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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 | 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
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