import OAI.Probability.GaussianPropeller.SpanReduction

namespace OAI

universe uι

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

open Metric
namespace GaussianPropeller.Polar
open Translation

variable {ι : Type uι} [Fintype ι] [Nonempty ι]
local notation "E" => EuclideanSpace ℝ ι
local notation "S" => sphere (0 : E) 1
local notation "V" => (volume : Measure E)

lemma integral_volume_polar (f : E → ℝ) :
    ∫ x, f x = ∫ p : S × Ioi (0 : ℝ), f (p.2.1 • p.1.1)
      ∂(V).toSphere.prod (Measure.volumeIoiPow (Fintype.card ι - 1)) := by
  have hh := (V).measurePreserving_homeomorphUnitSphereProd.integral_comp
    (homeomorphUnitSphereProd E).measurableEmbedding
    (fun p => f ((homeomorphUnitSphereProd E).symm p).val)
  simp only [Homeomorph.symm_apply_apply, homeomorphUnitSphereProd_symm_apply_coe,
    finrank_euclideanSpace] at hh
  rw [integral_subtype_comap (measurableSet_singleton (0:E)).compl,
    restrict_compl_singleton] at hh
  exact hh

lemma integral_stdGaussian_polar (f : E → ℝ) :
    ∫ x, f x ∂stdGaussian E = (Real.sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι *
      ∫ p : S × Ioi (0 : ℝ), Real.exp (-p.2.1^2/2) * f (p.2.1 • p.1.1)
        ∂(V).toSphere.prod (Measure.volumeIoiPow (Fintype.card ι - 1)) := by
  rw [stdGaussian_density]
  simp_rw [ENNReal.ofReal]
  rw [integral_withDensity_eq_integral_smul measurable_stdDensity.real_toNNReal]
  have heq : (fun x : E => (stdDensity x).toNNReal • f x) =
      fun x => stdDensity x * f x := by
    funext x
    simp only [NNReal.smul_def, Real.coe_toNNReal _ (stdDensity_pos x).le, smul_eq_mul]
  rw [heq, integral_volume_polar, ← integral_const_mul]
  apply integral_congr_ae
  filter_upwards [] with p
  rw [stdDensity_eq]
  have hn : ‖p.2.1 • p.1.1‖ = p.2.1 := by
    rw [norm_smul, Real.norm_eq_abs, abs_of_pos p.2.2,
      mem_sphere_zero_iff_norm.mp p.1.2, mul_one]
  rw [hn]
  ring

lemma integral_volumeIoiPow (d : ℕ) (g : ℝ → ℝ) :
    (∫ r : Ioi (0:ℝ), g r.1 ∂Measure.volumeIoiPow d) =
      ∫ r in Ioi (0:ℝ), r^d * g r := by
  rw [Measure.volumeIoiPow]
  simp_rw [ENNReal.ofReal]
  rw [integral_withDensity_eq_integral_smul
    (measurable_subtype_coe.pow_const d).real_toNNReal]
  have heq : (fun r : Ioi (0:ℝ) => (r.1^d).toNNReal • g r.1) =
      fun r => r.1^d * g r.1 := by
    funext r
    simp only [NNReal.smul_def, Real.coe_toNNReal _ (pow_nonneg r.2.le d), smul_eq_mul]
  rw [heq]
  exact integral_subtype_comap measurableSet_Ioi (fun r : ℝ => r^d * g r)

lemma integral_homogeneous (f : E → ℝ) (q : ℕ)
    (hf : ∀ (r : ℝ), 0 < r → ∀ x, f (r • x) = r^q * f x) :
    ∫ x, f x ∂stdGaussian E =
      ((Real.sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι *
      ∫ r in Ioi (0:ℝ), r^(Fintype.card ι - 1 + q) * Real.exp (-r^2/2)) *
      ∫ θ : S, f θ.1 ∂(V).toSphere := by
  rw [integral_stdGaussian_polar]
  have heq : (fun p : S × Ioi (0:ℝ) => Real.exp (-p.2.1^2/2) * f (p.2.1 • p.1.1)) =
      fun p => f p.1.1 * (p.2.1^q * Real.exp (-p.2.1^2/2)) := by
    funext p
    rw [hf _ p.2.2]
    ring
  rw [heq, integral_prod_mul (fun θ : S => f θ.1)
    (fun r : Ioi (0:ℝ) => r.1^q * Real.exp (-r.1^2/2)), integral_volumeIoiPow (Fintype.card ι - 1)
      (fun r : ℝ => r^q * Real.exp (-r^2/2))]
  have heq' : (fun r : ℝ => r^(Fintype.card ι-1) * (r^q * Real.exp (-r^2/2))) =
      fun r => r^(Fintype.card ι-1+q) * Real.exp (-r^2/2) := by
    funext r
    rw [pow_add]
    ring
  rw [heq']
  ring

end GaussianPropeller.Polar

namespace GaussianPropeller.Polar
open Real
noncomputable def radial (n : ℕ) (b : ℝ) : ℝ := ∫ r in Ioi (0:ℝ), r^n * exp (-b*r^2)

lemma radial_eq (n : ℕ) {b : ℝ} (hb : 0<b) :
    radial n b = b ^ (-((n:ℝ)+1)/2) * (1/2) * Gamma (((n:ℝ)+1)/2) := by
  have hh := integral_rpow_mul_exp_neg_mul_rpow (p:=2) (q:=n) (b:=b)
    (by norm_num) (by linarith [Nat.cast_nonneg (α:=ℝ) n]) hb
  simpa only [Real.rpow_natCast, Real.rpow_two, neg_mul, radial] using hh

lemma radial_integrable (n : ℕ) {b : ℝ} (hb : 0<b) :
    IntegrableOn (fun r : ℝ => r^n*exp (-b*r^2)) (Ioi 0) := by
  have h := integrableOn_rpow_mul_exp_neg_mul_sq hb (s:=(n:ℝ)) (by linarith [Nat.cast_nonneg (α:=ℝ) n])
  simpa only [Real.rpow_natCast] using h

lemma radial_pos (n : ℕ) {b : ℝ} (hb : 0<b) : 0<radial n b := by
  rw [radial_eq n hb]
  exact mul_pos (mul_pos (rpow_pos_of_pos hb _) (by norm_num))
    (Gamma_pos_of_pos (by positivity))

lemma radial_scale (n : ℕ) {b : ℝ} (hb : 0<b) :
    radial n b = (sqrt (2*b))⁻¹ ^ (n+1) * radial n (1/2) := by
  have hp : 0<sqrt (2*b) := by positivity
  have hs := sq_sqrt (show 0≤2*b by positivity)
  have hh := integral_comp_mul_left_Ioi (fun r : ℝ => r^n*exp (-(1/2:ℝ)*r^2)) 0 hp
  rw [mul_zero] at hh
  have heq : (fun r : ℝ => (sqrt (2*b)*r)^n * exp (-(1/2:ℝ)*(sqrt (2*b)*r)^2)) =
      fun r => sqrt (2*b)^n * (r^n*exp (-b*r^2)) := by
    funext r
    rw [mul_pow, show -(1/2:ℝ)*(sqrt (2*b)*r)^2 = -b*r^2 by rw [mul_pow,hs]; ring]
    ring
  change (∫ r in Ioi (0:ℝ), (sqrt (2*b)*r)^n * exp (-(1/2:ℝ)*(sqrt (2*b)*r)^2)) = _ at hh
  rw [heq, integral_const_mul] at hh
  change sqrt (2*b)^n*radial n b = (sqrt (2*b))⁻¹*radial n (1/2) at hh
  apply (mul_left_cancel₀ (pow_ne_zero n hp.ne'))
  rw [hh, pow_succ, mul_assoc, ← mul_assoc (sqrt (2*b)^n)]
  simp only [inv_pow, mul_inv_cancel₀ (pow_ne_zero n hp.ne'), one_mul]

lemma radial_half_zero : radial 0 (1/2) = sqrt (2*Real.pi)/2 := by
  simpa [radial, sqrt_mul (by norm_num : (0:ℝ)≤2), mul_comm] using integral_gaussian_Ioi (1/2:ℝ)

lemma radial_half_one : radial 1 (1/2) = 1 := by
  rw [radial_eq 1 (by norm_num)]
  norm_num

lemma radial_half_two : radial 2 (1/2) = sqrt (2*Real.pi)/2 := by
  rw [radial_eq 2 (by norm_num)]
  have hg := Gamma_nat_add_one_add_half 0
  norm_num at hg ⊢
  rw [hg]
  rw [show (-((3:ℝ)/2))=-(1+(1/2:ℝ)) by ring, rpow_neg (by norm_num : (0:ℝ)≤1/2),
    rpow_add (by norm_num : (0:ℝ)<1/2), rpow_one, ← sqrt_eq_rpow]
  have h1 := sq_sqrt (by norm_num : (0:ℝ)≤1/2)
  have h2 := sq_sqrt (by norm_num : (0:ℝ)≤2)
  have hp1 := sqrt_pos.mpr (by norm_num : (0:ℝ)<1/2)
  have hp2 := sqrt_pos.mpr (by norm_num : (0:ℝ)<2)
  have hh : sqrt (1/2:ℝ)*sqrt 2=1 := by rw [← sqrt_mul (by norm_num : (0:ℝ)≤1/2)]; norm_num
  field_simp
  nlinarith only [hh]

lemma radial_half_three : radial 3 (1/2) = 2 := by
  rw [radial_eq 3 (by norm_num)]
  norm_num [Gamma_nat_eq_factorial, rpow_neg, rpow_natCast]

lemma radial_half_four : radial 4 (1/2) = 3*sqrt (2*Real.pi)/2 := by
  have h2 := radial_half_two
  rw [radial_eq 2 (by norm_num)] at h2
  rw [radial_eq 4 (by norm_num)]
  have hg := Gamma_add_one (s:=(3/2:ℝ)) (by norm_num)
  norm_num at hg h2 ⊢
  rw [hg]
  have he : (1/2:ℝ)^(-((5:ℝ)/2)) = 2*(1/2:ℝ)^(-((3:ℝ)/2)) := by
    rw [show (-((5:ℝ)/2))=-(1:ℝ)+(-((3:ℝ)/2)) by ring, rpow_add (by norm_num : (0:ℝ)<1/2)]
    norm_num
  rw [he]
  nlinarith only [h2]

variable {ι : Type uι} [Fintype ι] [Nonempty ι]
local notation "E" => EuclideanSpace ℝ ι
local notation "S" => Metric.sphere (0:E) 1

lemma integral_radial (f : ℝ → ℝ) :
    ∫ x : E, f ‖x‖ ∂stdGaussian E =
      ((sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι * ∫ _ : S, (1:ℝ) ∂(volume : Measure E).toSphere) *
      ∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)*f r := by
  rw [integral_stdGaussian_polar]
  have heq : (fun p : S × Ioi (0:ℝ) => exp (-p.2.1^2/2)*f ‖p.2.1•p.1.1‖) =
      fun p => (1:ℝ)*(exp (-p.2.1^2/2)*f p.2.1) := by
    funext p
    simp [norm_smul, Real.norm_eq_abs, abs_of_pos (show 0<p.2.1 from p.2.2),
      mem_sphere_zero_iff_norm.mp p.1.2]
  rw [heq, integral_prod_mul (fun θ : S => (1:ℝ))
    (fun r : Ioi (0:ℝ) => exp (-r.1^2/2)*f r.1), integral_volumeIoiPow (Fintype.card ι-1) (fun r => exp (-r^2/2)*f r)]
  simp_rw [mul_assoc]

lemma integral_radial_normalized (f : ℝ → ℝ) :
    ∫ x : E, f ‖x‖ ∂stdGaussian E =
      (∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)*f r) /
      radial (Fintype.card ι-1) (1/2) := by
  have h1 := integral_radial (ι:=ι) (fun _ => (1:ℝ))
  simp only [integral_const, probReal_univ, smul_eq_mul, mul_one] at h1
  have hi : (∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)) = radial (Fintype.card ι-1) (1/2) := by
    unfold radial
    congr 1; funext r; congr 2; ring
  rw [hi] at h1
  rw [integral_radial]
  rw [← one_div_mul_eq_div]
  congr 1
  apply (eq_div_iff (radial_pos _ (by norm_num)).ne').mpr
  simpa using h1.symm

lemma integral_norm_pow_exp (q : ℕ) {a : ℝ} (_ha : 0≤a) :
    ∫ x : E, ‖x‖^q*exp (-a*‖x‖^2) ∂stdGaussian E =
      radial (Fintype.card ι-1+q) (a+1/2)/radial (Fintype.card ι-1) (1/2) := by
  rw [integral_radial_normalized (fun r => r^q*exp (-a*r^2))]
  congr 1
  unfold radial
  apply setIntegral_congr_fun measurableSet_Ioi
  intro r hr
  dsimp only
  rw [pow_add]
  rw [show r^(Fintype.card ι-1)*exp (-r^2/2)*(r^q*exp (-a*r^2)) =
    r^(Fintype.card ι-1)*r^q*(exp (-r^2/2)*exp (-a*r^2)) by ring, ← exp_add]
  congr 2
  ring

lemma integral_norm (hd : Fintype.card ι = 3) :
    ∫ x : E, ‖x‖ ∂stdGaussian E = 4/sqrt (2*Real.pi) := by
  have h := integral_norm_pow_exp (ι:=ι) 1 (a:=0) (by norm_num)
  convert h using 1 <;> simp only [hd, zero_add, neg_zero, zero_mul, exp_zero, mul_one, pow_one,
    show (3-1+1:ℕ)=3 by decide, show (3-1:ℕ)=2 by decide,
    radial_half_three,radial_half_two]
  ring

end GaussianPropeller.Polar

end OAI