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import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Angular.AngularWordOperatorBound
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
open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical
variable {d : ℕ}
noncomputable def actualDescriptorSmooth (ρ : Measure (EuclideanSpace ℝ (Fin d)))
(m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) (b : PolicyDescriptor) :
SmoothFunctionND (EuclideanSpace ℝ (Fin d)) :=
⟨fun w => (actualDescriptorMask ρ m N sig C J i P b w : ℂ),
Complex.ofRealCLM.contDiff.comp (actualDescriptorMask_smooth ρ m N sig C J i P b)⟩
lemma actualEndpointSmooth_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) :
actualEndpointSmooth ρ m N sig C J i b x=actualDescriptorSmooth ρ m N sig C J i P b := by
apply Subtype.ext
funext w
exact congrArg Complex.ofReal (congrFun (actualEndpointMask_of_mem ρ m N sig C J i b hP hx) w)
lemma actualEndpointSmooth_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) :
actualEndpointSmooth ρ m N sig C J i b x=0 := by
apply Subtype.ext
funext w
change (actualEndpointMask ρ m N sig C J i b x w : ℂ)=0
rw [actualEndpointMask_of_absent ρ m N sig C J i b hx]
rfl
lemma actualEndpointAngular_partition
(L : 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) (b : PolicyDescriptor) (x : EuclideanSpace ℝ (Fin d)) :
actualEndpointAngular L ρ m N sig C J i b x=
∑ P ∈ dyadicPaths (descriptorAnchor b),
if x ∈ treeCell P then
normalizedAngularWordND (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig) L
(actualDescriptorSmooth ρ m N sig C J i P b) else 0 := by
classical
by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P
· obtain ⟨P,hP,hx⟩ := hx
unfold actualEndpointAngular
rw [actualEndpointSmooth_of_mem ρ m N sig C J i b hP hx,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
· have hx' : ∀ P ∈ dyadicPaths (descriptorAnchor b), x ∉ treeCell P := by
simpa only [not_exists,not_and] using hx
unfold actualEndpointAngular normalizedAngularWordND
rw [actualEndpointSmooth_of_absent ρ m N sig C J i b hx',map_zero,smul_zero]
exact (Finset.sum_eq_zero fun P hP => ite_eq_right (hx' P hP)).symm
lemma actualEndpointAngular_measurable
(L : 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) (b : PolicyDescriptor) :
Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) =>
(actualEndpointAngular L ρ m N sig C J i b z.1).1 z.2) := by
classical
have he : (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) =>
(actualEndpointAngular L ρ m N sig C J i b z.1).1 z.2)=
fun z => ∑ P ∈ dyadicPaths (descriptorAnchor b), if z.1 ∈ treeCell P then
(normalizedAngularWordND (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig) L
(actualDescriptorSmooth ρ m N sig C J i P b)).1 z.2 else 0 := by
funext z
rw [actualEndpointAngular_partition]
simp only [Submodule.coe_sum,Finset.sum_apply]
apply Finset.sum_congr rfl
intro P _hP
split_ifs <;> rfl
rw [he]
apply Finset.measurable_sum
intro P _hP
exact Measurable.ite ((measurableSet_treeCell P).preimage measurable_fst)
((smoothFunctionND_contDiff _).continuous.measurable.comp measurable_snd) measurable_const
lemma actualEndpointSmooth_even (ρ : Measure (EuclideanSpace ℝ (Fin d)))
(m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor)
(x : EuclideanSpace ℝ (Fin d)) : Function.Even (actualEndpointSmooth ρ m N sig C J i b x).1 := by
classical
intro w
change (actualEndpointMask ρ m N sig C J i b x (-w):ℂ)=_
unfold actualEndpointMask
congr 1
apply Finset.sum_congr rfl
intro P _hP
split_ifs
· exact actualDescriptorMask_even ρ m N sig C J i P b w
· rfl
lemma actualEndpointAngular_even
(L : 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) (b : PolicyDescriptor) (x : EuclideanSpace ℝ (Fin d)) :
Function.Even (actualEndpointAngular L ρ m N sig C J i b x).1 :=
normalizedAngularWordND_even _ L _ (actualEndpointSmooth_even ρ m N sig C J i b x)
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI
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