openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /CriticalFragility.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.BernoulliThinning | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical ENNReal Topology | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def thinPercolatesEvent (ω : Set E) : Set (Set E) := | |
| {η | ω∩η∈G.percolates} | |
| lemma measurableSet_thinPercolatesEvent [Countable V] [Countable E] (ω : Set E) : | |
| MeasurableSet (G.thinPercolatesEvent ω) := | |
| G.measurableSet_percolates.preimage | |
| (BernoulliProduct.measurable_intersection.comp (measurable_const.prodMk measurable_id)) | |
| lemma thinPercolatesEvent_upper (ω : Set E) : IsUpperSet (G.thinPercolatesEvent ω) := | |
| fun _ _ h hη => G.percolates_mono (Set.inter_subset_inter_right _ h) hη | |
| lemma law_thinPercolates_mono [Countable V] [Countable E] (ω : Set E) | |
| {p q : unitInterval} (hpq : p≤q) : | |
| G.law p (G.thinPercolatesEvent ω)≤G.law q (G.thinPercolatesEvent ω) := by | |
| simpa only [BernoulliProduct.law_const,law] using | |
| BernoulliProduct.law_mono_parameters hpq (G.thinPercolatesEvent ω) | |
| (G.measurableSet_thinPercolatesEvent ω) (G.thinPercolatesEvent_upper ω) | |
| lemma law_map_intersection (p r : unitInterval) : | |
| ((G.law p).prod (G.law r)).map (fun z : Set E×Set E => z.1∩z.2)=G.law (p*r) := by | |
| simpa only [BernoulliProduct.law_const,law] using | |
| BernoulliProduct.law_map_intersection (E := E) p r | |
| lemma thinning_null_ae_of_product_null [Countable V] [Countable E] | |
| (p r : unitInterval) (h : G.law (p*r) G.percolates=0) : | |
| ∀ᵐ ω ∂G.law p, G.law r (G.thinPercolatesEvent ω)=0 := by | |
| have hn : (G.law p).prod (G.law r) | |
| ((fun z : Set E×Set E => z.1∩z.2) ⁻¹' G.percolates)=0 := by | |
| rw [←Measure.map_apply BernoulliProduct.measurable_intersection G.measurableSet_percolates, | |
| G.law_map_intersection,h] | |
| exact Measure.measure_ae_null_of_prod_null hn | |
| lemma critical_thinning_null_ae_fixed [Countable V] [Countable E] | |
| (r : unitInterval) (hr : r<1) : | |
| ∀ᵐ ω ∂G.law G.criticalProbability, G.law r (G.thinPercolatesEvent ω)=0 := by | |
| apply G.thinning_null_ae_of_product_null | |
| by_cases hp : G.criticalProbability=0 | |
| · rw [hp,zero_mul,G.law_zero_percolates] | |
| · apply G.law_percolates_eq_zero_of_lt_critical | |
| have hp' : (0 : ℝ)<G.criticalProbability := | |
| (show (0 : unitInterval)<G.criticalProbability from lt_of_le_of_ne bot_le (Ne.symm hp)) | |
| have hr' : (r : ℝ)<1 := hr | |
| change (G.criticalProbability : ℝ)*(r : ℝ)<(G.criticalProbability : ℝ) | |
| nlinarith | |
| theorem critical_thinning_null_ae | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) : | |
| ∀ᵐ ω ∂G.law G.criticalProbability, ∀ r : unitInterval, r<1 → | |
| G.law r (G.thinPercolatesEvent ω)=0 := by | |
| let := G.countable_vertices hc hl | |
| let := G.countable_bonds hc hl | |
| obtain ⟨r,hmono,hr,hlim⟩ := exists_seq_strictMono_tendsto' | |
| (show (0 : unitInterval)<1 by norm_num) | |
| have hn : ∀ᵐ ω ∂G.law G.criticalProbability, ∀ n : ℕ, | |
| G.law (r n) (G.thinPercolatesEvent ω)=0 := | |
| ae_all_iff.mpr (fun n => G.critical_thinning_null_ae_fixed (r n) (hr n).2) | |
| filter_upwards [hn] with ω hω | |
| intro q hq | |
| obtain ⟨n,hn⟩ := (hlim.eventually (lt_mem_nhds hq)).exists | |
| exact le_antisymm ((G.law_thinPercolates_mono ω hn.le).trans_eq (hω n)) bot_le | |
| theorem critical_not_robust_ae | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) : | |
| ∀ᵐ ω ∂G.law G.criticalProbability, ¬∃ r : unitInterval, | |
| r<1 ∧ 0<G.law r (G.thinPercolatesEvent ω) := by | |
| filter_upwards [G.critical_thinning_null_ae hc hl] with ω hω | |
| rintro ⟨r,hr,hp⟩ | |
| exact hp.ne' (hω r hr) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |