| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| namespace ConnectedHull | |
| def root (d : ℕ) : TreeVertex d := ⟨[], List.IsChain.nil⟩ | |
| def takeVertex {d : ℕ} (v : TreeVertex d) (n : ℕ) : TreeVertex d := | |
| ⟨v.1.take n, v.2.take n⟩ | |
| @[simp] theorem prefix_zero {d : ℕ} (v : TreeVertex d) : takeVertex v 0 = root d := by | |
| apply Subtype.ext | |
| simp [takeVertex, root] | |
| @[simp] theorem prefix_length {d : ℕ} (v : TreeVertex d) : | |
| takeVertex v v.1.length = v := by | |
| apply Subtype.ext | |
| simp [takeVertex] | |
| theorem prefix_adj {d : ℕ} (v : TreeVertex d) {n : ℕ} (hn : n < v.1.length) : | |
| treeAdj (takeVertex v n) (takeVertex v (n + 1)) := by | |
| left | |
| exact ⟨v.1[n], List.take_succ_eq_append_getElem hn⟩ | |
| def vertices {d : ℕ} (v : TreeVertex d) : Finset (TreeVertex d) := by | |
| classical | |
| exact (Finset.range (v.1.length + 1)).image (takeVertex v) | |
| def hull {d : ℕ} (D : Finset (TreeVertex d)) : Finset (TreeVertex d) := by | |
| classical | |
| exact insert (root d) (D.biUnion vertices) | |
| theorem prefix_mem_hull {d : ℕ} {D : Finset (TreeVertex d)} {v : TreeVertex d} | |
| (hv : v ∈ D) {n : ℕ} (hn : n ≤ v.1.length) : takeVertex v n ∈ hull D := by | |
| classical | |
| apply Finset.mem_insert_of_mem | |
| apply Finset.mem_biUnion.mpr | |
| refine ⟨v, hv, ?_⟩ | |
| exact Finset.mem_image.mpr ⟨n, Finset.mem_range.mpr (by omega), rfl⟩ | |
| theorem subset_hull {d : ℕ} (D : Finset (TreeVertex d)) : D ⊆ hull D := by | |
| intro v hv | |
| simpa using prefix_mem_hull hv (Nat.le_refl v.1.length) | |
| theorem prefix_paths {d : ℕ} (E : Finset (TreeVertex d)) (v : TreeVertex d) | |
| (hE : ∀ n ≤ v.1.length, takeVertex v n ∈ E) {n : ℕ} (hn : n ≤ v.1.length) : | |
| Relation.ReflTransGen (fun x y : TreeVertex d => x ∈ E ∧ y ∈ E ∧ treeAdj x y) | |
| (root d) (takeVertex v n) ∧ | |
| Relation.ReflTransGen (fun x y : TreeVertex d => x ∈ E ∧ y ∈ E ∧ treeAdj x y) | |
| (takeVertex v n) (root d) := by | |
| induction n with | |
| | zero => simp only [prefix_zero]; exact ⟨.refl, .refl⟩ | |
| | succ n ih => | |
| have hn' : n ≤ v.1.length := by omega | |
| have hlt : n < v.1.length := by omega | |
| have hadj := prefix_adj v hlt | |
| have hrev : treeAdj (takeVertex v (n + 1)) (takeVertex v n) := by | |
| rcases hadj with h | h | |
| · exact Or.inr h | |
| · exact Or.inl h | |
| obtain ⟨hforward, hbackward⟩ := ih hn' | |
| exact ⟨hforward.trans (.single ⟨hE n hn', hE (n + 1) hn, hadj⟩), | |
| (Relation.ReflTransGen.single ⟨hE (n + 1) hn, hE n hn', hrev⟩).trans hbackward⟩ | |
| theorem hull_paths {d : ℕ} (D : Finset (TreeVertex d)) {v : TreeVertex d} | |
| (hv : v ∈ hull D) : | |
| Relation.ReflTransGen | |
| (fun x y : TreeVertex d => x ∈ hull D ∧ y ∈ hull D ∧ treeAdj x y) | |
| (root d) v ∧ | |
| Relation.ReflTransGen | |
| (fun x y : TreeVertex d => x ∈ hull D ∧ y ∈ hull D ∧ treeAdj x y) | |
| v (root d) := by | |
| classical | |
| rcases Finset.mem_insert.mp hv with rfl | hv | |
| · exact ⟨.refl, .refl⟩ | |
| rcases Finset.mem_biUnion.mp hv with ⟨w, hw, hv⟩ | |
| rcases Finset.mem_image.mp hv with ⟨n, hn, rfl⟩ | |
| apply prefix_paths | |
| · intro k hk | |
| exact prefix_mem_hull hw hk | |
| · have := Finset.mem_range.mp hn | |
| omega | |
| theorem connected_hull {d : ℕ} (D : Finset (TreeVertex d)) : | |
| ConnectedFinset (hull D) := by | |
| classical | |
| refine ⟨⟨root d, Finset.mem_insert_self _ _⟩, ?_⟩ | |
| intro u hu v hv | |
| exact (hull_paths D hu).2.trans (hull_paths D hv).1 | |
| end ConnectedHull | |
| theorem exists_connectedFinset_superset {d : ℕ} (D : Finset (TreeVertex d)) : | |
| ∃ E : Finset (TreeVertex d), D ⊆ E ∧ ConnectedFinset E := | |
| ⟨ConnectedHull.hull D, ConnectedHull.subset_hull D, ConnectedHull.connected_hull D⟩ | |
| end Problem367 | |
| end | |
| end OAI | |