openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /OpenFrontierDivergence.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Gluing.SelectedFresh | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability | |
| import OAI.Probability.CriticalPercolation.Criticality.FewOpenBonds | |
| import OAI.Probability.CriticalPercolation.Exploration.FrontierDivergence | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical ENNReal Topology | |
| namespace CriticalPercolation.BernoulliProduct | |
| variable {E : Type*} [Countable E] | |
| lemma measurableSet_fewOpenBonds (S : Set E) (N : ℕ) : | |
| MeasurableSet[blockSigma S] {ω : Set E | (S∩ω).ncard ≤ N} := by | |
| simpa only [Set.inter_comm,Set.preimage_ofPred_eq,Set.Iic,Function.comp_apply] using | |
| (measurableSet_Iic.preimage (measurable_ncard.comp (measurable_inter_block S))) | |
| end CriticalPercolation.BernoulliProduct | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| def fewOpenFreshBoundary (X : Set E) (v : V) (N : ℕ) : Set (Set E) := | |
| {ω | (G.cluster (ω∩X) v).Finite ∧ (G.freshClusterBoundary X v ω ∩ ω).ncard ≤ N} | |
| lemma fewOpenFreshBoundary_probability_le (hl : G.LocallyFinite) (p : unitInterval) | |
| (X : Set E) (v : V) (N L : ℕ) : | |
| (G.law p).real (G.infiniteEvent v ∩ G.fewOpenFreshBoundary X v N) ≤ | |
| (G.law p).real (G.infiniteEvent v ∩ G.smallFreshBoundary X v ((N+1)*L)) + | |
| (N+1 : ℝ)*(1-(p:ℝ))^L := by | |
| let M := (N+1)*L | |
| let C : Set (Set E) := {ω | (G.cluster (ω∩X) v).Finite ∧ | |
| M ≤ (G.freshClusterBoundary X v ω).ncard} | |
| let d : Set E → Finset E := fun ω => finiteSetChoice (G.freshClusterBoundary X v ω) | |
| let A : Finset E → Set (Set E) := fun S => | |
| {ω | (((S : Set E)∩Xᶜ)∩ω).ncard ≤ N} | |
| have hC : MeasurableSet[blockSigma X] C := by | |
| have hf : MeasurableSet[blockSigma X] {ω | (G.cluster (ω∩X) v).Finite} := by | |
| let : MeasurableSpace (Set E) := blockSigma X | |
| simpa only [Function.comp_apply,Set.not_infinite] using | |
| (((G.measurable_infinite_cluster v).comp (measurable_inter_block X)).not).setOf | |
| exact hf.inter (measurableSet_Ici.preimage | |
| (measurable_ncard.comp (G.measurable_freshClusterBoundary X v))) | |
| have hd (S : Finset E) : MeasurableSet[blockSigma X] {ω | d ω=S} := | |
| @measurableSet_finiteSetChoice (Set E) E (blockSigma X) inferInstance | |
| (G.freshClusterBoundary X v) (G.measurable_freshClusterBoundary X v) S | |
| have hA (S : Finset E) : MeasurableSet[blockSigma Xᶜ] (A S) := | |
| blockSigma_mono Set.inter_subset_right _ (measurableSet_fewOpenBonds _ N) | |
| have he (ω : Set E) (hω : (G.cluster (ω∩X) v).Finite) : | |
| ((d ω : Set E)∩Xᶜ)=G.freshClusterBoundary X v ω := by | |
| change (finiteSetChoice (G.freshClusterBoundary X v ω) : Set E)∩Xᶜ = _ | |
| have hf : (G.freshClusterBoundary X v ω).Finite := | |
| (G.finite_cutEdges hl hω).subset Set.sdiff_subset | |
| rw [finiteSetChoice_coe hf] | |
| exact Set.inter_eq_left.mpr (Set.sdiff_subset_compl _ _) | |
| let r : ℝ := (N+1 : ℝ)*(1-(p:ℝ))^L | |
| have hr : 0 ≤ r := mul_nonneg (by positivity) (pow_nonneg (sub_nonneg.mpr p.2.2) _) | |
| have hb (ω : Set E) (hω : ω∈C) : (BernoulliProduct.law (fun _ : E => p)).real (A (d ω)) ≤ r := by | |
| change (BernoulliProduct.law (fun _ : E => p)).real | |
| {τ | (((d ω : Set E)∩Xᶜ)∩τ).ncard ≤ N} ≤ _ | |
| rw [he ω hω.1] | |
| exact fewOpenBonds_probability_le p _ | |
| ((G.finite_cutEdges hl hω.1).subset Set.sdiff_subset) N L hω.2 | |
| have ht := selected_fresh_bound (fun _ : E => p) X Xᶜ disjoint_compl_right | |
| d A hd hA C hC r hr hb | |
| simp only [law_const] at ht | |
| change (G.law p).real (C∩selectedEvent d A) ≤ r*(G.law p).real C at ht | |
| have ht' : (G.law p).real (C∩selectedEvent d A) ≤ r := | |
| ht.trans ((mul_le_mul_of_nonneg_left measureReal_le_one hr).trans_eq (mul_one r)) | |
| have hsub : G.infiniteEvent v ∩ G.fewOpenFreshBoundary X v N ⊆ | |
| (G.infiniteEvent v ∩ G.smallFreshBoundary X v M) ∪ (C∩selectedEvent d A) := by | |
| intro ω hω | |
| by_cases h : (G.freshClusterBoundary X v ω).ncard ≤ M | |
| · exact Or.inl ⟨hω.1,hω.2.1,h⟩ | |
| · right | |
| refine ⟨⟨hω.2.1,(Nat.le_of_lt (Nat.lt_of_not_ge h))⟩,?_⟩ | |
| change (((d ω : Set E)∩Xᶜ)∩ω).ncard ≤ N | |
| rw [he ω hω.2.1] | |
| exact hω.2.2 | |
| exact ((measureReal_mono hsub).trans (measureReal_union_le _ _)).trans | |
| (add_le_add_right ht' _) | |
| theorem fewOpenFreshBoundary_probability_tendsto_zero | |
| (hl : G.LocallyFinite) (p : unitInterval) (hp : 0 < (p:ℝ)) (hp1 : (p:ℝ)<1) | |
| (X : ℕ → Set E) (hX : ∀ S : Finset E, ∀ᶠ n in atTop, (S : Set E) ⊆ X n) | |
| (v : V) (N : ℕ) : | |
| Tendsto (fun n => G.law p (G.infiniteEvent v ∩ G.fewOpenFreshBoundary (X n) v N)) | |
| atTop (𝓝 0) := by | |
| have hpow : Tendsto (fun L : ℕ => (N+1 : ℝ)*(1-(p:ℝ))^L) atTop (𝓝 0) := by | |
| simpa only [mul_zero] using (tendsto_pow_atTop_nhds_zero_of_lt_one | |
| (sub_nonneg.mpr p.2.2) (by linarith : 1-(p:ℝ)<1)).const_mul (N+1 : ℝ) | |
| have ht : Tendsto (fun n => (G.law p).real | |
| (G.infiniteEvent v ∩ G.fewOpenFreshBoundary (X n) v N)) atTop (𝓝 0) := by | |
| apply tendsto_order.mpr | |
| constructor | |
| · intro a ha | |
| exact Eventually.of_forall (fun _ => lt_of_lt_of_le ha measureReal_nonneg) | |
| · intro b hb | |
| obtain ⟨L,hL⟩ := (hpow.eventually (gt_mem_nhds (half_pos hb))).exists | |
| have hs := G.smallFreshBoundary_probability_tendsto_zero hl p hp1 X hX v ((N+1)*L) | |
| have hsreal := ENNReal.tendsto_toReal ENNReal.zero_ne_top |>.comp hs | |
| simp only [ENNReal.toReal_zero,Function.comp_def] at hsreal | |
| change Tendsto (fun n => (G.law p).real | |
| (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v ((N+1)*L))) atTop (𝓝 0) at hsreal | |
| filter_upwards [hsreal.eventually (gt_mem_nhds (half_pos hb))] with n hn | |
| exact lt_of_le_of_lt (G.fewOpenFreshBoundary_probability_le hl p (X n) v N L) | |
| (by linarith) | |
| have hh := ENNReal.tendsto_ofReal ht | |
| simpa only [ENNReal.ofReal_zero,Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using hh | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |