| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| private theorem edge_adj_symm {d : ℕ} {u v : TreeVertex d} | |
| (h : treeAdj u v) : treeAdj v u := Or.symm h | |
| private theorem edge_adj_ne {d : ℕ} {u v : TreeVertex d} | |
| (h : treeAdj u v) : u ≠ v := by | |
| intro heq | |
| subst v | |
| rcases h with ⟨c, hc⟩ | ⟨c, hc⟩ <;> | |
| have hl := congrArg List.length hc <;> simp at hl | |
| theorem connectedFinset_pair_of_adj {d : ℕ} {u v : TreeVertex d} | |
| (h : treeAdj u v) : ConnectedFinset ({u, v} : Finset (TreeVertex d)) := by | |
| classical | |
| constructor | |
| · simp | |
| · intro a ha b hb | |
| simp only [Finset.mem_insert, Finset.mem_singleton] at ha hb | |
| rcases ha with rfl | rfl <;> rcases hb with rfl | rfl | |
| · exact Relation.ReflTransGen.refl | |
| · exact Relation.ReflTransGen.single ⟨by simp, by simp, h⟩ | |
| · exact Relation.ReflTransGen.single ⟨by simp, by simp, edge_adj_symm h⟩ | |
| · exact Relation.ReflTransGen.refl | |
| theorem inducedEdges_pair_of_length_lt {d : ℕ} {u v : TreeVertex d} | |
| (h : treeAdj u v) (hl : u.1.length < v.1.length) : | |
| inducedEdges ({u, v} : Finset (TreeVertex d)) = {(u, v)} := by | |
| classical | |
| ext ⟨a,b⟩ | |
| rw [show inducedEdges ({u,v} : Finset (TreeVertex d)) = | |
| (({u,v} : Finset (TreeVertex d)).product {u,v}).filter | |
| (fun e => treeAdj e.1 e.2 ∧ e.1.1.length < e.2.1.length) from rfl] | |
| rw [Finset.mem_filter] | |
| rw [Finset.product_eq_sprod, Finset.mem_product] | |
| simp only [Finset.mem_insert, Finset.mem_singleton, Prod.mk.injEq] | |
| constructor | |
| · rintro ⟨⟨ha, hb⟩, hab, hlen⟩ | |
| rcases ha with rfl | rfl <;> rcases hb with rfl | rfl | |
| · omega | |
| · exact ⟨rfl, rfl⟩ | |
| · omega | |
| · omega | |
| · rintro ⟨rfl, rfl⟩ | |
| exact ⟨⟨Or.inl rfl, Or.inr rfl⟩, h, hl⟩ | |
| theorem freeCylinderWeight_pair_of_length_lt {d : ℕ} (theta : ℝ) | |
| {u v : TreeVertex d} (h : treeAdj u v) (hl : u.1.length < v.1.length) | |
| (eta : TreeVertex d → IsingSpin) : | |
| freeCylinderWeight theta {u,v} eta = | |
| ENNReal.ofReal ((1 + (if eta u = eta v then theta else -theta)) / 4) := by | |
| classical | |
| rw [freeCylinderWeight, inducedEdges_pair_of_length_lt h hl] | |
| simp only [Finset.prod_singleton] | |
| congr 1 | |
| ring | |
| private theorem edge_adj_length_lt_or {d : ℕ} {u v : TreeVertex d} | |
| (h : treeAdj u v) : u.1.length < v.1.length ∨ v.1.length < u.1.length := by | |
| rcases h with ⟨c, hc⟩ | ⟨c, hc⟩ | |
| · left | |
| simp [hc] | |
| · right | |
| simp [hc] | |
| theorem freeCylinderWeight_pair {d : ℕ} (theta : ℝ) | |
| {u v : TreeVertex d} (h : treeAdj u v) (eta : TreeVertex d → IsingSpin) : | |
| freeCylinderWeight theta {u,v} eta = | |
| ENNReal.ofReal ((1 + (if eta u = eta v then theta else -theta)) / 4) := by | |
| classical | |
| rcases edge_adj_length_lt_or h with hl | hl | |
| · exact freeCylinderWeight_pair_of_length_lt theta h hl eta | |
| · rw [Finset.pair_comm] | |
| simpa only [eq_comm] using | |
| freeCylinderWeight_pair_of_length_lt theta (edge_adj_symm h) hl eta | |
| theorem HasFreeIsingLaw.edge_cylinder {d : ℕ} {theta : ℝ} | |
| {mu : Measure (TreeVertex d → IsingSpin)} (hmu : HasFreeIsingLaw theta mu) | |
| {u v : TreeVertex d} (huv : treeAdj u v) (a b : IsingSpin) : | |
| mu {x | x u = a ∧ x v = b} = | |
| ENNReal.ofReal ((1 + (if a = b then theta else -theta)) / 4) := by | |
| classical | |
| let eta : TreeVertex d → IsingSpin := fun w => if w = u then a else b | |
| have h := hmu {u,v} (connectedFinset_pair_of_adj huv) eta | |
| rw [freeCylinderWeight_pair theta huv eta] at h | |
| have hne : v ≠ u := (edge_adj_ne huv).symm | |
| simpa [eta, hne] using h | |
| end Problem367 | |
| end | |
| end OAI | |