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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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