import OAI.Probability.ContinuumTransition.KacEnergy

namespace OAI

section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature

theorem familyTerm_product_bound (F G H : ConfigFamily)
    (hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space}
    (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B)
    (hGF : GrandFinite G A) (hHF : GrandFinite H B) (n : ℕ)
    (hFi : IntegrableOn (F (Fin n)) {x | ∀ i, x i ∈ A ∪ B})
    (hbound : ∀ (s : Finset (Fin n)) (x : Fin n → Space),
      x ∈ binaryRegion A B s → F (Fin n) x ≤
        G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) :
    familyTerm F (A ∪ B) n ≤ ∑ k ∈ Finset.range (n+1), familyTerm G A k*familyTerm H B (n-k) := by
  have hb := integral_binary_bound hA hB hd
    (fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k))
    hFi ?_
  swap
  · intro s
    have hprod := integrableOn_binary_product volume A B s
      (by simpa only [← volume_pi] using hGF.integrable {i : Fin n // i ∈ s})
      (by simpa only [← volume_pi] using hHF.integrable {i : Fin n // i ∉ s})
    have hi := setIntegral_mono_on
      (hFi.mono_set (binaryRegion_subset_union A B s))
      (by simpa only [← volume_pi] using hprod)
      (measurableSet_binaryRegion hA hB s) (hbound s)
    rw [show volume = Measure.pi (fun _ : Fin n => (volume : Measure Space)) by exact volume_pi,
      setIntegral_binary_product volume A B s (G {i : Fin n // i ∈ s})
        (H {i : Fin n // i ∉ s})] at hi
    simp only [← volume_pi] at hi
    have heg := familyIntegral_eq_Fin G hG A {i : Fin n // i ∈ s}
    have heh := familyIntegral_eq_Fin H hH B {i : Fin n // i ∉ s}
    have hc : Fintype.card {i : Fin n // i ∈ s} = s.card := Fintype.card_coe s
    rw [hc] at heg
    have hcc : Fintype.card {i : Fin n // i ∉ s} = n-s.card := by
      have hh := Fintype.card_subtype_compl (fun i : Fin n => i ∈ s)
      simpa only [Fintype.card_fin, hc] using hh
    rw [hcc] at heh
    change (∫ x in binaryRegion A B s, F (Fin n) x) ≤ _
    calc
      _ ≤ familyIntegral G A {i : Fin n // i ∈ s}*
          familyIntegral H B {i : Fin n // i ∉ s} := hi
      _ = _ := by rw [heg, heh, Fintype.card_fin]
  have hh := div_le_div_of_nonneg_right hb (Nat.cast_nonneg (n.factorial) : (0 : ℝ) ≤ n.factorial)
  rw [Fintype.card_fin, factorial_binomial_sum
    (fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k)) n] at hh
  exact hh

theorem GrandFinite.union_product {F G H : ConfigFamily}
    (hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space}
    (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B)
    (hGF : GrandFinite G A) (hHF : GrandFinite H B)
    (hFi : ∀ (ι : Type) [Fintype ι], IntegrableOn (F ι) {x | ∀ i, x i ∈ A ∪ B})
    (h0 : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x)
    (hbound : ∀ (n : ℕ) (s : Finset (Fin n)) (x : Fin n → Space),
      x ∈ binaryRegion A B s → F (Fin n) x ≤
        G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) :
    GrandFinite F (A ∪ B) := by
  refine ⟨hFi, ?_⟩
  have hab : Summable (fun p : ℕ × ℕ => familyTerm G A p.1*familyTerm H B p.2) :=
    summable_mul_of_summable_norm (summable_norm_iff.mpr hGF.summable)
      (summable_norm_iff.mpr hHF.summable)
  exact Summable.of_nonneg_of_le (familyTerm_nonneg F (A ∪ B) h0)
    (fun n => familyTerm_product_bound F G H hG hH hA hB hd hGF hHF n (hFi (Fin n)) (hbound n))
    (summable_sum_mul_range_of_summable_mul hab)

end ContinuumTemperature
end

end OAI