| import OAI.Probability.FreeIsing.Gibbs.Weights | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367.GibbsLeaf | |
| def energy {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) | |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : ℝ := | |
| (∑ x ∈ D, h x * spinSign (eta x)) + | |
| beta * ∑ e ∈ inducedEdges D, spinSign (eta e.1) * spinSign (eta e.2) | |
| def weight {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) | |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : ℝ := | |
| Real.exp (energy beta h D eta) | |
| theorem energy_congr_on {d : ℕ} (beta : ℝ) (h : TreeVertex d → ℝ) | |
| (D : Finset (TreeVertex d)) {eta xi : TreeVertex d → IsingSpin} | |
| (heq : ∀ x ∈ D, eta x = xi x) : energy beta h D eta = energy beta h D xi := by | |
| classical | |
| unfold energy | |
| congr 1 | |
| · exact Finset.sum_congr rfl (fun x hx => by rw [heq x hx]) | |
| · congr 1 | |
| apply Finset.sum_congr rfl | |
| intro e he | |
| have hm := (Finset.mem_filter.mp he).1 | |
| rw [Finset.product_eq_sprod] at hm | |
| obtain ⟨hu, hv⟩ := Finset.mem_product.mp hm | |
| rw [heq e.1 hu, heq e.2 hv] | |
| theorem energy_update_eq_erase_add | |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} | |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hpne : p ≠ v) | |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) | |
| (eta : TreeVertex d → IsingSpin) (s : IsingSpin) : | |
| energy beta h D (Function.update eta v s) = energy beta h (D.erase v) eta + | |
| (h v + beta * spinSign (eta p)) * spinSign s := by | |
| classical | |
| have hnot : (p, v) ∉ inducedEdges (D.erase v) := by | |
| simp [inducedEdges, Finset.product_eq_sprod] | |
| have hfield : (∑ x ∈ D, h x * spinSign (Function.update eta v s x)) = | |
| (∑ x ∈ D.erase v, h x * spinSign (eta x)) + h v * spinSign s := by | |
| rw [← Finset.sum_erase_add D _ hv] | |
| simp only [Function.update_self] | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro x hx | |
| rw [Function.update_of_ne (Finset.mem_erase.mp hx).1] | |
| have hedge : (∑ e ∈ inducedEdges (D.erase v), | |
| spinSign (Function.update eta v s e.1) * spinSign (Function.update eta v s e.2)) = | |
| ∑ e ∈ inducedEdges (D.erase v), spinSign (eta e.1) * spinSign (eta e.2) := by | |
| apply Finset.sum_congr rfl | |
| intro e he | |
| have hm := (Finset.mem_filter.mp he).1 | |
| rw [Finset.product_eq_sprod] at hm | |
| obtain ⟨hu, hw⟩ := Finset.mem_product.mp hm | |
| rw [Function.update_of_ne (Finset.mem_erase.mp hu).1, | |
| Function.update_of_ne (Finset.mem_erase.mp hw).1] | |
| unfold energy | |
| rw [hfield, hedges, Finset.sum_insert hnot, hedge] | |
| simp only [Function.update_of_ne hpne, Function.update_self] | |
| ring | |
| theorem sum_weight_update_of_edges | |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} | |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hpne : p ≠ v) | |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) | |
| (eta : TreeVertex d → IsingSpin) : | |
| (∑ s : IsingSpin, weight beta h D (Function.update eta v s)) = | |
| 2 * Real.cosh (h v + beta * spinSign (eta p)) * weight beta h (D.erase v) eta := by | |
| simp only [weight, energy_update_eq_erase_add beta h hv hpne hedges, Real.exp_add] | |
| rw [← Finset.mul_sum, sum_isingSpin] | |
| simp only [spinSign_neg, spinSign_pos, mul_neg_one, mul_one] | |
| rw [Real.cosh_eq] | |
| ring | |
| theorem sum_weight_update_leaf | |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} | |
| (beta : ℝ) (h : TreeVertex d → ℝ) | |
| (hv : v ∈ D) (hp : p ∈ D) (hpv : treeAdj p v) | |
| (hplen : p.1.length < v.1.length) | |
| (hmax : ∀ x ∈ D, x.1.length ≤ v.1.length) | |
| (eta : TreeVertex d → IsingSpin) : | |
| (∑ s : IsingSpin, weight beta h D (Function.update eta v s)) = | |
| 2 * Real.cosh (h v + beta * spinSign (eta p)) * weight beta h (D.erase v) eta := by | |
| apply sum_weight_update_of_edges beta h hv | |
| · intro heq | |
| subst p | |
| exact lt_irrefl _ hplen | |
| · exact inducedEdges_eq_insert_erase_of_max_length hv hp hpv hplen hmax | |
| theorem sum_weight_eq_erase_sum | |
| {d : ℕ} {D : Finset (TreeVertex d)} {v p : TreeVertex d} | |
| (beta : ℝ) (h : TreeVertex d → ℝ) (hv : v ∈ D) (hpne : p ≠ v) | |
| (hedges : inducedEdges D = insert (p, v) (inducedEdges (D.erase v))) : | |
| (∑ eta : D → IsingSpin, weight beta h D (extendFiniteSpin D eta)) = | |
| ∑ eta : D.erase v → IsingSpin, | |
| 2 * Real.cosh (h v + beta * spinSign (extendFiniteSpin (D.erase v) eta p)) * | |
| weight beta h (D.erase v) (extendFiniteSpin (D.erase v) eta) := by | |
| classical | |
| calc | |
| _ = ∑ z : (D.erase v → IsingSpin) × IsingSpin, | |
| weight beta h D (extendFiniteSpin D ((finiteSpinEraseEquiv hv).symm z)) := | |
| ((finiteSpinEraseEquiv hv).symm.sum_comp | |
| (fun eta => weight beta h D (extendFiniteSpin D eta))).symm | |
| _ = _ := by | |
| rw [Fintype.sum_prod_type] | |
| apply Finset.sum_congr rfl | |
| intro eta _ | |
| simp only [extendFiniteSpin_eraseEquiv_symm] | |
| exact sum_weight_update_of_edges beta h hv hpne hedges _ | |
| end Problem367.GibbsLeaf | |
| end | |
| end OAI | |