import OAI.Probability.FreeIsing.Spins.ObservationLaw
import OAI.Probability.FreeIsing.Spins.PosteriorCertification
import OAI.Probability.FreeIsing.Spins.PosteriorFusion

namespace OAI

noncomputable section

open MeasureTheory ProbabilityTheory

namespace Problem367

def observationFusionJointLaw {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
    (mu : Measure A) (nu : Measure B) (q : A → ℝ) (r : B → ℝ) : Measure (A × B) :=
  (mu.prod nu).withDensity (fun p => ENNReal.ofReal (1 + q p.1 * r p.2))

private theorem ae_observation_pair_bounds
    {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
    {mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
    {q : A → ℝ} {r : B → ℝ}
    (hq : ∀ᵐ x ∂mu, |q x| < 1) (hr : ∀ᵐ y ∂nu, |r y| < 1) :
    ∀ᵐ p ∂mu.prod nu, |q p.1| < 1 ∧ |r p.2| < 1 := by
  have hq' : ∀ᵐ p ∂mu.prod nu, |q p.1| < 1 :=
    measurePreserving_fst.quasiMeasurePreserving.tendsto_ae hq
  have hr' : ∀ᵐ p ∂mu.prod nu, |r p.2| < 1 :=
    measurePreserving_snd.quasiMeasurePreserving.tendsto_ae hr
  exact hq'.and hr'

theorem observationFusionJointLaw_eq_mixture
    {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
    {mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
    {q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
    (hqb : ∀ᵐ x ∂mu, |q x| < 1) (hrb : ∀ᵐ y ∂nu, |r y| < 1) :
    (1 / 2 : ENNReal) • ((linearSignalTilt mu q 1).prod (linearSignalTilt nu r 1)) +
      (1 / 2 : ENNReal) • ((linearSignalTilt mu q (-1)).prod (linearSignalTilt nu r (-1))) =
        observationFusionJointLaw mu nu q r := by
  unfold linearSignalTilt observationFusionJointLaw
  simp only [one_mul, neg_one_mul, ← sub_eq_add_neg]
  rw [prod_withDensity (by fun_prop) (by fun_prop),
    prod_withDensity (by fun_prop) (by fun_prop),
    ← withDensity_smul (1 / 2) (by fun_prop),
    ← withDensity_smul (1 / 2) (by fun_prop),
    ← withDensity_add_left (by fun_prop)]
  apply withDensity_congr_ae
  filter_upwards [ae_observation_pair_bounds hqb hrb] with p hp
  exact binaryFusion_density_eq_mixture hp.1 hp.2

theorem observationFusion_positive_product_eq_tilt
    {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]
    {mu : Measure A} {nu : Measure B} [IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
    {q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
    (hqb : ∀ᵐ x ∂mu, |q x| < 1) (hrb : ∀ᵐ y ∂nu, |r y| < 1) :
    (linearSignalTilt mu q 1).prod (linearSignalTilt nu r 1) =
      linearSignalTilt (observationFusionJointLaw mu nu q r)
        (fun p => binaryFusion (q p.1) (r p.2)) 1 := by
  unfold linearSignalTilt observationFusionJointLaw
  rw [prod_withDensity (by fun_prop) (by fun_prop),
    ← withDensity_mul _ (by fun_prop) (by unfold binaryFusion; fun_prop)]
  apply withDensity_congr_ae
  filter_upwards [ae_observation_pair_bounds hqb hrb] with p hp
  simp only [one_mul, Pi.mul_apply]
  have hpq : 0 ≤ 1 + q p.1 := by have := (abs_lt.mp hp.1).1; linarith
  have hden := binaryFusion_denominator_pos hp.1 hp.2
  rw [← ENNReal.ofReal_mul hpq, ← ENNReal.ofReal_mul hden.le]
  congr 1
  unfold binaryFusion
  field_simp
  ring

theorem condExp_spin_given_observation_pair_eq_binaryFusion
    {Ω A B : Type*} {mΩ : MeasurableSpace Ω} {mA : MeasurableSpace A}
    {mB : MeasurableSpace B} {rho : Measure Ω} [IsProbabilityMeasure rho]
    {S : Ω → IsingSpin} (hS : Measurable S)
    {X : Ω → A} {Y : Ω → B} (hX : Measurable X) (hY : Measurable Y)
    {q : A → ℝ} {r : B → ℝ} (hq : Measurable q) (hr : Measurable r)
    (hpostX : rho[fun x => spinSign (S x) | mA.comap X] =ᵐ[rho] fun x => q (X x))
    (hpostY : rho[fun x => spinSign (S x) | mB.comap Y] =ᵐ[rho] fun x => r (Y x))
    (hqb : ∀ᵐ x ∂rho, |q (X x)| < 1) (hrb : ∀ᵐ x ∂rho, |r (Y x)| < 1)
    (hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
    (hInd : ∀ a : IsingSpin, IndepFun X Y (cond rho {x | S x = a})) :
    rho[fun x => spinSign (S x) |
      (inferInstance : MeasurableSpace (A × B)).comap (fun x => (X x, Y x))] =ᵐ[rho]
      fun x => binaryFusion (q (X x)) (r (Y x)) := by
  let mu := Measure.map X rho
  let nu := Measure.map Y rho
  have hqm : ∀ᵐ x ∂mu, |q x| < 1 :=
    (ae_map_iff hX.aemeasurable (measurableSet_lt hq.abs measurable_const)).2 hqb
  have hrm : ∀ᵐ y ∂nu, |r y| < 1 :=
    (ae_map_iff hY.aemeasurable (measurableSet_lt hr.abs measurable_const)).2 hrb
  have hpair (a : IsingSpin) : Measure.map (fun x => (X x, Y x)) (cond rho {x | S x = a}) =
      (linearSignalTilt mu q (spinSign a)).prod (linearSignalTilt nu r (spinSign a)) := by
    let : IsProbabilityMeasure (cond rho {x | S x = a}) :=
      cond_isProbabilityMeasure (by rw [hfair]; norm_num)
    rw [(hInd a).map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,
      map_cond_spin_observation_eq_linearSignalTilt hS hX hq hpostX a (hfair a),
      map_cond_spin_observation_eq_linearSignalTilt hS hY hr hpostY a (hfair a)]
  have hjoint : Measure.map (fun x => (X x, Y x)) rho =
      observationFusionJointLaw mu nu q r := by
    calc
      _ = Measure.map (fun x => (X x, Y x))
          ((1 / 2 : ENNReal) • cond rho {x | S x = .pos} +
           (1 / 2 : ENNReal) • cond rho {x | S x = .neg}) := by
        rw [← measure_eq_fair_spin_mixture hS hfair]
      _ = _ := by
        rw [Measure.map_add _ _ (hX.prodMk hY),
          Measure.map_smul _ (hX.prodMk hY).aemeasurable,
          Measure.map_smul _ (hX.prodMk hY).aemeasurable,
          hpair .pos, hpair .neg]
        exact observationFusionJointLaw_eq_mixture hq hr hqm hrm
  apply condExp_spin_eq_of_conditional_linearTilt hS (hX.prodMk hY)
    (show Measurable (fun p : A × B => binaryFusion (q p.1) (r p.2)) by
      unfold binaryFusion
      fun_prop)
  · rw [hjoint]
    have hb : ∀ᵐ p ∂observationFusionJointLaw mu nu q r, |q p.1| < 1 ∧ |r p.2| < 1 :=
      (withDensity_absolutelyContinuous (mu.prod nu)
        (fun p => ENNReal.ofReal (1 + q p.1 * r p.2))).ae_le
          (ae_observation_pair_bounds hqm hrm)
    filter_upwards [hb] with p hp
    exact (abs_binaryFusion_lt_one hp.1 hp.2).le
  · exact hfair .pos
  · rw [hpair .pos, hjoint]
    exact observationFusion_positive_product_eq_tilt hq hr hqm hrm

end Problem367

end

end OAI