import Mathlib

namespace OAI

open MeasureTheory

namespace Problem367.SimpleIntegral

variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}

theorem integral_mul_eq_zero_of_condExp_eq_zero [IsFiniteMeasure μ]
    {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {H D : Ω → ℝ}
    (hH : StronglyMeasurable[m] H) (hD : Integrable D μ)
    (hHD : Integrable (H * D) μ) (hzero : μ[D | m] =ᵐ[μ] 0) :
    ∫ ω, H ω * D ω ∂μ = 0 := by
  have hpull := condExp_mul_of_stronglyMeasurable_left hH hHD hD
  have hCE : μ[H * D | m] =ᵐ[μ] 0 := by
    filter_upwards [hpull, hzero] with ω hp hz
    simp only [Pi.mul_apply, Pi.zero_apply] at hp hz ⊢
    rw [hp, hz, mul_zero]
  calc
    ∫ ω, H ω * D ω ∂μ = ∫ ω, μ[H * D | m] ω ∂μ :=
      (integral_condExp hm).symm
    _ = 0 := by rw [integral_congr_ae hCE]; simp

theorem integral_sq_mul_of_condExp_sq [IsFiniteMeasure μ]
    {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {H D : Ω → ℝ} {v : ℝ}
    (hH : StronglyMeasurable[m] H)
    (hDsq : Integrable (fun ω => D ω ^ 2) μ)
    (hHDsq : Integrable (fun ω => (H ω * D ω) ^ 2) μ)
    (hvar : μ[(fun ω => D ω ^ 2) | m] =ᵐ[μ] fun _ => v) :
    ∫ ω, (H ω * D ω) ^ 2 ∂μ = v * ∫ ω, H ω ^ 2 ∂μ := by
  have hHsq : StronglyMeasurable[m] (fun ω => H ω ^ 2) := by
    simpa only [pow_two, Pi.mul_def] using hH.mul hH
  have hprod : Integrable ((fun ω => H ω ^ 2) * (fun ω => D ω ^ 2)) μ := by
    simpa only [Pi.mul_def, mul_pow] using hHDsq
  have hpull := condExp_mul_of_stronglyMeasurable_left hHsq hprod hDsq
  calc
    ∫ ω, (H ω * D ω) ^ 2 ∂μ =
        ∫ ω, μ[(fun ω => H ω ^ 2) * (fun ω => D ω ^ 2) | m] ω ∂μ := by
      simpa only [Pi.mul_def, mul_pow] using (integral_condExp hm).symm
    _ = ∫ ω, v * H ω ^ 2 ∂μ := by
      apply integral_congr_ae
      filter_upwards [hpull, hvar] with ω hp hv
      simp only [Pi.mul_apply] at hp
      rw [hp, hv, mul_comm]
    _ = v * ∫ ω, H ω ^ 2 ∂μ := integral_const_mul _ _

theorem integral_sq_sum_of_orthogonal {ι : Type*} (s : Finset ι)
    (Y : ι → Ω → ℝ)
    (hint : ∀ i ∈ s, ∀ j ∈ s, Integrable (fun ω => Y i ω * Y j ω) μ)
    (horth : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → ∫ ω, Y i ω * Y j ω ∂μ = 0) :
    ∫ ω, (∑ i ∈ s, Y i ω) ^ 2 ∂μ = ∑ i ∈ s, ∫ ω, Y i ω ^ 2 ∂μ := by
  classical
  calc
    ∫ ω, (∑ i ∈ s, Y i ω) ^ 2 ∂μ =
        ∫ ω, ∑ i ∈ s, ∑ j ∈ s, Y i ω * Y j ω ∂μ := by
      congr 1
      funext ω
      simp only [pow_two, Finset.sum_mul_sum]
    _ = ∑ i ∈ s, ∑ j ∈ s, ∫ ω, Y i ω * Y j ω ∂μ := by
      rw [integral_finsetSum s (fun i hi =>
        integrable_finsetSum s (fun j hj => hint i hi j hj))]
      apply Finset.sum_congr rfl
      intro i hi
      exact integral_finsetSum s (fun j hj => hint i hi j hj)
    _ = ∑ i ∈ s, ∫ ω, Y i ω ^ 2 ∂μ := by
      apply Finset.sum_congr rfl
      intro i hi
      rw [Finset.sum_eq_single i]
      · simp only [pow_two]
      · intro j hj hji
        exact horth i hi j hj hji.symm
      · exact fun h => (h hi).elim

theorem predictable_increments_orthogonal [IsFiniteMeasure μ]
    (ℱ : Filtration ℕ mΩ) (H D : ℕ → Ω → ℝ)
    (hH : ∀ i, StronglyMeasurable[ℱ i] (H i))
    (hD : ∀ i, StronglyMeasurable[ℱ (i + 1)] (D i))
    (hDint : ∀ i, Integrable (D i) μ)
    (hHD : ∀ i, MemLp (fun ω => H i ω * D i ω) 2 μ)
    (hzero : ∀ i, μ[D i | ℱ i] =ᵐ[μ] 0)
    {i j : ℕ} (hij : i < j) :
    ∫ ω, (H i ω * D i ω) * (H j ω * D j ω) ∂μ = 0 := by
  have hcoeff : StronglyMeasurable[ℱ j]
      (fun ω => (H i ω * D i ω) * H j ω) := by
    simpa only [Pi.mul_def] using
      (((hH i).mono (ℱ.mono hij.le)).mul
        ((hD i).mono (ℱ.mono (Nat.succ_le_of_lt hij)))).mul (hH j)
  have hprod : Integrable
      ((fun ω => (H i ω * D i ω) * H j ω) * D j) μ := by
    simpa only [Pi.mul_def, mul_assoc] using (hHD i).integrable_mul (hHD j)
  simpa only [mul_assoc] using
    integral_mul_eq_zero_of_condExp_eq_zero (ℱ.le j) hcoeff (hDint j) hprod (hzero j)

theorem predictable_sum_isometry [IsFiniteMeasure μ]
    (ℱ : Filtration ℕ mΩ) (H D : ℕ → Ω → ℝ) (v : ℕ → ℝ)
    (hH : ∀ i, StronglyMeasurable[ℱ i] (H i))
    (hD : ∀ i, StronglyMeasurable[ℱ (i + 1)] (D i))
    (hD2 : ∀ i, MemLp (D i) 2 μ)
    (hHD : ∀ i, MemLp (fun ω => H i ω * D i ω) 2 μ)
    (hzero : ∀ i, μ[D i | ℱ i] =ᵐ[μ] 0)
    (hvar : ∀ i, μ[(fun ω => D i ω ^ 2) | ℱ i] =ᵐ[μ] fun _ => v i)
    (s : Finset ℕ) :
    ∫ ω, (∑ i ∈ s, H i ω * D i ω) ^ 2 ∂μ =
      ∑ i ∈ s, v i * ∫ ω, H i ω ^ 2 ∂μ := by
  have hint : ∀ i ∈ s, ∀ j ∈ s,
      Integrable (fun ω => (H i ω * D i ω) * (H j ω * D j ω)) μ := by
    intro i _ j _
    simpa only [Pi.mul_def] using (hHD i).integrable_mul (hHD j)
  have hDint : ∀ i, Integrable (D i) μ := fun i =>
    MemLp.integrable (by norm_num) (hD2 i)
  have horth : ∀ i ∈ s, ∀ j ∈ s, i ≠ j →
      ∫ ω, (H i ω * D i ω) * (H j ω * D j ω) ∂μ = 0 := by
    intro i _ j _ hij
    rcases lt_or_gt_of_ne hij with hij | hji
    · exact predictable_increments_orthogonal ℱ H D hH hD hDint hHD hzero hij
    · simpa only [mul_comm] using
        predictable_increments_orthogonal ℱ H D hH hD hDint hHD hzero hji
  rw [integral_sq_sum_of_orthogonal s (fun i ω => H i ω * D i ω) hint horth]
  apply Finset.sum_congr rfl
  intro i _
  exact integral_sq_mul_of_condExp_sq (ℱ.le i) (hH i)
    (hD2 i).integrable_sq (hHD i).integrable_sq (hvar i)

end Problem367.SimpleIntegral

end OAI