File size: 2,647 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 | import OAI.Probability.SATVariance.FiniteVariance
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
lemma integrated_tail_subcritical {F : ℝ → ℝ} (hF : Antitone F)
{A θ : ℝ} (hθ : θ < 1)
(hbound : ∀ a ∈ Set.Ioo (0 : ℝ) 1, F a ≤ A * a^(-θ)) :
(∫ a in (0 : ℝ)..1, F a) ≤ A / (1-θ) := by
have hp : -1 < -θ := by linarith
calc
(∫ a in (0 : ℝ)..1, F a) ≤ ∫ a in (0 : ℝ)..1, A*a^(-θ) := by
apply intervalIntegral.integral_mono_on_of_le_Ioo (by norm_num)
hF.intervalIntegrable ((intervalIntegral.intervalIntegrable_rpow' hp).const_mul A)
exact hbound
_ = A/(1-θ) := by
rw [intervalIntegral.integral_const_mul, integral_rpow (Or.inl hp),
Real.one_rpow, Real.zero_rpow (by linarith : -θ+1 ≠ 0)]
ring
lemma integrated_tail_critical {F : ℝ → ℝ} (hF : Antitone F)
{A L : ℝ} (hA : 1 ≤ A) (hL : 1 ≤ L)
(hmass : ∀ a, F a ≤ L)
(hbound : ∀ a ∈ Set.Ioo (0 : ℝ) 1, F a ≤ A/a) :
(∫ a in (0 : ℝ)..1, F a) ≤ A * (1+Real.log L) := by
have hL₀ : 0 < L := by linarith
have hd : 0 < L⁻¹ := inv_pos.mpr hL₀
have hd1 : L⁻¹ ≤ 1 := (inv_le_one₀ hL₀).mpr hL
have hleft : (∫ a in (0 : ℝ)..L⁻¹, F a) ≤ 1 := by
calc
_ ≤ ∫ _a in (0 : ℝ)..L⁻¹, L := by
exact intervalIntegral.integral_mono_on hd.le hF.intervalIntegrable
intervalIntegrable_const (fun a _ => hmass a)
_ = 1 := by simp [intervalIntegral.integral_const, hL₀.ne']
have hIi : IntervalIntegrable (fun a : ℝ => a⁻¹) volume L⁻¹ 1 := by
apply intervalIntegral.intervalIntegrable_inv _ continuous_id.continuousOn
intro a ha
rw [Set.uIcc_of_le hd1] at ha
exact (hd.trans_le ha.1).ne'
have hright : (∫ a in L⁻¹..(1 : ℝ), F a) ≤ A*Real.log L := by
calc
_ ≤ ∫ a in L⁻¹..(1 : ℝ), A*a⁻¹ := by
apply intervalIntegral.integral_mono_on_of_le_Ioo hd1 hF.intervalIntegrable
(hIi.const_mul A)
intro a ha
simpa only [div_eq_mul_inv] using hbound a ⟨lt_trans hd ha.1, ha.2⟩
_ = A*Real.log L := by
rw [intervalIntegral.integral_const_mul, integral_inv_of_pos hd (by norm_num)]
simp
calc
_ = (∫ a in (0 : ℝ)..L⁻¹, F a) + (∫ a in L⁻¹..(1 : ℝ), F a) :=
(intervalIntegral.integral_add_adjacent_intervals hF.intervalIntegrable
hF.intervalIntegrable).symm
_ ≤ 1 + A*Real.log L := add_le_add hleft hright
_ ≤ A*(1+Real.log L) := by nlinarith
end RandomKSAT
end
end OAI
|