| import OAI.MeasureTheory.Falconer.Geometry.FarNormalization | |
| namespace OAI | |
| /-! # Exact conversion from positive collision to far-shell normalization -/ | |
| namespace Falconer | |
| theorem profile_far_collision_normalization {N : ℕ} (hN : 0 < N) | |
| (d : ℕ) (m : ℝ → ℝ) (a b : ℕ) (v lam : ℝ) : | |
| ((2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N a)⁻¹^4* | |
| (2:ℝ)^(-((d:ℝ)-1)*(v-b)) / | |
| ((2:ℝ)^(v-b)*(profileModelMass m N b)⁻¹^4)) * | |
| ((2:ℝ)^N)^(2*(m ((b:ℝ)/N)-m ((a:ℝ)/N))+4*((d:ℝ)+1)*lam) = | |
| ((2:ℝ)^N)^(-2*((m ((b:ℝ)/N)-((d:ℝ)/2)*(b:ℝ)/N)- | |
| (m ((a:ℝ)/N)-((d:ℝ)/2)*(a:ℝ)/N))+4*((d:ℝ)+1)*lam) := by | |
| have ha : profileModelMass m N a ≠ 0 := by unfold profileModelMass; positivity | |
| have hb : profileModelMass m N b ≠ 0 := by unfold profileModelMass; positivity | |
| have hv : (2:ℝ)^(v-b) ≠ 0 := by positivity | |
| have hi : ((2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N a)⁻¹^4* | |
| (2:ℝ)^(-((d:ℝ)-1)*(v-b)) / | |
| ((2:ℝ)^(v-b)*(profileModelMass m N b)⁻¹^4)) = | |
| (2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N b/profileModelMass m N a)^4* | |
| (2:ℝ)^(-((d:ℝ)-1)*(v-b))*(2:ℝ)^(-(v-b)) := by | |
| rw [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)] | |
| field_simp | |
| rw [hi] | |
| exact profile_far_prefactor hN d m a b v lam | |
| end Falconer | |
| end OAI | |