import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density

namespace OAI

section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators

namespace CriticalPercolation.FiniteProduct
open scoped Classical

variable {E : Type*} [Fintype E]

lemma density_modular (w : E → Prop → ℝ) (x y : E → Prop) :
    density w x * density w y = density w (x ⊓ y) * density w (x ⊔ y) := by
  unfold density
  rw [← Finset.prod_mul_distrib, ← Finset.prod_mul_distrib]
  apply Finset.prod_congr rfl
  intro e _
  simp only [Pi.inf_apply, Pi.sup_apply]
  by_cases hx : x e <;> by_cases hy : y e <;> simp [hx, hy, mul_comm]

lemma expect_mul_ge (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
    (hpos : ∀ e b, 0 ≤ w e b) (f g : (E → Prop) → ℝ)
    (hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω)
    (hf : Monotone f) (hg : Monotone g) :
    expect w f * expect w g ≤ expect w (fun ω => f ω * g ω) := by
  have h := fkg f g (density w) (fun ω => density_nonneg hpos ω) hf0 hg0 hf hg
    (fun x y => le_of_eq (density_modular w x y))
  simpa only [expect, sum_density w hw, one_mul] using h

end CriticalPercolation.FiniteProduct

end
end

end OAI