| import OAI.Analysis.FiniteFactor.Sampling.Periodization |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory |
| noncomputable section |
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| / |
| def samplingDistance (s : ℝ) : ℝ := min s (1-s) |
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| 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 |
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| 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 _ _ |
|
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| 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 |
|
|
| / |
| 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⟩ |
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| / |
| 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]⟩⟩ |
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|
| / |
| 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] |
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|
| end |
| end FiniteFactor |
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| end OAI |
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