import OAI.Analysis.FiniteFactor.Coupling.FiniteLaw
import OAI.Analysis.FiniteFactor.ProductDensityLimit

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Topology ENNReal
noncomputable section

/-- Finite-coordinate factorization under the actual infinite product measure. -/
theorem integral_infinitePi_finset_prod {ι : Type*} {X : ι → Type*}
    [∀ i, MeasurableSpace (X i)] (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
    (I : Finset ι) (f : ∀ i, X i → ℝ) (hf : ∀ i, Measurable (f i)) :
    (∫ x, ∏ i ∈ I, f i (x i) ∂Measure.infinitePi μ) = ∏ i ∈ I, ∫ t, f i t ∂μ i := by
  classical
  have he (x : ∀ i, X i) : (∏ i ∈ I, f i (x i)) = ∏ i : I, f i (x i) :=
    (Finset.prod_coe_sort I (fun i => f i (x i))).symm
  simp_rw [he]
  change (∫ x, (∏ i : I, f i (I.restrict x i)) ∂Measure.infinitePi μ) = _
  rw [integral_restrict_infinitePi μ
    (f := fun x : ∀ i : I, X i => ∏ i : I, f i (x i))
    ((Finset.measurable_prod _ fun (i : I) _ => (hf i).comp (measurable_pi_apply i)).aestronglyMeasurable)]
  rw [integral_fintype_prod_eq_prod]
  exact Finset.prod_coe_sort I (fun i => ∫ t, f i t ∂μ i)

namespace CouplingParameters
variable {ι : Type*}

def prefixDensity (p : ι → CouplingParameters) (right : ι → Bool)
    (I : Finset ι) (x : ι → ℝ) : ℝ := ∏ i ∈ I, (p i).endpointDensity (right i) (x i)

theorem prefixDensity_measurable (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) :
    Measurable (prefixDensity p right I) :=
  Finset.measurable_prod _ fun i _ => ((p i).endpointDensity_measurable (right i)).comp (measurable_pi_apply i)

theorem prefixDensity_nonneg (p : ι → CouplingParameters) (right : ι → Bool)
    (I : Finset ι) (x : ι → ℝ) : 0 ≤ prefixDensity p right I x :=
  Finset.prod_nonneg fun i _ => ((p i).diagonalMass_pos.trans_le ((p i).endpointDensity_lower (right i) (x i))).le

theorem prefixDensity_memLp (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) :
    MemLp (prefixDensity p right I) 2 (Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) := by
  apply MemLp.of_bound (prefixDensity_measurable p right I).aestronglyMeasurable (10 ^ I.card)
  apply Eventually.of_forall
  intro x
  rw [Real.norm_of_nonneg (prefixDensity_nonneg p right I x)]
  calc
    _ ≤ ∏ i ∈ I, (10 : ℝ) := Finset.prod_le_prod₀
      (fun i _ => ((p i).diagonalMass_pos.trans_le ((p i).endpointDensity_lower (right i) (x i))).le)
      (fun i _ => ((p i).endpointDensity_lt_ten (right i) (x i)).le)
    _ = _ := by simp

theorem prefixDensity_integral (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) :
    (∫ x, prefixDensity p right I x ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = 1 := by
  unfold prefixDensity
  rw [integral_infinitePi_finset_prod _ _ _ (fun i => (p i).endpointDensity_measurable (right i))]
  simp [endpointDensity_integral]

theorem prefixDensity_sq_integral (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) :
    (∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) =
      ∏ i ∈ I, ∫ t, (p i).endpointDensity (right i) t ^ 2 ∂unitIntervalMeasure := by
  simp only [prefixDensity, ← Finset.prod_pow]
  exact integral_infinitePi_finset_prod _ _ _ (fun i => ((p i).endpointDensity_measurable (right i)).pow_const 2)

theorem prefixDensity_energy (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) :
    (∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) ≤
      Real.exp (∑ i ∈ I, 20*(2*(p i).kappa^2/(p i).m + (p i).tagMass)) := by
  classical
  rw [prefixDensity_sq_integral, ← Finset.prod_coe_sort I,
    ← Finset.sum_coe_sort I]
  exact product_energy_le_exp _ _ (fun i => integral_nonneg (fun _ => sq_nonneg _))
    (fun i => (p i).endpointDensity_source_energy (right i))

/-- Exact orthogonality identity for nested finite product densities. -/
theorem prefixDensity_inner (p : ι → CouplingParameters) (right : ι → Bool)
    {I J : Finset ι} (hIJ : I ⊆ J) :
    (∫ x, prefixDensity p right I x * prefixDensity p right J x
      ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) =
    ∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure) := by
  classical
  have he (x : ι → ℝ) : prefixDensity p right I x * prefixDensity p right J x =
      ∏ i ∈ J, if i ∈ I then (p i).endpointDensity (right i) (x i)^2
        else (p i).endpointDensity (right i) (x i) := by
    have hi : prefixDensity p right I x = ∏ i ∈ J,
        if i ∈ I then (p i).endpointDensity (right i) (x i) else 1 := by
      rw [← Finset.prod_subset hIJ]
      · apply Finset.prod_congr rfl
        intro i hi
        simp [hi]
      · intro i _ hi
        simp [hi]
    rw [hi, prefixDensity, ← Finset.prod_mul_distrib]
    apply Finset.prod_congr rfl
    intro i _
    split <;> simp_all [sq]
  simp_rw [he]
  rw [integral_infinitePi_finset_prod (fun _ : ι => unitIntervalMeasure) J
    (fun i t => if i ∈ I then (p i).endpointDensity (right i) t ^ 2
      else (p i).endpointDensity (right i) t) (fun i => by
    split
    · exact ((p i).endpointDensity_measurable (right i)).pow_const 2
    · exact (p i).endpointDensity_measurable (right i)), prefixDensity_sq_integral]
  calc
    _ = ∏ i ∈ I, ∫ t, (if i ∈ I then (p i).endpointDensity (right i) t ^ 2
        else (p i).endpointDensity (right i) t) ∂unitIntervalMeasure := by
      symm
      apply Finset.prod_subset hIJ
      intro i _ hi
      simp [hi, endpointDensity_integral]
    _ = _ := by
      apply Finset.prod_congr rfl
      intro i hi
      simp [hi]

/-- The lifted prefix density has exactly the finite endpoint law on every
Borel cylinder at that prefix. -/
theorem prefixDensity_cylinder (p : ι → CouplingParameters) (right : ι → Bool)
    (I : Finset ι) {S : Set (I → ℝ)} (hS : MeasurableSet S) :
    ENNReal.ofReal (∫ x in (I.restrict (π := fun _ : ι => ℝ)) ⁻¹' S, prefixDensity p right I x
      ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) =
      Measure.pi (fun i : I => (p i).selectedLaw true (right i)) S := by
  classical
  rw [finiteProductLaw_density, withDensity_apply _ hS]
  rw [← Measure.infinitePi_map_restrict (fun _ : ι => unitIntervalMeasure),
    setLIntegral_map hS (finiteProductDensity_measurable
      (fun i : I => p i) (fun _ => true) (fun i => right i)).ennreal_ofReal (measurable_restrict I)]
  have he (x : ι → ℝ) : finiteProductDensity (fun i : I => p i)
      (fun _ => true) (fun i => right i) ((I.restrict (π := fun _ : ι => ℝ)) x) = prefixDensity p right I x := by
    exact Finset.prod_coe_sort I (fun i => (p i).endpointDensity (right i) (x i))
  simp_rw [he]
  exact ofReal_integral_eq_lintegral_ofReal
    ((prefixDensity_memLp p right I).integrable (by norm_num)).restrict
    (Eventually.of_forall (prefixDensity_nonneg p right I))

/-- Enlarging the prefix does not change any earlier cylinder law. -/
theorem prefixDensity_cylinder_of_subset (p : ι → CouplingParameters) (right : ι → Bool)
    {J I : Finset ι} (hJI : J ⊆ I) {S : Set (J → ℝ)} (hS : MeasurableSet S) :
    ENNReal.ofReal (∫ x in (J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S, prefixDensity p right I x
      ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) =
      Measure.pi (fun i : J => (p i).selectedLaw true (right i)) S := by
  have he : (J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S = (I.restrict (π := fun _ : ι => ℝ)) ⁻¹' ((Finset.restrict₂ (π := fun _ : ι => ℝ) hJI : (I → ℝ) → (J → ℝ)) ⁻¹' S) := by
    rw [← Set.preimage_comp, Finset.restrict₂_comp_restrict hJI]
  rw [he, prefixDensity_cylinder p right I (hS.preimage (measurable_restrict₂ hJI))]
  have hh := congrArg (fun ν : Measure (J → ℝ) => ν S)
    (isProjectiveMeasureFamily_pi (fun i => (p i).selectedLaw true (right i)) I J hJI)
  rw [Measure.map_apply (show Measurable (Finset.restrict₂ (π := fun _ : ι => ℝ) hJI) from measurable_restrict₂ hJI) hS] at hh
  exact hh.symm

end CouplingParameters
end
end FiniteFactor

end OAI