| 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 |
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|
