File size: 3,284 Bytes
135052a | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Probability.SATVariance.Subcube
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal Topology
namespace RandomKSAT
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)
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
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
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
end RandomKSAT
end
end OAI
|