openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /ProductEmbedding.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {A E : Type*} | |
| lemma law_map_preimage (q : E → unitInterval) (i : A → E) (hi : Injective i) : | |
| (law q).map (fun ω : Set E => i ⁻¹' ω)=law (q∘i) := by | |
| have hind : iIndepFun (fun a (ω : Set E) => i a∈ω) (law q) := | |
| iIndepFun.precomp hi (independent_coordinates q) | |
| have hh := hind.map_fun_eq_infinitePi_map (fun a => measurable_set_mem (i a)) | |
| change (law q).map (fun ω : Set E => i ⁻¹' ω)=_ at hh | |
| rw [hh] | |
| rw [law_eq_infinitePi (q∘i)] | |
| apply congrArg Measure.infinitePi | |
| funext a | |
| rw [law_eq_infinitePi] | |
| exact Measure.infinitePi_map_eval _ (i a) | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end | |
| end | |
| end OAI | |