| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.NormNum |
|
|
| namespace OAI |
|
|
| namespace GammaFPP.FiniteLocal |
|
|
| noncomputable def concentrationKappa (shape rate : ℝ) : ℝ := |
| min (rate/4) (1/(4*((16*(2 : ℝ)^shape/rate^2)+1))) |
|
|
| lemma concentrationKappa_bounds {shape rate : ℝ} (hr : 0 < rate) : |
| 0 < concentrationKappa shape rate ∧ |
| concentrationKappa shape rate ≤ rate/4 ∧ |
| concentrationKappa shape rate ≤ 1/4 ∧ |
| (16*(2 : ℝ)^shape/rate^2)*(concentrationKappa shape rate)^2 ≤ 1/16 := by |
| let B : ℝ := 16*(2 : ℝ)^shape/rate^2 |
| let k := concentrationKappa shape rate |
| have hB : 0 ≤ B := by dsimp [B]; positivity |
| have hk : 0 < k := by dsimp [k, concentrationKappa]; positivity |
| have hkr : k ≤ rate/4 := min_le_left _ _ |
| have hkd : k ≤ 1/(4*(B+1)) := min_le_right _ _ |
| have hd : 0 < 4*(B+1) := by linarith |
| have hm : k*(4*(B+1)) ≤ 1 := (le_div_iff₀ hd).mp hkd |
| have hBk : 0 ≤ B*k := mul_nonneg hB hk.le |
| have hkquarter : k ≤ 1/4 := by nlinarith |
| have hBkquarter : B*k ≤ 1/4 := by nlinarith |
| have hprod := mul_le_mul hBkquarter hkquarter hk.le (by norm_num : (0 : ℝ) ≤ 1/4) |
| refine ⟨hk, hkr, hkquarter, ?_⟩ |
| change B*k^2 ≤ 1/16 |
| nlinarith |
|
|
| end GammaFPP.FiniteLocal |
|
|
| end OAI |
|
|
