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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 37 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Prob
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 prob_or (w : E → Prop → ℝ) (P Q : Set E → Prop)
(hd : ∀ ω, ¬(P ω ∧ Q ω)) :
prob w (fun ω => P ω ∨ Q ω) = prob w P + prob w Q := by
unfold prob
rw [← expect_add]
apply congrArg (expect w)
funext ω
by_cases hp : P ω <;> by_cases hq : Q ω <;> simp [hp,hq]
exact (hd ω ⟨hp,hq⟩).elim
lemma prob_diff (w : E → Prop → ℝ) (P Q : Set E → Prop) :
prob w (fun ω => P ω ∧ ¬Q ω) = prob w P - prob w (fun ω => P ω ∧ Q ω) := by
have hh := prob_split w P Q
linarith
end CriticalPercolation.FiniteProduct
end
end
end OAI
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