import OAI.Probability.PosteriorReplicas.Gaussian.RadialIntegrals
import OAI.Probability.PosteriorReplicas.Geometry.BallJacobian

namespace OAI

open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Metric
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace PosteriorReplicas

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

lemma gaussian_orthogonalProjectionOnto (U : Submodule ℝ E) :
    (stdGaussian E).map U.orthogonalProjectionOnto = stdGaussian U := by
  apply Measure.ext_of_charFun
  ext t
  rw [charFun_apply, integral_map (by fun_prop) (by fun_prop)]
  have he (x : E) : ⟪U.orthogonalProjectionOnto x, t⟫ = ⟪x, (t : E)⟫ := by
    change ⟪U.starProjection x, (t : E)⟫ = _
    rw [real_inner_comm, pairing_starProjection U t.2, real_inner_comm]
  simp_rw [he]
  change charFun (stdGaussian E) (t : E) = _
  rw [charFun_stdGaussian, charFun_stdGaussian]
  rfl

lemma gaussian_subspace_map (U : Submodule ℝ E) :
    (stdGaussian U).map (Subtype.val : U → E) = (stdGaussian E).map U.starProjection := by
  rw [← gaussian_orthogonalProjectionOnto U, Measure.map_map (by fun_prop) (by fun_prop)]
  rfl

lemma gaussian_subspace_sum (U : Submodule ℝ E) :
    ((stdGaussian U).prod (stdGaussian Uᗮ)).map (fun z => (z.1 : E) + (z.2 : E)) =
      stdGaussian E := by
  rw [← gaussian_orthogonal_sum U, ← gaussian_subspace_map, ← gaussian_subspace_map,
    Measure.map_prod_map _ _ (by fun_prop) (by fun_prop),
    Measure.map_map (by fun_prop) (by fun_prop)]
  rfl

omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma subspace_sum_norm_sq [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (U : Submodule ℝ E) (x : U) (z : Uᗮ) :
    ‖(x : E) + (z : E)‖^2 = ‖x‖^2 + ‖z‖^2 := by
  have ho : ⟪(x : E), (z : E)⟫ = 0 :=
    (real_inner_comm _ _).trans ((Submodule.mem_orthogonal' U (z : E)).mp z.2 _ x.2)
  simp [norm_add_sq_real, ho]

omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma normalDirection_pos_smul [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {r : ℝ} (hr : 0 < r) (x : E) :
    normalDirection (r • x) = normalDirection x := by
  simp [normalDirection, norm_smul, Real.norm_eq_abs, abs_of_pos hr,
    _root_.mul_inv_rev, smul_smul, hr.ne']

lemma sphere_subspace_gaussian_integral (U : Submodule ℝ E) [Nontrivial E]
    {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ x, g x ∂uniformDirection E =
      ∫⁻ x : U, ∫⁻ z : Uᗮ, g (normalDirection ((x : E) + (z : E)))
        ∂stdGaussian Uᗮ ∂stdGaussian U := by
  rw [← gaussian_normalDirection, lintegral_map hg (by fun_prop),
    ← gaussian_subspace_sum U, lintegral_map (by fun_prop) (by fun_prop),
    lintegral_prod _ (by fun_prop)]

lemma sphere_subspace_polar_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
    {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ y, g y ∂uniformDirection E =
      ∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ r : Ioi (0 : ℝ), ∫⁻ x : U,
        gaussianRadialDensity (Module.finrank ℝ U) ‖x‖ *
          g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E)))
          ∂volume ∂gaussianPolarRadius Uᗮ ∂(volume : Measure Uᗮ).toSphere := by
  rw [sphere_subspace_gaussian_integral U hg]
  have hp (x : U) : (∫⁻ z : Uᗮ, g (normalDirection ((x : E) + (z : E))) ∂stdGaussian Uᗮ) =
      ∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ r : Ioi (0 : ℝ),
        g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E)))
          ∂gaussianPolarRadius Uᗮ ∂(volume : Measure Uᗮ).toSphere := by
    rw [← gaussian_polar_reconstruction (E := Uᗮ), lintegral_map (by fun_prop) (by fun_prop),
      lintegral_prod _ (by fun_prop)]
    rfl
  simp_rw [hp]
  rw [lintegral_lintegral_swap (by fun_prop)]
  apply lintegral_congr
  intro z
  rw [lintegral_lintegral_swap (by fun_prop)]
  apply lintegral_congr
  intro r
  rw [stdGaussian_density, lintegral_withDensity_eq_lintegral_mul _ (by fun_prop) (by fun_prop)]
  rfl

lemma radial_subspace_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ] (s : ℝ) :
    ∫⁻ r : Ioi (0 : ℝ), ENNReal.ofReal (r.1 ^ Module.finrank ℝ U) *
        gaussianRadialDensity (Module.finrank ℝ U) (r.1 * s) ∂gaussianPolarRadius Uᗮ =
      ENNReal.ofReal ((1 + s^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
        ((volume : Measure E).toSphere univ)⁻¹ := by
  rw [lintegral_gaussianPolarRadius (E := Uᗮ) (g := fun r : ℝ =>
    ENNReal.ofReal (r ^ Module.finrank ℝ U) *
      gaussianRadialDensity (Module.finrank ℝ U) (r * s)) (by fun_prop)]
  calc
    _ = ∫⁻ r in Ioi (0 : ℝ), ENNReal.ofReal (r ^ (Module.finrank ℝ E - 1)) *
        ENNReal.ofReal ((2 * Real.pi) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2) *
          Real.exp (-(1 + s^2) * r^2 / 2)) := by
      apply setLIntegral_congr_fun measurableSet_Ioi
      intro r hr
      exact radial_power_gaussian_product Module.finrank_pos U.finrank_add_finrank_orthogonal hr.le s
    _ = _ := radial_gaussian_integral (by positivity)

lemma scaled_subspace_integral (U : Submodule ℝ E) (z : sphere (0 : Uᗮ) 1)
    (r : Ioi (0 : ℝ)) {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ x : U, gaussianRadialDensity (Module.finrank ℝ U) ‖x‖ *
        g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E))) =
      ∫⁻ x : U, ENNReal.ofReal (r.1 ^ Module.finrank ℝ U) *
        gaussianRadialDensity (Module.finrank ℝ U) (r.1 * ‖x‖) *
          g (normalDirection ((x : E) + ((z : Uᗮ) : E))) := by
  rw [lintegral_pos_smul r.2 (by fun_prop), ← lintegral_const_mul _ (by fun_prop)]
  apply lintegral_congr
  intro x
  rw [norm_smul, Real.norm_eq_abs, abs_of_pos r.2]
  have he : ((r.1 • x : U) : E) + r.1 • ((z : Uᗮ) : E) =
      r.1 • ((x : E) + ((z : Uᗮ) : E)) := by simp [smul_add]
  rw [he, normalDirection_pos_smul r.2, mul_assoc]

lemma sphere_subspace_slope_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
    {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ y, g y ∂uniformDirection E =
      ((volume : Measure E).toSphere univ)⁻¹ *
        ∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ x : U,
          ENNReal.ofReal ((1 + ‖x‖^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
            g (normalDirection ((x : E) + ((z : Uᗮ) : E)))
          ∂volume ∂(volume : Measure Uᗮ).toSphere := by
  rw [sphere_subspace_polar_integral U hg, ← lintegral_const_mul _ (by fun_prop)]
  apply lintegral_congr
  intro z
  simp_rw [scaled_subspace_integral U z _ hg]
  rw [lintegral_lintegral_swap (by fun_prop), ← lintegral_const_mul _ (by fun_prop)]
  apply lintegral_congr
  intro x
  rw [lintegral_mul_const _ (by fun_prop), radial_subspace_integral]
  ac_rfl

lemma normalDirection_ballStretch (U : Submodule ℝ E) {x : U}
    (hx : x ∈ ball (0 : U) 1) (z : sphere (0 : Uᗮ) 1) :
    normalDirection (((ballStretch x : U) : E) + ((z : Uᗮ) : E)) =
      (x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E) := by
  have hb := one_sub_norm_sq_pos hx
  have hs := Real.sqrt_pos.mpr hb
  have hz : ‖(z : Uᗮ)‖ = 1 := mem_sphere_zero_iff_norm.mp z.2
  have hn : ‖(x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)‖ = 1 := by
    have h := subspace_sum_norm_sq U x (Real.sqrt (1 - ‖x‖^2) • (z : Uᗮ))
    simp only [Submodule.coe_smul_of_tower, norm_smul, Real.norm_eq_abs,
      abs_of_pos hs, hz, mul_one, Real.sq_sqrt hb.le] at h
    nlinarith [norm_nonneg ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E))]
  have he : ((ballStretch x : U) : E) + ((z : Uᗮ) : E) =
      (Real.sqrt (1 - ‖x‖^2))⁻¹ •
        ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)) := by
    rw [ballStretch_eq_inv_sqrt hx]
    change (Real.sqrt (1 - ‖x‖^2))⁻¹ • (x : E) + ((z : Uᗮ) : E) = _
    rw [smul_add, smul_smul, inv_mul_cancel₀ hs.ne', one_smul]
  rw [he, normalDirection_smul_sphere (inv_pos.mpr hs) hn]

lemma sphere_subspace_surface_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
    {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ y, g y ∂uniformDirection E =
      ((volume : Measure E).toSphere univ)⁻¹ *
        ∫⁻ x : U in ball 0 1,
          ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
            ∫⁻ z : sphere (0 : Uᗮ) 1,
              g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E))
              ∂(volume : Measure Uᗮ).toSphere := by
  rw [sphere_subspace_slope_integral U hg]
  congr 1
  have ht (z : sphere (0 : Uᗮ) 1) :
      (∫⁻ x : U, ENNReal.ofReal ((1 + ‖x‖^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
        g (normalDirection ((x : E) + ((z : Uᗮ) : E)))) =
      ∫⁻ x : U in ball 0 1,
        ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
          g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)) := by
    rw [lintegral_ballStretch]
    apply setLIntegral_congr_fun measurableSet_ball
    intro x hx
    dsimp only
    rw [normalDirection_ballStretch U hx z, ← mul_assoc,
      ← U.finrank_add_finrank_orthogonal, ballStretch_weight hx]
  simp_rw [ht]
  rw [lintegral_lintegral_swap (by fun_prop)]
  apply setLIntegral_congr_fun measurableSet_ball
  intro x hx
  dsimp only
  rw [lintegral_const_mul _ (by fun_prop)]

lemma surface_lintegral_eq_gaussian [Nontrivial E] {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ z : sphere (0 : E) 1, g (z : E) ∂(volume : Measure E).toSphere =
      (volume : Measure E).toSphere univ * ∫⁻ x, g (normalDirection x) ∂stdGaussian E := by
  have he : (∫⁻ x, g (normalDirection x) ∂stdGaussian E) = ∫⁻ x, g x ∂uniformDirection E := by
    rw [← gaussian_normalDirection, lintegral_map hg (by fun_prop)]
  rw [he, uniformDirection, lintegral_map hg (by fun_prop), lintegral_smul_measure]
  change _ = (volume : Measure E).toSphere univ *
    (((volume : Measure E).toSphere univ)⁻¹ *
      ∫⁻ z : sphere (0 : E) 1, g (z : E) ∂(volume : Measure E).toSphere)
  rw [← mul_assoc, ENNReal.mul_inv_cancel
    (Measure.measure_univ_ne_zero.mpr (Measure.toSphere_ne_zero volume)) (measure_ne_top _ _), one_mul]

lemma sphere_subspace_gaussian_fiber_integral (U : Submodule ℝ E)
    [Nontrivial E] [Nontrivial Uᗮ] {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ y, g y ∂uniformDirection E =
      ((volume : Measure E).toSphere univ)⁻¹ * (volume : Measure Uᗮ).toSphere univ *
        ∫⁻ x : U in ball 0 1,
          ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
            ∫⁻ z : E, g ((x : E) + Real.sqrt (1 - ‖x‖^2) •
              normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E := by
  rw [sphere_subspace_surface_integral U hg, mul_assoc]
  congr 1
  rw [← lintegral_const_mul _ (by fun_prop)]
  apply setLIntegral_congr_fun measurableSet_ball
  intro x hx
  dsimp only
  rw [surface_lintegral_eq_gaussian (E := Uᗮ) (g := fun z =>
    g ((x : E) + Real.sqrt (1 - ‖x‖^2) • (z : E))) (by fun_prop)]
  have he : (∫⁻ z : Uᗮ, g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((normalDirection z : Uᗮ) : E))
      ∂stdGaussian Uᗮ) = ∫⁻ z : E, g ((x : E) + Real.sqrt (1 - ‖x‖^2) •
        normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E := by
    rw [← gaussian_orthogonalProjectionOnto Uᗮ, lintegral_map (by fun_prop) (by fun_prop)]
    rfl
  rw [he]
  ac_rfl

lemma sphere_area_ratio (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ] :
    ((volume : Measure E).toSphere univ)⁻¹ * (volume : Measure Uᗮ).toSphere univ =
      ENNReal.ofReal (Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
        (Real.pi ^ (((Module.finrank ℝ U : ℕ) : ℝ) / 2) *
          Real.Gamma (((Module.finrank ℝ Uᗮ : ℕ) : ℝ) / 2))) := by
  have hd : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
  have hq : 0 < ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
  have he : ((Module.finrank ℝ E : ℕ) : ℝ) =
      ((Module.finrank ℝ U : ℕ) : ℝ) + ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) := by
    exact_mod_cast U.finrank_add_finrank_orthogonal.symm
  have hp : Real.pi ^ (((Module.finrank ℝ E : ℕ) : ℝ) / 2) =
      Real.pi ^ (((Module.finrank ℝ U : ℕ) : ℝ) / 2) *
        Real.pi ^ (((Module.finrank ℝ Uᗮ : ℕ) : ℝ) / 2) := by
    rw [he, add_div, Real.rpow_add Real.pi_pos]
  rw [sphere_area, sphere_area, ← ENNReal.ofReal_inv_of_pos (by positivity),
    ← ENNReal.ofReal_mul (by positivity)]
  congr 1
  rw [hp]
  field_simp

end PosteriorReplicas
end

end OAI