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