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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 | import OAI.Probability.FreeIsing.Estimates.LabelDecomposition
import OAI.Probability.FreeIsing.Trees.BroadcastLocality
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
lemma indepFun_disjoint_subarrays {ι A : Type*} [MeasurableSpace A]
(μ : ι → Measure A) [∀ i, IsProbabilityMeasure (μ i)]
(S T : Set ι) (hST : Disjoint S T) :
IndepFun (fun U : ι → A => fun i : S => U i.val)
(fun U : ι → A => fun i : T => U i.val) (Measure.infinitePi μ) := by
have hi : iIndepFun (fun i (U : ι → A) => U i) (Measure.infinitePi μ) :=
iIndepFun_infinitePi (X := fun _ a => a) (fun _ => measurable_id)
have hle : ∀ i : ι,
MeasurableSpace.comap (fun U : ι → A => U i) inferInstance ≤
(inferInstance : MeasurableSpace (ι → A)) :=
fun i => (measurable_pi_apply i).comap_le
have h := indep_iSup_of_disjoint hle hi hST
change Indep
(MeasurableSpace.comap (fun U : ι → A => fun i : S => U i.val) inferInstance)
(MeasurableSpace.comap (fun U : ι → A => fun i : T => U i.val) inferInstance)
(Measure.infinitePi μ)
simpa only [MeasurableSpace.comap_process_pi, iSup_subtype] using h
lemma map_disjoint_subarrays {ι A : Type*} [MeasurableSpace A]
(μ : ι → Measure A) [∀ i, IsProbabilityMeasure (μ i)]
(S T : Set ι) (hST : Disjoint S T) :
(Measure.infinitePi μ).map (fun U : ι → A =>
((fun i : S => U i.val), (fun i : T => U i.val))) =
(Measure.infinitePi (fun i : S => μ i.val)).prod
(Measure.infinitePi (fun i : T => μ i.val)) := by
rw [(indepFun_disjoint_subarrays μ S T hST).map_prod_eq_prod_map_map
((Measurable.of_eval (fun i : S => measurable_pi_apply i.val)).aemeasurable)
((Measurable.of_eval (fun i : T => measurable_pi_apply i.val)).aemeasurable)]
rw [Measure.map_infinitePi_infinitePi_of_inj Subtype.val_injective,
Measure.map_infinitePi_infinitePi_of_inj Subtype.val_injective]
namespace RootedBroadcast
lemma branch_disjoint {d : ℕ} {c e : Fin d} (hce : c ≠ e) :
Disjoint {v : TreeVertex d | [c] <+: v.val} {v | [e] <+: v.val} := by
apply Set.disjoint_left.mpr
intro v hc he
have hprefix : [c] <+: [e] := List.prefix_of_prefix_length_le hc he (by simp)
have heq : [c] = [e] := hprefix.eq_of_length (by simp)
exact hce (by simpa only [List.singleton_inj] using heq)
def branchEmbedding {d : ℕ} (c : Fin d) (v : BranchVertex c) : NonrootVertex d := by
refine ⟨v.val, ?_⟩
intro heq
have hlen := v.property.length_le
rw [heq, root_val] at hlen
simp at hlen
lemma map_branch_labels {d : ℕ} (c e : Fin d) (hce : c ≠ e) :
(labelMeasure d).map (fun U =>
((fun v : BranchVertex c => U v.val), (fun v : BranchVertex e => U v.val))) =
(Measure.infinitePi (fun _ : BranchVertex c => uniform01)).prod
(Measure.infinitePi (fun _ : BranchVertex e => uniform01)) :=
map_disjoint_subarrays (fun _ : TreeVertex d => uniform01) _ _ (branch_disjoint hce)
lemma map_root_two_branches {d : ℕ} (c e : Fin d) (hce : c ≠ e) :
(labelMeasure d).map (fun U => (fairSpin (U (root d)),
((fun v : BranchVertex c => U v.val), (fun v : BranchVertex e => U v.val)))) =
(uniform01.map fairSpin).prod
((Measure.infinitePi (fun _ : BranchVertex c => uniform01)).prod
(Measure.infinitePi (fun _ : BranchVertex e => uniform01))) := by
let : IsProbabilityMeasure (labelMeasure d) := by
unfold labelMeasure
infer_instance
let F : (NonrootVertex d → ℝ) → (BranchVertex c → ℝ) × (BranchVertex e → ℝ) :=
fun Y => ((fun v => Y (branchEmbedding c v)), (fun v => Y (branchEmbedding e v)))
have hF : Measurable F := by fun_prop
have hind := (LabelDecomposition.indepFun_fairSpin_rest d (root d)).comp measurable_id hF
have hB : Measurable (fun U : TreeVertex d → ℝ => fairSpin (U (root d))) :=
measurable_fairSpin.comp (measurable_pi_apply (root d))
have hY : Measurable (fun U : TreeVertex d → ℝ =>
((fun v : BranchVertex c => U v.val), (fun v : BranchVertex e => U v.val))) := by fun_prop
have hfair : (labelMeasure d).map (fun U => fairSpin (U (root d))) =
uniform01.map fairSpin := by
change (labelMeasure d).map (fairSpin ∘ (fun U : TreeVertex d → ℝ => U (root d))) = _
rw [← Measure.map_map measurable_fairSpin
(measurable_pi_apply (X := fun _ : TreeVertex d => ℝ) (root d))]
congr 1
exact Measure.infinitePi_map_eval (fun _ : TreeVertex d => uniform01) (root d)
change IndepFun (fun U : TreeVertex d → ℝ => fairSpin (U (root d)))
(fun U => ((fun v : BranchVertex c => U v.val), (fun v : BranchVertex e => U v.val)))
(labelMeasure d) at hind
rw [hind.map_prod_eq_prod_map_map hB.aemeasurable hY.aemeasurable,
hfair, map_branch_labels c e hce]
end RootedBroadcast
end Problem367
end
end OAI
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