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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 82 83 84 85 86 87 | import OAI.Probability.SATVariance.MarkovReplacement
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
lemma real_rate_pos {k : ℕ} (hk : 1 ≤ k) : 0 < 1-((2 : ℝ)^k)⁻¹ := by
have hp : (2 : ℝ) ≤ (2 : ℝ)^k := by
simpa using (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) hk)
exact sub_pos.mpr ((inv_lt_one₀ (by positivity)).mpr (by linarith))
lemma real_rate_lt_one (k : ℕ) : 1-((2 : ℝ)^k)⁻¹ < 1 := by
have h : 0 < ((2 : ℝ)^k)⁻¹ := by positivity
linarith
lemma measurable_natH (n k : ℕ) :
Measurable (fun ω : Stream n k => (firstFailure ω).toNat) :=
(measurable_of_countable fun x : ℕ∞ => x.toNat).comp (measurable_firstFailure n k)
lemma natH_mass_bound {n k : ℕ} (hk : 1 ≤ k) (hkn : k ≤ n) (m : ℕ) :
((streamLaw n k).map (fun ω => (firstFailure ω).toNat) {m}).toReal ≤
((2 : ℝ)^n/(1-((2 : ℝ)^k)⁻¹))*(1-((2 : ℝ)^k)⁻¹)^m := by
let := streamLaw_probability n k hkn
have hq := real_rate_pos hk
rw [Measure.map_apply (measurable_natH n k) (by measurability)]
cases m with
| zero =>
simp only [pow_zero, mul_one]
apply (measureReal_le_one (μ := streamLaw n k)).trans
apply (le_div_iff₀ hq).2
have hp := one_le_pow₀ (n := n) (by norm_num : (1 : ℝ) ≤ 2)
have hr := real_rate_lt_one k
linarith
| succ m =>
have hs : (fun ω : Stream n k => (firstFailure ω).toNat) ⁻¹' {m+1} ⊆
{ω | (m : ℕ∞) < firstFailure ω} := by
intro ω hω
have he : firstFailure ω = ((m+1 : ℕ) : ℕ∞) :=
(ENat.toNat_eq_iff (by omega)).mp hω
change (m : ℕ∞) < firstFailure ω
rw [he]
exact_mod_cast Nat.lt_succ_self m
have hh := ENNReal.toReal_mono (by finiteness)
((measure_mono hs).trans (first_moment_tail hkn m))
simp only [ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_ofNat, rate_toReal] at hh
apply hh.trans_eq
rw [pow_succ, ← mul_assoc, mul_right_comm, div_mul_cancel₀ _ hq.ne']
lemma H_integrable_pow {n k : ℕ} (hk : 1 ≤ k) (hkn : k ≤ n) (p : ℕ) :
Integrable (fun ω : Stream n k => (H ω)^p) (streamLaw n k) := by
let := streamLaw_probability n k hkn
let ν := (streamLaw n k).map (fun ω => (firstFailure ω).toNat)
have hp : IsProbabilityMeasure ν := inferInstance
let q := 1-((2 : ℝ)^k)⁻¹
have hq := real_rate_pos hk
have hqNorm : ‖q‖ < 1 := by rw [Real.norm_eq_abs, abs_of_pos hq]; exact real_rate_lt_one k
have hs : Summable (fun m : ℕ => (ν {m}).toReal * ‖(m : ℝ)^p‖) := by
apply Summable.of_nonneg_of_le (fun _ => by positivity) _
((summable_pow_mul_geometric_of_norm_lt_one p hqNorm).mul_left ((2 : ℝ)^n/q))
intro m
rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : (0 : ℝ) ≤ (m : ℝ)^p)]
calc
_ ≤ (((2 : ℝ)^n/q)*q^m)*(m : ℝ)^p :=
mul_le_mul_of_nonneg_right (natH_mass_bound hk hkn m) (by positivity)
_ = _ := by ring
have hi : Integrable (fun m : ℕ => (m : ℝ)^p) ν := by
rw [← Measure.sum_smul_dirac ν]
exact integrable_sum_dirac (fun _ => measure_ne_top _ _) hs
exact hi.comp_measurable (measurable_natH n k)
lemma H_memLp_two {n k : ℕ} (hk : 1 ≤ k) (hkn : k ≤ n) :
MemLp (H : Stream n k → ℝ) 2 (streamLaw n k) :=
(memLp_two_iff_integrable_sq (measurable_H n k).aestronglyMeasurable).mpr
(H_integrable_pow hk hkn 2)
end RandomKSAT
end
end OAI
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