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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 | 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
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