kernelAtRefK_eq

formal source

  83noncomputable def kernelAtRefK (P : KernelParams) (a : ℝ) : ℝ :=
  84  1 + P.C * (max 0.01 (a / P.tau0)) ^ P.alpha
  85
  86@[simp] lemma kernelAtRefK_eq (P : KernelParams) (a : ℝ) :
  87    kernelAtRefK P a = kernel P 1 a := by
  88  simp [kernelAtRefK, kernel, one_mul]
  89
  90/-! ## Basic Properties -/
  91
  92/-- Kernel is always positive for valid parameters. -/
  93theorem kernel_pos (P : KernelParams) (k a : ℝ) : 0 < kernel P k a := by
  94  unfold kernel
  95  have hmax_pos : 0 < max 0.01 (a / (k * P.tau0)) := by
  96    apply lt_max_of_lt_left
  97    norm_num
  98  have hpow_nonneg : 0 ≤ (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
  99    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
 100  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
 101    mul_nonneg P.C_nonneg hpow_nonneg
 102  linarith
 103
 104/-- Kernel is at least 1. -/
 105theorem kernel_ge_one (P : KernelParams) (k a : ℝ) : 1 ≤ kernel P k a := by
 106  unfold kernel
 107  have hmax_pos : 0 < max 0.01 (a / (k * P.tau0)) := by
 108    apply lt_max_of_lt_left
 109    norm_num
 110  have hpow_nonneg : 0 ≤ (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
 111    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
 112  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (a / (k * P.tau0))) ^ P.alpha :=
 113    mul_nonneg P.C_nonneg hpow_nonneg
 114  linarith
 115
 116/-- Kernel equals 1 + C when the ratio a/(k τ₀) = 1 and α = 0. -/