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67aee1d | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | import OAI.Analysis.FiniteFactor.Sampling.Transport
import Mathlib.MeasureTheory.Integral.Prod
namespace OAI
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]
/-- Actual averaged density of the uncut diagonal transport. -/
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
/-- Fubini gives the exact mass identity for the averaged transport density. -/
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
/-- The source's quadratic cancellation for the actual averaged uncut
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
end OAI
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