openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /FiniteInterpolationMeasure.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteInterpolationUniform | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper | |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterTail | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| namespace CriticalPercolation | |
| namespace BernoulliProduct | |
| variable {E : Type*} [Fintype E] | |
| lemma prob_eq_real (q : E → unitInterval) (A : Set (Set E)) : | |
| FiniteProduct.prob (fun e => bitWeight (q e)) (fun ω => ω ∈ A) = (law q).real A := by | |
| rw [real_eq_expect_indicator] | |
| unfold FiniteProduct.prob | |
| rfl | |
| end BernoulliProduct | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def twoClusterEvent (u v : V) (s : ℕ) : Set (Set E) := | |
| {ω | ¬ (G.openGraph ω).Reachable u v} ∩ G.clusterTail u s ∩ G.clusterTail v s | |
| lemma measurableSet_twoClusterEvent [Countable V] [Countable E] (u v : V) (s : ℕ) : | |
| MeasurableSet (G.twoClusterEvent u v s) := by | |
| exact ((G.measurable_reachable u v).setOf.compl.inter | |
| (G.measurableSet_clusterTail u s)).inter (G.measurableSet_clusterTail v s) | |
| lemma clusterTail_iff_card [Fintype V] (ω : Set E) (u : V) (s : ℕ) : | |
| ω ∈ G.clusterTail u s ↔ s ≤ (G.cluster ω u).toFinset.card := by | |
| constructor | |
| · rintro ⟨K,hK,hsub,hs⟩ | |
| exact hs.trans (by simpa only [Set.ncard_eq_toFinset_card'] using Set.ncard_le_ncard hsub) | |
| · intro hs | |
| exact ⟨G.cluster ω u, Set.toFinite _, Set.Subset.rfl, by simpa only [Set.ncard_eq_toFinset_card'] using hs⟩ | |
| lemma twoClusterEvent_iff_endpoint [Fintype V] (ω : Set E) (u v : V) (s : ℕ) : | |
| ω ∈ G.twoClusterEvent u v s ↔ BinaryLaw.endpointEvent G u v s ω := by | |
| simp only [twoClusterEvent, Set.mem_inter_iff, Set.mem_ofPred_eq, | |
| clusterTail_iff_card, BinaryLaw.endpointEvent, and_assoc] | |
| lemma finite_endpoint_prob [Fintype V] [Fintype E] (p : unitInterval) (u v : V) (s : ℕ) : | |
| FiniteProduct.prob (fun _ : E => BernoulliProduct.bitWeight p) | |
| (BinaryLaw.endpointEvent G u v s) = (G.law p).real (G.twoClusterEvent u v s) := by | |
| rw [law, ← BernoulliProduct.law_const, ← BernoulliProduct.prob_eq_real] | |
| congr 1 | |
| funext ω | |
| exact propext (G.twoClusterEvent_iff_endpoint ω u v s).symm | |
| theorem finite_interpolation_measure [Fintype V] [Fintype E] | |
| (d : ℕ) (x : ℕ → V) (hd : 2≤d) (p : unitInterval) | |
| (hp : (p : ℝ) ∈ Set.Ioo 0 1) {c ε : ℝ} (hc : 0<c) (hε : 0<ε) | |
| (hεp : ε≤p) (hεq : ε≤1-(p:ℝ)) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ (G.law p).real {ω | (G.openGraph ω).Reachable (x j) (x k)}) | |
| (s : ℕ) (hδ : 0 < (G.law p).real (G.twoClusterEvent (x 0) (x d) s)) : | |
| ∃ j<d, 0 < (G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s) ∧ | |
| -Real.log ((G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s)) ≤ | |
| interpolationConstant c ε * | |
| (Real.log (1/(G.law p).real (G.twoClusterEvent (x 0) (x d) s))+Nat.clog 2 d+1) := by | |
| have h := BinaryLaw.finite_interpolation_uniform G d x hd hp | |
| (fun _ : E => BernoulliProduct.bitWeight p) | |
| (fun _ => by simp [BernoulliProduct.bitWeight]) | |
| (fun _ => by simp [BernoulliProduct.bitWeight]) hc hε hεp hεq | |
| simp_rw [G.finite_endpoint_prob p] at h | |
| apply h | |
| intro j k hj hk hjk | |
| change c ≤ FiniteProduct.prob (fun _ : E => BernoulliProduct.bitWeight p) | |
| (fun ω => ω ∈ {η | (G.openGraph η).Reachable (x j) (x k)}) | |
| rw [BernoulliProduct.prob_eq_real, BernoulliProduct.law_const] | |
| exact hconn j k hj hk hjk | |
| exact hδ | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end | |
| end OAI | |