openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /FreshRegionLaw.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} | |
| lemma trim_law_eq (q r : E → unitInterval) (S : Set E) (h : ∀ e∈S, q e=r e) : | |
| (law q).map (forceOn Sᶜ ∅)=(law r).map (forceOn Sᶜ ∅) := by | |
| rw [forceOn_map_law,forceOn_map_law] | |
| congr 1 | |
| funext e | |
| by_cases he : e∈S | |
| · simp [pinnedParameters,he,h e he] | |
| · simp [pinnedParameters,he] | |
| lemma fresh_region_lower (q r : E → unitInterval) (S : Set E) (h : ∀ e∈S, q e=r e) | |
| (B : Set (Set E)) (hm : MeasurableSet B) (hu : IsUpperSet B) : | |
| (law r).real ((forceOn Sᶜ ∅) ⁻¹' B) ≤ (law q).real B := by | |
| have he : (law r) ((forceOn Sᶜ ∅) ⁻¹' B)=(law q) ((forceOn Sᶜ ∅) ⁻¹' B) := by | |
| rw [← Measure.map_apply (measurable_forceOn _ _) hm, | |
| ← Measure.map_apply (measurable_forceOn _ _) hm,trim_law_eq q r S h] | |
| rw [measureReal_def,he] | |
| change (law q).real ((forceOn Sᶜ ∅) ⁻¹' B) ≤ _ | |
| apply measureReal_mono _ (measure_ne_top _ _) | |
| intro ω hω | |
| apply hu (a := forceOn Sᶜ ∅ ω) (b := ω) ?_ hω | |
| intro e he | |
| by_cases hes : e∈S | |
| · simpa [forceOn,hes] using he | |
| · simp [hes] at he | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end | |
| end | |
| end OAI | |