| import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalIntegral |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
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| / |
| theorem samplingTestedFirstDensity_whole (b : Bool) (h α s : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) : |
| samplingTestedFirstDensity b h α (Set.Icc 0 1) (Set.Icc 0 1) s = |
| if b then 1-s else if α ≤ samplingDistance s then 1 else 0 := by |
| have he : samplingTestedFirstDensity b h α (Set.Icc 0 1) (Set.Icc 0 1) s = |
| ∫ t, samplingWeight b h α s t ∂unitIntervalMeasure := by |
| apply integral_congr_ae |
| filter_upwards [] with t |
| have hs' : s ∈ Set.Icc (0 : ℝ) 1 := ⟨hs.1,hs.2.le⟩ |
| have ht' : Int.fract (s+t) ∈ Set.Icc (0 : ℝ) 1 := ⟨Int.fract_nonneg _,(Int.fract_lt_one _).le⟩ |
| simp only [samplingTestedFirstWeight,hs',ht',and_self,ite_true] |
| exact he.trans (samplingWeight_first_integral b h α s hh hh1 hα hs) |
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| / |
| theorem sampling_diagonal_whole_mass (h α : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2) : |
| (∫ s, samplingTestedFirstDensity false h α (Set.Icc 0 1) (Set.Icc 0 1) s ∂unitIntervalMeasure) = 1-2*α := by |
| have he : samplingTestedFirstDensity false h α (Set.Icc 0 1) (Set.Icc 0 1) =ᵐ[unitIntervalMeasure] |
| (Set.Icc α (1-α)).indicator (fun _ => (1 : ℝ)) := by |
| filter_upwards [ae_restrict_mem measurableSet_Ico] with s hs |
| rw [samplingTestedFirstDensity_whole false h α s hh hh1 hα hs] |
| have hc : α ≤ samplingDistance s ↔ s ∈ Set.Icc α (1-α) := by |
| simp only [samplingDistance,le_min_iff,Set.mem_Icc] |
| constructor <;> intro h <;> constructor <;> linarith [h.1,h.2] |
| simp only [Bool.false_eq_true,ite_false,hc,Set.indicator_apply] |
| rw [integral_congr_ae he,integral_indicator measurableSet_Icc,integral_const] |
| have hsub : Set.Icc α (1-α) ⊆ Set.Ico (0 : ℝ) 1 := by |
| intro s hs |
| exact ⟨hα.le.trans hs.1,hs.2.trans_lt (by linarith)⟩ |
| simp only [measureReal_def,Measure.restrict_apply MeasurableSet.univ,Set.univ_inter, |
| smul_eq_mul,mul_one] |
| unfold unitIntervalMeasure |
| rw [Measure.restrict_apply measurableSet_Icc,Set.inter_eq_left.mpr hsub,Real.volume_Icc, |
| ENNReal.toReal_ofReal (by linarith)] |
| ring |
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| / |
| theorem sourceSampling_cutoff_budget_tendsto : |
| Tendsto (fun stage => 2*((sourceSamplingPrefix stage).card : ℝ)*sourceSamplingCutoff stage) |
| atTop (𝓝 0) := by |
| have ht : Tendsto (fun stage => (101/50 : ℝ)*(sourceBlockSize stage : ℝ)⁻¹) atTop (𝓝 0) := by |
| simpa using (tendsto_inv_atTop_zero.comp |
| (tendsto_natCast_atTop_atTop.comp sourceBlockSize_tendsto)).const_mul (101/50 : ℝ) |
| apply squeeze_zero (fun stage => by have := (sourceSampling_parameters stage).2.2; positivity) _ ht |
| intro stage |
| have hm := sourceBlockSize_pos stage |
| calc |
| _ ≤ 2*((101/100 : ℝ)*sourceBlockSize stage)*sourceSamplingCutoff stage := by |
| exact mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left (sourceSamplingPrefix_card_bound stage) (by norm_num)) |
| (sourceSampling_parameters stage).2.2.le |
| _ = (101/50 : ℝ)*(sourceBlockSize stage : ℝ)⁻¹ := by |
| simp only [sourceSamplingCutoff,Real.rpow_neg hm.le,Real.rpow_two] |
| field_simp |
| ring |
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| / |
| The estimate uses its actual varying cardinality, bounded by the prefix. -/ |
| theorem sourceSampling_untested_mass_tendsto (k : ℕ → ℕ) |
| (hk : ∀ stage, k stage ≤ (sourceSamplingPrefix stage).card) : |
| Tendsto (fun stage => (1-2*sourceSamplingCutoff stage)^(k stage)) atTop (𝓝 1) := by |
| have he : ∀ stage, |(1-2*sourceSamplingCutoff stage)^(k stage)-1| ≤ |
| 2*((sourceSamplingPrefix stage).card : ℝ)*sourceSamplingCutoff stage := by |
| intro stage |
| have hα := (sourceSampling_parameters stage).2.2 |
| have hα1 := sourceSamplingCutoff_le_half stage |
| have hlo : 0 ≤ 1-2*sourceSamplingCutoff stage := by linarith |
| have hup : (1-2*sourceSamplingCutoff stage)^(k stage) ≤ 1 := pow_le_one₀ hlo (by linarith) |
| have hb := one_add_mul_le_pow (show (-2 : ℝ) ≤ -(2*sourceSamplingCutoff stage) by linarith) (k stage) |
| have hk' : (k stage : ℝ) ≤ (sourceSamplingPrefix stage).card := by exact_mod_cast hk stage |
| rw [abs_of_nonpos (by linarith)] |
| have hm := mul_le_mul_of_nonneg_right hk' (show 0 ≤ 2*sourceSamplingCutoff stage by positivity) |
| simp only [mul_neg,← sub_eq_add_neg] at hb |
| nlinarith |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| simp only [Real.norm_eq_abs] |
| exact squeeze_zero (fun stage => abs_nonneg _) he sourceSampling_cutoff_budget_tendsto |
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| end |
| end FiniteFactor |
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| end OAI |
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