openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /BernoulliSingleton.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning | |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} | |
| lemma force_singleton_inter (q : E → unitInterval) (e : E) (σ : Set E) | |
| (A : Set (Set E)) (hA : MeasurableSet A) : | |
| law q (A ∩ {ω : Set E | (e∈ω)=(e∈σ)}) = | |
| bitMeasure (q e) {e∈σ} * law q (forceOn {e} σ ⁻¹' A) := by | |
| let C := {ω : Set E | (e∈ω)=(e∈σ)} | |
| let B := forceOn {e} σ ⁻¹' A | |
| have hmC : MeasurableSet[blockSigma {e}] C := by | |
| have hc : MeasurableSet[coordinateSigma e] C := | |
| ⟨{e∈σ},MeasurableSet.singleton _,rfl⟩ | |
| exact (le_iSup_of_le e (le_iSup_of_le (Set.mem_singleton e) le_rfl) : | |
| coordinateSigma e≤blockSigma {e}) _ hc | |
| have hmB : MeasurableSet[blockSigma ({e} : Set E)ᶜ] B := | |
| hA.preimage (measurable_forceOn_block {e} σ) | |
| have hmul := ((independent_blocks q {e}).indepSet_of_measurableSet hmC hmB).measure_inter_eq_mul | |
| have heq : A∩C = C∩B := by | |
| ext ω | |
| have hf (hω : ω∈C) : forceOn {e} σ ω=ω := by | |
| ext f | |
| by_cases he : f=e | |
| · subst f | |
| simpa only [mem_forceOn,Set.mem_singleton_iff,ite_true] using Iff.of_eq hω.symm | |
| · simp only [mem_forceOn,Set.mem_singleton_iff,ite_eq_right he] | |
| constructor | |
| · rintro ⟨ha,hc⟩ | |
| exact ⟨hc,by change forceOn {e} σ ω∈A; rwa [hf hc]⟩ | |
| · rintro ⟨hc,hb⟩ | |
| exact ⟨by change forceOn {e} σ ω∈A at hb; rwa [hf hc] at hb,hc⟩ | |
| change law q (A∩C)=_ | |
| rw [heq,hmul] | |
| change law q {ω : Set E | (e∈ω)=(e∈σ)} * _ = _ | |
| rw [law_coordinate] | |
| lemma law_singleton_decomposition (q : E → unitInterval) (e : E) : | |
| law q = bitMeasure (q e) {True} • (law q).map (forceOn {e} Set.univ) + | |
| bitMeasure (q e) {False} • (law q).map (forceOn {e} ∅) := by | |
| ext A hA | |
| rw [Measure.add_apply,Measure.smul_apply,Measure.smul_apply, | |
| Measure.map_apply (measurable_forceOn _ _) hA, | |
| Measure.map_apply (measurable_forceOn _ _) hA] | |
| simp only [smul_eq_mul] | |
| have ht := force_singleton_inter q e Set.univ A hA | |
| have hf := force_singleton_inter q e ∅ A hA | |
| simp only [Set.mem_univ,Set.mem_empty_iff_false] at ht hf | |
| rw [←ht,←hf] | |
| have he : {ω : Set E | (e∈ω)=False} = {ω : Set E | (e∈ω)=True}ᶜ := by | |
| ext ω | |
| simp | |
| rw [he,←Set.sdiff_eq] | |
| exact (measure_inter_add_sdiff A (measurableSet_eq_fun (measurable_set_mem e) measurable_const)).symm | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end | |
| end OAI | |