| import OAI.Analysis.FiniteFactor.Sampling.TestedFirstMoment |
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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 samplingFirstLength (b : Bool) (h s : ℝ) : ℝ := |
| if b then 1-s else h*samplingDistance s |
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| def samplingFirstHeight (b : Bool) (h α s : ℝ) : ℝ := |
| if b then 1 else if α ≤ samplingDistance s then 1/(h*samplingDistance s) else 0 |
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| / |
| theorem samplingWeight_first_interval (b : Bool) (h α s t : ℝ) : |
| samplingWeight b h α s t = |
| (Set.Ioo 0 (samplingFirstLength b h s)).indicator |
| (fun _ => samplingFirstHeight b h α s) t := by |
| cases b <;> unfold samplingWeight samplingFirstLength samplingFirstHeight <;> |
| simp only [Bool.false_eq_true,ite_false,ite_true,Set.indicator,Set.mem_Ioo]; |
| split_ifs <;> simp_all |
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| / |
| no bounded-variation theorem or independent pair density is assumed. -/ |
| theorem samplingTestedFirstWeight_interval (b : Bool) (h α : ℝ) (E : Set ℝ) (a c s t : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) : |
| samplingTestedFirstWeight b h α E (Set.Icc a c) s t = |
| (Set.Ioo 0 (samplingFirstLength b h s) ∩ Set.Icc (a-s) (c-s)).indicator |
| (fun _ => if s ∈ E then samplingFirstHeight b h α s else 0) t := by |
| classical |
| have he := samplingWeight_first_interval b h α s t |
| by_cases ht : t ∈ Set.Ioo 0 (samplingFirstLength b h s) |
| · rw [Set.indicator_of_mem ht] at he |
| by_cases hc : samplingFirstHeight b h α s = 0 |
| · simp [samplingTestedFirstWeight,he,hc,Set.indicator] |
| · have hw : samplingWeight b h α s t ≠ 0 := by rwa [he] |
| have hf := (samplingWeight_orders b h α s t hh hh1 hα hs hw).1 |
| have heq : Int.fract (s+t) ∈ Set.Icc a c ↔ t ∈ Set.Icc (a-s) (c-s) := by |
| rw [hf] |
| simp only [Set.mem_Icc] |
| constructor <;> intro H <;> constructor <;> linarith [H.1,H.2] |
| simp only [samplingTestedFirstWeight,he,heq,Set.indicator,Set.mem_inter_iff,ht,true_and] |
| split_ifs <;> simp_all |
| · rw [Set.indicator_of_notMem ht] at he |
| simp [samplingTestedFirstWeight,he,Set.indicator,ht] |
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| / |
| when the tested interval is empty or degenerates to a singleton. -/ |
| theorem samplingTestedFirstWeight_closed_interval (b : Bool) (h α : ℝ) (E : Set ℝ) (a c s t : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) |
| (ht0 : t ≠ 0) (htL : t ≠ samplingFirstLength b h s) : |
| samplingTestedFirstWeight b h α E (Set.Icc a c) s t = |
| (Set.Icc (max 0 (a-s)) (min (samplingFirstLength b h s) (c-s))).indicator |
| (fun _ => if s ∈ E then samplingFirstHeight b h α s else 0) t := by |
| rw [samplingTestedFirstWeight_interval b h α E a c s t hh hh1 hα hs] |
| have heq : t ∈ Set.Ioo 0 (samplingFirstLength b h s) ∩ Set.Icc (a-s) (c-s) ↔ |
| t ∈ Set.Icc (max 0 (a-s)) (min (samplingFirstLength b h s) (c-s)) := by |
| simp only [Set.mem_inter_iff,Set.mem_Ioo,Set.mem_Icc,max_le_iff,le_min_iff] |
| constructor |
| · intro H |
| exact ⟨⟨H.1.1.le,H.2.1⟩,⟨H.1.2.le,H.2.2⟩⟩ |
| · intro H |
| exact ⟨⟨lt_of_le_of_ne H.1.1 (Ne.symm ht0),lt_of_le_of_ne H.2.1 htL⟩,⟨H.1.2,H.2.2⟩⟩ |
| simp only [Set.indicator,heq] |
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| / |
| theorem samplingTestedFirstWeight_closed_interval_ae (μ : Measure ℝ) [NullSingletonClass μ] |
| (b : Bool) (h α : ℝ) (E : Set ℝ) (a c s : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) : |
| samplingTestedFirstWeight b h α E (Set.Icc a c) s =ᵐ[μ] |
| (Set.Icc (max 0 (a-s)) (min (samplingFirstLength b h s) (c-s))).indicator |
| (fun _ => if s ∈ E then samplingFirstHeight b h α s else 0) := by |
| filter_upwards [ae_iff.mpr (show μ {t | ¬t ≠ 0} = 0 by simp), |
| ae_iff.mpr (show μ {t | ¬t ≠ samplingFirstLength b h s} = 0 by simp)] with t ht0 htL |
| exact samplingTestedFirstWeight_closed_interval b h α E a c s t hh hh1 hα hs ht0 htL |
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| end |
| end FiniteFactor |
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| end OAI |
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