| import OAI.NumberTheory.TwoPoint.Walks.ForestCodeAsymptotics |
| import OAI.NumberTheory.TwoPoint.Walks.ColumnDecoder |
|
|
| namespace OAI |
|
|
| /-! Exponential bound for the universal full-column decoder. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Filter |
|
|
| / |
| sublinear segment budget only change the lower threshold for `L`. -/ |
| theorem eventually_column_decoder_exponential (C : ℝ) (hC : 0 ≤ C) : |
| ∀ᶠ L : ℝ in atTop, ∀ k N segments omitted imperfect : ℕ, |
| L / 2 ≤ (k : ℝ) → (k : ℝ) ≤ L → 1 ≤ N → N ≤ 4 * k → |
| ((2 * segments + omitted + imperfect : ℕ) : ℝ) ≤ C * L ^ (0.92 : ℝ) → |
| (Fintype.card (ColumnDecoderCode N segments omitted imperfect) : ℝ) ≤ |
| Real.exp (52 * k) := by |
| filter_upwards [eventually_ge_atTop 4, eventually_forest_endpoint_cost C hC] with L hL hcost |
| intro k N segments omitted imperfect hklo hkhi hN hNk hM |
| let M := 2 * segments + omitted + imperfect |
| have hNpos : 0 < (N : ℝ) := by exact_mod_cast (by omega : 0 < N) |
| have hNreal : (N : ℝ) ≤ 4 * L := by |
| have h := (show (N : ℝ) ≤ 4 * k by exact_mod_cast hNk) |
| linarith |
| have hlogN : Real.log N ≤ Real.log (4 * L) := Real.log_le_log hNpos hNreal |
| have hlogpos : 0 ≤ Real.log (4 * L) := Real.log_nonneg (by linarith) |
| have hMcost : (M : ℝ) * Real.log N ≤ L := by |
| calc |
| _ ≤ (M : ℝ) * Real.log (4 * L) := mul_le_mul_of_nonneg_left hlogN (by positivity) |
| _ ≤ (C * L ^ (0.92 : ℝ)) * Real.log (4 * L) := |
| mul_le_mul_of_nonneg_right hM hlogpos |
| _ ≤ L := hcost |
| have htwo : (2 : ℝ) ≤ Real.exp 1 := by linarith [Real.add_one_le_exp (1 : ℝ)] |
| have hbits : (2 : ℝ) ^ (6 * N + 1) ≤ Real.exp (6 * N + 1) := by |
| calc |
| _ ≤ (Real.exp 1) ^ (6 * N + 1) := pow_le_pow_left₀ (by norm_num) htwo _ |
| _ = _ := by rw [← Real.exp_nat_mul]; congr 1; push_cast; ring |
| have hcard : (Fintype.card (ColumnDecoderCode N segments omitted imperfect) : ℝ) ≤ |
| (2 : ℝ) ^ (6 * N + 1) * (N : ℝ) ^ M := by |
| exact_mod_cast card_columnDecoderCode_le N segments omitted imperfect |
| calc |
| _ ≤ (2 : ℝ) ^ (6 * N + 1) * (N : ℝ) ^ M := hcard |
| _ ≤ Real.exp (6 * N + 1) * Real.exp ((M : ℝ) * Real.log N) := by |
| rw [Real.exp_nat_mul, Real.exp_log hNpos] |
| exact mul_le_mul_of_nonneg_right hbits (by positivity) |
| _ = Real.exp ((6 * N + 1) + (M : ℝ) * Real.log N) := (Real.exp_add _ _).symm |
| _ ≤ Real.exp (52 * k) := by |
| apply Real.exp_le_exp.mpr |
| have hNbound : (N : ℝ) ≤ 4 * k := by exact_mod_cast hNk |
| linarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
