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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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Shells.TransportedAngularField
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
open Set Metric MeasureTheory PlanarFalconer
open scoped ContDiff
variable {d : ℕ}
lemma tensorCapND_compact {δ : ℝ} (hδ : 0 < δ) (a : Fin d → ℤ) :
HasCompactSupport (tensorCapND δ a) := by
apply (isCompact_closedBall (capGridPointND δ a) ((d+1:ℝ)*δ)).of_isClosed_subset
isClosed_closure
exact fun w hw => tensorCapND_tsupport_dist hδ a hw
lemma bufferedMaskND_compact {δ : ℝ} (hδ : 0 < δ)
(K : Set (EuclideanSpace ℝ (Fin d))) : HasCompactSupport (bufferedMaskND δ K) := by
have h : HasCompactSupport (∑ a ∈ maskCapSelectionND δ K, tensorCapND δ a) :=
HasCompactSupport.finset_sum (fun a _ => tensorCapND_compact hδ a)
have he : (∑ a ∈ maskCapSelectionND δ K, tensorCapND δ a) = bufferedMaskND δ K := by
funext w
simp only [Finset.sum_apply,bufferedMaskND,selectedCapND]
rw [he] at h
exact h
lemma evenBufferedMaskND_compact {δ : ℝ} (hδ : 0 < δ)
(K : Set (EuclideanSpace ℝ (Fin d))) : HasCompactSupport (evenBufferedMaskND δ K) := by
have h := bufferedMaskND_compact hδ K
have hn : HasCompactSupport (fun w => bufferedMaskND δ K (-w)) :=
h.comp_homeomorph (Homeomorph.neg _)
exact (h.add hn).mul_left
namespace TreeND
lemma actualDescriptorSmooth_compact (ρ : Measure (EuclideanSpace ℝ (Fin d)))
(m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d)
(b : PolicyDescriptor) : HasCompactSupport (actualDescriptorSmooth ρ m N sig C J i P b).1 := by
exact (evenBufferedMaskND_compact (sourceMaskRadiusND_pos _ _ _ _)
(descriptorGood ρ m N sig C J i P b)).comp_left (by simp : Complex.ofReal 0=0)
lemma actualEndpointAngular_global_bound
(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) :
∃ B : ℝ, 0 ≤ B ∧ ∀ x w, ‖(actualEndpointAngular L ρ m N sig C J i b x).1 w‖ ≤ B := by
classical
let D := sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig
let f := fun P : CellPath d => normalizedAngularWordND D L
(actualDescriptorSmooth ρ m N sig C J i P b)
have hf : ∀ P, ∃ B : ℝ, 0 ≤ B ∧ ∀ w, ‖(f P).1 w‖ ≤ B := by
intro P
have hc : HasCompactSupport (f P).1 :=
(actualDescriptorSmooth_compact ρ m N sig C J i P b).of_isClosed_subset
isClosed_closure (normalizedAngularWordND_support _ _ _)
obtain ⟨B,hB⟩ := hc.exists_bound_of_continuous (smoothFunctionND_contDiff (f P)).continuous
exact ⟨max 0 B,le_max_left _ _,fun w => (hB w).trans (le_max_right _ _)⟩
choose B hB hbound using hf
refine ⟨∑ P ∈ dyadicPaths (descriptorAnchor b), B P,
Finset.sum_nonneg (fun P _ => hB P),?_⟩
intro x w
by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P
· obtain ⟨P,hP,hx⟩ := hx
have he : actualEndpointSmooth ρ m N sig C J i b x=
actualDescriptorSmooth ρ m N sig C J i P b := by
apply Subtype.ext
exact congrArg (fun g => fun u => (g u:ℂ)) (actualEndpointMask_of_mem ρ m N sig C J i b hP hx)
unfold actualEndpointAngular
rw [he]
exact (hbound P w).trans (Finset.single_le_sum (fun Q _ => hB Q) hP)
· have he : actualEndpointSmooth ρ m N sig C J i b x=0 := by
apply Subtype.ext
have hzero := actualEndpointMask_of_absent ρ m N sig C J i b
(by simpa only [not_exists,not_and] using hx)
funext u
change (actualEndpointMask ρ m N sig C J i b x u:ℂ)=0
rw [hzero]
simp
simp only [actualEndpointAngular,he,normalizedAngularWordND,map_zero,smul_zero]
change ‖(0:ℂ)‖ ≤ _
rw [norm_zero]
exact Finset.sum_nonneg (fun P _ => hB P)
lemma actualAngularProduct_global_bound {ι : Type*} (S : Finset ι)
(b : ι → PolicyDescriptor)
(L : ι → 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 : ℝ, 0 ≤ B ∧ ∀ x w, ‖(actualAngularProduct S b L ρ m N sig C J i x).1 w‖ ≤ B := by
classical
choose B hB hb using (fun a => actualEndpointAngular_global_bound (L a) ρ m N sig C J i (b a))
refine ⟨∏ a ∈ S, B a,Finset.prod_nonneg (fun a _ => hB a),?_⟩
intro x w
change ‖∏ a ∈ S, (actualEndpointAngular (L a) ρ m N sig C J i (b a) x).1 w‖ ≤ _
rw [norm_prod]
exact Finset.prod_le_prod₀ (fun _ _ => norm_nonneg _) (fun a _ => hb a x w)
namespace PositiveDistanceState
variable {n : ℕ} (e : PositiveDistanceState (n+1))
lemma prefixInverse_global_bound (p : PrefixAssignment n) :
∃ B : ℝ, 0 ≤ B ∧ ∀ x w, ‖e.prefixInverseLeft p x w‖ ≤ B ∧
‖e.prefixInverseRight p x w‖ ≤ B := by
obtain ⟨B,hB,hleft⟩ := actualAngularProduct_global_bound e.prefixDescriptors id (prefixLeftWords p)
e.ρ e.mass e.N e.sig e.C e.J e.i
obtain ⟨C,hC,hright⟩ := actualAngularProduct_global_bound e.prefixDescriptors id (prefixRightWords p)
e.ρ e.mass e.N e.sig e.C e.J e.i
refine ⟨max B C,hB.trans (le_max_left _ _),?_⟩
intro x w
rw [e.prefixInverseLeft_eq,e.prefixInverseRight_eq]
exact ⟨(hleft x w).trans (le_max_left _ _),(hright x w).trans (le_max_right _ _)⟩
end PositiveDistanceState
end TreeND
end CurrentFalconer
end GraphLiteralSupport
end
end
end OAI
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