| import OAI.MeasureTheory.Falconer.Graphs.GraphAmplitudeDerivatives |
| import OAI.MeasureTheory.Falconer.Geometry.InactiveBinScales |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| noncomputable def graphNonedgeDerivativeScale (N : ℕ) (sig u : ℝ) : ℝ := |
| (2:ℝ)^u * ((2:ℝ)^N)^(4*sig) |
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|
| theorem graphNonedgeDerivativeScale_one_le (N : ℕ) {sig u : ℝ} |
| (hsig : 0 ≤ sig) (hu : 0 ≤ u) : 1 ≤ graphNonedgeDerivativeScale N sig u := by |
| apply one_le_mul_of_one_le_of_one_le |
| · exact Real.one_le_rpow (by norm_num) hu |
| · exact Real.one_le_rpow (one_le_pow₀ (by norm_num)) (by positivity) |
|
|
| theorem source_symbol_le_nonedge_scale (N L : ℕ) {sig u : ℝ} |
| (hsig : 0 ≤ sig) (hu : 0 ≤ u) (hL : (L:ℝ) ≤ u+2*sig*(N:ℝ)) : |
| sourceSymbolScale N sig L ≤ graphNonedgeDerivativeScale N sig u := by |
| have hN : (0:ℝ) ≤ N := Nat.cast_nonneg _ |
| unfold sourceSymbolScale graphNonedgeDerivativeScale |
| rw [mul_max_of_nonneg _ _ (by positivity)] |
| apply max_le |
| · simp only [mul_one,dyadic_power_real,← Real.rpow_add (by norm_num : (0:ℝ)<2)] |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| nlinarith only [hu, mul_nonneg hsig hN] |
| · rw [← Real.rpow_natCast (2:ℝ) L] |
| simp only [dyadic_power_real,← Real.rpow_add (by norm_num : (0:ℝ)<2)] |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| nlinarith [mul_nonneg hsig hN] |
|
|
| theorem graph_bin_le_nonedge_scale (N : ℕ) {sig t u c₀ : ℝ} |
| (hsig : 0 ≤ sig) (hu : 0 ≤ u) (_hc : 0 < c₀) |
| (ht : t ≤ u+2*sig*(N:ℝ)) : |
| max 1 (graphAngularScale N sig t c₀)⁻¹ ≤ |
| max 1 c₀⁻¹ * graphNonedgeDerivativeScale N sig u := by |
| have hE := graphNonedgeDerivativeScale_one_le N hsig hu |
| have hN : (0:ℝ) ≤ N := Nat.cast_nonneg _ |
| have hunc : (graphUncappedAngularScale N sig t)⁻¹ ≤ graphNonedgeDerivativeScale N sig u := by |
| simp only [graphUncappedAngularScale,graphNonedgeDerivativeScale,dyadic_power_real, |
| ← Real.rpow_add (by norm_num : (0:ℝ)<2),← Real.rpow_neg_one, |
| ← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| nlinarith [mul_nonneg hsig hN] |
| have hM : 1 ≤ max 1 c₀⁻¹ := le_max_left _ _ |
| have hEM : graphNonedgeDerivativeScale N sig u ≤ max 1 c₀⁻¹ * graphNonedgeDerivativeScale N sig u := |
| le_mul_of_one_le_left (zero_le_one.trans hE) hM |
| apply max_le (hE.trans hEM) |
| unfold graphAngularScale |
| by_cases hcap : c₀ ≤ graphUncappedAngularScale N sig t |
| · rw [min_eq_left hcap] |
| exact (le_max_right 1 c₀⁻¹).trans (le_mul_of_one_le_right (zero_le_one.trans hM) hE) |
| · rw [min_eq_right (le_of_not_ge hcap)] |
| exact hunc.trans hEM |
|
|
| theorem graph_total_amplitude_scale (N L : ℕ) {sig t u c₀ : ℝ} |
| (hsig : 0 ≤ sig) (hu : 0 ≤ u) (hc : 0 < c₀) |
| (ht : t ≤ u+2*sig*(N:ℝ)) (hL : (L:ℝ) ≤ u+2*sig*(N:ℝ)) : |
| 1+4*max 1 (graphAngularScale N sig t c₀)⁻¹+sourceSymbolScale N sig L ≤ |
| (2+4*max 1 c₀⁻¹)*graphNonedgeDerivativeScale N sig u := by |
| have hE := graphNonedgeDerivativeScale_one_le N hsig hu |
| have hB := graph_bin_le_nonedge_scale N hsig hu hc ht |
| have hD := source_symbol_le_nonedge_scale N L hsig hu hL |
| nlinarith |
|
|
| theorem graph_nonedge_scale_gain (N : ℕ) (sig v p : ℝ) : |
| graphNonedgeDerivativeScale N sig (v-p) / |
| ((2:ℝ)^v*((2:ℝ)^(-p)*((2:ℝ)^N)^(30*sig))) = ((2:ℝ)^N)^(-26*sig) := by |
| simp only [graphNonedgeDerivativeScale,dyadic_power_real, |
| ← Real.rpow_add (by norm_num : (0:ℝ)<2),← Real.rpow_sub (by norm_num : (0:ℝ)<2)] |
| congr 1 |
| ring |
|
|
| end PlanarFalconer |
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|
| end OAI |
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