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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 | import OAI.Analysis.FiniteFactor.Sampling.Periodization
namespace OAI
namespace FiniteFactor
open MeasureTheory
noncomputable section
/-- Distance of a representative to the two ends of the unit interval. -/
def samplingDistance (s : ℝ) : ℝ := min s (1-s)
def samplingWeight (triangular : Bool) (h α s t : ℝ) : ℝ :=
if triangular then
if 0 < t ∧ t < 1-s then 1 else 0
else
if α ≤ samplingDistance s ∧ 0 < t ∧ t < h*samplingDistance s then
1/(h*samplingDistance s)
else 0
theorem samplingWeight_measurable (triangular : Bool) (h α : ℝ) :
Measurable (fun p : ℝ × ℝ => samplingWeight triangular h α p.1 p.2) := by
cases triangular <;> unfold samplingWeight samplingDistance <;> simp only [Bool.false_eq_true,ite_false,ite_true]
· apply Measurable.ite _ (by fun_prop) measurable_const
exact (measurableSet_le measurable_const (measurable_fst.min (measurable_const.sub measurable_fst))).inter
((measurableSet_lt measurable_const measurable_snd).inter
(measurableSet_lt measurable_snd (measurable_const.mul (measurable_fst.min (measurable_const.sub measurable_fst)))))
· exact Measurable.ite ((measurableSet_lt measurable_const measurable_snd).inter
(measurableSet_lt measurable_snd (measurable_const.sub measurable_fst))) measurable_const measurable_const
theorem samplingWeight_bounds (triangular : Bool) (h α s t : ℝ) (hh : 0 < h) (hα : 0 < α) :
0 ≤ samplingWeight triangular h α s t ∧
samplingWeight triangular h α s t ≤ max 1 (1/(h*α)) := by
cases triangular
· unfold samplingWeight
simp only [Bool.false_eq_true,ite_false]
split_ifs with hc
· have hd : 0 < samplingDistance s := hα.trans_le hc.1
exact ⟨by positivity, (one_div_le_one_div_of_le (mul_pos hh hα)
(mul_le_mul_of_nonneg_left hc.1 hh.le)).trans (le_max_right _ _)⟩
· exact ⟨le_rfl,le_trans (by norm_num) (le_max_left _ _)⟩
· unfold samplingWeight
simp only [ite_true]
split_ifs <;> constructor <;> first | positivity | exact le_max_left _ _
theorem samplingWeight_support (triangular : Bool) (h α s t : ℝ)
(_hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(_hs : s ∈ Set.Ico (0 : ℝ) 1) (hw : samplingWeight triangular h α s t ≠ 0) :
0 < t ∧ t < 1-s := by
cases triangular
· unfold samplingWeight at hw
simp only [Bool.false_eq_true,ite_false] at hw
split_ifs at hw with hc
· have hd : 0 < samplingDistance s := hα.trans_le hc.1
have hd1 : samplingDistance s ≤ 1-s := min_le_right _ _
exact ⟨hc.2.1, hc.2.2.trans_le ((by simpa using
(mul_lt_mul_of_pos_right hh1 hd).le : h * samplingDistance s ≤ samplingDistance s).trans hd1)⟩
· exact False.elim (hw rfl)
· simpa only [samplingWeight,ite_true,ne_eq,ite_eq_right_iff,one_ne_zero,imp_false,not_not] using hw
/-- The positive weight already orders each prefix coordinate strictly,
with no wrap of the orbit endpoint. -/
theorem samplingWeight_orders (triangular : Bool) (h α s t : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hs : s ∈ Set.Ico (0 : ℝ) 1) (hw : samplingWeight triangular h α s t ≠ 0) :
Int.fract (s+t) = s+t ∧ s < Int.fract (s+t) := by
have ht := samplingWeight_support triangular h α s t hh hh1 hα hs hw
have he := Int.fract_eq_self.mpr (show s+t ∈ Set.Ico (0 : ℝ) 1 from ⟨by linarith [hs.1],by linarith⟩)
exact ⟨he,by rw [he]; linarith⟩
/-- In the second diagonal orientation the nonzero weight has no wrap and
the original coordinate is exactly s-t, as in the source calculation. -/
theorem samplingWeight_second_no_wrap (h α s t : ℝ) (hh : 0 < h) (hh1 : h < 1)
(hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) (ht : t ∈ Set.Ico (0 : ℝ) 1)
(hw : samplingWeight false h α (Int.fract (s-t)) t ≠ 0) :
t < s ∧ Int.fract (s-t) = s-t := by
have hsp := samplingWeight_support false h α (Int.fract (s-t)) t hh hh1 hα
⟨Int.fract_nonneg _,Int.fract_lt_one _⟩ hw
have hcut : α ≤ samplingDistance (Int.fract (s-t)) := by
unfold samplingWeight at hw
simp only [Bool.false_eq_true,ite_false] at hw
split_ifs at hw with hc
· exact hc.1
· exact False.elim (hw rfl)
have hts : t < s := by
by_contra hn
have hle : s ≤ t := le_of_not_gt hn
by_cases he : s = t
· subst t
simp [samplingDistance] at hcut
linarith
· have hst : s-t < 0 := sub_neg.mpr (lt_of_le_of_ne hle he)
have hf : ⌊s-t⌋ = (-1 : ℤ) := Int.floor_eq_iff.mpr
⟨by norm_num; linarith [hs.1,ht.2],by norm_num; linarith⟩
rw [Int.fract,hf] at hsp
norm_num at hsp
linarith [hsp.2,hs.1]
exact ⟨hts,Int.fract_eq_self.mpr ⟨by linarith,by linarith [hs.2,ht.1]⟩⟩
/-- Exact first marginal of either untested one-coordinate weight. -/
theorem samplingWeight_first_integral (triangular : Bool) (h α s : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
(∫ t, samplingWeight triangular h α s t ∂unitIntervalMeasure) =
if triangular then 1-s else if α ≤ samplingDistance s then 1 else 0 := by
have hvol (L : ℝ) (hL : 0 ≤ L) (hL1 : L ≤ 1) :
unitIntervalMeasure (Set.Ioo (0 : ℝ) L) = ENNReal.ofReal L := by
unfold unitIntervalMeasure
rw [Measure.restrict_apply measurableSet_Ioo,Set.inter_eq_left.mpr (by
intro t ht; exact ⟨ht.1.le,ht.2.trans_le hL1⟩),Real.volume_Ioo,sub_zero]
cases triangular
· simp only [Bool.false_eq_true,ite_false]
by_cases hc : α ≤ samplingDistance s
· have hd : 0 < samplingDistance s := hα.trans_le hc
have hL : 0 < h*samplingDistance s := mul_pos hh hd
have hL1 : h*samplingDistance s ≤ 1 := by
have := min_le_right s (1-s)
change samplingDistance s ≤ 1-s at this
nlinarith [hs.1]
have he : samplingWeight false h α s =
(Set.Ioo 0 (h*samplingDistance s)).indicator (fun _ => 1/(h*samplingDistance s)) := by
funext t
simp [samplingWeight,hc,Set.indicator,Set.mem_Ioo]
rw [he,integral_indicator measurableSet_Ioo]
simp [hc,hvol (h*samplingDistance s) hL.le hL1,ENNReal.toReal_ofReal hL.le,measureReal_def]
field_simp
· have he : samplingWeight false h α s = fun _ => 0 := by funext t; simp [samplingWeight,hc]
simp [he,hc]
· simp only [ite_true]
have he : samplingWeight true h α s = (Set.Ioo 0 (1-s)).indicator (fun _ => (1 : ℝ)) := by
funext t
simp [samplingWeight,Set.indicator,Set.mem_Ioo]
rw [he,integral_indicator measurableSet_Ioo]
simp [hvol (1-s) (by linarith [hs.2]) (by linarith [hs.1]),ENNReal.toReal_ofReal,show 0 ≤ 1-s by linarith [hs.2],measureReal_def]
end
end FiniteFactor
end OAI
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