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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Angular.AngularOperator2
import OAI.MeasureTheory.FalconerCompletion.Angular.SymbolMulDerivativeBound
import OAI.MeasureTheory.FalconerCompletion.Graphs.AllMaskProductAdmissible
import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale2
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
open _root_.OAI.PlanarFalconer
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport.CurrentFalconer.TreeND
open Set Metric MeasureTheory PlanarFalconer Filter
open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical ComplexConjugate
variable {d : ℕ}
noncomputable def actualListSourceSymbol
(S : Finset PolicyDescriptor)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))
(ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (N : ℕ)
(sig C : ℝ) (J i : ℕ) (f : EuclideanSpace ℝ (Fin d) → ℂ)
(Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) (x : EuclideanSpace ℝ (Fin d)) :
SmoothFunctionND (EuclideanSpace ℝ (Fin d)) :=
⟨fun w => f x*(Q.1 w*(actualAngularProduct S id L ρ m N sig C J i x).1 w),
contDiff_const.mul ((show ContDiff ℝ ∞ Q.1 from Q.2).mul
(actualAngularProduct S id L ρ m N sig C J i x).2)⟩
lemma actualListSourceSymbol_measurable
(S : Finset PolicyDescriptor)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))
(ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (N : ℕ)
(sig C : ℝ) (J i : ℕ) (f : EuclideanSpace ℝ (Fin d) → ℂ) (hf : Measurable f)
(Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) :
Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) =>
(actualListSourceSymbol S L ρ m N sig C J i f Q z.1).1 z.2) :=
(hf.comp measurable_fst).mul (((show ContDiff ℝ ∞ Q.1 from Q.2).continuous.measurable.comp measurable_snd).mul
(actualAngularProduct_measurable S id L ρ m N sig C J i))
lemma validDescriptorList_card {N : ℕ} (hN : 1 ≤ N) (S : Finset PolicyDescriptor)
(hS : ∀ b ∈ S, ValidDescriptor N b) : (S.card:ℝ) ≤ 8*(N:ℝ)^2 := by
let K := (Finset.range (N+1)).product (Finset.range (N+1))
let U := K.image (fun z => PolicyDescriptor.parallel z.1 z.2) ∪
K.image (fun z => PolicyDescriptor.projection z.1 z.2)
have hsub : S ⊆ U := by
intro b hb
have hv := hS b hb
cases b with
| parallel g p =>
apply Finset.mem_union_left
exact Finset.mem_image.mpr ⟨(g,p),Finset.mem_product.mpr
⟨Finset.mem_range.mpr (by dsimp [ValidDescriptor] at hv; omega),
Finset.mem_range.mpr (by exact Nat.lt_succ_of_le hv.2)⟩,rfl⟩
| projection p t =>
apply Finset.mem_union_right
exact Finset.mem_image.mpr ⟨(p,t),Finset.mem_product.mpr
⟨Finset.mem_range.mpr (by dsimp [ValidDescriptor] at hv; omega),
Finset.mem_range.mpr (by exact Nat.lt_succ_of_le hv.2)⟩,rfl⟩
have hc : S.card ≤ 2*(N+1)^2 := by
calc
_ ≤ U.card := Finset.card_le_card hsub
_ ≤ (K.image (fun z => PolicyDescriptor.parallel z.1 z.2)).card+
(K.image (fun z => PolicyDescriptor.projection z.1 z.2)).card := Finset.card_union_le _ _
_ ≤ K.card+K.card := add_le_add Finset.card_image_le Finset.card_image_le
_ = _ := by simp [K,Finset.product_eq_sprod,Finset.card_product]; ring
have hNr : (1:ℝ) ≤ N := by exact_mod_cast hN
calc
_ ≤ 2*((N:ℝ)+1)^2 := by exact_mod_cast hc
_ ≤ 8*(N:ℝ)^2 := by nlinarith
lemma uniform_actualListSourceSymbol_jets
(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)
(Cq : ℝ) (hCq : 0≤Cq) :
∃ E : ℝ, 0≤E ∧ ∀ᶠ N : ℕ in atTop, ∀ (S : Finset PolicyDescriptor)
(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 l : ℕ)
(f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))),
(∀ e ∈ S, ValidDescriptor N e) → (∀ e ∈ S, descriptorLength e ≤ l) →
(∀ e ∈ S, L e ∈ W) → (∑ e ∈ S, (L e).length ≤ total) →
Measurable f → (∀ x, ‖f x‖≤1) →
(∀ j≤M, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) →
_root_.OAI.CurrentFalconer.GraphSymbolJet
(fun x w => (actualListSourceSymbol S L ρ m N sig C J i f Q x).1 w) M
(((2:ℝ)^N)^sig*sourceMaskScaleND l ((2:ℝ)^N) sig) E := by
obtain ⟨D,hD,hDb⟩ := uniform_actualAngularProduct_derivatives (ι:=PolicyDescriptor)
W M B total 2 hW 8 sig hsig
refine ⟨Cq*D^total*2^M,by positivity,?_⟩
filter_upwards [hDb,Filter.eventually_ge_atTop 1] with N hDb hN
intro S L ρ m C J i l f Q hValid hl hL htotal hfm hf hQ
refine ⟨actualListSourceSymbol_measurable S L ρ m N sig C J i f hfm Q,
fun x => (actualListSourceSymbol S L ρ m N sig C J i f Q x).2,by positivity,?_⟩
intro x j hj w hw
let Z := actualAngularProduct S id L ρ m N sig C J i x
let scale := ((2:ℝ)^N)^sig*sourceMaskScaleND l ((2:ℝ)^N) sig
have hscale : 1 ≤ scale := one_le_mul_of_one_le_of_one_le
(Real.one_le_rpow (one_le_pow₀ (by norm_num : (1:ℝ)≤2)) hsig.le)
(sourceMaskScaleND_one_le _ _ _)
have hb := symbol_mul_derivative_boundND Q Z j hCq (pow_nonneg (zero_le_one.trans hD) total)
(zero_le_one.trans hscale) w (fun k hk =>
(hQ k (hk.trans hj) w hw).trans (le_mul_of_one_le_right hCq (one_le_pow₀ hscale)))
(fun k hk => hDb S id L ρ m C J i l (validDescriptorList_card hN S hValid)
hL htotal hl x k (hk.trans hj) w hw)
have hh := symbol_const_mul_derivative_boundND (smoothMulND Q Z) (f x) (hf x) j w
apply hh.trans (hb.trans ?_)
rw [mul_pow]
change Cq*D^total*(2^j*scale^j) ≤ Cq*D^total*2^M*scale^j
rw [mul_assoc (Cq*D^total) (2^M)]
exact mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hj)
(pow_nonneg (zero_le_one.trans hscale) j)) (mul_nonneg hCq (pow_nonneg (zero_le_one.trans hD) total))
lemma uniform_actualListSourceSymbol_test {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) (S : Finset PolicyDescriptor),
(∀ e ∈ S, ValidDescriptor N e) →
∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))
(f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d)))
(x w : EuclideanSpace ℝ (Fin d)), ‖w‖=1 →
(actualListSourceSymbol S L ρ m N sig C J i f Q x).1 w ≠ 0 →
∀ e ∈ S, (x,w) ∈ endpointGood ρ m N sig C J (i+1) e := by
obtain ⟨N₀,h₀⟩ := uniform_actualAngularProduct_support (d:=d) (ι:=PolicyDescriptor) hsig hJ
refine ⟨N₀,?_⟩
intro N hN ρ hρ m C i S hValid L f Q x w hw hn e he
have hp : (actualAngularProduct S id L ρ m N sig C J i x).1 w ≠ 0 := by
intro hz
apply hn
simp [actualListSourceSymbol,hz]
apply h₀ N hN ρ hρ m C i S id hValid L 0 x w hw _ e he
apply subset_closure
rw [Function.mem_support,←norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff]
exact hp
end GraphLiteralSupport.CurrentFalconer.TreeND
end
end
end OAI
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