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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Potential
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E : Type*} [Fintype E]
lemma prob_requires_bit (w : E → Prop → ℝ) (e : E) (b : Prop) (P : Set E → Prop)
(hb : ∀ ω, P ω → ω e = b) : prob w P = w e b*prob (pin w e b) P := by
have hh := expect_prop_mask_pin w e b (fun ω => if P ω then 1 else 0)
have he : (fun ω => if ω e = b then (if P ω then (1:ℝ) else 0) else 0) =
fun ω => if P ω then 1 else 0 := by
funext ω
by_cases hp : P ω
· simp [hp,hb ω hp]
· simp [hp]
erw [he] at hh
exact hh
end CriticalPercolation.FiniteProduct
end
end
end OAI
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