| import OAI.Analysis.FiniteFactor.Sampling.Transport |
| import Mathlib.MeasureTheory.Integral.Prod |
|
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| namespace OAI |
|
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| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| theorem samplingTransportDensity_bounds (h v s : ℝ) (hh : 0 ≤ h) (hh1 : h < 1) |
| (hv : 0 ≤ v) (hvh : v ≤ h) : |
| 0 ≤ samplingTransportDensity v s ∧ samplingTransportDensity v s ≤ 1/(1-h) ∧ |
| |samplingTransportDensity v s-1| ≤ h/(1-h) := by |
| have hmh : 0 < 1-h := by linarith |
| have hph : 0 < 1+h := by linarith |
| have hmv : 0 < 1-v := by linarith |
| have hpv : 0 < 1+v := by linarith |
| have hlo : 1/(1+h) ≤ samplingTransportDensity v s := by |
| unfold samplingTransportDensity |
| split_ifs |
| · exact one_div_le_one_div_of_le hpv (by linarith) |
| · exact one_div_le_one_div_of_le hmv (by linarith) |
| have hhi : samplingTransportDensity v s ≤ 1/(1-h) := by |
| unfold samplingTransportDensity |
| split_ifs <;> exact one_div_le_one_div_of_le hmh (by linarith) |
| have hl : 1-h/(1-h) ≤ 1/(1+h) := by |
| field_simp |
| nlinarith |
| have hr : 1/(1-h)-1 = h/(1-h) := by field_simp; ring |
| exact ⟨(by positivity : 0 ≤ 1/(1+h)).trans hlo,hhi, |
| abs_le.mpr ⟨by linarith,by linarith⟩⟩ |
|
|
| theorem samplingTransportDensity_integral (v : ℝ) (hv : 0 ≤ v) (hv1 : v < 1) : |
| (∫ s, samplingTransportDensity v s ∂unitIntervalMeasure) = 1 := by |
| have hm : Measurable (samplingTransportDensity v) := |
| samplingTransportDensity_measurable.comp (measurable_const.prodMk measurable_id) |
| have hi : Integrable (samplingTransportDensity v) unitIntervalMeasure := |
| (integrable_const (1/(1-v))).mono' hm.aestronglyMeasurable |
| (Eventually.of_forall (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTransportDensity_bounds v v s hv hv1 hv le_rfl).1] |
| exact (samplingTransportDensity_bounds v v s hv hv1 hv le_rfl).2.1)) |
| have hmap := congrArg (fun μ : Measure ℝ => μ Set.univ) (samplingTransport_density v hv hv1) |
| have htm : Measurable (samplingTransport v) := by unfold samplingTransport samplingDistance; fun_prop |
| rw [Measure.map_apply htm MeasurableSet.univ,withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ] at hmap |
| simp only [Set.preimage_univ,measure_univ] at hmap |
| have he := ofReal_integral_eq_lintegral_ofReal hi |
| (Eventually.of_forall (fun s => (samplingTransportDensity_bounds v v s hv hv1 hv le_rfl).1)) |
| rw [← hmap] at he |
| have hn : 0 ≤ ∫ s, samplingTransportDensity v s ∂unitIntervalMeasure := integral_nonneg (fun s => (samplingTransportDensity_bounds v v s hv hv1 hv le_rfl).1) |
| have hh := congrArg ENNReal.toReal he |
| simpa [ENNReal.toReal_ofReal hn] using hh |
|
|
| def samplingVelocityMeasure (h : ℝ) : Measure ℝ := cond volume (Set.Ioc (0 : ℝ) h) |
|
|
| theorem samplingVelocity_probability (h : ℝ) (hh : 0 < h) : IsProbabilityMeasure (samplingVelocityMeasure h) := by |
| apply cond_isProbabilityMeasure_of_finite |
| · simp [Real.volume_Ioc,hh] |
| · simp [Real.volume_Ioc] |
|
|
| / |
| def samplingIdealSecond (h s : ℝ) : ℝ := |
| ∫ v, samplingTransportDensity v s ∂samplingVelocityMeasure h |
|
|
| theorem samplingIdealSecond_bounds (h s : ℝ) (hh : 0 < h) (hh1 : h < 1) : |
| 0 ≤ samplingIdealSecond h s ∧ samplingIdealSecond h s ≤ 1/(1-h) ∧ |
| |samplingIdealSecond h s-1| ≤ h/(1-h) := by |
| have := samplingVelocity_probability h hh |
| have hv : ∀ᵐ v ∂samplingVelocityMeasure h, v ∈ Set.Ioc (0 : ℝ) h := ae_cond_mem measurableSet_Ioc |
| have hm : Measurable (fun v => samplingTransportDensity v s) := |
| samplingTransportDensity_measurable.comp (measurable_id.prodMk measurable_const) |
| have hqi : Integrable (fun v => samplingTransportDensity v s) (samplingVelocityMeasure h) := |
| (integrable_const (1/(1-h))).mono' hm.aestronglyMeasurable (by |
| filter_upwards [hv] with v hv |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).1] |
| exact (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).2.1) |
| have hnon : 0 ≤ samplingIdealSecond h s := integral_nonneg_of_ae (hv.mono (fun v hv => |
| (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).1)) |
| have hu := integral_mono_ae hqi (integrable_const (1/(1-h))) (hv.mono (fun v hv => |
| (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).2.1)) |
| refine ⟨hnon,by simpa [samplingIdealSecond] using hu,?_⟩ |
| have he : samplingIdealSecond h s-1 = ∫ v, samplingTransportDensity v s-1 ∂samplingVelocityMeasure h := by |
| rw [integral_sub hqi (integrable_const 1)] |
| simp [samplingIdealSecond] |
| rw [he] |
| calc |
| _ ≤ ∫ v, |samplingTransportDensity v s-1| ∂samplingVelocityMeasure h := by |
| simpa [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun v => samplingTransportDensity v s-1) |
| _ ≤ ∫ _v, h/(1-h) ∂samplingVelocityMeasure h := by |
| apply integral_mono_ae |
| · simpa [Real.norm_eq_abs] using (hqi.sub (integrable_const 1)).norm |
| · exact integrable_const _ |
| · exact hv.mono (fun v hv => (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).2.2) |
| _ = _ := by simp |
|
|
| theorem samplingIdealSecond_measurable (h : ℝ) (hh : 0 < h) : Measurable (samplingIdealSecond h) := by |
| have := samplingVelocity_probability h hh |
| exact samplingTransportDensity_measurable.stronglyMeasurable.integral_prod_left.measurable |
|
|
| / |
| theorem samplingIdealSecond_integral (h : ℝ) (hh : 0 < h) (hh1 : h < 1) : |
| (∫ s, samplingIdealSecond h s ∂unitIntervalMeasure) = 1 := by |
| have := samplingVelocity_probability h hh |
| have hv : ∀ᵐ v ∂samplingVelocityMeasure h, v ∈ Set.Ioc (0 : ℝ) h := ae_cond_mem measurableSet_Ioc |
| have hvp : ∀ᵐ p : ℝ × ℝ ∂(samplingVelocityMeasure h).prod unitIntervalMeasure, |
| p.1 ∈ Set.Ioc (0 : ℝ) h := by |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_Ioc.preimage measurable_fst)).mpr |
| exact hv.mono (fun v hv => Eventually.of_forall (fun _ => hv)) |
| have hqi : Integrable (fun p : ℝ × ℝ => samplingTransportDensity p.1 p.2) |
| ((samplingVelocityMeasure h).prod unitIntervalMeasure) := |
| (integrable_const (1/(1-h))).mono' samplingTransportDensity_measurable.aestronglyMeasurable (by |
| filter_upwards [hvp] with p hp |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTransportDensity_bounds h p.1 p.2 hh.le hh1 hp.1.le hp.2).1] |
| exact (samplingTransportDensity_bounds h p.1 p.2 hh.le hh1 hp.1.le hp.2).2.1) |
| have hswap := integral_integral_swap (f := fun v s => samplingTransportDensity v s) hqi |
| calc |
| _ = ∫ v, (∫ s, samplingTransportDensity v s ∂unitIntervalMeasure) ∂samplingVelocityMeasure h := hswap.symm |
| _ = ∫ _v, (1 : ℝ) ∂samplingVelocityMeasure h := integral_congr_ae |
| (hv.mono (fun v hv => samplingTransportDensity_integral v hv.1.le (hv.2.trans_lt hh1))) |
| _ = 1 := by simp |
|
|
| / |
| transport density. The exact mass-one identity eliminates the linear term. -/ |
| theorem samplingIdealSecond_square (h : ℝ) (hh : 0 < h) (hh1 : h < 1) : |
| (∫ s, samplingIdealSecond h s^2 ∂unitIntervalMeasure) = |
| 1+(∫ s, (samplingIdealSecond h s-1)^2 ∂unitIntervalMeasure) ∧ |
| (∫ s, samplingIdealSecond h s^2 ∂unitIntervalMeasure) ≤ 1+h^2/(1-h)^2 := by |
| let q := samplingIdealSecond h |
| let C := h/(1-h) |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hm := samplingIdealSecond_measurable h hh |
| have hqi : Integrable q unitIntervalMeasure := |
| (integrable_const (1/(1-h))).mono' hm.aestronglyMeasurable (Eventually.of_forall (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingIdealSecond_bounds h s hh hh1).1] |
| exact (samplingIdealSecond_bounds h s hh hh1).2.1)) |
| have hdev (s : ℝ) : (q s-1)^2 ≤ C^2 := by |
| have hb := abs_le.mp (samplingIdealSecond_bounds h s hh hh1).2.2 |
| dsimp [C,q] |
| nlinarith |
| have hdi : Integrable (fun s => (q s-1)^2) unitIntervalMeasure := |
| (integrable_const (C^2)).mono' (by fun_prop) (Eventually.of_forall (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact hdev s)) |
| have heq : (fun s => q s^2) = (fun s => ((q s-1)^2+2*q s)-1) := by funext s; ring |
| have hmass : (∫ s, q s ∂unitIntervalMeasure) = 1 := samplingIdealSecond_integral h hh hh1 |
| have he : (∫ s, q s^2 ∂unitIntervalMeasure) = 1+∫ s, (q s-1)^2 ∂unitIntervalMeasure := by |
| rw [heq] |
| have hsum : Integrable (fun s => (q s-1)^2+2*q s) unitIntervalMeasure := hdi.add (hqi.const_mul 2) |
| rw [integral_sub hsum (integrable_const 1),integral_add hdi (hqi.const_mul 2),integral_const_mul,hmass] |
| simp |
| ring |
| refine ⟨he,?_⟩ |
| have hu := integral_mono_ae hdi (integrable_const (C^2)) (Eventually.of_forall hdev) |
| simp at hu |
| have hCeq : C^2 = h^2/(1-h)^2 := by dsimp [C]; rw [div_pow] |
| rw [he] |
| linarith |
|
|
| end |
| end FiniteFactor |
|
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| end OAI |
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