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62291ee | 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 | import OAI.NumberTheory.Ostmann.Conclusion.Scales
namespace OAI
noncomputable section
namespace Ostmann.Conclusion
open scoped BigOperators
lemma bulkScale_one_le {k : ℕ} (hk : 1 ≤ k) : 1 ≤ bulkScale k := by
unfold bulkScale
exact one_le_pow₀ (by exact_mod_cast hk)
theorem frequencyBudget_nonneg {Bs BD Bz : ℝ} (hBs : 0 ≤ Bs) (hBD : 0 ≤ BD) (hBz : 0 ≤ Bz)
{k : ℕ} (hk : 1 ≤ k) (L : ℝ) (l : ℕ) : 0 ≤ frequencyBudget Bs BD Bz k L l := by
have hlog := Real.log_nonneg (bulkScale_one_le hk)
have hgap : 0 ≤ initialGap Bs k L := by unfold initialGap; positivity
have hstep (j : ℕ) : 0 ≤ stepGap BD Bz k L j := by
have hlogj := Real.log_nonneg (show (1:ℝ) ≤ (2:ℝ)^j from one_le_pow₀ (by norm_num))
unfold stepGap
positivity
unfold frequencyBudget
exact add_nonneg (add_nonneg hgap (Finset.sum_nonneg fun j _=>hstep j)) (by positivity)
theorem frequency_card_le_exp {Bs BD Bz : ℝ} (hBs : 0 ≤ Bs) (hBD : 0 ≤ BD) (hBz : 0 ≤ Bz)
{k : ℕ} (hk : 1 ≤ k) (L : ℝ) (l : ℕ) :
2*(frequencyBound Bs BD Bz k L l:ℝ)+1 ≤ 3*Real.exp (frequencyBudget Bs BD Bz k L l) := by
have he := Real.one_le_exp (frequencyBudget_nonneg hBs hBD hBz hk L l)
have hf := Nat.floor_le (Real.exp_pos (frequencyBudget Bs BD Bz k L l)).le
change (frequencyBound Bs BD Bz k L l:ℝ) ≤ _ at hf
linarith
theorem final_bad_exponent_le {Bs BD Bz : ℝ} (hBs : 0 ≤ Bs) (hBD : 0 ≤ BD) (hBz : 0 ≤ Bz)
(C : ℝ) {k : ℕ} (hk : 1 ≤ k) (L : ℝ) :
frequencyBudget Bs BD Bz k L k +
(2-(3/4:ℝ)*Real.log ((2:ℝ)^k))*(2:ℝ)^k*(bulkSize k L:ℝ) +
((2:ℝ)^k*(initialGap Bs k L+C*(bulkSize k L:ℝ))+(bulkSize k L:ℝ)) ≤
finalRate BD Bs Bz (C+2) k*(2:ℝ)^k*(bulkSize k L:ℝ)+
(2:ℝ)^k*Real.sqrt (bulkSize k L)+(bulkSize k L:ℝ) := by
have hlog := Real.log_nonneg (bulkScale_one_le hk)
have hr : (1:ℝ) ≤ (2:ℝ)^k := one_le_pow₀ (by norm_num)
have hlog2 : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
have hnon : 0 ≤ ((2:ℝ)^k-1)*(Bs+8*Real.log (bulkScale k))*(bulkSize k L:ℝ)+
(BD+Bz*Real.log (bulkScale k))*(bulkSize k L:ℝ)+
(4/5:ℝ)*((2:ℝ)^k-1)*Real.log 2*(bulkSize k L:ℝ) := by positivity
rw [frequencyBudget_eq]
unfold finalRate initialGap
nlinarith [hnon]
end Ostmann.Conclusion
end
end OAI
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