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b224933 | 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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Estimates.FullParallelCountFilter
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
open _root_.OAI.PlanarFalconer
section
section
namespace GraphLiteralSupport.CurrentFalconer.TreeND
open Set Metric MeasureTheory PlanarFalconer
open scoped ENNReal
variable {d : ℕ}
local notation "V" => EuclideanSpace ℝ (Fin d)
def descriptorValid (N : ℕ) : PolicyDescriptor → Prop
| .parallel g p => g ≤ p ∧ p ≤ N
| .projection p t => p ≤ t ∧ t ≤ N
lemma endpointGood_at_cell (ρ : Measure V) (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ)
(J i : ℕ) (b : PolicyDescriptor) {P : CellPath d}
(hP : P ∈ dyadicPaths (descriptorAnchor b)) {x : V} (hx : x∈treeCell P) (w : V) :
(x,w)∈endpointGood ρ m N sig C J i b ↔ w∈descriptorGood ρ m N sig C J i P b := by
refine ⟨fun h => h P hP hx,?_⟩
intro hw Q hQ hxQ
have he : Q=P := by
by_contra hn
exact Set.disjoint_left.mp (treeCell_disjoint
(((mem_dyadicPaths _ _).mp hQ).trans ((mem_dyadicPaths _ _).mp hP).symm) hn) hxQ hx
simpa only [he] using hw
lemma ae_occupied_anchor (ρ : Measure V) [IsFiniteMeasure ρ] (p : ℕ)
(hroot : ρ (treeCell [])ᶜ=0) :
∀ᵐ x ∂ρ, ∃ P∈dyadicPaths p, x∈treeCell P ∧ 0<ρ (treeCell P) := by
have ha : ∀ P : CellPath d, ∀ᵐ x ∂ρ, x∈treeCell P → 0<ρ (treeCell P) := by
intro P
by_cases hp : 0<ρ (treeCell P)
· exact Filter.Eventually.of_forall (fun _ _ => hp)
· have hz : ρ (treeCell P)=0 := nonpos_iff_eq_zero.mp (le_of_not_gt hp)
filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with x hx
exact fun h => (hx h).elim
have hall := ae_all_iff.mpr ha
have hr : ∀ᵐ x ∂ρ, x∈treeCell [] := by
simpa only [mem_compl_iff,not_not] using measure_eq_zero_iff_ae_notMem.mp hroot
filter_upwards [hall,hr] with x hx hxroot
rw [←treeCell_cover_root p] at hxroot
obtain ⟨P,hP,hxp⟩ := mem_iUnion₂.mp hxroot
exact ⟨P,hP,hxp,hx P hxp⟩
theorem uniform_original_fiber_test_bound (sig : ℝ) (hsig : 0<sig) :
∃ N₀ : ℕ, ∀ N≥N₀, ∀ (ρ : Measure V) (m : ℝ → ℝ) (B : ℕ) (s : ℝ),
RegularizedProfileData ρ N B s sig m →
∀ (ν : Measure V) [IsProbabilityMeasure ν] (S : Set V), MeasurableSet S →
∀ dir : V → V, Measurable dir → (∀ y∈S, ‖dir y‖=1) →
(∀ w : V, ‖w‖=1 → ∀ r : ℝ, ((2:ℝ)^N)⁻¹ ≤ r → r ≤ 1 →
ν {y | y∈S ∧ dir y∈ball w r} ≤ ENNReal.ofReal (((2:ℝ)^N)^sig*r^((d:ℝ)/2))) →
∀ (J : ℕ) (b : PolicyDescriptor), descriptorValid N b →
∀ x : V, (∃ P∈dyadicPaths (descriptorAnchor b), x∈treeCell P ∧ 0<ρ (treeCell P)) →
ν {y | y∈S ∧ (x,dir y)∉endpointGood ρ m N sig (1000*(d+1:ℝ)) J 0 b} ≤
ENNReal.ofReal (((2:ℝ)^N)^(-sig)) := by
obtain ⟨N₁,hparallel⟩ := uniform_actual_parallel_exceptional d sig hsig
obtain ⟨N₂,hprojection⟩ := uniform_actual_projection_exceptional d sig hsig
refine ⟨max N₁ N₂,?_⟩
intro N hN ρ m B s hr ν _ S hS dir hdir hunit hcap J b hb x hx
obtain ⟨P,hP,hx,hpos⟩ := hx
let ξ : Measure V := (ν.restrict S).map dir
have hfinite : IsFiniteMeasure ξ := by dsimp [ξ]; infer_instance
have hmass : ξ univ ≤ 1 := by
dsimp only [ξ]
rw [Measure.map_apply hdir MeasurableSet.univ]
simpa only [preimage_univ,Measure.restrict_apply MeasurableSet.univ,univ_inter] using (prob_le_one (μ:=ν) (s:=S))
have hsphere : ∀ᵐ w ∂ξ, ‖w‖=1 := by
dsimp only [ξ]
rw [ae_map_iff hdir.aemeasurable (measurableSet_eq_fun measurable_norm measurable_const)]
filter_upwards [ae_restrict_mem hS] with y hy
exact hunit y hy
have hc : ∀ w : V, ‖w‖=1 → ∀ r : ℝ, ((2:ℝ)^N)⁻¹ ≤ r → r ≤ 1 →
ξ (ball w r) ≤ ENNReal.ofReal (((2:ℝ)^N)^sig*r^((d:ℝ)/2)) := by
intro w hw r hr0 hr1
dsimp only [ξ]
rw [Measure.map_apply hdir measurableSet_ball,Measure.restrict_apply (hdir measurableSet_ball)]
convert hcap w hw r hr0 hr1 using 1
congr 1
ext y
simp only [mem_inter_iff,mem_preimage,mem_ofPred_eq,and_comm]
have ht : ξ (descriptorGood ρ m N sig (1000*(d+1:ℝ)) J 0 P b)ᶜ ≤
ENNReal.ofReal (((2:ℝ)^N)^(-sig)) := by
cases b with
| parallel g p =>
exact hparallel N (by omega) ρ m B s hr ξ hfinite hmass hsphere hc J g p hb.1 hb.2
P ((mem_dyadicPaths _ _).mp hP) hpos
| projection p t =>
exact hprojection N (by omega) ρ m B s hr ξ hfinite hmass hsphere hc J p t hb.1 hb.2
P ((mem_dyadicPaths _ _).mp hP) hpos
let := hr.probability
dsimp only [ξ] at ht
rw [Measure.map_apply hdir (measurable_descriptorGood _ _ _ _ _ _ _ _ _).compl,
Measure.restrict_apply (hdir (measurable_descriptorGood _ _ _ _ _ _ _ _ _).compl)] at ht
convert ht using 1
congr 1
ext y
simp only [mem_ofPred_eq,mem_inter_iff,mem_preimage,mem_compl_iff,endpointGood_at_cell ρ m N sig
(1000*(d+1:ℝ)) J 0 b hP hx,and_comm]
end GraphLiteralSupport.CurrentFalconer.TreeND
end
end
end OAI
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