| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AllMaskProductAdmissible |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set Metric MeasureTheory PlanarFalconer Filter |
| open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical |
| variable {d : ℕ} |
|
|
| noncomputable def actualPolicyAngular |
| (A : ℕ → ℝ) (N q a t : ℕ) |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) |
| (x : EuclideanSpace ℝ (Fin d)) : SmoothFunctionND (EuclideanSpace ℝ (Fin d)) := |
| actualAngularProduct (policyDescriptorList A N q a t) id L ρ m N sig C J i x |
|
|
| lemma actualPolicyAngular_measurable |
| (A : ℕ → ℝ) (N q a t : ℕ) |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) : |
| Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) => |
| (actualPolicyAngular A N q a t L ρ m sig C J i z.1).1 z.2) := |
| actualAngularProduct_measurable _ _ _ _ _ _ _ _ _ _ |
|
|
| lemma actualPolicyAngular_even |
| (A : ℕ → ℝ) (N q a t : ℕ) |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) |
| (x : EuclideanSpace ℝ (Fin d)) : |
| Function.Even (actualPolicyAngular A N q a t L ρ m sig C J i x).1 := |
| actualAngularProduct_even _ _ _ _ _ _ _ _ _ _ _ |
|
|
| lemma actualPolicy_card_bound (A : ℕ → ℝ) {N q a t : ℕ} (hN : 1 ≤ N) (hq : 0<q) : |
| ((policyDescriptorList A N q a t).card:ℝ) ≤ 8*(N:ℝ)^3 := by |
| have hh : ((policyDescriptorList A N q a t).card:ℝ) ≤ ((N:ℝ)+1)^3 := by |
| exact_mod_cast policyDescriptorList_card (a:=a) (t:=t) A hq |
| have hN' : (1:ℝ) ≤ N := by exact_mod_cast hN |
| calc |
| _ ≤ ((N:ℝ)+1)^3 := hh |
| _ ≤ (2*(N:ℝ))^3 := pow_le_pow_left₀ (by positivity) (by linarith) 3 |
| _ = _ := by ring |
|
|
| lemma uniform_actualPolicyAngular_derivatives |
| (W : Finset (List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))) |
| (M B total : ℕ) (hW : ∀ L ∈ W, L.length ≤ B) (sig : ℝ) (hsig : 0<sig) : |
| ∃ D : ℝ, 1 ≤ D ∧ ∀ᶠ N : ℕ in atTop, ∀ (A : ℕ → ℝ) (q a t : ℕ), 0<q → |
| ∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (C : ℝ) (J i : ℕ), |
| (∀ b ∈ policyDescriptorList A N q a t, L b ∈ W) → |
| (∑ b ∈ policyDescriptorList A N q a t, (L b).length ≤ total) → |
| ∀ (x : EuclideanSpace ℝ (Fin d)), ∀ j ≤ M, ∀ w, ‖w‖=1 → |
| ‖iteratedFDeriv ℝ j (actualPolicyAngular A N q a t L ρ m sig C J i x).1 w‖ ≤ |
| D^total*(((2:ℝ)^N)^sig* |
| sourceMaskScaleND ((policyDescriptorList A N q a t).sup descriptorLength) ((2:ℝ)^N) sig)^j := by |
| obtain ⟨D,hD,hDb⟩ := uniform_actualAngularProduct_derivatives (ι:=PolicyDescriptor) W M B total 3 |
| hW 8 sig hsig |
| refine ⟨D,hD,?_⟩ |
| filter_upwards [hDb,Filter.eventually_ge_atTop 1] with N hDb hN |
| intro A q a t hq L ρ m C J i hL htotal x j hj w hw |
| exact hDb (policyDescriptorList A N q a t) id L ρ m C J i _ |
| (actualPolicy_card_bound A hN hq) hL htotal (fun b hb => Finset.le_sup hb) x j hj w hw |
|
|
| lemma uniform_actualPolicyAngular_support {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))), IsFiniteMeasure ρ → |
| ∀ (A : ℕ → ℝ) (q a t : ℕ), 0<q → t≤N → |
| ∀ (m : ℝ → ℝ) (C : ℝ) (i : ℕ) |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) |
| (k : ℕ) (x w : EuclideanSpace ℝ (Fin d)), ‖w‖=1 → |
| w ∈ tsupport (iteratedFDeriv ℝ k (actualPolicyAngular A N q a t L ρ m sig C J i x).1) → |
| ∀ b ∈ policyDescriptorList A N q a t, (x,w) ∈ endpointGood ρ m N sig C J (i+1) b := by |
| obtain ⟨N₀,h₀⟩ := uniform_actualAngularProduct_support (d:=d) (ι:=PolicyDescriptor) hsig hJ |
| refine ⟨N₀,?_⟩ |
| intro N hN ρ hf A q a t hq ht m C i L k x w hw hs b hb |
| exact h₀ N hN ρ hf m C i (policyDescriptorList A N q a t) id |
| (fun e he => policyDescriptor_valid A hq ht he) L k x w hw hs b hb |
|
|
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
