kernel_ge_one

formal source

 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. -/
 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)