openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /CylinderApproximation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical symmDiff ENNReal | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} | |
| def finiteCoordinateEvents : Set (Set (Set E)) := | |
| {A | ∃ S : Finset E, MeasurableSet[blockSigma (S : Set E)] A} | |
| lemma finiteCoordinateEvents_ring : IsSetRing (finiteCoordinateEvents (E := E)) where | |
| empty_mem := ⟨∅,@MeasurableSet.empty (Set E) (blockSigma ((∅ : Finset E) : Set E))⟩ | |
| union_mem := by | |
| rintro A B ⟨S,hS⟩ ⟨T,hT⟩ | |
| refine ⟨S ∪ T,?_⟩ | |
| exact @MeasurableSet.union (Set E) (blockSigma ((S ∪ T : Finset E) : Set E)) _ _ | |
| (blockSigma_mono (by simp) _ hS) (blockSigma_mono (by simp) _ hT) | |
| sdiff_mem := by | |
| rintro A B ⟨S,hS⟩ ⟨T,hT⟩ | |
| refine ⟨S ∪ T,?_⟩ | |
| exact @MeasurableSet.diff (Set E) (blockSigma ((S ∪ T : Finset E) : Set E)) _ _ | |
| (blockSigma_mono (by simp) _ hS) (blockSigma_mono (by simp) _ hT) | |
| lemma generateFrom_finiteCoordinateEvents : | |
| MeasurableSpace.generateFrom (finiteCoordinateEvents (E := E)) = | |
| (Set.instMeasurableSpace : MeasurableSpace (Set E)) := by | |
| apply le_antisymm | |
| · exact MeasurableSpace.generateFrom_le fun A ⟨S,hS⟩ => blockSigma_le _ _ hS | |
| · change (⨆ e : E, coordinateSigma e) ≤ _ | |
| refine iSup_le fun e => ?_ | |
| intro A hA | |
| apply MeasurableSpace.measurableSet_generateFrom | |
| refine ⟨{e},?_⟩ | |
| exact (le_iSup_of_le e (le_iSup_of_le (Finset.mem_singleton_self e) le_rfl) : | |
| coordinateSigma e ≤ blockSigma (({e} : Finset E) : Set E)) _ hA | |
| lemma exists_finiteCoordinate_approx (q : E → unitInterval) | |
| {A : Set (Set E)} (hA : MeasurableSet A) {ε : ℝ≥0∞} (hε : 0 < ε) : | |
| ∃ (S : Finset E) (B : Set (Set E)), MeasurableSet[blockSigma (S : Set E)] B ∧ | |
| law q (B ∆ A) < ε := by | |
| obtain ⟨B,⟨S,hB⟩,h⟩ := exists_measure_symmDiff_lt_of_generateFrom_isSetRing | |
| finiteCoordinateEvents_ring (μ := law q) | |
| (show ∃ D : Set (Set (Set E)), D.Countable ∧ D ⊆ finiteCoordinateEvents ∧ | |
| law q (⋃₀ D)ᶜ = 0 from | |
| ⟨{Set.univ},by simp,by rintro A rfl; exact ⟨∅,@MeasurableSet.univ (Set E) (blockSigma ((∅ : Finset E) : Set E))⟩,by simp⟩) | |
| generateFrom_finiteCoordinateEvents.symm hA hε | |
| exact ⟨S,B,hB,h⟩ | |
| lemma measurable_image_equiv_blocks (e : E ≃ E) (S : Set E) : | |
| @Measurable (Set E) (Set E) (blockSigma (e.symm '' S)) (blockSigma S) | |
| (fun ω => e '' ω) := by | |
| apply measurable_iff_comap_le.mpr | |
| simp only [blockSigma, MeasurableSpace.comap_iSup] | |
| refine iSup₂_le fun x hx => ?_ | |
| have hm : @Measurable (Set E) Prop (coordinateSigma (e.symm x)) inferInstance | |
| (fun ω => x ∈ e '' ω) := by | |
| have hh : (fun ω : Set E => x ∈ e '' ω) = (fun ω => e.symm x ∈ ω) := by | |
| funext ω | |
| exact propext (by simp) | |
| rw [hh] | |
| exact measurable_iff_comap_le.mpr le_rfl | |
| have hc := hm.comap_le | |
| rw [coordinateSigma,MeasurableSpace.comap_comp] | |
| change MeasurableSpace.comap (fun ω : Set E => x ∈ e '' ω) inferInstance ≤ _ | |
| exact hc.trans (le_iSup_of_le (e.symm x) (le_iSup_of_le ⟨x,hx,rfl⟩ le_rfl)) | |
| end CriticalPercolation.BernoulliProduct | |
| end | |
| end | |
| end OAI | |