File size: 3,607 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 66 67 68 69 70 71 72 73 74 75 76 77 | 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)
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
end OAI
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