openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /FiniteClusterCover.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| import OAI.Probability.CriticalPercolation.Clusters.PercolationTail | |
| import OAI.Probability.CriticalPercolation.Clusters.InfiniteUniqueness | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def infiniteClusterCover (ω : Set E) (S : Set V) : Prop := | |
| ∀ v, (G.cluster ω v).Infinite → ∃ s ∈ S, (G.openGraph ω).Reachable v s | |
| lemma measurable_infiniteClusterCover [Countable V] [Countable E] (S : Set V) : | |
| Measurable (fun ω => G.infiniteClusterCover ω S) := by | |
| unfold infiniteClusterCover | |
| exact .forall fun v => (G.measurable_infinite_cluster v).imp | |
| (.exists fun s => measurable_const.and (G.measurable_reachable v s)) | |
| lemma infinite_cluster_union_has_old (ω J : Set E) (hJ : J.Finite) (o : V) | |
| (ho : (G.cluster (ω ∪ J) o).Infinite) : | |
| ∃ v, (G.openGraph (ω ∪ J)).Reachable o v ∧ (G.cluster ω v).Infinite := by | |
| by_contra h | |
| have hf (v : V) (hv : (G.openGraph (ω ∪ J)).Reachable o v) : | |
| (G.cluster ω v).Finite := by | |
| apply Set.not_infinite.mp | |
| intro hi | |
| exact h ⟨v,hv,hi⟩ | |
| have hσ (v : V) : (G.cluster ω v ∩ G.cluster (ω ∪ J) o).Finite := by | |
| by_cases hv : (G.openGraph (ω ∪ J)).Reachable o v | |
| · exact (hf v hv).inter_of_left _ | |
| · have he : G.cluster ω v ∩ G.cluster (ω ∪ J) o = ∅ := by | |
| apply Set.eq_empty_iff_forall_notMem.mpr | |
| rintro x ⟨hx,hox⟩ | |
| exact hv (hox.trans (G.cluster_mono Set.subset_union_left v hx).symm) | |
| rw [he] | |
| exact Set.finite_empty | |
| apply ho | |
| apply ((hf o (.refl _)).union (hJ.biUnion fun e _ => | |
| (G.ends e).toFinset.finite_toSet.biUnion fun v _ => hσ v)).subset | |
| intro z hz | |
| rcases G.cluster_union_subset ω J o hz with hh | hh | |
| · exact Or.inl hh | |
| · obtain ⟨e,he,hh⟩ := Set.mem_iUnion₂.mp hh | |
| obtain ⟨v,hv,hvz⟩ := Set.mem_iUnion₂.mp hh | |
| exact Or.inr (Set.mem_iUnion₂.mpr ⟨e,he,Set.mem_iUnion₂.mpr ⟨v,hv,hvz,hz⟩⟩) | |
| lemma finite_bonds_connect_set (hc : G.fullGraph.Connected) (o : V) | |
| (S : Set V) (hS : S.Finite) : | |
| ∃ J : Set E, J.Finite ∧ ∀ s ∈ S, (G.openGraph J).Reachable o s := by | |
| obtain ⟨J,hJ,_,hconn⟩ := G.finite_subset_cluster_finite_witness hS | |
| (show S ⊆ G.cluster Set.univ o from fun s _ => hc o s) | |
| exact ⟨J,hJ,fun s hs => hconn hs⟩ | |
| lemma atMostOneInfinite_union_of_cover (ω J : Set E) (hJ : J.Finite) | |
| (o : V) (S : Set V) (hconn : ∀ s ∈ S, (G.openGraph J).Reachable o s) | |
| (hcover : G.infiniteClusterCover ω S) : ω ∪ J ∈ G.atMostOneInfinite := by | |
| have hroot (x : V) (hx : (G.cluster (ω ∪ J) x).Infinite) : | |
| (G.openGraph (ω ∪ J)).Reachable x o := by | |
| obtain ⟨v,hxv,hv⟩ := G.infinite_cluster_union_has_old ω J hJ x hx | |
| obtain ⟨s,hs,hvs⟩ := hcover v hv | |
| exact hxv.trans ((G.cluster_mono Set.subset_union_left v hvs).trans | |
| (G.cluster_mono Set.subset_union_right o (hconn s hs)).symm) | |
| intro x y hx hy | |
| exact (hroot x hx).trans (hroot y hy).symm | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |