| import OAI.NumberTheory.TwoPoint.Bounds.BoundedColumnCodes |
| import Mathlib.Data.Nat.Cast.Order.Field |
|
|
| namespace OAI |
|
|
| /-! The actual short-block budgets have the exponent required by the column code. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Filter |
|
|
| lemma eventually_short_block_floor : |
| ∀ᶠ L : ℝ in atTop, L ^ (1 / 10 : ℝ) / 2 ≤ (⌊L ^ (1 / 10 : ℝ)⌋₊ : ℝ) ∧ |
| 0 < ⌊L ^ (1 / 10 : ℝ)⌋₊ := by |
| have h := (tendsto_rpow_atTop (show 0 < (1 / 10 : ℝ) by norm_num)).eventually |
| (eventually_ge_atTop 2) |
| filter_upwards [h] with L hL |
| have hf := Nat.lt_floor_add_one (L ^ (1 / 10 : ℝ)) |
| have hp : (0 : ℝ) < (⌊L ^ (1 / 10 : ℝ)⌋₊ : ℝ) := by linarith |
| exact ⟨by linarith, by exact_mod_cast hp⟩ |
|
|
| / |
| remaining constants are absolute, and the exponent is exactly `0.92`. -/ |
| theorem column_budget_bound (L : ℝ) (hL : 1 ≤ L) |
| (n s imperfect omittedLabels blocks segments omittedRuns : ℕ) |
| (hn : (n : ℝ) ≤ 2 * L) |
| (hs : L ^ (1 / 10 : ℝ) / 2 ≤ (s : ℝ)) |
| (hI : (imperfect : ℝ) ≤ 2 * L ^ (1 / 4 : ℝ)) |
| (hO : (omittedLabels : ℝ) ≤ 2 * L ^ (1 / 50 : ℝ)) |
| (hB : blocks ≤ n / s + imperfect + 2) |
| (hS : segments ≤ (omittedLabels + 1) * blocks) |
| (hR : omittedRuns ≤ omittedLabels * blocks) : |
| ((2 * segments + omittedRuns + imperfect : ℕ) : ℝ) ≤ 66 * L ^ (0.92 : ℝ) := by |
| have hLp : 0 < L := lt_of_lt_of_le zero_lt_one hL |
| have hspos : (0 : ℝ) < s := (by positivity : (0 : ℝ) < L ^ (1 / 10 : ℝ) / 2).trans_le hs |
| have hr : 1 ≤ L ^ (1 / 50 : ℝ) := Real.one_le_rpow hL (by norm_num) |
| have hd : 1 ≤ L ^ (9 / 10 : ℝ) := Real.one_le_rpow hL (by norm_num) |
| have hId : (imperfect : ℝ) ≤ 2 * L ^ (9 / 10 : ℝ) := hI.trans |
| (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hL (by norm_num)) (by norm_num)) |
| have hprod : L ^ (9 / 10 : ℝ) * L ^ (1 / 10 : ℝ) = L := by |
| rw [← Real.rpow_add hLp] |
| norm_num |
| have hdiv : ((n / s : ℕ) : ℝ) ≤ 4 * L ^ (9 / 10 : ℝ) := by |
| apply Nat.cast_div_le.trans |
| apply (div_le_iff₀ hspos).mpr |
| have hm := mul_le_mul_of_nonneg_left hs (show 0 ≤ 4 * L ^ (9 / 10 : ℝ) by positivity) |
| calc |
| (n : ℝ) ≤ 2 * L := hn |
| _ = 2 * (L ^ (9 / 10 : ℝ) * L ^ (1 / 10 : ℝ)) := by rw [hprod] |
| _ = 4 * L ^ (9 / 10 : ℝ) * (L ^ (1 / 10 : ℝ) / 2) := by ring |
| _ ≤ _ := hm |
| have hB' : (blocks : ℝ) ≤ 8 * L ^ (9 / 10 : ℝ) := by |
| have hb : (blocks : ℝ) ≤ ((n / s : ℕ) : ℝ) + imperfect + 2 := by exact_mod_cast hB |
| linarith |
| have hS' : (segments : ℝ) ≤ ((omittedLabels : ℝ) + 1) * blocks := by exact_mod_cast hS |
| have hR' : (omittedRuns : ℝ) ≤ (omittedLabels : ℝ) * blocks := by exact_mod_cast hR |
| have hfactor : 3 * (omittedLabels : ℝ) + 2 ≤ 8 * L ^ (1 / 50 : ℝ) := by linarith |
| have hcost : (2 * (segments : ℝ) + omittedRuns + imperfect) ≤ |
| (3 * (omittedLabels : ℝ) + 2) * blocks + imperfect := by nlinarith |
| have hm := mul_le_mul hfactor hB' (Nat.cast_nonneg blocks) (by positivity) |
| have hI' : (imperfect : ℝ) ≤ 2 * (L ^ (1 / 50 : ℝ) * L ^ (9 / 10 : ℝ)) := by |
| have ht := mul_le_mul_of_nonneg_right hr (Real.rpow_nonneg hLp.le (9 / 10 : ℝ)) |
| nlinarith |
| have hp : L ^ (1 / 50 : ℝ) * L ^ (9 / 10 : ℝ) = L ^ (0.92 : ℝ) := by |
| rw [← Real.rpow_add hLp] |
| norm_num |
| push_cast |
| nlinarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
