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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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.InteriorPolicy
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Estimates.FarChildMaximum
import OAI.MeasureTheory.FalconerCompletion.Graphs.ResidualAngular
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 MeasureTheory Filter PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar
open scoped ENNReal Topology Classical
variable {n : ℕ}
lemma policyDescriptor_anchor_gt_start (A : ℕ → ℝ) {N q a t : ℕ}
(hq : 0<q) {b : PolicyDescriptor} (hb : b∈policyDescriptorList A N q a t) :
a<descriptorAnchor b := by
obtain ⟨z,hz,hb⟩ := Finset.mem_biUnion.mp hb
obtain ⟨hz,ha,ht,hwide⟩ := Finset.mem_filter.mp hz
have huN : z.2≤N := Nat.le_of_lt_succ (Finset.mem_range.mp (Finset.mem_product.mp hz).2)
rw [originDescriptor_anchor A hb]
have hp := (policyDepth_interior A hq (policyBlocks_nonempty hq huN hwide)).1
omega
lemma policyDescriptorList_subset_unreached
(A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q)
(ha : a₀≤a) (hab : a<b) (ht : t≤t₀) :
policyDescriptorList A N q b t ⊆ unreachedDescriptorList A N q a₀ t₀ a := by
intro e he
exact Finset.mem_filter.mpr ⟨policyDescriptorList_mono A (ha.trans hab.le) ht he,
hab.trans (policyDescriptor_anchor_gt_start A hq he)⟩
lemma actualResidualAmbientGood_subset_farChild
(A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q)
(ha : a₀≤a) (hab : a<b) (ht : t≤t₀)
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ)
(sig C : ℝ) (J i : ℕ) (x : EuclideanSpace ℝ (Fin (n+1))) :
actualResidualAmbientGood A N q a₀ t₀ a ρ m sig C J i x ⊆
actualPolicyAmbientGood A N q b t ρ m sig C J i x := by
intro w hw
rcases hw with hw | hw
· exact Or.inl hw
· exact Or.inr (fun e he => hw e (policyDescriptorList_subset_unreached A hq ha hab ht he))
lemma actualResidualPairAmbientGood_subset_farChild
(A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q)
(ha : a₀≤a) (hab : a<b) (ht : t≤t₀)
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ)
(sig C : ℝ) (J i : ℕ) (z : EuclideanSpace ℝ (Fin (n+1))×EuclideanSpace ℝ (Fin (n+1))) :
actualResidualPairAmbientGood A N q a₀ t₀ a ρ m sig C J i z ⊆
actualPairAmbientGood A N q b t ρ m sig C J i z := by
intro w hw
exact ⟨actualResidualAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i z.1 hw.1,
actualResidualAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i z.2 hw.2⟩
lemma positiveDistanceCountND_mono
{V : Type*} [PseudoMetricSpace V] [MeasurableSpace V]
(μ ν : Measure (V×V)) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(hμ : μ≤ν) (d : ℕ) (v a M : ℝ) :
positiveDistanceCountND μ d v a M ≤ positiveDistanceCountND ν d v a M := by
exact mul_le_mul' le_rfl ((Measure.prod_mono hμ hμ) _)
lemma farChildMaximumND_mono
(ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
(a : ℕ) (P Q : Finset ((Fin (n+1) → ℤ)×(Fin (n+1) → ℤ))) (v M : ℝ)
(U V U' V' : (EuclideanSpace ℝ (Fin (n+1))×EuclideanSpace ℝ (Fin (n+1))) → Set (EuclideanSpace ℝ (Fin (n+1))))
(hU : ∀ z, U z⊆U' z) (hV : ∀ z, V z⊆V' z) :
farChildMaximumND ρ κ a P Q v M U V ≤ farChildMaximumND ρ κ a P Q v M U' V' := by
apply sup_le_sup
· apply Finset.sup_mono_fun
intro p hp
exact positiveDistanceCountND_mono
((gridPairMeasureND ρ a p).restrict {z | euclideanPairDirectionND z∈U z})
((gridPairMeasureND ρ a p).restrict {z | euclideanPairDirectionND z∈U' z})
(Measure.restrict_mono (fun z hz => hU z hz) le_rfl) _ _ _ _
· apply Finset.sup_mono_fun
intro p hp
exact positiveDistanceCountND_mono
((gridPairMeasureND κ a p).restrict {z | euclideanPairDirectionND z∈V z})
((gridPairMeasureND κ a p).restrict {z | euclideanPairDirectionND z∈V' z})
(Measure.restrict_mono (fun z hz => hV z hz) le_rfl) _ _ _ _
lemma actualResidual_farChildMaximum_le
(ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
(A : ℕ → ℝ) {N q a₀ t₀ a b t : ℕ} (hq : 0<q)
(ha : a₀≤a) (hab : a<b) (ht : t≤t₀)
(m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ)
(P Q : Finset ((Fin (n+1) → ℤ)×(Fin (n+1) → ℤ))) (v M : ℝ) :
farChildMaximumND ρ κ b P Q v M
(actualResidualPairAmbientGood A N q a₀ t₀ a ρ m sig C J i)
(actualResidualPairAmbientGood A N q a₀ t₀ a κ m sig C J i) ≤
farChildMaximumND ρ κ b P Q v M
(actualPairAmbientGood A N q b t ρ m sig C J i)
(actualPairAmbientGood A N q b t κ m sig C J i) :=
farChildMaximumND_mono ρ κ b P Q v M _ _ _ _
(actualResidualPairAmbientGood_subset_farChild A hq ha hab ht ρ m sig C J i)
(actualResidualPairAmbientGood_subset_farChild A hq ha hab ht κ m sig C J i)
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI
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