| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.DescriptorGoodNeg | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set Metric MeasureTheory PlanarFalconer | |
| open scoped ENNReal RealInnerProductSpace ContDiff Topology | |
| variable {d : ℕ} | |
| noncomputable def actualEndpointMask (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| (x w : EuclideanSpace ℝ (Fin d)) : ℝ := by | |
| classical | |
| exact ∑ P ∈ dyadicPaths (descriptorAnchor b), | |
| if x ∈ treeCell P then actualDescriptorMask ρ m N sig C J i P b w else 0 | |
| lemma actualEndpointMask_of_mem (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {P : CellPath d} (hP : P ∈ dyadicPaths (descriptorAnchor b)) | |
| {x : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) : | |
| actualEndpointMask ρ m N sig C J i b x=actualDescriptorMask ρ m N sig C J i P b := by | |
| classical | |
| funext w | |
| unfold actualEndpointMask | |
| rw [Finset.sum_eq_single P] | |
| · simp only [hx,ite_true] | |
| · intro Q hQ hQP | |
| have hnot : x ∉ treeCell Q := fun hxQ => | |
| Set.disjoint_left.mp (treeCell_disjoint | |
| (((mem_dyadicPaths _ _).mp hQ).trans ((mem_dyadicPaths _ _).mp hP).symm) hQP) hxQ hx | |
| simp only [hnot,ite_false] | |
| · exact fun h => (h hP).elim | |
| lemma actualEndpointMask_of_absent (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {x : EuclideanSpace ℝ (Fin d)} | |
| (hx : ∀ P ∈ dyadicPaths (descriptorAnchor b), x ∉ treeCell P) : | |
| actualEndpointMask ρ m N sig C J i b x=0 := by | |
| classical | |
| funext w | |
| exact Finset.sum_eq_zero (fun P hP => ite_eq_right (hx P hP)) | |
| lemma actualEndpointMask_smooth (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| (x : EuclideanSpace ℝ (Fin d)) : ContDiff ℝ ∞ (actualEndpointMask ρ m N sig C J i b x) := by | |
| classical | |
| by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P | |
| · obtain ⟨P,hP,hx⟩ := hx | |
| rw [actualEndpointMask_of_mem ρ m N sig C J i b hP hx] | |
| exact actualDescriptorMask_smooth _ _ _ _ _ _ _ _ _ | |
| · rw [actualEndpointMask_of_absent ρ m N sig C J i b (by simpa only [not_exists,not_and] using hx)] | |
| exact contDiff_const | |
| lemma actualEndpointMask_measurable (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) : | |
| Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) => | |
| actualEndpointMask ρ m N sig C J i b z.1 z.2) := by | |
| classical | |
| unfold actualEndpointMask | |
| apply Finset.measurable_sum | |
| intro P hP | |
| exact Measurable.ite ((measurableSet_treeCell P).preimage measurable_fst) | |
| ((actualDescriptorMask_smooth ρ m N sig C J i P b).continuous.measurable.comp measurable_snd) | |
| measurable_const | |
| lemma actualEndpointMask_bounds (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| (x : EuclideanSpace ℝ (Fin d)) {w : EuclideanSpace ℝ (Fin d)} (hw : ‖w‖≤1) : | |
| 0≤actualEndpointMask ρ m N sig C J i b x w ∧ actualEndpointMask ρ m N sig C J i b x w ≤ 1 := by | |
| classical | |
| by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P | |
| · obtain ⟨P,hP,hx⟩ := hx | |
| rw [actualEndpointMask_of_mem ρ m N sig C J i b hP hx] | |
| exact actualDescriptorMask_bounds _ _ _ _ _ _ _ _ _ hw | |
| · rw [actualEndpointMask_of_absent ρ m N sig C J i b (by simpa only [not_exists,not_and] using hx)] | |
| simp | |
| lemma actualEndpointMask_one (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {x w : EuclideanSpace ℝ (Fin d)} | |
| (hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P) | |
| (hg : (x,w) ∈ endpointGood ρ m N sig C J i b) : | |
| actualEndpointMask ρ m N sig C J i b x w=1 := by | |
| obtain ⟨P,hP,hx⟩ := hx | |
| rw [actualEndpointMask_of_mem ρ m N sig C J i b hP hx] | |
| exact actualDescriptorMask_one ρ m N sig C J i P b (hg P hP hx) | |
| lemma uniform_actualEndpointMask_support {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))), IsFiniteMeasure ρ → | |
| ∀ (m : ℝ → ℝ) (C : ℝ) (i : ℕ) (b : PolicyDescriptor), ValidDescriptor N b → | |
| ∀ (k : ℕ) (x w : EuclideanSpace ℝ (Fin d)), ‖w‖=1 → | |
| w ∈ tsupport (iteratedFDeriv ℝ k (actualEndpointMask ρ m N sig C J i b x)) → | |
| (x,w) ∈ endpointGood ρ m N sig C J (i+1) b := by | |
| obtain ⟨N₀,h₀⟩ := uniform_actualDescriptorMask_support (d := d) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ρ hf m C i b hb k x w hw hs P hP hx | |
| rw [actualEndpointMask_of_mem ρ m N sig C J i b hP hx] at hs | |
| exact h₀ N hN ρ hf m C i b hb P ((mem_dyadicPaths _ _).mp hP) k w hw hs | |
| lemma actualEndpointMask_derivatives (d B : ℕ) : | |
| ∃ C₀ : ℝ, 0 ≤ C₀ ∧ ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| (k : ℕ), k ≤ B → ∀ x w, | |
| ‖iteratedFDeriv ℝ k (actualEndpointMask ρ m N sig C J i b x) w‖ ≤ | |
| C₀*(sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig)^k := by | |
| classical | |
| obtain ⟨C₀,hC,h₀⟩ := actualDescriptorMask_derivatives d B | |
| refine ⟨C₀,hC,?_⟩ | |
| intro ρ m N sig C J i b k hk x w | |
| by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P | |
| · obtain ⟨P,hP,hx⟩ := hx | |
| rw [actualEndpointMask_of_mem ρ m N sig C J i b hP hx] | |
| exact h₀ ρ m N sig C J i P b k hk w | |
| · rw [actualEndpointMask_of_absent ρ m N sig C J i b (by simpa only [not_exists,not_and] using hx)] | |
| change ‖iteratedFDeriv ℝ k (fun _ : EuclideanSpace ℝ (Fin d) => (0:ℝ)) w‖ ≤ _ | |
| simp only [iteratedFDeriv_fun_zero,Pi.zero_apply,norm_zero] | |
| exact mul_nonneg hC (pow_nonneg (sourceMaskScaleND_pos _ _ _).le k) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |