openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /LevelContactGeometry.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical BigOperators ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def exteriorCluster (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set V := | |
| G.cluster (ω∩G.inducedEdges {v | j ≤ π v}) o | |
| def levelContacts (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set V := | |
| G.exteriorCluster π j o ω∩{v | π v=j} | |
| def contactDrop (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : Set E := | |
| {e | ∃ v∈G.levelContacts π j o ω, ∃ w, π w<j ∧ G.ends e=s(v,w)} | |
| lemma contactDrop_subset_incident (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : | |
| G.contactDrop π j o ω ⊆ G.incidentEdges (G.levelContacts π j o ω) := by | |
| rintro e ⟨v,hv,w,hw,he⟩ | |
| exact ⟨v,hv,by rw [he]; simp⟩ | |
| lemma contactDrop_disjoint_exterior (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) : | |
| Disjoint (G.contactDrop π j o ω) (G.inducedEdges {v | j ≤ π v}) := by | |
| apply Set.disjoint_left.mpr | |
| rintro e ⟨v,hv,w,hw,he⟩ he' | |
| exact (not_le.mpr hw) (he' w (by rw [he]; simp)) | |
| lemma exteriorCluster_subset_level (π : V → ℤ) (j : ℤ) (o : V) | |
| (ho : j ≤ π o) (ω : Set E) : G.exteriorCluster π j o ω ⊆ {v | j ≤ π v} := by | |
| intro v hv | |
| have hc : ∀ ⦃x y⦄, x∈{v | j ≤ π v} → | |
| (G.openGraph (ω∩G.inducedEdges {v | j ≤ π v})).Adj x y → y∈{v | j ≤ π v} := by | |
| intro x y hx hxy | |
| obtain ⟨⟨e,he,hend⟩,_⟩ := hxy | |
| exact he.2 y (by rw [hend]; simp) | |
| exact ClusterDeletion.reachable_mem_of_closed hc ho hv | |
| lemma cluster_eq_exterior_of_drop_closed (π : V → ℤ) (j : ℤ) (o : V) | |
| (ho : j ≤ π o) | |
| (hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j) | |
| (ω : Set E) (hclosed : Disjoint (G.contactDrop π j o ω) ω) : | |
| G.cluster ω o=G.exteriorCluster π j o ω := by | |
| apply Set.Subset.antisymm | |
| · intro v hv | |
| have hc : ∀ ⦃x y⦄, x∈G.exteriorCluster π j o ω → | |
| (G.openGraph ω).Adj x y → y∈G.exteriorCluster π j o ω := by | |
| intro x y hx hxy | |
| obtain ⟨⟨e,he,hend⟩,hne⟩ := hxy | |
| have hxj := G.exteriorCluster_subset_level π j o ho ω hx | |
| by_cases hy : j ≤ π y | |
| · apply hx.trans | |
| apply SimpleGraph.Adj.reachable | |
| refine ⟨⟨e,⟨he,?_⟩,hend⟩,hne⟩ | |
| intro z hz | |
| rw [hend] at hz | |
| rcases Sym2.mem_iff.mp hz with rfl|rfl | |
| · exact hxj | |
| · exact hy | |
| · have hy' := lt_of_not_ge hy | |
| exact False.elim (Set.disjoint_left.mp hclosed | |
| ⟨x,⟨hx,hcross e x y hend hxj hy'⟩,y,hy',hend⟩ he) | |
| exact ClusterDeletion.reachable_mem_of_closed hc (SimpleGraph.Reachable.refl o) hv | |
| · exact G.cluster_mono Set.inter_subset_left o | |
| def minimumLevel (π : V → ℤ) (o : V) (j : ℤ) : Set (Set E) := | |
| G.hitEvent o {v | π v=j}\G.hitEvent o {v | π v<j} | |
| lemma minimumLevel_of_drop_closed (π : V → ℤ) (j : ℤ) (o : V) | |
| (ho : j ≤ π o) | |
| (hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j) | |
| (ω : Set E) (hcontact : (G.levelContacts π j o ω).Nonempty) | |
| (hclosed : Disjoint (G.contactDrop π j o ω) ω) : | |
| ω∈G.minimumLevel π o j := by | |
| have he := G.cluster_eq_exterior_of_drop_closed π j o ho hcross ω hclosed | |
| constructor | |
| · obtain ⟨v,hv,hj⟩ := hcontact | |
| exact ⟨v,he.symm ▸ hv,hj⟩ | |
| · rintro ⟨v,hv,hj⟩ | |
| have hlt : π v < j := hj | |
| have hle : j ≤ π v := G.exteriorCluster_subset_level π j o ho ω (he ▸ hv) | |
| omega | |
| lemma minimumLevel_pairwiseDisjoint (π : V → ℤ) (o : V) : | |
| Pairwise (fun i j => Disjoint (G.minimumLevel π o i) (G.minimumLevel π o j)) := by | |
| intro i j hij | |
| apply Set.disjoint_left.mpr | |
| rintro ω ⟨⟨v,hv,hvi⟩,hnoti⟩ ⟨⟨w,hw,hwj⟩,hnotj⟩ | |
| rcases lt_or_gt_of_ne hij with hlt|hgt | |
| · exact hnotj ⟨v,hv,by change π v < j; rw [hvi]; exact hlt⟩ | |
| · exact hnoti ⟨w,hw,by change π w < i; rw [hwj]; exact hgt⟩ | |
| lemma measurableSet_minimumLevel [Countable V] [Countable E] (π : V → ℤ) (o : V) (j : ℤ) : | |
| MeasurableSet (G.minimumLevel π o j) := | |
| (G.measurableSet_hitEvent o _).diff (G.measurableSet_hitEvent o _) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |