| import OAI.Probability.SATVariance.Subcube |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped Classical ENNReal Topology |
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| namespace RandomKSAT |
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| lemma lifetime_ge_survival {u k : ℕ} (hku : k ≤ u) (S : Finset (Assignment u)) (t : ℕ) : |
| (t : ℝ) * (1 - blockKill u k S t) ≤ lifetime u k S := by |
| let := streamLaw_probability u k hku |
| have he : 1 - blockKill u k S t = survival u k S t := by |
| rw [blockKill_compl hku, survival_eq_prob hku] |
| ring |
| have hm (m : ℕ) (hmt : m ≤ t) : survival u k S t ≤ survival u k S m := by |
| rw [survival_eq_prob hku, survival_eq_prob hku] |
| apply ENNReal.toReal_mono (measure_ne_top _ _) |
| apply measure_mono |
| rintro ω ⟨a, ha, hcs⟩ |
| exact ⟨a, ha, fun i hi => hcs i (hi.trans_le hmt)⟩ |
| rw [he, lifetime] |
| calc |
| _ = ∑ _m ∈ Finset.range t, survival u k S t := by simp |
| _ ≤ ∑ m ∈ Finset.range t, survival u k S m := |
| Finset.sum_le_sum fun m hm' => hm m (Finset.mem_range.mp hm').le |
| _ ≤ ∑' m, survival u k S m := |
| (survival_summable hku S).sum_le_tsum (Finset.range t) (fun m _ => survival_nonneg u k S m) |
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| lemma tendsto_nat_pow_real (m : ℕ) (hm : 0 < m) : |
| Tendsto (fun N : ℕ => (N : ℝ)^m) atTop atTop := |
| (tendsto_pow_atTop (Nat.ne_of_gt hm)).comp tendsto_natCast_atTop_atTop |
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| lemma subcube_factor_limit {m : ℕ} (hm : 0 < m) (s : ℕ) : |
| Tendsto (fun N : ℕ => (N : ℝ) * ((N^m : ℕ) - (s : ℝ)) / |
| (2 * ((N^(m+1) : ℕ) - (s : ℝ)))) atTop (𝓝 (1/2 : ℝ)) := by |
| have h1 : Tendsto (fun N : ℕ => (1 : ℝ) - s/(N : ℝ)^m) atTop (𝓝 1) := by |
| simpa using tendsto_const_nhds.sub ((tendsto_const_nhds (x := (s : ℝ))).div_atTop (tendsto_nat_pow_real m hm)) |
| have h2 : Tendsto (fun N : ℕ => (2 : ℝ) * (1 - s/(N : ℝ)^(m+1))) atTop (𝓝 2) := by |
| simpa using (tendsto_const_nhds (x := (2 : ℝ))).mul ((tendsto_const_nhds (x := (1 : ℝ))).sub |
| ((tendsto_const_nhds (x := (s : ℝ))).div_atTop (tendsto_nat_pow_real (m+1) (by omega)))) |
| apply (h1.div h2 (by norm_num)).congr' |
| filter_upwards [eventually_ge_atTop (s+1)] with N hN |
| have hN0 : 0 < N := by omega |
| have hx : (N : ℝ) ≠ 0 := by positivity |
| have hsN : s < N^(m+1) := lt_of_lt_of_le (by omega) (Nat.le_pow (by omega)) |
| have hden : (N : ℝ)^(m+1) - s ≠ 0 := by |
| have hh : (s : ℝ) < (N^(m+1) : ℕ) := by exact_mod_cast hsN |
| push_cast at hh |
| linarith |
| change (1 - (s : ℝ)/(N : ℝ)^m) / (2 * (1 - (s : ℝ)/(N : ℝ)^(m+1))) = _ |
| push_cast |
| field_simp |
| ring |
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| lemma killRatio_limit {m : ℕ} (hm : 0 < m) (s : ℕ) : |
| Tendsto (fun N : ℕ => (N : ℝ)^s * killRatio (N^(m+1)) (N^m) s) |
| atTop (𝓝 (((2 : ℝ)^s)⁻¹)) := by |
| induction s with |
| | zero => simp [killRatio_zero] |
| | succ s ih => |
| have hh := ih.mul (subcube_factor_limit hm s) |
| have hc : ((2 : ℝ)^s)⁻¹ * (1/2) = ((2 : ℝ)^(s+1))⁻¹ := by |
| rw [pow_succ, mul_inv_rev] |
| ring |
| rw [hc] at hh |
| apply hh.congr' |
| filter_upwards [eventually_ge_atTop (s+1)] with N hN |
| have hsb : s ≤ N^m := (by omega : s ≤ N).trans (Nat.le_pow hm) |
| have hsu : s < N^(m+1) := lt_of_lt_of_le (by omega) (Nat.le_pow (by omega)) |
| rw [killRatio_succ hsu hsb, pow_succ] |
| ring |
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| end RandomKSAT |
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| end |
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| end OAI |
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