| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace CriticalPercolation |
|
|
| lemma collar_error_budget (κ η shells closed : ℝ) (hκ : 0<κ) (hκ1 : κ≤1) |
| (hη : η ≤ κ*(κ/4)/(4*(Real.log (4/κ)+1))) |
| (hclosed : 0<closed) (hshells : 4/(κ*closed)<shells) : |
| 1/(shells*closed)+Real.exp (-Real.log (4/κ))+ |
| (Real.log (4/κ)+1)*η/(κ/4)+κ/4 ≤ κ := by |
| have hratio : 1 ≤ 4/κ := (le_div_iff₀ hκ).mpr (by linarith) |
| have hK : 0 ≤ Real.log (4/κ) := Real.log_nonneg hratio |
| have hK1 : 0<Real.log (4/κ)+1 := by linarith |
| have hlam : 0<κ/4 := by linarith |
| have hden : 0<κ*closed := mul_pos hκ hclosed |
| have hs : 0<shells := (div_pos (by norm_num) hden).trans hshells |
| have hfirst : 1/(shells*closed) ≤ κ/4 := by |
| apply (div_le_iff₀ (mul_pos hs hclosed)).mpr |
| have hh := (div_lt_iff₀ hden).mp hshells |
| nlinarith |
| have hexp : Real.exp (-Real.log (4/κ))=κ/4 := by |
| rw [Real.exp_neg,Real.exp_log (div_pos (by norm_num) hκ)] |
| simp |
| have hthird : (Real.log (4/κ)+1)*η/(κ/4) ≤ κ/4 := by |
| apply (div_le_iff₀ hlam).mpr |
| have hh := (le_div_iff₀ (mul_pos (by norm_num : (0:ℝ)<4) hK1)).mp hη |
| nlinarith |
| rw [hexp] |
| linarith |
|
|
| lemma positive_trial_power (p : ℝ) (hp : 0<p) (m VM : ℕ) : |
| 0<(p/2)^(m+1+VM) := pow_pos (by linarith) _ |
|
|
| lemma trial_ceiling_budget (K s : ℝ) (hs : 0<s) : |
| K ≤ (⌈K/s⌉₊ : ℝ)*s := by |
| have hh : K/s ≤ (⌈K/s⌉₊ : ℝ) := Nat.le_ceil _ |
| exact (div_le_iff₀ hs).mp hh |
|
|
| end CriticalPercolation |
| end |
| end |
| end |
|
|
| end OAI |
|
|
