| import OAI.Analysis.FiniteFactor.Sampling.TestedSecondMoment |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical |
| noncomputable section |
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| def samplingSecondLength (h s : ℝ) : ℝ := min (h*s/(1+h)) (h*(1-s)/(1-h)) |
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| theorem samplingSecondWeight_support_iff (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) : |
| samplingSecondWeight h α s t ≠ 0 ↔ |
| 0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α := by |
| have hplus : 0 < 1+h := by linarith |
| have hminus : 0 < 1-h := by linarith |
| constructor |
| · intro hw |
| have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2 |
| have hc : α ≤ samplingDistance (s-t) ∧ 0 < t ∧ t < h*samplingDistance (s-t) := by |
| unfold samplingSecondWeight samplingWeight at hw |
| simp only [Bool.false_eq_true,ite_false,hf] at hw |
| split_ifs at hw with hc |
| · exact hc |
| · exact False.elim (hw rfl) |
| have hcut := le_min_iff.mp hc.1 |
| have hleft : t < h*(s-t) := hc.2.2.trans_le |
| (mul_le_mul_of_nonneg_left (min_le_left _ _) hh.le) |
| have hright : t < h*(1-(s-t)) := hc.2.2.trans_le |
| (mul_le_mul_of_nonneg_left (min_le_right _ _) hh.le) |
| refine ⟨hc.2.1,?_,hcut.1,by linarith [hcut.2]⟩ |
| unfold samplingSecondLength |
| rw [lt_min_iff,lt_div_iff₀ hplus,lt_div_iff₀ hminus] |
| constructor <;> nlinarith |
| · rintro ⟨ht0,htL,hcut,hcut'⟩ |
| have hz : s-t ∈ Set.Ico (0 : ℝ) 1 := ⟨hα.le.trans hcut,by linarith⟩ |
| have hf := Int.fract_eq_self.mpr hz |
| have hL := lt_min_iff.mp htL |
| have hl := (lt_div_iff₀ hplus).mp hL.1 |
| have hr := (lt_div_iff₀ hminus).mp hL.2 |
| have htl : t < h*(s-t) := by nlinarith |
| have htr : t < h*(1-(s-t)) := by nlinarith |
| have hd : α ≤ samplingDistance (s-t) := le_min hcut (by linarith) |
| have htd : t < h*samplingDistance (s-t) := by |
| unfold samplingDistance |
| rw [mul_min_of_nonneg _ _ hh.le] |
| exact lt_min htl htr |
| have hp : 0 < h*samplingDistance (s-t) := mul_pos hh (hα.trans_le hd) |
| simp [samplingSecondWeight,samplingWeight,hf,hd,ht0,htd,(hα.trans_le hd).ne',hh.ne'] |
| |
| /-- Exact reciprocal on the actual second-oriented support. -/ |
| theorem samplingSecondWeight_interval (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) : |
| samplingSecondWeight h α s t = |
| if 0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α |
| then 1/(h*samplingDistance (s-t)) else 0 := by |
| by_cases hw : samplingSecondWeight h α s t ≠ 0 |
| · have hc := (samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mp hw |
| have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2 |
| rw [ite_eq_left hc] |
| unfold samplingSecondWeight samplingWeight at * |
| simp only [Bool.false_eq_true,ite_false,hf] at * |
| split_ifs at * <;> simp_all |
| · rw [not_ne_iff] at hw |
| have hc : ¬ (0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α) := by |
| intro hc |
| exact ((samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mpr hc) hw |
| rw [ite_eq_right hc,hw] |
| |
| /-- The x-test intersects that support with a reflected interval, while the |
| y-test is constant in the phase variable. -/ |
| theorem samplingTestedSecondWeight_interval (h α : ℝ) (G : Set ℝ) (a c 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) : |
| samplingTestedSecondWeight h α (Set.Icc a c) G s t = |
| (Set.Ioo 0 (samplingSecondLength h s) ∩ |
| Set.Icc (max (s-(1-α)) (s-c)) (min (s-α) (s-a))).indicator |
| (fun t => if s ∈ G then 1/(h*samplingDistance (s-t)) else 0) t := by |
| have heq : t ∈ Set.Ioo 0 (samplingSecondLength h s) ∩ |
| Set.Icc (max (s-(1-α)) (s-c)) (min (s-α) (s-a)) ↔ |
| (0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α) ∧ s-t ∈ Set.Icc a c := by |
| simp only [Set.mem_inter_iff,Set.mem_Ioo,Set.mem_Icc,max_le_iff,le_min_iff] |
| constructor |
| · rintro ⟨ht0,hL⟩ |
| exact ⟨⟨ht0.1,ht0.2,by linarith [hL.2.1],by linarith [hL.1.1]⟩, |
| ⟨by linarith [hL.2.2],by linarith [hL.1.2]⟩⟩ |
| · rintro ⟨⟨ht0,hL,hcut,hcut'⟩,ha,hc⟩ |
| exact ⟨⟨ht0,hL⟩,⟨⟨by linarith,by linarith⟩,⟨by linarith,by linarith⟩⟩⟩ |
| by_cases hw : samplingSecondWeight h α s t ≠ 0 |
| · have hc := (samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mp hw |
| have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2 |
| rw [samplingTestedSecondWeight,samplingSecondWeight_interval h α s t hh hh1 hα hs ht, |
| ite_eq_left hc,hf] |
| simp only [Set.indicator,heq,hc,true_and] |
| split_ifs <;> simp_all |
| · have hc : ¬ (0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α) := by |
| intro hc |
| exact hw ((samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mpr hc) |
| have hz := not_ne_iff.mp hw |
| simp only [samplingTestedSecondWeight,hz,ite_self,Set.indicator,heq,hc,false_and,ite_false] |
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|
| end |
| end FiniteFactor |
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| end OAI |
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