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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 | import OAI.Probability.FreeIsing.Trees.RootedBroadcastLaw
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
namespace RootedBroadcast
lemma ofNoise_congr_prefix {d : ℕ} (noise₁ noise₂ : TreeVertex d → IsingSpin)
(v : TreeVertex d)
(heq : ∀ w : TreeVertex d, w.val <+: v.val → noise₁ w = noise₂ w) :
ofNoise noise₁ v = ofNoise noise₂ v := by
have h : ∀ n : ℕ, ∀ v : TreeVertex d, v.val.length = n →
(∀ w : TreeVertex d, w.val <+: v.val → noise₁ w = noise₂ w) →
ofNoise noise₁ v = ofNoise noise₂ v := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro v hv heq
by_cases hnil : v.val = []
· calc
ofNoise noise₁ v = noise₁ v := by rw [ofNoise]; simp [hnil]
_ = noise₂ v := heq v (List.prefix_refl _)
_ = ofNoise noise₂ v := by rw [ofNoise]; simp [hnil]
· have hlt : (parent v).val.length < n := by
simp only [parent, parentVertex, List.length_dropLast, ← hv]
have : 0 < v.val.length := List.length_pos_iff.mpr hnil
omega
rw [ofNoise_nonroot noise₁ v hnil, ofNoise_nonroot noise₂ v hnil]
apply congrArg₂ spinMul (heq v (List.prefix_refl _))
apply ih _ hlt (parent v) rfl
intro w hw
exact heq w (hw.trans (List.dropLast_prefix v.val))
exact h _ v rfl heq
lemma factor_congr_prefix {d : ℕ} (theta : ℝ) (U V : TreeVertex d → ℝ)
(v : TreeVertex d) (heq : ∀ w : TreeVertex d, w.val <+: v.val → U w = V w) :
factor theta U v = factor theta V v := by
apply ofNoise_congr_prefix
intro w hw
simp only [inputNoise, heq w hw]
abbrev BranchVertex {d : ℕ} (c : Fin d) := {v : TreeVertex d // [c] <+: v.val}
lemma prefix_in_branch_or_root {d : ℕ} (c : Fin d)
{u v : TreeVertex d} (hv : [c] <+: v.val) (huv : u.val <+: v.val) :
u.val = [] ∨ [c] <+: u.val := by
by_cases hu : u.val = []
· exact Or.inl hu
· right
apply List.prefix_of_prefix_length_le hv huv
have : 0 < u.val.length := List.length_pos_iff.mpr hu
simpa only [List.length_singleton] using Nat.succ_le_of_lt this
def branchNoise {d : ℕ} (theta : ℝ) (c : Fin d)
(s : IsingSpin) (Y : BranchVertex c → ℝ) (v : TreeVertex d) : IsingSpin :=
if v.val = [] then s
else if hv : [c] <+: v.val then edgeNoise theta (Y ⟨v, hv⟩)
else IsingSpin.pos
lemma measurable_branchNoise {d : ℕ} (theta : ℝ) (c : Fin d) :
Measurable (fun p : IsingSpin × (BranchVertex c → ℝ) => branchNoise theta c p.1 p.2) := by
apply Measurable.of_eval
intro v
by_cases hnil : v.val = []
· simpa only [branchNoise, ite_eq_left hnil] using
(measurable_fst : Measurable (Prod.fst : IsingSpin × (BranchVertex c → ℝ) → IsingSpin))
· by_cases hv : [c] <+: v.val
· simpa only [branchNoise, ite_eq_right hnil, dite_eq_left hv, Function.comp_def] using
(measurable_edgeNoise theta).comp
((measurable_pi_apply (X := fun _ : BranchVertex c => ℝ) ⟨v,hv⟩).comp measurable_snd)
· simpa only [branchNoise, ite_eq_right hnil, dite_eq_right hv] using
(measurable_const : Measurable (fun _ : IsingSpin × (BranchVertex c → ℝ) => IsingSpin.pos))
def branchFactor {d : ℕ} (theta : ℝ) (c : Fin d)
(p : IsingSpin × (BranchVertex c → ℝ)) : BranchVertex c → IsingSpin :=
fun v => ofNoise (branchNoise theta c p.1 p.2) v.val
lemma measurable_branchFactor {d : ℕ} (theta : ℝ) (c : Fin d) :
Measurable (branchFactor theta c) := by
apply Measurable.of_eval
intro v
exact (measurable_ofNoise_apply v.val).comp (measurable_branchNoise theta c)
lemma factor_branch_representation {d : ℕ} (theta : ℝ) (c : Fin d)
(U : TreeVertex d → ℝ) :
(fun v : BranchVertex c => factor theta U v.val) =
branchFactor theta c (fairSpin (U (root d)), fun v : BranchVertex c => U v.val) := by
funext v
apply ofNoise_congr_prefix
intro w hw
rcases prefix_in_branch_or_root c v.property hw with hnil | hbranch
· have heq : w = root d := (eq_root_iff w).mpr hnil
subst w
simp [inputNoise, branchNoise]
· have hnil : w.val ≠ [] := by
intro heq
rw [heq] at hbranch
have hl := hbranch.length_le
simp at hl
simp [inputNoise, branchNoise, hnil, hbranch]
end RootedBroadcast
end Problem367
end
end OAI
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