| 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 | |