lean/OAI/Probability/GaussianPropeller/Analytic.lean

import OAI.Probability.GaussianPropeller.ProbabilityBounds

namespace OAI

universe uE

open MeasureTheory ProbabilityTheory
open scoped ENNReal
open scoped RealInnerProductSpace
open scoped RealInnerProductSpace
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal RealInnerProductSpace
open Filter
open scoped Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
open scoped RealInnerProductSpace
open Set Filter
open scoped Topology RealInnerProductSpace
open scoped NNReal
open Set Filter
open scoped Topology RealInnerProductSpace NNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology BigOperators

open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology

namespace GaussianPropeller.Analytic

theorem integral_pos_mul_exp {b : ℝ} (hb : 0 < b) :
    ∫ r : ℝ in Ioi 0, r * Real.exp (-b * r ^ 2) = (2 * b)⁻¹ := by
  have hb' : b ≠ 0 := ne_of_gt hb
  have hd (x : ℝ) : HasDerivAt (fun x => -(2 * b)⁻¹ * Real.exp (-b * x ^ 2))
      (x * Real.exp (-b * x ^ 2)) x := by
    apply (((hasDerivAt_pow 2 x).const_mul (-b)).exp.const_mul (-(2 * b)⁻¹)).congr_deriv
    field_simp [hb']
    ring
  have ht : Tendsto (fun y : ℝ => -(2 * b)⁻¹ * Real.exp (-b * y ^ 2))
      atTop (𝓝 (-(2 * b)⁻¹ * 0)) := by
    apply Tendsto.const_mul
    exact Real.tendsto_exp_atBot.comp
      ((tendsto_pow_atTop two_ne_zero).const_mul_atTop_of_neg (neg_lt_zero.mpr hb))
  convert integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ => hd x)
    (integrable_mul_exp_neg_mul_sq hb).integrableOn ht using 1
  simp

theorem integral_abs_mul_exp {b : ℝ} (hb : 0 < b) :
    ∫ r : ℝ, |r| * Real.exp (-b * r ^ 2) = b⁻¹ := by
  let f : ℝ → ℝ := fun r => |r| * Real.exp (-b * r ^ 2)
  have hf : Integrable f := by
    simpa only [f, abs_mul, abs_of_pos (Real.exp_pos _)] using
      (integrable_mul_exp_neg_mul_sq hb).abs
  have hpos : ∫ r in Ioi 0, f r = (2 * b)⁻¹ := by
    rw [← integral_pos_mul_exp hb]
    exact setIntegral_congr_fun measurableSet_Ioi (fun r hr => by
      simp only [f, abs_of_pos (show (0 : ℝ) < r from hr)])
  have hneg : ∫ r in Iic 0, f r = ∫ r in Ioi 0, f r := by
    calc
      ∫ r in Iic 0, f r = ∫ r in Ioi 0, f (-r) := by
        simpa [f] using (integral_comp_neg_Ioi 0 f).symm
      _ = ∫ r in Ioi 0, f r := by simp only [f, abs_neg, neg_sq]
  change (∫ r, f r) = _
  rw [← integral_add_compl measurableSet_Ioi hf, compl_Ioi, hneg, hpos]
  field_simp
  ring

theorem integral_abs_gaussianReal_std :
    ∫ x : ℝ, |x| ∂gaussianReal 0 1 = 2 / Real.sqrt (2 * Real.pi) := by
  rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1 : NNReal) ≠ 0)]
  have hfun : (fun x : ℝ => gaussianPDFReal 0 1 x • |x|) =
      (fun x : ℝ => (Real.sqrt (2 * Real.pi))⁻¹ *
        (|x| * Real.exp (-(1 / 2 : ℝ) * x ^ 2))) := by
    ext x
    simp only [gaussianPDFReal, NNReal.coe_one, sub_zero, mul_one, smul_eq_mul]
    rw [show -(x ^ 2) / (2 : ℝ) = -(1 / 2 : ℝ) * x ^ 2 by ring]
    ring
  rw [hfun, integral_const_mul, integral_abs_mul_exp (by norm_num : (0 : ℝ) < 1/2)]
  norm_num
  ring

theorem integral_abs_dual_stdGaussian {E : Type uE} [NormedAddCommGroup E]
    [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (L : StrongDual ℝ E) :
    ∫ x, |L x| ∂stdGaussian E = ‖L‖ * (2 / Real.sqrt (2 * Real.pi)) := by
  have hm : (stdGaussian E).map L = (gaussianReal 0 1).map (‖L‖ * ·) := by
    rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian,
      variance_dual_stdGaussian, gaussianReal_map_const_mul]
    simp only [mul_zero, mul_one, Real.toNNReal_of_nonneg (sq_nonneg ‖L‖)]
  rw [← integral_map L.continuous.aemeasurable (by fun_prop :
    AEStronglyMeasurable (fun x : ℝ => |x|) ((stdGaussian E).map L)), hm,
    integral_map (by fun_prop) (by fun_prop)]
  simp_rw [abs_mul, abs_of_nonneg (norm_nonneg L)]
  rw [integral_const_mul, integral_abs_gaussianReal_std]

theorem integral_abs_inner_stdGaussian {E : Type uE} [NormedAddCommGroup E]
    [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (v : E) :
    ∫ x, |⟪v, x⟫| ∂stdGaussian E = ‖v‖ * (2 / Real.sqrt (2 * Real.pi)) := by
  simpa only [InnerProductSpace.toDual_apply_apply, LinearIsometryEquiv.norm_map] using
    integral_abs_dual_stdGaussian (InnerProductSpace.toDual ℝ E v)

end GaussianPropeller.Analytic

end OAI