| import OAI.Geometry.Convex.GeneralMahler.Density |
|
|
| namespace OAI |
| /-! One-dimensional Gaussian layers, preliminary identities and bounds (§§01--02). -/ |
| noncomputable section |
| open MeasureTheory MeasureTheory.Measure Real Metric Filter Topology Set ProbabilityTheory |
| open scoped ENNReal NNReal |
| namespace GeneralMahler |
| namespace Layers |
|
|
| def p (z : ℝ) := ∫ x in Iic z, phi x |
| def a (z : ℝ) := phi z + z*p z |
|
|
| lemma d_phi (z : ℝ) : HasDerivAt phi (-z*phi z) z := by |
| simp_rw [funext phi_apply] |
| have he := ((hasDerivAt_id' z).pow 2).neg.div_const 2 |
| convert he.exp.const_mul (√(2 * π))⁻¹ using 1 |
| all_goals first | rfl | (simp; ring) |
|
|
| lemma c_phi : Continuous phi := |
| continuous_iff_continuousAt.mpr fun x => (d_phi x).continuousAt |
| lemma i_phi : Integrable phi := integrable_gaussianPDFReal _ _ |
| lemma e_phi : ∫ x, phi x = 1 := integral_gaussianPDFReal_eq_one _ one_ne_zero |
|
|
| lemma int_Ioi (z : ℝ) : ∫ x in Ioi z, phi x = 1-p z := by |
| have h := integral_add_compl measurableSet_Iic (μ := volume) i_phi (s := Iic z) |
| rw [compl_Iic,e_phi] at h |
| dsimp [p] |
| linarith |
|
|
| lemma neg_phi (x : ℝ) : phi (-x) = phi x := by simp [phi_apply] |
|
|
| lemma neg_p (z : ℝ) : p (-z) = 1-p z := by |
| rw [← int_Ioi, ← integral_Ici_eq_integral_Ioi] |
| unfold p |
| let e := (Homeomorph.neg ℝ).toMeasurableEquiv |
| have he := setIntegral_map_equiv (μ := volume) e phi (Iic (-z)) |
| have hm : Measure.map e volume = volume := measurePreserving_neg _ |>.map_eq |
| rw [hm] at he |
| rw [he] |
| have he' : e ⁻¹' Iic (-z) = Ici z := by ext; change (-_ ≤ -z) ↔ _; simp |
| rw [he'] |
| exact integral_congr_ae (ae_of_all _ fun x => neg_phi x) |
|
|
| lemma p_half : p 0 = 1/2 := by |
| have he := neg_p 0 |
| rw [neg_zero] at he; linarith |
|
|
| lemma d_p (z : ℝ) : HasDerivAt p (phi z) z := by |
| have H : p = fun x => p 0 + ∫ t in (0:ℝ)..x, phi t := by |
| ext x |
| rw [p,p, ← intervalIntegral.integral_Iic_sub_Iic i_phi.integrableOn i_phi.integrableOn] |
| ring |
| rw [H] |
| exact ((intervalIntegral.integral_hasDerivAt_right (c_phi.intervalIntegrable ..) |
| (c_phi.stronglyMeasurableAtFilter _ _) c_phi.continuousAt).const_add _) |
|
|
| lemma d_a (z : ℝ) : HasDerivAt a (p z) z := by |
| have he := (d_phi z).add ((hasDerivAt_id' z).mul (d_p z)) |
| convert he using 1 |
| all_goals first | rfl | simp |
|
|
| lemma cp : Continuous p := continuous_iff_continuousAt.mpr fun x => (d_p x).continuousAt |
| lemma ca : Continuous a := continuous_iff_continuousAt.mpr fun x => (d_a x).continuousAt |
| lemma p_pos (z : ℝ) : 0 < p z := by |
| apply (setIntegral_pos_iff_support_of_nonneg_ae |
| (ae_of_all _ fun x => (phi_pos _).le) i_phi.integrableOn).mpr |
| apply lt_of_lt_of_le (isOpen_Iio.measure_pos volume (Set.nonempty_Iio (a := z))) |
| apply measure_mono |
| exact fun x hx => ⟨(phi_pos x).ne',show x ≤ z from le_of_lt hx⟩ |
|
|
| lemma p_lt_one (z : ℝ) : p z < 1 := by |
| have h := p_pos (-z) |
| rw [neg_p] at h; linarith |
|
|
| lemma p_mono : StrictMono p := strictMono_of_deriv_pos fun z => by |
| rw [(d_p _).deriv]; apply p_pos_phi |
| where p_pos_phi := phi_pos |
| lemma a_mono : StrictMono a := strictMono_of_deriv_pos fun z => by |
| rw [(d_a _).deriv]; apply p_pos |
|
|
| lemma phi_bdd (z : ℝ) : phi z ≤ (√(2*π))⁻¹ := by |
| rw [phi_apply] |
| apply mul_le_of_le_one_right (by positivity) |
| exact exp_le_one_iff.mpr (by nlinarith) |
|
|
| lemma int_moment1 : Integrable (fun z => z * phi z) := by |
| have h := (integrable_mul_exp_neg_mul_sq (b := 1/2) (by norm_num)).const_mul (√(2*π))⁻¹ |
| convert h using 1 |
| ext x |
| rw [phi_apply] |
| ring_nf |
|
|
| lemma moment1 (z : ℝ) : ∫ x in Iic z, -(x * phi x) = phi z := by |
| have hlim : Tendsto phi atBot (𝓝 0) := by |
| rw [funext phi_apply] |
| have he : Tendsto (fun x:ℝ => -(x ^ 2)/2) atBot atBot := by |
| refine tendsto_atBot.mpr fun a => ?_ |
| filter_upwards [eventually_le_atBot (-|2*a|-1)] with x hx |
| nlinarith [neg_abs_le (2*a), abs_nonneg (2*a)] |
| convert (Real.tendsto_exp_atBot.comp he).const_mul ((√(2*π))⁻¹) using 1 <;> simp |
| refine Eq.trans (integral_Iic_of_hasDerivAt_of_tendsto' ?_ int_moment1.neg.integrableOn hlim) (sub_zero _) |
| exact fun x hx => by simpa using d_phi x |
|
|
| lemma int_gap (z:ℝ) : Integrable (fun x => (z-x)*phi x) := by |
| simp_rw [sub_mul] |
| exact (i_phi.const_mul z).sub int_moment1 |
|
|
| lemma a_eq (z : ℝ) : a z = ∫ x in Iic z, (z-x)*phi x := by |
| simp_rw [sub_mul] |
| rw [integral_sub (i_phi.const_mul _).integrableOn int_moment1.integrableOn, integral_const_mul] |
| have hh := moment1 z |
| rw [integral_neg] at hh |
| dsimp [a,p]; rw [← hh]; ring |
|
|
| lemma a_pos (z : ℝ) : 0 < a z := by |
| rw [a_eq] |
| apply (setIntegral_pos_iff_support_of_nonneg_ae ?_ (int_gap z).integrableOn).mpr |
| · apply lt_of_lt_of_le (isOpen_Iio.measure_pos volume (Set.nonempty_Iio (a := z))) |
| apply measure_mono |
| intro x hx |
| exact ⟨ne_of_gt (mul_pos (sub_pos.mpr hx) (phi_pos x)),show x ≤ z from le_of_lt hx⟩ |
| filter_upwards [ae_restrict_mem measurableSet_Iic] with x hx |
| exact mul_nonneg (sub_nonneg.mpr hx) (phi_pos x).le |
|
|
| -- Two-sided estimates, exact constants unimportant here. |
| lemma a_high (s : ℝ) (hs : 0 ≤ s) : a (-s) ≤ phi s := by |
| rw [a, neg_phi] |
| have h := p_pos (-s); nlinarith |
|
|
| lemma a_low (s : ℝ) (hs : 0 ≤ s) : |
| phi s * exp (-4) / (1+s)^2 ≤ a (-s) := by |
| let b : ℝ := (1+s)⁻¹ |
| have hp : 0 < 1+s := by linarith |
| have hb : 0 < b := inv_pos.mpr hp |
| have hbr : b * (1+s) = 1 := inv_mul_cancel₀ hp.ne' |
| have hbs : b*s ≤ 1 := by nlinarith |
| have hb1 : b ≤ 1 := by nlinarith |
| let S := Ioc (-s-2*b) (-s-b) |
| have hm : MeasurableSet S := measurableSet_Ioc |
| have hsub : S ⊆ Iic (-s) := fun x hx => by |
| change x ≤ -s; exact hx.2.trans (by linarith) |
| have hlow (x) (hx : x ∈ S) : phi s*exp (-4)*b ≤ (-s-x)*phi x := by |
| have hh : -s-2*b < x ∧ x ≤ -s-b := hx |
| have heQ : x ^ 2 ≤ s ^ 2 + 8 := by |
| have h1 : x ^ 2 ≤ (-s-2*b)^2 := by nlinarith |
| nlinarith |
| have hf : phi s * exp (-4) ≤ phi x := by |
| rw [phi_apply,phi_apply,mul_assoc,← Real.exp_add] |
| gcongr |
| linarith |
| rw [mul_comm _ (phi x)] |
| exact mul_le_mul hf (by linarith) hb.le (phi_pos x).le |
| calc |
| _ = ∫ x in S, phi s*exp (-4)*b := by |
| rw [setIntegral_const] |
| have he : volume.real S = b := by |
| rw [measureReal_def,Real.volume_Ioc,ENNReal.toReal_ofReal] |
| · ring |
| linarith |
| rw [he,smul_eq_mul] |
| dsimp [b]; field_simp |
| _ ≤ ∫ x in S, (-s-x)*phi x := setIntegral_mono_on |
| (integrableOn_const (hs := measure_Ioc_lt_top.ne)) ((int_gap _).integrableOn) hm hlow |
| _ ≤ a (-s) := by |
| rw [a_eq] |
| apply setIntegral_mono_set (int_gap _).integrableOn _ (Filter.Eventually.of_forall hsub) |
| filter_upwards [ae_restrict_mem measurableSet_Iic] with x hx |
| exact mul_nonneg (sub_nonneg.mpr hx) (phi_pos _).le |
|
|
| end Layers |
| end GeneralMahler |
|
|
| end |
|
|
| end OAI |
|
|
