openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /ProbDisjoint.lean
| 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_disjoint_le (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b) | |
| (P Q R : Set E → Prop) (hP : ∀ ω, P ω → R ω) (hQ : ∀ ω, Q ω → R ω) | |
| (hPQ : ∀ ω, ¬(P ω ∧ Q ω)) : prob w P + prob w Q ≤ prob w R := by | |
| unfold prob | |
| rw [← expect_add] | |
| apply expect_mono hp | |
| intro ω | |
| by_cases hp' : P ω <;> by_cases hq' : Q ω <;> by_cases hr' : R ω <;> | |
| simp [hp',hq',hr'] | |
| · exact (hPQ ω ⟨hp',hq'⟩).elim | |
| · exact (hr' (hP ω hp')).elim | |
| · exact (hr' (hP ω hp')).elim | |
| · exact (hr' (hQ ω hq')).elim | |
| end CriticalPercolation.FiniteProduct | |
| end | |
| end | |
| end OAI | |