| import OAI.Analysis.FiniteFactor.Sampling.Fourier |
| import OAI.Analysis.FiniteFactor.OrbitSampling.Powers |
|
|
| namespace OAI |
|
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| noncomputable section |
|
|
| theorem sourceBlockSize_monotone : Monotone sourceBlockSize := by |
| apply monotone_nat_of_le_succ |
| intro s |
| have hp := (sourceStage (s+1)).previous_size |
| have hd : (sourceBlockSize s : ℝ) ≤ (sourceStageState (s+1)).d := by |
| change (sourceBlockSize s : ℝ) ≤ ((sourceStageState s).d+(sourceStage s).m : ℕ) |
| change ((sourceStage s).m : ℝ) ≤ ((sourceStageState s).d+(sourceStage s).m : ℕ) |
| push_cast |
| exact le_add_of_nonneg_left (Nat.cast_nonneg _) |
| have hm : (sourceBlockSize s : ℝ) ≤ sourceBlockSize (s+1) := by |
| change ((sourceStageState (s+1)).d : ℝ) ≤ (sourceBlockSize (s+1) : ℝ)/100 at hp |
| have := Nat.cast_nonneg (α := ℝ) (sourceBlockSize (s+1)) |
| linarith |
| exact_mod_cast hm |
|
|
| theorem sourceFrequencyScale_monotone : |
| Monotone (fun s => sourceBlockSize s * (sourceStageState s).K) := by |
| have hK : Monotone (fun s => (sourceStageState s).K) := by |
| apply monotone_nat_of_le_succ |
| intro s |
| have := sourceStage_cutoff_growth s |
| omega |
| intro s t hst |
| exact Nat.mul_le_mul (sourceBlockSize_monotone hst) (hK hst) |
|
|
| / |
| Fourier estimate; no independence between two phases is asserted. -/ |
| theorem sourceAngleLaw_pair_fourier_le (i : SourceCoordinate) (v v' : ℤ) (n n' : ℝ) |
| (hw : (v : ℝ)*n+(v' : ℝ)*n' ≠ 0) : |
| ‖∫ θ, Complex.exp (2 * Real.pi * Complex.I * |
| ((v : ℂ) * (Int.fract (n*θ i) : ℝ) + (v' : ℂ) * (Int.fract (n'*θ i) : ℝ))) |
| ∂sourceAngleLaw‖ ≤ |
| 1/(Real.pi*sourceAngleUnit i*|(v : ℝ)*n+(v' : ℝ)*n'|) := by |
| have hh := sourceAngle_pair_fourier_le i v v' n n' hw |
| simp_rw [pair_fourier_fract] at hh ⊢ |
| rw [← sourceAngleLaw_coordinate i, |
| integral_map (measurable_pi_apply i).aemeasurable (by fun_prop)] at hh |
| exact hh |
|
|
| theorem sourceSampling_pair_fourier_bound (s : ℕ) (hs : 200000 ≤ sourceBlockSize s) |
| (i : SourceCoordinate) (hi : i.1 ≤ s) (n n' : ℤ) |
| (hn : n ∈ sourceSamplingPowers s) (hn' : n' ∈ sourceSamplingPowers s) (hne : n ≠ n') |
| (v v' : ℤ) (hv : |v| ≤ (samplingH (sourceBlockSize s) : ℤ)) |
| (hv' : |v'| ≤ (samplingH (sourceBlockSize s) : ℤ)) (hvv' : v ≠ 0 ∨ v' ≠ 0) : |
| ‖∫ θ, Complex.exp (2 * Real.pi * Complex.I * |
| ((v : ℂ) * (Int.fract ((n : ℝ)*θ i) : ℝ) + |
| (v' : ℂ) * (Int.fract ((n' : ℝ)*θ i) : ℝ))) ∂sourceAngleLaw‖ ≤ |
| Real.pi⁻¹*(sourceBlockSize s : ℝ)^(-1000 : ℝ) := by |
| have hsep := (sourceSamplingPowers_spec s hs).2.2 hn hn' hne |
| have hfreq : (samplingR (sourceBlockSize s) (sourceStageState s).K : ℤ) < |v*n+v'*n'| := by |
| by_contra hh |
| exact hsep ⟨v,mem_Icc.mpr (abs_le.mp hv),v',mem_Icc.mpr (abs_le.mp hv'),hvv',le_of_not_gt hh⟩ |
| let w : ℝ := (v : ℝ)*(n : ℝ)+(v' : ℝ)*(n' : ℝ) |
| have hwR : (samplingR (sourceBlockSize s) (sourceStageState s).K : ℝ) < |w| := by |
| dsimp [w] |
| exact_mod_cast hfreq |
| have hwpos : 0 < |w| := lt_of_le_of_lt (Nat.cast_nonneg _) hwR |
| have hw : w ≠ 0 := abs_pos.mp hwpos |
| apply (sourceAngleLaw_pair_fourier_le i v v' (n : ℝ) (n' : ℝ) hw).trans |
| have hN : (sourceBlockSize i.1 : ℝ)*(sourceStageState i.1).K ≤ |
| (sourceBlockSize s : ℝ)*(sourceStageState s).K := by |
| exact_mod_cast sourceFrequencyScale_monotone hi |
| have hmpos := sourceBlockSize_pos s |
| have hNpos := mul_pos (sourceBlockSize_pos i.1) (sourceCutoff_pos i.1) |
| have hscale : (sourceBlockSize s : ℝ)^1000 ≤ sourceAngleUnit i * |w| := by |
| unfold samplingR at hwR |
| push_cast at hwR |
| unfold sourceAngleUnit |
| rw [one_div, inv_mul_eq_div, le_div_iff₀ hNpos] |
| have hh := mul_le_mul_of_nonneg_right hN (pow_nonneg hmpos.le 1000) |
| nlinarith |
| change 1/(Real.pi*sourceAngleUnit i*|w|) ≤ _ |
| calc |
| _ = Real.pi⁻¹*(1/(sourceAngleUnit i*|w|)) := by ring |
| _ ≤ Real.pi⁻¹*(1/(sourceBlockSize s : ℝ)^1000) := by |
| exact mul_le_mul_of_nonneg_left |
| (one_div_le_one_div_of_le (pow_pos hmpos 1000) hscale) (inv_nonneg.mpr Real.pi_pos.le) |
| _ = _ := by rw [Real.rpow_neg, Real.rpow_ofNat, one_div]; positivity |
|
|
| end |
| end FiniteFactor |
|
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| end OAI |
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