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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 | import OAI.Probability.FreeIsing.Trees.ConeCoding
import OAI.Probability.FreeIsing.Trees.SubcriticalClusters
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.TreeConeCoding
def rootedPathOpen {b : ℕ} (p : ℝ) (U : List (Fin b) → ℝ) (w : List (Fin b)) : Prop :=
∀ k : ℕ, 0 < k → k ≤ w.length → U (w.take k) ≤ p
theorem measurableSet_rootedPathOpen {b : ℕ} (p : ℝ) (w : List (Fin b)) :
MeasurableSet {U : List (Fin b) → ℝ | rootedPathOpen p U w} := by
simp only [rootedPathOpen, Set.ofPred_forall]
exact MeasurableSet.iInter fun k : ℕ =>
MeasurableSet.iInter fun _ : 0 < k =>
MeasurableSet.iInter fun _ : k ≤ w.length =>
measurableSet_le (measurable_pi_apply _) measurable_const
theorem downwardOpen_vertex_iff {b : ℕ} (p : ℝ)
(U : TreeVertex (b + 1) → ℝ) (r : TreeVertex (b + 1)) (hr : r.1 ≠ [])
(w : List (Fin b)) :
SubcriticalClusters.DownwardOpen p U r (vertex r hr w) ↔
rootedPathOpen p (labels r hr U) w := by
constructor
· intro h k hk hklen
have hc := h.2 (r.1.length + k) (by omega) (by simp; omega)
simpa only [← vertex_take, labels] using hc
· intro h
refine ⟨vertex_prefix r hr w, fun k hk hklen => ?_⟩
have hkle : k - r.1.length ≤ w.length := by
rw [length_vertex] at hklen
omega
have hpos : 0 < k - r.1.length := by omega
have hc := h (k - r.1.length) hpos hkle
have he : r.1.length + (k - r.1.length) = k := Nat.add_sub_of_le hk.le
simpa only [labels, vertex_take, he] using hc
theorem survival_iff {b : ℕ} (p : ℝ)
(U : TreeVertex (b + 1) → ℝ) (r : TreeVertex (b + 1)) (hr : r.1 ≠ []) (n : ℕ) :
(∃ v : TreeVertex (b + 1), v.1.length = r.1.length + n ∧
SubcriticalClusters.DownwardOpen p U r v) ↔
∃ w : List (Fin b), w.length = n ∧ rootedPathOpen p (labels r hr U) w := by
constructor
· rintro ⟨v, hvlen, hv⟩
obtain ⟨w, rfl⟩ := vertex_surjective_cone r hr v hv.1
refine ⟨w, ?_, (downwardOpen_vertex_iff p U r hr w).mp hv⟩
rw [length_vertex] at hvlen
exact Nat.add_left_cancel hvlen
· rintro ⟨w, hwlen, hw⟩
refine ⟨vertex r hr w, ?_, (downwardOpen_vertex_iff p U r hr w).mpr hw⟩
rw [length_vertex, hwlen]
theorem measurableSet_rootedSurvival {b : ℕ} (p : ℝ) (n : ℕ) :
MeasurableSet {U : List (Fin b) → ℝ |
∃ w : List (Fin b), w.length = n ∧ rootedPathOpen p U w} := by
simp only [Set.ofPred_exists]
apply MeasurableSet.iUnion
intro w
by_cases hw : w.length = n
· simpa only [hw, true_and] using measurableSet_rootedPathOpen p w
· simp only [hw, false_and, Set.ofPred_false, MeasurableSet.empty]
theorem survival_probability {b : ℕ} (p : ℝ)
(r : TreeVertex (b + 1)) (hr : r.1 ≠ []) (n : ℕ) :
labelMeasure (b + 1) {U | ∃ v : TreeVertex (b + 1),
v.1.length = r.1.length + n ∧ SubcriticalClusters.DownwardOpen p U r v} =
(Measure.infinitePi (fun _ : List (Fin b) => uniform01))
{W | ∃ w : List (Fin b), w.length = n ∧ rootedPathOpen p W w} := by
have he : {U : TreeVertex (b + 1) → ℝ | ∃ v : TreeVertex (b + 1),
v.1.length = r.1.length + n ∧ SubcriticalClusters.DownwardOpen p U r v} =
(labels r hr) ⁻¹' {W | ∃ w : List (Fin b),
w.length = n ∧ rootedPathOpen p W w} := by
ext U
exact survival_iff p U r hr n
rw [he]
exact (measurePreserving_labels uniform01 r hr).measure_preimage
(measurableSet_rootedSurvival p n).nullMeasurableSet
theorem ae_finite_downwardOpen_of_rooted {b : ℕ} (p : ℝ)
(r : TreeVertex (b + 1)) (hr : r.1 ≠ [])
(hfin : ∀ᵐ W ∂Measure.infinitePi (fun _ : List (Fin b) => uniform01),
{w | rootedPathOpen p W w}.Finite) :
∀ᵐ U ∂labelMeasure (b + 1),
{v | SubcriticalClusters.DownwardOpen p U r v}.Finite := by
filter_upwards [(measurePreserving_labels uniform01 r hr).quasiMeasurePreserving.ae hfin]
with U hU
apply (hU.image (vertex r hr)).subset
intro v hv
obtain ⟨w, hw⟩ := vertex_surjective_cone r hr v hv.1
refine ⟨w, ?_, hw⟩
apply (downwardOpen_vertex_iff p U r hr w).mp
rwa [hw]
end Problem367.TreeConeCoding
end
end OAI
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