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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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Estimates.AssemblyEndpointSmooth
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphSymbolJetConj
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 MeasureTheory Filter PlanarFalconer
open scoped ContDiff Classical Topology ComplexConjugate
variable {n : ℕ}
local notation "V" => EuclideanSpace ℝ (Fin (n+1))
lemma assemblyLists_card (e₁ e₂ : RecursiveEnvironmentND n) (hN : 1≤e₁.N)
(heN : e₂.N=e₁.N) (hq₁ : 0<e₁.q) (hq₂ : 0<e₂.q) {c : ℕ} (hc : c≤e₁.N) :
max 1 (((assemblyDescriptorList e₁.profile e₁.N e₁.q c).card+
(assemblyDescriptorList e₂.profile e₂.N e₂.q c).card:ℕ):ℝ) ≤ 16*(e₁.N:ℝ)^2 := by
have h₁ := validDescriptorList_card hN (assemblyDescriptorList e₁.profile e₁.N e₁.q c)
(fun _ hb => assemblyDescriptorList_valid e₁.profile hq₁ hc hb)
have h₂ := validDescriptorList_card (by omega : 1≤e₂.N) (assemblyDescriptorList e₂.profile e₂.N e₂.q c)
(fun _ hb => assemblyDescriptorList_valid e₂.profile hq₂ (by omega) hb)
rw [heN] at h₂
have hN' : (1:ℝ)≤e₁.N := by exact_mod_cast hN
apply max_le
· nlinarith only [hN',sq_nonneg ((e₁.N:ℝ)-1)]
· push_cast
rw [heN]
linarith only [h₁,h₂]
lemma assemblyDescriptor_scale (e : RecursiveEnvironmentND n) (hq : 0<e.q)
{c : ℕ} (hc : 2*c≤e.N) {b : PolicyDescriptor}
(hb : b∈assemblyDescriptorList e.profile e.N e.q c) :
1 ≤ sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ∧
sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ≤
Attempt4PrefixShell.sourceMaskScaleND (e.N/2) ((2:ℝ)^e.N) e.sig := by
refine ⟨sourceMaskScaleND_one_le _ _ _,?_⟩
change sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ≤
sourceMaskScaleND (e.N/2) ((2:ℝ)^e.N) e.sig
apply sourceMaskScaleND_mono _ (by positivity)
have h := assemblyDescriptorList_length e.profile hq hc
have hb' := Finset.le_sup (f:=descriptorLength) hb
omega
lemma assembly_pairMask_eq (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ) (z : V×V) (w : V) :
(Attempt4PrefixShell.pairMaskProductND
(assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c)
(assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1 w =
((∏ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c,
actualEndpointMask e₁.ρ e₁.mass e₁.N e₁.sig e₁.C e₁.J 0 b z.1 w)*
(∏ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c,
actualEndpointMask e₂.ρ e₂.mass e₂.N e₂.sig e₂.C e₂.J 0 b z.2 w):ℝ) := by
simp only [Attempt4PrefixShell.pairMaskProductND,Attempt4PrefixShell.smoothFinsetProductND,
Finset.prod_disjSum,Sum.elim_inl,Sum.elim_inr,Attempt4PrefixShell.smoothStarND,
assemblyEndpointSmooth,Complex.star_def,Complex.conj_ofReal,Complex.ofReal_mul,Complex.ofReal_prod]
lemma assembly_endpoint_radial_left (e : RecursiveEnvironmentND n) (b : PolicyDescriptor) :
Measurable (fun z : V×V => actualEndpointMask e.ρ e.mass e.N e.sig e.C e.J 0 b z.1
((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by
simpa only [radialVector,dist_eq_norm,Function.comp_def] using
(actualEndpointMask_measurable e.ρ e.mass e.N e.sig e.C e.J 0 b).comp
(measurable_fst.prodMk measurable_radialVector)
lemma assembly_endpoint_radial_right (e : RecursiveEnvironmentND n) (b : PolicyDescriptor) :
Measurable (fun z : V×V => actualEndpointMask e.ρ e.mass e.N e.sig e.C e.J 0 b z.2
((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by
simpa only [radialVector,dist_eq_norm,Function.comp_def] using
(actualEndpointMask_measurable e.ρ e.mass e.N e.sig e.C e.J 0 b).comp
(measurable_snd.prodMk measurable_radialVector)
lemma assembly_pairMask_radial_measurable (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ) :
Measurable (fun z : V×V =>
(Attempt4PrefixShell.pairMaskProductND
(assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c)
(assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1
((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by
simp only [assembly_pairMask_eq]
apply Complex.continuous_ofReal.measurable.comp
exact (Finset.measurable_prod _ (fun b _ => assembly_endpoint_radial_left e₁ b)).mul
(Finset.measurable_prod _ (fun b _ => assembly_endpoint_radial_right e₂ b))
lemma assembly_pairMask_measurable (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ)
{f g : V → ℂ} (hf : Measurable f) (hg : Measurable g) :
Measurable (fun z : V×V => (f z.1*conj (g z.2))*
(Attempt4PrefixShell.pairMaskProductND
(assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c)
(assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1
((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by
exact ((hf.comp measurable_fst).mul (Complex.continuous_conj.measurable.comp (hg.comp measurable_snd))).mul
(assembly_pairMask_radial_measurable e₁ e₂ c)
end GraphLiteralSupport.CurrentFalconer.TreeND
end
end
end OAI
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