| 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 | |