| import OAI.Probability.FreeIsing.Trees.BroadcastLocality |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace Problem367.RootedBroadcast |
|
|
| abbrev OutsideConeVertex {d : ℕ} (b : TreeVertex d) := |
| {v : TreeVertex d |
|
|
| abbrev OutsideConeNonroot {d : ℕ} (b : TreeVertex d) := |
| {v : TreeVertex d |
|
|
| def outsideConeNoise {d : ℕ} (theta : ℝ) (b : TreeVertex d) |
| (Y : OutsideConeNonroot b → ℝ) (v : TreeVertex d) : IsingSpin := |
| if hv : v.1 ≠ [] ∧ ¬ b.1 <+: v.1 then edgeNoise theta (Y ⟨v, hv⟩) |
| else IsingSpin.pos |
|
|
| theorem measurable_outsideConeNoise {d : ℕ} (theta : ℝ) (b : TreeVertex d) : |
| Measurable (outsideConeNoise theta b) := by |
| apply Measurable.of_eval |
| intro v |
| by_cases hv : v.1 ≠ [] ∧ ¬ b.1 <+: v.1 |
| · simpa only [outsideConeNoise, dite_eq_left hv, Function.comp_def] using |
| (measurable_edgeNoise theta).comp |
| (measurable_pi_apply (X := fun _ : OutsideConeNonroot b => ℝ) ⟨v, hv⟩) |
| · simpa only [outsideConeNoise, dite_eq_right hv] using |
| (measurable_const : Measurable (fun _ : OutsideConeNonroot b → ℝ => IsingSpin.pos)) |
|
|
| theorem outsideConeNoise_ofNoise_eq_rootless {d : ℕ} (theta : ℝ) |
| (b : TreeVertex d) (U : TreeVertex d → ℝ) (v : OutsideConeVertex b) : |
| ofNoise (outsideConeNoise theta b (fun w : OutsideConeNonroot b => U w.1)) v.1 = |
| rootlessFactor theta (fun w : NonrootVertex d => U w.1) v.1 := by |
| apply ofNoise_congr_prefix |
| intro w hw |
| have hb : ¬ b.1 <+: w.1 := fun hbw => v.2 (hbw.trans hw) |
| by_cases hr : w = root d |
| · subst w |
| simp [outsideConeNoise, nonrootNoise] |
| · have hnil : w.1 ≠ [] := fun h => hr ((eq_root_iff w).mpr h) |
| simp [outsideConeNoise, nonrootNoise, hnil, hb, hr] |
|
|
| def outsideConeFactor {d : ℕ} (theta : ℝ) (b : TreeVertex d) |
| (a : OutsideConeVertex b) (p : IsingSpin × (OutsideConeNonroot b → ℝ)) : |
| OutsideConeVertex b → IsingSpin := |
| fun v => spinMul p.1 |
| (spinMul (ofNoise (outsideConeNoise theta b p.2) a.1) |
| (ofNoise (outsideConeNoise theta b p.2) v.1)) |
|
|
| theorem measurable_outsideConeFactor {d : ℕ} (theta : ℝ) |
| (b : TreeVertex d) (a : OutsideConeVertex b) : |
| Measurable (outsideConeFactor theta b a) := by |
| apply Measurable.of_eval |
| intro v |
| have ha : Measurable (fun p : IsingSpin × (OutsideConeNonroot b → ℝ) => |
| ofNoise (outsideConeNoise theta b p.2) a.1) := |
| (measurable_ofNoise_apply a.1).comp |
| ((measurable_outsideConeNoise theta b).comp measurable_snd) |
| have hv : Measurable (fun p : IsingSpin × (OutsideConeNonroot b → ℝ) => |
| ofNoise (outsideConeNoise theta b p.2) v.1) := |
| (measurable_ofNoise_apply v.1).comp |
| ((measurable_outsideConeNoise theta b).comp measurable_snd) |
| exact measurable_spinMul.comp |
| (measurable_fst.prodMk (measurable_spinMul.comp (ha.prodMk hv))) |
|
|
| theorem factor_outsideCone_representation {d : ℕ} (theta : ℝ) |
| (b : TreeVertex d) (a : OutsideConeVertex b) (U : TreeVertex d → ℝ) : |
| (fun v : OutsideConeVertex b => factor theta U v.1) = |
| outsideConeFactor theta b a (factor theta U a.1, |
| fun w : OutsideConeNonroot b => U w.1) := by |
| funext v |
| change factor theta U v.1 = spinMul (factor theta U a.1) |
| (spinMul (ofNoise (outsideConeNoise theta b _) a.1) |
| (ofNoise (outsideConeNoise theta b _) v.1)) |
| rw [outsideConeNoise_ofNoise_eq_rootless theta b U a, |
| outsideConeNoise_ofNoise_eq_rootless theta b U v, |
| factor_root_decomposition theta U a.1, factor_root_decomposition theta U v.1] |
| simp only [spinMul_assoc, spinMul_cancel] |
|
|
| end Problem367.RootedBroadcast |
|
|
| end |
|
|
| end OAI |
|
|
