import OAI.Analysis.FiniteFactor.Coupling.Limit
import OAI.Analysis.FiniteFactor.Coupling.Infinite

namespace OAI

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

/-- The source coordinates, retaining the block and the position in that block. -/
abbrev BlockCoordinate (m : ℕ → ℕ) := Σ s : ℕ, Fin (m s)

def blockPrefix (m : ℕ → ℕ) (n : ℕ) : Finset (BlockCoordinate m) :=
  (Finset.range n).sigma (fun _ => Finset.univ)

theorem mem_blockPrefix (m : ℕ → ℕ) (n : ℕ) (i : BlockCoordinate m) :
    i ∈ blockPrefix m n ↔ i.1 < n := by simp [blockPrefix]

theorem blockPrefix_mono (m : ℕ → ℕ) : Monotone (blockPrefix m) := by
  intro n k hnk i hi
  exact (mem_blockPrefix m k i).2 (lt_of_lt_of_le ((mem_blockPrefix m n i).1 hi) hnk)

theorem blockPrefix_cover (m : ℕ → ℕ) (J : Finset (BlockCoordinate m)) :
    ∃ n, J ⊆ blockPrefix m n := by
  refine ⟨J.sup (fun i => i.1) + 1, fun i hi => ?_⟩
  exact (mem_blockPrefix m _ i).2 (Nat.lt_succ_of_le (Finset.le_sup (f := fun i => i.1) hi))

theorem blockPrefix_energy (p : ℕ → CouplingParameters) (m : ℕ → ℕ)
    (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ)
    (htag : ∀ s, (p s).tagged = decide (s=j))
    (hκ : Summable (fun s => (p s).kappa^2))
    (right : BlockCoordinate m → Bool) (n : ℕ) :
    (∫ x, prefixDensity (fun i : BlockCoordinate m => p i.1) right (blockPrefix m n) x ^ 2
      ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤
      Real.exp (20*(2*(∑' s, (p s).kappa^2)+1)) := by
  apply (prefixDensity_energy _ right _).trans
  apply Real.exp_le_exp.mpr
  have heq : (∑ i ∈ blockPrefix m n, 20*(2*(p i.1).kappa^2/(p i.1).m + (p i.1).tagMass)) =
      20 * ∑ s ∈ Finset.range n, (p s).m * (2*(p s).kappa^2/(p s).m + (p s).tagMass) := by
    rw [blockPrefix, Finset.sum_sigma]
    simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
    rw [Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro s _
    rw [hm s]
    ring
  rw [heq]
  exact mul_le_mul_of_nonneg_left (uniform_block_budget p j htag hκ (Finset.range n)) (by norm_num)

theorem infinite_block_source_density (p : ℕ → CouplingParameters) (m : ℕ → ℕ)
    (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ)
    (htag : ∀ s, (p s).tagged = decide (s=j))
    (hκ : ∀ s, (p s).kappa = sourceKappa s) (right : BlockCoordinate m → Bool) :
    ∃ g : (BlockCoordinate m → ℝ) → ℝ,
      MemLp g 2 (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ∧
      (∀ᵐ x ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure), 0 ≤ g x) ∧
      (∫ x, g x ^ 2 ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤
        Real.exp (20*(2*(∑' s, sourceKappa s ^ 2)+1)) ∧
      Measure.infinitePi (fun i : BlockCoordinate m => (p i.1).selectedLaw true (right i)) =
        (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)).withDensity
          (fun x => ENNReal.ofReal (g x)) := by
  apply infiniteEndpoint_density_of_prefix_bound _ right (blockPrefix m)
    (blockPrefix_mono m) (blockPrefix_cover m)
  intro n
  have hs : Summable (fun s => (p s).kappa^2) := by simpa only [hκ] using sourceKappa_sq_summable
  simpa only [hκ] using blockPrefix_energy p m hm j htag hs right n

/-- Actual augmented law on the block-indexed coordinates. -/
def blockProductCoupling (p : ℕ → CouplingParameters) (m : ℕ → ℕ) :
    Measure (BlockCoordinate m → Fin 6 × (ℝ × ℝ)) :=
  Measure.infinitePi (fun i => (p i.1).labelledCoupling)

instance blockProductCoupling_probability (p : ℕ → CouplingParameters) (m : ℕ → ℕ) :
    IsProbabilityMeasure (blockProductCoupling p m) := by unfold blockProductCoupling; infer_instance

def blockProductEndpoint {m : ℕ → ℕ} (right : BlockCoordinate m → Bool)
    (ω : BlockCoordinate m → Fin 6 × (ℝ × ℝ)) (i : BlockCoordinate m) : ℝ :=
  endpoint (right i) (ω i).2

theorem blockProductEndpoint_law (p : ℕ → CouplingParameters) (m : ℕ → ℕ)
    (right : BlockCoordinate m → Bool) :
    (blockProductCoupling p m).map (blockProductEndpoint right) =
      Measure.infinitePi (fun i : BlockCoordinate m => (p i.1).selectedLaw true (right i)) := by
  unfold blockProductCoupling blockProductEndpoint
  rw [Measure.infinitePi_map_pi _ (f := fun (i : BlockCoordinate m) (z : Fin 6 × (ℝ × ℝ)) => endpoint (right i) z.2)
    (fun i => (endpoint_measurable (right i)).comp measurable_snd)]
  simp_rw [labelled_endpoint_density]
  rfl

/-- The density belongs to the actual law of the selected endpoint map, rather
than only to an abstract measure with matching finite moments. -/
theorem blockProductEndpoint_L2_density (p : ℕ → CouplingParameters) (m : ℕ → ℕ)
    (hm : ∀ s, (p s).m = (m s : ℝ)) (j : ℕ)
    (htag : ∀ s, (p s).tagged = decide (s=j))
    (hκ : ∀ s, (p s).kappa = sourceKappa s) (right : BlockCoordinate m → Bool) :
    ∃ g : (BlockCoordinate m → ℝ) → ℝ,
      MemLp g 2 (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ∧
      (∀ᵐ x ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure), 0 ≤ g x) ∧
      (∫ x, g x ^ 2 ∂Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)) ≤
        Real.exp (20*(2*(∑' s, sourceKappa s ^ 2)+1)) ∧
      (blockProductCoupling p m).map (blockProductEndpoint right) =
        (Measure.infinitePi (fun _ : BlockCoordinate m => unitIntervalMeasure)).withDensity
          (fun x => ENNReal.ofReal (g x)) := by
  rw [blockProductEndpoint_law]
  exact infinite_block_source_density p m hm j htag hκ right

end CouplingParameters
end
end FiniteFactor

end OAI