import OAI.Probability.PosteriorReplicas.Model

namespace OAI

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace
noncomputable section
universe u
namespace PosteriorRegression
open scoped Topology
open InnerProductSpace
section GramFactorization
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  {ι : Type*} [Fintype ι]
lemma gram_linear_combination (w : ι → E) (T : Matrix ι ι ℝ) :
    Matrix.gram ℝ (fun j => ∑ i, (T i j) • w i) =
      T.transpose * Matrix.gram ℝ w * T := by
  ext a b
  simp only [Matrix.gram_apply, sum_inner, inner_sum, real_inner_smul_left,
    real_inner_smul_right, Matrix.mul_apply, Matrix.transpose_apply]
  simp only [Finset.sum_mul]
  congr 1
  ext j
  congr 1
  ext i
  ring

variable (n : ℕ)
def gsUpper (v : Fin n → E) : Matrix (Fin n) (Fin n) ℝ := fun i j =>
  (if i = j then 1 else 0) +
    if i < j then ⟪gramSchmidt ℝ v i, v j⟫ / ‖gramSchmidt ℝ v i‖ ^ 2 else 0

lemma gsUpper_expansion (v : Fin n → E) (j : Fin n) :
    ∑ i, (gsUpper n v i j) • gramSchmidt ℝ v i = v j := by
  simp only [gsUpper, add_smul, Finset.sum_add_distrib]
  simp only [ite_smul, one_smul, zero_smul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
  conv_rhs => rw [gramSchmidt_def'' ℝ v j]
  congr 1
  rw [← Finset.sum_filter]
  congr 1
  ext i
  simp

lemma gsUpper_det (v : Fin n → E) : (gsUpper n v).det = 1 := by
  have htri : (gsUpper n v).IsUpperTriangular := by
    intro i j hij
    change j < i at hij
    simp [gsUpper, (ne_of_gt hij), (not_lt_of_ge hij.le)]
  rw [Matrix.det_of_isUpperTriangular htri]
  simp [gsUpper]

lemma gram_gramSchmidt_diagonal (v : Fin n → E) :
    Matrix.gram ℝ (gramSchmidt ℝ v) = Matrix.diagonal (fun i => ‖gramSchmidt ℝ v i‖ ^ 2) := by
  ext i j
  by_cases hij : i = j
  · subst j
    simp [Matrix.gram_apply]
  · simp [Matrix.gram_apply, hij,
      gramSchmidt_orthogonal ℝ v hij]

theorem det_gram_eq_prod_gramSchmidt (v : Fin n → E) :
    (Matrix.gram ℝ v).det = ∏ i, ‖gramSchmidt ℝ v i‖ ^ 2 := by
  have hv : (fun j => ∑ i, (gsUpper n v i j) • gramSchmidt ℝ v i) = v :=
    funext (gsUpper_expansion n v)
  have h := gram_linear_combination (gramSchmidt ℝ v) (gsUpper n v)
  rw [hv] at h
  rw [h, Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose, gsUpper_det,
    one_mul, mul_one, gram_gramSchmidt_diagonal, Matrix.det_diagonal]
end GramFactorization

end PosteriorRegression
end

end OAI