openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /MonotoneIndicatorUpper.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Modular.DensityModular | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness | |
| namespace OAI | |
| section | |
| section | |
| open scoped BigOperators | |
| open scoped Classical ENNReal | |
| open Set | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| open Filter Topology | |
| open scoped Topology Classical | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} | |
| lemma monotone_indicator_upper (A : Set (Set E)) (hA : IsUpperSet A) : | |
| Monotone (fun ω : E → Prop => A.indicator (1 : Set E → ℝ) {e | ω e}) := by | |
| intro ω ω' h | |
| change (if {e | ω e} ∈ A then (1 : ℝ) else 0) ≤ | |
| (if {e | ω' e} ∈ A then (1 : ℝ) else 0) | |
| split_ifs with ha hb | |
| · rfl | |
| · exact (hb (hA h ha)).elim | |
| · norm_num | |
| · rfl | |
| lemma real_eq_expect_indicator [Fintype E] (q : E → unitInterval) (A : Set (Set E)) : | |
| (law q).real A = FiniteProduct.expect (fun e => bitWeight (q e)) | |
| (fun ω : E → Prop => A.indicator (1 : Set E → ℝ) {e | ω e}) := by | |
| rw [← integral_law_eq_expect] | |
| exact (integral_indicator_one (.of_discrete)).symm | |
| lemma finite_harris [Fintype E] (q : E → unitInterval) | |
| (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) : | |
| (law q).real A * (law q).real B ≤ (law q).real (A ∩ B) := by | |
| simp_rw [real_eq_expect_indicator] | |
| have he : (fun ω : E → Prop => (A ∩ B).indicator (1 : Set E → ℝ) {e | ω e}) = | |
| (fun ω => A.indicator (1 : Set E → ℝ) {e | ω e} * | |
| B.indicator (1 : Set E → ℝ) {e | ω e}) := by | |
| funext ω | |
| by_cases ha : {e | ω e} ∈ A <;> by_cases hb : {e | ω e} ∈ B <;> simp [ha, hb] | |
| rw [he] | |
| exact FiniteProduct.expect_mul_ge _ (fun e => bitWeight_sum (q e)) | |
| (fun e b => bitWeight_nonneg (q e) b) _ _ | |
| (Set.indicator_nonneg (fun _ _ => zero_le_one)) | |
| (Set.indicator_nonneg (fun _ _ => zero_le_one)) | |
| (monotone_indicator_upper A hA) (monotone_indicator_upper B hB) | |
| lemma real_law_restrict_general (q : E → unitInterval) (s : Set E) (hs : s.Finite) | |
| (A : Set (Set E)) : | |
| (law q).real {ω | s ∩ ω ∈ A} = | |
| (law (fun e : s => q e)).real {ω | Subtype.val '' ω ∈ A} := by | |
| let := hs.fintype | |
| have hi (ω : Set E) : Subtype.val '' (Subtype.val ⁻¹' ω : Set s) = s ∩ ω := by | |
| ext e | |
| simp | |
| rw [← law_map_restrict q s] | |
| have hf : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) := | |
| Measurable.of_eval (fun edge => measurable_set_mem edge.val) | |
| simp only [Measure.real] | |
| rw [Measure.map_apply hf (.of_discrete)] | |
| simp only [Set.preimage_ofPred_eq, hi] | |
| lemma finite_restriction_harris (q : E → unitInterval) (s : Set E) (hs : s.Finite) | |
| (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) : | |
| (law q).real {ω | s ∩ ω ∈ A} * (law q).real {ω | s ∩ ω ∈ B} ≤ | |
| (law q).real {ω | s ∩ ω ∈ A ∩ B} := by | |
| let := hs.fintype | |
| simp_rw [real_law_restrict_general _ s hs] | |
| exact finite_harris _ _ _ (fun _ _ h => hA (Set.image_mono h)) | |
| (fun _ _ h => hB (Set.image_mono h)) | |
| lemma HasFiniteOpenWitness.inter {A B : Set (Set E)} | |
| (hA : HasFiniteOpenWitness A) (hB : HasFiniteOpenWitness B) | |
| (hAm : IsUpperSet A) (hBm : IsUpperSet B) : HasFiniteOpenWitness (A ∩ B) := by | |
| rintro ω ⟨hωA, hωB⟩ | |
| obtain ⟨s, hs, hsω, hsA⟩ := hA ω hωA | |
| obtain ⟨t, ht, htω, htB⟩ := hB ω hωB | |
| exact ⟨s ∪ t, hs.union ht, Set.union_subset hsω htω, | |
| hAm Set.subset_union_left hsA, hBm Set.subset_union_right htB⟩ | |
| lemma tendsto_real_finiteWitness_general (q : E → unitInterval) (A : Set (Set E)) | |
| (hmono : IsUpperSet A) (hw : HasFiniteOpenWitness A) | |
| (K : Set.FiniteExhaustion (Set.univ : Set E)) : | |
| Tendsto (fun n => (law q).real {ω | K n ∩ ω ∈ A}) | |
| atTop (𝓝 ((law q).real A)) := by | |
| have h := tendsto_measure_iUnion_atTop (μ := law q) | |
| (finiteWitness_exhaustion_mono A hmono K) | |
| rw [finiteWitness_exhaustion A hmono hw K] at h | |
| exact (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp h | |
| theorem harris_finiteWitness [Countable E] (q : E → unitInterval) | |
| (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) | |
| (hwA : HasFiniteOpenWitness A) (hwB : HasFiniteOpenWitness B) : | |
| (law q).real A * (law q).real B ≤ (law q).real (A ∩ B) := by | |
| let K := (Set.to_countable (Set.univ : Set E)).finiteExhaustion | |
| exact le_of_tendsto_of_tendsto | |
| ((tendsto_real_finiteWitness_general q A hA hwA K).mul | |
| (tendsto_real_finiteWitness_general q B hB hwB K)) | |
| (tendsto_real_finiteWitness_general q (A ∩ B) (hA.inter hB) | |
| (hwA.inter hwB hA hB) K) | |
| (Eventually.of_forall fun n => finite_restriction_harris q (K n) (K.finite n) A B hA hB) | |
| end CriticalPercolation.BernoulliProduct | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| end | |
| end | |
| end OAI | |