| import OAI.MeasureTheory.Falconer.Graphs.ActualDistanceDegrees |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set MeasureTheory |
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| 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) |
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| 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⟩) |
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| 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] |
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| 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 _ _ _ _ _ _ _ _ _) |
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| end PlanarFalconer |
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| end OAI |
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