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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
/-- The exponent constant is absolute. Fixed coefficients in the
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
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