| import OAI.NumberTheory.Ostmann.Conclusion.Scales |
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| namespace OAI |
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| noncomputable section |
| namespace Ostmann.Conclusion |
| open scoped BigOperators |
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| lemma bulkScale_one_le {k : ℕ} (hk : 1 ≤ k) : 1 ≤ bulkScale k := by |
| unfold bulkScale |
| exact one_le_pow₀ (by exact_mod_cast hk) |
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| 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) |
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| 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 |
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| 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] |
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| end Ostmann.Conclusion |
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| end |
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| end OAI |
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