openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /FiniteInduced.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def inducedEdges (S : Set V) : Set E := {e | ∀ v ∈ G.ends e, v∈S} | |
| def induce (S : Set V) : BondGraph S (G.inducedEdges S) where | |
| ends e := (G.ends e.val).attachWith e.property | |
| lemma induce_ends_map (S : Set V) (e : G.inducedEdges S) : | |
| Sym2.map Subtype.val ((G.induce S).ends e) = G.ends e.val := | |
| Sym2.attachWith_map_subtypeVal _ | |
| lemma induce_ends_eq (S : Set V) (e : G.inducedEdges S) (u v : S) : | |
| (G.induce S).ends e = s(u,v) ↔ G.ends e.val = s(u.val,v.val) := by | |
| rw [← G.induce_ends_map S e,← Sym2.map_mk] | |
| exact (Sym2.map.injective Subtype.val_injective).eq_iff.symm | |
| lemma finite_incident (hG : G.LocallyFinite) {S : Set V} (hS : S.Finite) : | |
| (G.incidentEdges S).Finite := by | |
| have hh := hS.biUnion (fun v _ => hG v) | |
| convert hh using 1 | |
| ext e | |
| simp only [incidentEdges,Set.mem_ofPred_eq,Set.mem_iUnion,exists_prop] | |
| lemma finite_inducedEdges (hG : G.LocallyFinite) {S : Set V} (hS : S.Finite) : | |
| (G.inducedEdges S).Finite := by | |
| apply (G.finite_incident hG hS).subset | |
| intro e he | |
| obtain ⟨⟨u,v⟩,huv⟩ := Sym2.mk_surjective (G.ends e) | |
| have hu : u ∈ G.ends e := by rw [← huv]; simp | |
| exact ⟨u,he u hu,hu⟩ | |
| lemma induce_adj (S : Set V) (ω : Set (G.inducedEdges S)) (u v : S) : | |
| ((G.induce S).openGraph ω).Adj u v ↔ | |
| (G.openGraph (Subtype.val '' ω)).Adj u.val v.val := by | |
| simp only [openGraph,SimpleGraph.fromEdgeSet_adj,Set.mem_image] | |
| constructor | |
| · rintro ⟨⟨e,he,hend⟩,hne⟩ | |
| exact ⟨⟨e.val,⟨e,he,rfl⟩,(G.induce_ends_eq S e u v).mp hend⟩, | |
| fun h => hne (Subtype.ext h)⟩ | |
| · rintro ⟨⟨e,⟨a,ha,rfl⟩,hend⟩,hne⟩ | |
| exact ⟨⟨a,ha,(G.induce_ends_eq S a u v).mpr hend⟩, | |
| fun h => hne (congrArg Subtype.val h)⟩ | |
| lemma induced_adj_mem (S : Set V) (ω : Set (G.inducedEdges S)) {u v : V} | |
| (h : (G.openGraph (Subtype.val '' ω)).Adj u v) : u∈S ∧ v∈S := by | |
| obtain ⟨⟨e,⟨a,ha,rfl⟩,he⟩,_⟩ := h | |
| constructor | |
| · exact a.property u (by rw [he]; simp) | |
| · exact a.property v (by rw [he]; simp) | |
| lemma induce_reachable (S : Set V) (ω : Set (G.inducedEdges S)) (u v : S) : | |
| ((G.induce S).openGraph ω).Reachable u v ↔ | |
| (G.openGraph (Subtype.val '' ω)).Reachable u.val v.val := by | |
| constructor | |
| · intro h | |
| exact h.map ⟨Subtype.val, fun {a b} hab => (G.induce_adj S ω a b).mp hab⟩ | |
| · intro h | |
| have H : ∀ w, (G.openGraph (Subtype.val '' ω)).Reachable u.val w → | |
| ∀ hw : w∈S, ((G.induce S).openGraph ω).Reachable u ⟨w,hw⟩ := by | |
| intro w hh | |
| rw [SimpleGraph.reachable_iff_reflTransGen] at hh | |
| induction hh with | |
| | refl => intro _; exact SimpleGraph.Reachable.refl _ | |
| | @tail w z _ hz ih => | |
| intro hmem | |
| have hw := (G.induced_adj_mem S ω hz).1 | |
| exact (ih hw).trans ((G.induce_adj S ω ⟨w,hw⟩ ⟨z,hmem⟩).mpr hz).reachable | |
| exact H v.val h v.property | |
| lemma induced_cluster_subset (S : Set V) (ω : Set (G.inducedEdges S)) (u : S) : | |
| G.cluster (Subtype.val '' ω) u.val ⊆ S := by | |
| intro v hv | |
| rw [cluster,Set.mem_ofPred_eq,SimpleGraph.reachable_iff_reflTransGen] at hv | |
| induction hv with | |
| | refl => exact u.property | |
| | tail _ hadj _ => exact (G.induced_adj_mem S ω hadj).2 | |
| lemma induce_cluster_image (S : Set V) (ω : Set (G.inducedEdges S)) (u : S) : | |
| Subtype.val '' ((G.induce S).cluster ω u) = G.cluster (Subtype.val '' ω) u.val := by | |
| ext v | |
| constructor | |
| · rintro ⟨a,ha,rfl⟩ | |
| exact (G.induce_reachable S ω u a).mp ha | |
| · intro hv | |
| have hmem := G.induced_cluster_subset S ω u hv | |
| exact ⟨⟨v,hmem⟩,(G.induce_reachable S ω u ⟨v,hmem⟩).mpr hv,rfl⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |