File size: 3,210 Bytes
d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Graphs.ActualDistanceDegrees
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
noncomputable def graphSupportedDistanceIndices (I : Finset GraphNonedgeIndex) (θ axis : ℝ) :
Finset GraphNonedgeIndex := by
classical
exact I.filter (fun e => (tsupport (graphSourceWeight true e.2.2 axis
(angularBinCount θ) e.2.1.1 e.2.1.2)).Nonempty)
theorem graphOriginalDistanceSum_supported (I : Finset GraphNonedgeIndex)
(ρ : Fin 4 → Measure Plane) (Q : Fin 4 → DyadicSquare) (M : Fin 4 → ℝ)
(β : Fin 4 → Plane → UnitCircle → ℂ) (θ axis v : ℝ) (Ψ : ℝ → ℝ) :
(∑ e ∈ I, graphOriginalTerm ρ (fun l => (graphQuadCells Q e.1 l).carrier) M β
true e.2.2 axis θ v e.2.1.1 e.2.1.2 Ψ) =
∑ e ∈ graphSupportedDistanceIndices I θ axis,
graphOriginalTerm ρ (fun l => (graphQuadCells Q e.1 l).carrier) M β
true e.2.2 axis θ v e.2.1.1 e.2.1.2 Ψ := by
classical
symm
apply Finset.sum_subset (Finset.filter_subset _ _)
intro e he hn
apply graphOriginalTerm_zero_of_empty_weight_support
intro hs
exact hn (Finset.mem_filter.mpr ⟨he,hs⟩)
theorem activeDistanceMatrixNeighbors_eq_supported (I : Finset GraphNonedgeIndex)
(p j b : ℕ) (θ axis : ℝ) (plus reverse : Bool) (row : CellPath × CellPath) :
activeDistanceMatrixNeighbors I p j b θ axis plus reverse row =
distanceMatrixNeighbors (graphSupportedDistanceIndices I θ axis) p j b plus reverse row := by
classical
ext col
cases reverse <;> by_cases hs : (tsupport (graphSourceWeight true plus axis (angularBinCount θ) j b)).Nonempty <;>
simp [activeDistanceMatrixNeighbors,distanceMatrixNeighbors,graphSupportedDistanceIndices,distanceMatrixIndex,hs]
theorem sourceRetainedDistance_original_sum (Q₁ Q₂ : DyadicSquare) (A B : CellPath)
(N a n : ℕ) (sig t θ v : ℝ) (hθ : θ = graphUncappedAngularScale N sig t)
(ρ ν : Measure Plane) (M₁ M₂ : ℝ) (β γ : Plane → UnitCircle → ℂ) (Ψ : ℝ → ℝ)
(rep : (ℕ × ℕ) × Bool → UnitCircle × UnitCircle) :
let axis := sourceParentAxis (Q₁.descendant A) (Q₂.descendant B)
let I := graphRetainedIndices
(sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B)
N sig (a+n) true ![Q₁,Q₂,Q₂,Q₁] rep
let H := graphSupportedDistanceIndices
(sourceRetainedDistanceIndices Q₁ Q₂ ρ ν β γ A B a n N sig t θ rep) θ axis
let F := fun e : GraphNonedgeIndex => graphOriginalTerm ![ρ,ν,ν,ρ]
(fun l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier) ![M₁,M₂,M₂,M₁] ![β,γ,γ,β]
true e.2.2 axis θ v e.2.1.1 e.2.1.2 Ψ
(∑ e ∈ I, F e) = ∑ e ∈ H, F e := by
classical
dsimp only
let I := graphRetainedIndices
(sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B)
N sig ((a:ℝ)+n) true ![Q₁,Q₂,Q₂,Q₁] rep
have he := graphOriginalDistanceSum_eligible Q₁ Q₂ A B N n sig t θ v hθ ρ ν M₁ M₂ β γ Ψ I
(Finset.filter_subset _ _)
exact he.trans (graphOriginalDistanceSum_supported _ _ _ _ _ _ _ _ _)
end PlanarFalconer
end OAI
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