| import OAI.MeasureTheory.Falconer.Estimates.SelectedDistanceAssembly | |
| import OAI.MeasureTheory.Falconer.Graphs.GraphOriginalNonedgePower | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate | |
| theorem distanceBinIntegrand_cross (ρ ν : Measure Plane) (X Y X' Y' : Set Plane) | |
| (M₁ M₂ : ℝ) (b₁ b₂ : Plane → UnitCircle → ℂ) (axis : ℝ) (n j k : ℕ) | |
| (r : ℝ) (u w : UnitCircle) : | |
| distanceBinIntegrand ρ ν X Y M₁ M₂ b₁ b₂ axis n j (r,u) * | |
| conj (distanceBinIntegrand ρ ν X' Y' M₁ M₂ b₁ b₂ axis n k (r,w)) = | |
| sourceGraphFourTransform ![ρ,ν,ν,ρ] ![X,Y,Y',X'] ![M₁,M₂,M₂,M₁] ![b₁,b₂,b₂,b₁] | |
| (graphSourceWeight true true axis n j k) r u w := by | |
| simp only [sourceGraphFourTransform,graphSourceWeight,↓reduceIte,graphRotatedDistanceWeight_eq, | |
| Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_two,Matrix.cons_val_three, | |
| Matrix.head_cons,Matrix.tail_cons,distanceBinIntegrand,map_mul,starRingEnd_self_apply] | |
| ring | |
| theorem graphOriginalTerm_profile_normalization | |
| (ρ : Fin 4 → Measure Plane) (X : Fin 4 → Set Plane) (Mp Ma : Fin 4 → ℝ) | |
| (hM : ∀ i, Mp i ≠ 0) (b : Fin 4 → Plane → UnitCircle → ℂ) | |
| (distance plus : Bool) (axis δ v : ℝ) (j k : ℕ) (Ψ : ℝ → ℝ) : | |
| graphOriginalTerm ρ X Ma b distance plus axis δ v j k Ψ = | |
| ((Mp 0/Ma 0)*(Mp 1/Ma 1)*(Mp 2/Ma 2)*(Mp 3/Ma 3)) • | |
| graphOriginalTerm ρ X Mp b distance plus axis δ v j k Ψ := by | |
| unfold graphOriginalTerm | |
| simp_rw [sourceGraphFourTransform_profile_normalization ρ X Mp Ma hM b, | |
| smul_comm (Ψ _) ((Mp 0/Ma 0)*(Mp 1/Ma 1)*(Mp 2/Ma 2)*(Mp 3/Ma 3)), integral_smul] | |
| exact smul_comm _ _ _ | |
| end PlanarFalconer | |
| end OAI | |