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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Probability.FreeIsing.Trees.BroadcastLocality
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.RootedBroadcast
theorem ofNoise_congr_cone {d : ℕ} (noise₁ noise₂ : TreeVertex d → IsingSpin)
(c : TreeVertex d) (hbase : ofNoise noise₁ c = ofNoise noise₂ c)
(hedge : ∀ w : TreeVertex d, c.1 <+: w.1 → w ≠ c → noise₁ w = noise₂ w)
(v : TreeVertex d) (hcv : c.1 <+: v.1) :
ofNoise noise₁ v = ofNoise noise₂ v := by
have h : ∀ n : ℕ, ∀ v : TreeVertex d, v.1.length = n → c.1 <+: v.1 →
ofNoise noise₁ v = ofNoise noise₂ v := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro v hv hcv
by_cases hvc : v = c
· simpa only [hvc] using hbase
· have hlt : c.1.length < v.1.length := by
have hle := hcv.length_le
have hne : c.1.length ≠ v.1.length := by
intro heq
exact hvc (Subtype.ext (hcv.eq_of_length heq).symm)
omega
have hnil : v.1 ≠ [] := by
intro hh
simp only [hh, List.length_nil] at hlt
omega
have hcp : c.1 <+: (parent v).1 := by
apply List.prefix_of_prefix_length_le hcv (List.dropLast_prefix v.1)
change c.1.length ≤ (parentVertex v).1.length
rw [parentVertex_length]
omega
have hp : (parent v).1.length < n := by
change (parentVertex v).1.length < n
rw [parentVertex_length]
omega
rw [ofNoise_nonroot noise₁ v hnil, ofNoise_nonroot noise₂ v hnil]
exact congrArg₂ spinMul (hedge v hcv hvc) (ih _ hp _ rfl hcp)
exact h _ v rfl hcv
abbrev DescendantVertex {d : ℕ} (c : TreeVertex d) :=
{v : TreeVertex d // c.1 <+: v.1}
abbrev StrictDescendantVertex {d : ℕ} (c : TreeVertex d) :=
{v : TreeVertex d // c.1 <+: v.1 ∧ v ≠ c}
def coneNoise {d : ℕ} (theta : ℝ) (c : TreeVertex d)
(s : IsingSpin) (Y : StrictDescendantVertex c → ℝ) (v : TreeVertex d) : IsingSpin :=
if v.1 = [] then s
else if hv : c.1 <+: v.1 ∧ v ≠ c then edgeNoise theta (Y ⟨v, hv⟩)
else IsingSpin.pos
theorem measurable_coneNoise {d : ℕ} (theta : ℝ) (c : TreeVertex d) :
Measurable (fun p : IsingSpin × (StrictDescendantVertex c → ℝ) =>
coneNoise theta c p.1 p.2) := by
apply Measurable.of_eval
intro v
by_cases hnil : v.1 = []
· simpa only [coneNoise, ite_eq_left hnil] using
(measurable_fst : Measurable (Prod.fst :
IsingSpin × (StrictDescendantVertex c → ℝ) → IsingSpin))
· by_cases hv : c.1 <+: v.1 ∧ v ≠ c
· simpa only [coneNoise, ite_eq_right hnil, dite_eq_left hv, Function.comp_def] using
(measurable_edgeNoise theta).comp
((measurable_pi_apply (X := fun _ : StrictDescendantVertex c => ℝ)
⟨v, hv⟩).comp measurable_snd)
· simpa only [coneNoise, ite_eq_right hnil, dite_eq_right hv] using
(measurable_const : Measurable (fun _ :
IsingSpin × (StrictDescendantVertex c → ℝ) => IsingSpin.pos))
theorem ofNoise_coneNoise_at_root {d : ℕ} (theta : ℝ) (c : TreeVertex d)
(s : IsingSpin) (Y : StrictDescendantVertex c → ℝ) :
ofNoise (coneNoise theta c s Y) c = s := by
calc
ofNoise (coneNoise theta c s Y) c =
ofNoise (noiseFromSpins (fun _ : TreeVertex d => s)) c := by
apply ofNoise_congr_prefix
intro w hw
have hnot : ¬ (c.1 <+: w.1 ∧ w ≠ c) := by
rintro ⟨hcw, hne⟩
exact hne (Subtype.ext (hw.eq_of_length_le hcw.length_le))
by_cases hnil : w.1 = [] <;>
simp [coneNoise, noiseFromSpins, hnil, hnot]
_ = s := congrFun (ofNoise_noiseFromSpins (fun _ : TreeVertex d => s)) c
def coneFactor {d : ℕ} (theta : ℝ) (c : TreeVertex d)
(p : IsingSpin × (StrictDescendantVertex c → ℝ)) : DescendantVertex c → IsingSpin :=
fun v => ofNoise (coneNoise theta c p.1 p.2) v.1
theorem measurable_coneFactor {d : ℕ} (theta : ℝ) (c : TreeVertex d) :
Measurable (coneFactor theta c) := by
apply Measurable.of_eval
intro v
exact (measurable_ofNoise_apply v.1).comp (measurable_coneNoise theta c)
theorem factor_cone_representation {d : ℕ} (theta : ℝ) (c : TreeVertex d)
(U : TreeVertex d → ℝ) :
(fun v : DescendantVertex c => factor theta U v.1) =
coneFactor theta c (factor theta U c,
fun w : StrictDescendantVertex c => U w.1) := by
funext v
apply ofNoise_congr_cone (inputNoise theta U) _ c
· exact (ofNoise_coneNoise_at_root theta c (factor theta U c) _).symm
· intro w hcw hne
have hnil : w.1 ≠ [] := by
intro hw
have hc : c.1 = [] := by simpa only [hw, List.prefix_nil] using hcw
exact hne (Subtype.ext (hw.trans hc.symm))
simp [inputNoise, coneNoise, hnil, hcw, hne]
· exact v.2
end Problem367.RootedBroadcast
end
end OAI
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