| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| namespace CriticalPercolation.ClusterDeletion | |
| variable {V : Type*} | |
| lemma component_self (H : SimpleGraph V) (s : V) : s ∈ component H s := | |
| SimpleGraph.Reachable.refl s | |
| lemma mem_component_symm {H : SimpleGraph V} {s t : V} : | |
| t ∈ component H s ↔ s ∈ component H t := | |
| ⟨SimpleGraph.Reachable.symm, SimpleGraph.Reachable.symm⟩ | |
| lemma roots_singleton (H : SimpleGraph V) (s : V) : roots H {s} = component H s := by | |
| ext v | |
| simp [roots, component] | |
| lemma component_eq_of_mem {H : SimpleGraph V} {s t : V} | |
| (h : t ∈ component H s) : component H t = component H s := by | |
| ext v | |
| exact ⟨fun hv => h.trans hv, fun hv => h.symm.trans hv⟩ | |
| lemma not_mem_roots_iff_disjoint (H : SimpleGraph V) (S : Set V) (s : V) : | |
| s ∉ roots H S ↔ Disjoint (component H s) S := by | |
| rw [Set.disjoint_left] | |
| constructor | |
| · intro hs t ht hS | |
| exact hs ⟨t, hS, ht.symm⟩ | |
| · intro hd ⟨t, ht, hts⟩ | |
| exact hd hts.symm ht | |
| lemma component_delete_component (H : SimpleGraph V) (s t : V) | |
| (h : t ∉ component H s) : | |
| component (delete H (component H s)) t = component H t := by | |
| simpa only [roots_singleton] using | |
| (delete_roots_component H {s} t (by simpa only [roots_singleton] using h)) | |
| lemma roots_delete_component (H : SimpleGraph V) (S : Set V) (s : V) | |
| (h : s ∉ roots H S) : roots (delete H (component H s)) S = roots H S := by | |
| ext v | |
| constructor | |
| · intro hv | |
| exact roots_mono (delete_le H _) S hv | |
| · rintro ⟨t, ht, htv⟩ | |
| refine ⟨t, ht, ?_⟩ | |
| have hts : t ∉ component H s := fun hg => h ⟨t, ht, hg.symm⟩ | |
| change v ∈ component (delete H (component H s)) t | |
| rw [component_delete_component H s t hts] | |
| exact htv | |
| lemma delete_empty (H : SimpleGraph V) : delete H ∅ = H := by | |
| ext v w | |
| simp only [delete, Set.mem_empty_iff_false, not_false_eq_true, and_true] | |
| lemma component_delete_self (H : SimpleGraph V) (U : Set V) (s : V) | |
| (hs : s ∈ U) : component (delete H U) s = {s} := by | |
| ext v | |
| constructor | |
| · intro h | |
| have hadj : ∀ t, ¬ (delete H U).Adj s t := fun t ht => ht.2.1 hs | |
| have he : v = s := by | |
| rcases h with ⟨p⟩ | |
| cases p with | |
| | nil => rfl | |
| | cons hst _ => exact (hadj _ hst).elim | |
| exact he | |
| · rintro rfl | |
| exact component_self _ _ | |
| end CriticalPercolation.ClusterDeletion | |
| end | |
| end | |
| end OAI | |