| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem exp_quadratic_remainder (x : ℝ) : |
| Real.exp x ≤ 1+x+x^2*Real.exp |x| := by |
| have H := Complex.norm_exp_sub_sum_le_norm_mul_exp (x:ℂ) 2 |
| norm_num [Finset.sum_range_succ,← Complex.ofReal_exp,← Complex.ofReal_add, |
| ← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs,sq_abs] at H |
| have he : (↑(Real.exp x)-(1+(x:ℂ)))=((Real.exp x-(1+x):ℝ):ℂ) := by push_cast; ring |
| rw [he,Complex.norm_real,Real.norm_eq_abs] at H |
| have HL := le_abs_self (Real.exp x-(1+x)) |
| linarith |
|
|
| theorem fourth_pow_le_exp_square {c : ℝ} (hc : 0<c) (y : ℝ) : |
| y^4 ≤ 2/c^2*Real.exp (c*y^2) := by |
| have H := Real.pow_div_factorial_le_exp (c*y^2) (mul_nonneg hc.le (sq_nonneg y)) 2 |
| norm_num at H |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ (sq_pos_of_pos hc)).mpr |
| calc |
| y^4*c^2 ≤ 2*Real.exp (c*y^2) := by nlinarith |
| _ = (2*Real.exp (c*y^2)) := rfl |
|
|
| theorem log_integral_exp_quadratic_bound {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (Y Δ : Ω → ℝ) |
| (hY : AEStronglyMeasurable Y μ) (hΔ : Integrable Δ μ) |
| {c δ M : ℝ} (hc : 0<c) (hδ : 4*δ^2 ≤ c) |
| (hE : Integrable (fun z => Real.exp (c*(Y z)^2)) μ) |
| (hM : (∫ z, Real.exp (c*(Y z)^2) ∂μ) ≤ M) |
| (hb : ∀ᵐ z ∂μ, |Δ z| ≤ 2*δ^2*(Y z)^2) : |
| Real.log (∫ z, Real.exp (Δ z) ∂μ) ≤ (∫ z, Δ z ∂μ)+ |
| (32/c^2)*δ^4*M := by |
| have hE : Integrable (fun z => Real.exp (c*(Y z)^2)) μ := |
| ⟨Real.continuous_exp.comp_aestronglyMeasurable ((hY.pow 2).const_mul c), |
| hE.hasFiniteIntegral⟩ |
| have hη : ∀ z, 2*δ^2*(Y z)^2 ≤ c/2*(Y z)^2 := by |
| intro z |
| exact mul_le_mul_of_nonneg_right (by linarith) (sq_nonneg _) |
| have hdom : ∀ᵐ z ∂μ, Real.exp (Δ z) ≤ Real.exp (c*(Y z)^2) := by |
| filter_upwards [hb] with z hz |
| apply Real.exp_le_exp.mpr |
| exact (le_abs_self _).trans (hz.trans (by |
| have H := hη z |
| nlinarith [mul_nonneg hc.le (sq_nonneg (Y z))])) |
| have hi : Integrable (fun z => Real.exp (Δ z)) μ := |
| hE.mono' (Real.continuous_exp.comp_aestronglyMeasurable hΔ.aestronglyMeasurable) (by |
| filter_upwards [hdom] with z hz |
| simpa only [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] using hz) |
| have hpos : 0 < (∫ z, Real.exp (Δ z) ∂μ) := integral_pos_iff_support_of_nonneg_ae |
| (ae_of_all _ (fun z => (Real.exp_pos (Δ z)).le)) hi |>.mpr (by simp [Function.support,Real.exp_ne_zero]) |
| have hr : ∀ᵐ z ∂μ, Real.exp (Δ z) ≤ 1+Δ z+ |
| (32/c^2)*δ^4*Real.exp (c*(Y z)^2) := by |
| filter_upwards [hb] with z hz |
| apply (exp_quadratic_remainder (Δ z)).trans |
| apply add_le_add_right |
| have hsq : (Δ z)^2 ≤ 4*δ^4*(Y z)^4 := by |
| have H := pow_le_pow_left₀ (abs_nonneg (Δ z)) hz 2 |
| rw [sq_abs] at H |
| nlinarith [H] |
| have hp := fourth_pow_le_exp_square (half_pos hc) (Y z) |
| have hexp : Real.exp |Δ z| ≤ Real.exp (c/2*(Y z)^2) := |
| Real.exp_le_exp.mpr (hz.trans (hη z)) |
| calc |
| _ ≤ (4*δ^4*(Y z)^4)*Real.exp (c/2*(Y z)^2) := |
| mul_le_mul hsq hexp (Real.exp_nonneg _) (by positivity) |
| _ ≤ (4*δ^4*(2/(c/2)^2*Real.exp (c/2*(Y z)^2)))*Real.exp (c/2*(Y z)^2) := |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hp (by positivity)) (Real.exp_nonneg _) |
| _ = (4*δ^4*(2/(c/2)^2))*(Real.exp (c/2*(Y z)^2)*Real.exp (c/2*(Y z)^2)) := by ring |
| _ = _ := by |
| rw [← Real.exp_add,show c/2*(Y z)^2+c/2*(Y z)^2=c*(Y z)^2 by ring] |
| congr 1 |
| field_simp |
| ring |
| have H := integral_mono_ae hi ((integrable_const (1:ℝ)).add hΔ |>.add (hE.const_mul ((32/c^2)*δ^4))) hr |
| simp only [Pi.add_apply] at H |
| have Hsum := integral_add ((integrable_const (1:ℝ)).add hΔ) (hE.const_mul ((32/c^2)*δ^4)) |
| have Hsum2 := integral_add (integrable_const (1:ℝ)) hΔ |
| simp only [Pi.add_apply] at Hsum Hsum2 |
| rw [Hsum,Hsum2,integral_const_mul] at H |
| simp only [integral_const,probReal_univ,smul_eq_mul,one_mul] at H |
| have HM := mul_le_mul_of_nonneg_left hM (by positivity : 0 ≤ (32/c^2)*δ^4) |
| have HL := Real.log_le_sub_one_of_pos hpos |
| linarith |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end OAI |
| |
