| import OAI.Analysis.FiniteFactor.Sampling.Weights |
|
|
| namespace OAI |
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|
| namespace FiniteFactor |
| open MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
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|
| theorem unitInterval_split (c : ℝ) (hc : 0 ≤ c) (hc1 : c ≤ 1) : |
| unitIntervalMeasure = volume.restrict (Set.Ioc (0 : ℝ) c) + volume.restrict (Set.Ioc c 1) := by |
| unfold unitIntervalMeasure |
| rw [restrict_Ico_eq_restrict_Ioc,← Set.Ioc_union_Ioc_eq_Ioc hc hc1, |
| Measure.restrict_union (Set.Ioc_disjoint_Ioc_of_le le_rfl) measurableSet_Ioc] |
|
|
| theorem map_interval_affine (a b c d : ℝ) (hc : 0 < c) : |
| (volume.restrict (Set.Ioc a b)).map (fun x => c*x+d) = |
| ENNReal.ofReal c⁻¹ • volume.restrict (Set.Ioc (c*a+d) (c*b+d)) := by |
| have hp : (fun x : ℝ => c*x+d) ⁻¹' Set.Ioc (c*a+d) (c*b+d) = Set.Ioc a b := by |
| ext x |
| simp only [Set.mem_preimage,Set.mem_Ioc,add_lt_add_iff_right,add_le_add_iff_right, |
| mul_lt_mul_iff_right₀ hc,mul_le_mul_iff_right₀ hc] |
| have hmap : volume.map (fun x : ℝ => c*x+d) = ENNReal.ofReal c⁻¹ • volume := by |
| have he : (fun x : ℝ => c*x+d) = (fun x : ℝ => x+d) ∘ (fun x : ℝ => c*x) := rfl |
| rw [he,← Measure.map_map (by fun_prop) (by fun_prop),Real.map_volume_mul_left hc.ne', |
| Measure.map_smul _ (by fun_prop), |
| (measurePreserving_add_right (volume : Measure ℝ) d).map_eq,abs_of_pos (inv_pos.mpr hc)] |
| rw [← hp,← Measure.restrict_map (f := fun x : ℝ => c*x+d) (by fun_prop) measurableSet_Ioc, |
| hmap,Measure.restrict_smul] |
|
|
| /-- The uncut diagonal model moves z by v times its endpoint distance. -/ |
| def samplingTransport (v z : ℝ) : ℝ := z+v*samplingDistance z |
|
|
| /-- Exact density of the two affine branches of the uncut transport. -/ |
| def samplingTransportDensity (v s : ℝ) : ℝ := |
| if s ≤ (1+v)/2 then 1/(1+v) else 1/(1-v) |
|
|
| theorem samplingTransportDensity_measurable : |
| Measurable (fun p : ℝ × ℝ => samplingTransportDensity p.1 p.2) := by |
| unfold samplingTransportDensity |
| exact Measurable.ite (measurableSet_le measurable_snd ((measurable_const.add measurable_fst).div_const 2)) |
| (by fun_prop) (by fun_prop) |
|
|
| /-- This is the actual push-forward density, not an assumed ideal moment. -/ |
| theorem samplingTransport_density (v : ℝ) (hv : 0 ≤ v) (hv1 : v < 1) : |
| unitIntervalMeasure.map (samplingTransport v) = |
| unitIntervalMeasure.withDensity (fun s => ENNReal.ofReal (samplingTransportDensity v s)) := by |
| let c := (1+v)/2 |
| have hc : 0 ≤ c := by dsimp [c]; linarith |
| have hc1 : c ≤ 1 := by dsimp [c]; linarith |
| have hleft : (volume.restrict (Set.Ioc (0 : ℝ) (1/2))).map (samplingTransport v) = |
| ENNReal.ofReal (1/(1+v)) • volume.restrict (Set.Ioc (0 : ℝ) c) := by |
| have he : samplingTransport v =ᵐ[volume.restrict (Set.Ioc (0 : ℝ) (1/2))] (fun z => (1+v)*z+0) := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with z hz |
| unfold samplingTransport samplingDistance |
| rw [min_eq_left (by linarith [hz.2])] |
| ring |
| rw [Measure.map_congr he,map_interval_affine _ _ _ _ (by linarith)] |
| simp only [mul_zero,add_zero,one_div] |
| congr 3 |
| have hright : (volume.restrict (Set.Ioc (1/2 : ℝ) 1)).map (samplingTransport v) = |
| ENNReal.ofReal (1/(1-v)) • volume.restrict (Set.Ioc c (1 : ℝ)) := by |
| have he : samplingTransport v =ᵐ[volume.restrict (Set.Ioc (1/2 : ℝ) 1)] (fun z => (1-v)*z+v) := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with z hz |
| unfold samplingTransport samplingDistance |
| rw [min_eq_right (by linarith [hz.1])] |
| ring |
| rw [Measure.map_congr he,map_interval_affine _ _ _ _ (by linarith)] |
| have hc' : (1-v)*(1/2)+v = c := by dsimp [c]; ring |
| rw [hc',show (1-v)*1+v = 1 by ring] |
| simp [one_div] |
| have hdleft : (volume.restrict (Set.Ioc (0 : ℝ) c)).withDensity |
| (fun s => ENNReal.ofReal (samplingTransportDensity v s)) = |
| ENNReal.ofReal (1/(1+v)) • volume.restrict (Set.Ioc (0 : ℝ) c) := by |
| rw [← withDensity_const] |
| apply withDensity_congr_ae |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with z hz |
| simp [samplingTransportDensity,show z ≤ (1+v)/2 from hz.2] |
| have hdright : (volume.restrict (Set.Ioc c (1 : ℝ))).withDensity |
| (fun s => ENNReal.ofReal (samplingTransportDensity v s)) = |
| ENNReal.ofReal (1/(1-v)) • volume.restrict (Set.Ioc c (1 : ℝ)) := by |
| rw [← withDensity_const] |
| apply withDensity_congr_ae |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with z hz |
| simp [samplingTransportDensity,show ¬z ≤ (1+v)/2 from not_le.mpr hz.1] |
| conv_lhs => rw [unitInterval_split (1/2) (by norm_num) (by norm_num)] |
| have htm : Measurable (samplingTransport v) := by unfold samplingTransport samplingDistance; fun_prop |
| rw [Measure.map_add _ _ htm,hleft,hright] |
| conv_rhs => rw [unitInterval_split c hc hc1] |
| rw [withDensity_add_measure,hdleft,hdright] |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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