import OAI.Probability.PosteriorReplicas.Geometry.SphereSubspace

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 volume_pairings_on_span {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v) :
    let U := Submodule.span ℝ (Set.range v)
    (volume : Measure U).map (fun x : U => fun i => ⟪v i, (x : E)⟫) =
      ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) • volume := by
  let U := Submodule.span ℝ (Set.range v)
  let F : U → Fin n → ℝ := fun x i => ⟪v i, (x : E)⟫
  let h : (Fin n → ℝ) → ℝ≥0∞ := fun y => gaussianRadialDensity n ‖pairingLift v y‖
  have hF : Measurable F := by unfold F; fun_prop
  have hh : Measurable h := by unfold h; fun_prop
  have hj : 0 < Real.sqrt (Matrix.gram ℝ v).det :=
    Real.sqrt_pos.mpr (Matrix.posDef_gram_of_linearIndependent hv).det_pos
  have hU : Module.finrank ℝ U = n := by
    simpa only [Fintype.card_fin] using finrank_span_eq_card hv
  have hl (x : U) : pairingLift v (F x) = (x : E) := by
    rw [← starProjection_eq_pairingLift hv]
    exact U.starProjection_eq_self_iff.mpr x.2
  have hg : (stdGaussian U).map F = (stdGaussian E).map (fun x => fun i => ⟪v i, x⟫) := by
    rw [← gaussian_orthogonalProjectionOnto U, Measure.map_map hF (by fun_prop)]
    congr 1
    funext x i
    exact pairing_starProjection U (Submodule.subset_span (Set.mem_range_self i)) x
  have hd : (stdGaussian U) = (volume : Measure U).withDensity (h ∘ F) := by
    rw [stdGaussian_density, hU]
    congr 1
    funext x
    simp only [Function.comp_apply, h, hl, Submodule.norm_coe]
  have he : ((volume : Measure U).map F).withDensity h =
      (ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) •
        (volume : Measure (Fin n → ℝ))).withDensity h := by
    rw [← map_withDensity_comp _ hF hh, ← hd, hg, gaussian_pairings_density hv,
      withDensity_smul_measure, ← withDensity_smul _ hh]
    congr 1
    funext y
    change ENNReal.ofReal _ = ENNReal.ofReal _ * gaussianRadialDensity n ‖pairingLift v y‖
    rw [gaussianRadialDensity, ← ENNReal.ofReal_mul (by positivity)]
    congr 1
    ring
  have hz (y : Fin n → ℝ) : h y ≠ 0 := by
    dsimp [h, gaussianRadialDensity]
    positivity
  have ht (y : Fin n → ℝ) : h y ≠ ∞ := ENNReal.ofReal_ne_top
  have hinv := congrArg (fun μ : Measure (Fin n → ℝ) => μ.withDensity (fun y => (h y)⁻¹)) he
  rw [withDensity_inv_same hh (Filter.Eventually.of_forall hz) (Filter.Eventually.of_forall ht),
    withDensity_inv_same hh (Filter.Eventually.of_forall hz) (Filter.Eventually.of_forall ht)] at hinv
  exact hinv

lemma lintegral_pairingLift {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v)
    {g : E → ℝ≥0∞} (hg : Measurable g) :
    let U := Submodule.span ℝ (Set.range v)
    ∫⁻ x : U, g (x : E) ∂volume =
      ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) * ∫⁻ y, g (pairingLift v y) := by
  let U := Submodule.span ℝ (Set.range v)
  have he := congrArg (fun μ : Measure (Fin n → ℝ) => ∫⁻ y, g (pairingLift v y) ∂μ)
    (volume_pairings_on_span hv)
  rw [lintegral_map (by fun_prop) (by fun_prop), lintegral_smul_measure] at he
  change _ = ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) *
    ∫⁻ y, g (pairingLift v y) at he
  convert he using 1
  apply lintegral_congr
  intro x
  rw [← starProjection_eq_pairingLift hv, U.starProjection_eq_self_iff.mpr x.2]

def sphericalPairingsDensity {n : ℕ} (v : Fin n → E) (y : Fin n → ℝ) : ℝ≥0∞ :=
  if ‖pairingLift v y‖ < 1 then ENNReal.ofReal
    (Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
      (Real.pi ^ ((n : ℝ) / 2) * Real.Gamma ((((Module.finrank ℝ E : ℕ) : ℝ) - n) / 2)) *
        (1 - ‖pairingLift v y‖^2) ^ ((((Module.finrank ℝ E : ℕ) : ℝ) - n - 2) / 2) /
          Real.sqrt (Matrix.gram ℝ v).det) else 0

omit [FiniteDimensional ℝ E] in
@[fun_prop] lemma measurable_sphericalPairingsDensity [FiniteDimensional ℝ E]
    {n : ℕ} (v : Fin n → E) :
    Measurable (sphericalPairingsDensity v) := by
  unfold sphericalPairingsDensity
  exact Measurable.ite (measurableSet_lt (by fun_prop) measurable_const) (by fun_prop) measurable_const

lemma sphere_fiber_integral {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v)
    (hn : n < Module.finrank ℝ E) {g : E → ℝ≥0∞} (hg : Measurable g) :
    ∫⁻ x, g x ∂uniformDirection E = ∫⁻ y, sphericalPairingsDensity v y *
      ∫⁻ z, g (pairingLift v y + Real.sqrt (1 - ‖pairingLift v y‖^2) •
        normalDirection ((Submodule.span ℝ (Set.range v))ᗮ.starProjection z)) ∂stdGaussian E := by
  let U := Submodule.span ℝ (Set.range v)
  have hU : Module.finrank ℝ U = n := by
    simpa only [Fintype.card_fin] using finrank_span_eq_card hv
  have hV : Module.finrank ℝ Uᗮ = Module.finrank ℝ E - n := by
    have ha := U.finrank_add_finrank_orthogonal
    omega
  have hVr : ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) = ((Module.finrank ℝ E : ℕ) : ℝ) - n := by
    rw [hV, Nat.cast_sub hn.le]
  let : Nontrivial E := Module.nontrivial_of_finrank_pos (by omega : 0 < Module.finrank ℝ E)
  let : Nontrivial Uᗮ := Module.nontrivial_of_finrank_pos (by omega : 0 < Module.finrank ℝ Uᗮ)
  let C : ℝ := Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
    (Real.pi ^ ((n : ℝ) / 2) * Real.Gamma ((((Module.finrank ℝ E : ℕ) : ℝ) - n) / 2))
  let β : ℝ := (((Module.finrank ℝ E : ℕ) : ℝ) - n - 2) / 2
  let G : E → ℝ≥0∞ := fun x => ENNReal.ofReal ((1 - ‖x‖^2)^β) *
    ∫⁻ z, g (x + Real.sqrt (1 - ‖x‖^2) • normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E
  have hG : Measurable G := by unfold G; fun_prop
  have hJ : 0 < Real.sqrt (Matrix.gram ℝ v).det :=
    Real.sqrt_pos.mpr (Matrix.posDef_gram_of_linearIndependent hv).det_pos
  have hC : 0 < C := by
    dsimp [C]
    have hd : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
    have hq : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) - n := by
      rw [← hVr]
      exact Nat.cast_pos.mpr Module.finrank_pos
    positivity
  rw [sphere_subspace_gaussian_fiber_integral U hg, sphere_area_ratio, hU, hVr]
  change ENNReal.ofReal C * (∫⁻ x : U in ball 0 1, G (x : E)) = _
  have hi : (∫⁻ x : U in ball 0 1, G (x : E)) =
      ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) *
        ∫⁻ y, (ball (0 : E) 1).indicator G (pairingLift v y) := by
    have he := lintegral_pairingLift hv (g := (ball (0 : E) 1).indicator G)
      (hG.indicator measurableSet_ball)
    convert he using 1
    rw [← lintegral_indicator measurableSet_ball]
    apply lintegral_congr
    intro x
    rfl
  have hI : Measurable (fun y => (ball (0 : E) 1).indicator G (pairingLift v y)) :=
    (hG.indicator measurableSet_ball).comp (continuous_pairingLift v).measurable
  rw [hi, ← mul_assoc, ← lintegral_const_mul _ hI]
  apply lintegral_congr
  intro y
  by_cases hy : ‖pairingLift v y‖ < 1
  · rw [Set.indicator_of_mem (mem_ball_zero_iff.mpr hy)]
    dsimp [G, sphericalPairingsDensity]
    rw [ite_eq_left hy, ← mul_assoc, ← ENNReal.ofReal_mul hC.le, ← ENNReal.ofReal_mul (by positivity)]
    congr 1
    congr 1
    dsimp [C, β]
    ring
  · rw [Set.indicator_of_notMem (mt mem_ball_zero_iff.mp hy)]
    simp [sphericalPairingsDensity, hy]

end PosteriorReplicas
end

end OAI