_zero

formal source

 114  linarith
 115
 116/-- Kernel equals 1 + C when the ratio a/(k τ₀) = 1 and α = 0. -/
 117theorem kernel_at_ratio_one_alpha_zero (P : KernelParams) (hα : P.alpha = 0)
 118    (k a : ℝ) (hk : k ≠ 0) (hratio : a / (k * P.tau0) = 1) (h1ge : (0.01 : ℝ) ≤ 1) :
 119    kernel P k a = 1 + P.C := by
 120  unfold kernel
 121  have hmax : max 0.01 (a / (k * P.tau0)) = 1 := by
 122    rw [hratio]
 123    exact max_eq_right h1ge
 124  simp [hmax, hα, Real.rpow_zero]
 125
 126/-- Kernel equals 1 when C = 0 (no ILG modification). -/
 127theorem kernel_eq_one_of_C_zero (P : KernelParams) (hC : P.C = 0) (k a : ℝ) :
 128    kernel P k a = 1 := by
 129  simp [kernel, hC]
 130
 131/-! ## Monotonicity Properties -/
 132
 133/-- For fixed k and positive α, the kernel is monotonically increasing in a
 134    when a/(k τ₀) ≥ 0.01. -/
 135theorem kernel_mono_in_a (P : KernelParams) (hα_pos : 0 < P.alpha) (hC_pos : 0 < P.C)
 136    (k : ℝ) (hk : 0 < k) (a₁ a₂ : ℝ)
 137    (ha₁ : 0.01 * (k * P.tau0) ≤ a₁) (ha₁₂ : a₁ ≤ a₂) :
 138    kernel P k a₁ ≤ kernel P k a₂ := by
 139  unfold kernel
 140  -- When a ≥ 0.01 * (k τ₀), the max is just a / (k τ₀)
 141  have hktau_pos : 0 < k * P.tau0 := mul_pos hk P.tau0_pos
 142  have hr₁ : 0.01 ≤ a₁ / (k * P.tau0) := by
 143    rwa [le_div_iff₀ hktau_pos]
 144  have hmax₁ : max 0.01 (a₁ / (k * P.tau0)) = a₁ / (k * P.tau0) := max_eq_right hr₁
 145  have hr₂ : 0.01 ≤ a₂ / (k * P.tau0) := by
 146    have : a₁ / (k * P.tau0) ≤ a₂ / (k * P.tau0) := by
 147      apply div_le_div_of_nonneg_right ha₁₂