| 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 | |