| import OAI.Probability.InvariantIsing.Core.Variational |
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| namespace OAI |
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| /-! The upper-tail order used in the finite-spectrum upper comparison. -/ |
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| noncomputable section |
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| open MeasureTheory Set |
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| namespace |
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| def ℝ ℝ ∫ |
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| lemma ℝ |
| (∫ u, max ∂ ∫ |
| rw |
| rfl |
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| lemma ℝ ℝ |
| (hs ∈ ℝ |
| pathTail ≤ ∫ ∂ |
| let ℝ → ℝ |
| have |
| max_le_max |
| have |
| (hg.intervalIntegrable μ |
| (hg.intervalIntegrable μ |
| have ≤ ∫ |
| intervalIntegral.integral_nonneg_of_forall |
| have ∫ ≤ ∫ |
| intervalIntegral.integral_mono |
| (p.monotone.intervalIntegrable.sub |
| (fun |
| rw |
| intervalIntegral.integral_const, smul_eq_mul] at |
| rw |
| change ∫ ≤ ∫ |
| linarith |
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| /-- A monotone quantile crosses any horizontal level at a cutoff, allowing a |
| jump or a nontrivial atom at that level. Endpoint values play no role. -/ |
| lemma ℝ |
| ∃ s ∈ ℝ |
| (∀ u ∈ ≤ ∧ ∀ ∈ ≤ |
| let ℝ | ∈ ℝ ∧ ∪ |
| have ⟨⟩ |
| have |
| refine ⟨⟩ |
| intro |
| rcases | |
| · exact |
| · rw ; norm_num |
| have hs0 : 0 ≤ sInf A := le_csInf hne (by |
| intro u hu |
| rcases hu with hu | hu |
| · exact hu.1.1 |
| · rw [Set.mem_singleton_iff.mp hu]; norm_num) |
| have hs1 : sInf A ≤ 1 := csInf_le hb (Or.inr rfl) |
| refine ⟨sInf A, ⟨hs0, hs1⟩, ?_, ?_⟩ |
| · intro u hu |
| by_contra h |
| have humem : u ∈ A := Or.inl ⟨⟨hu.1.le, hu.2.le.trans hs1⟩, lt_of_not_ge h⟩ |
| exact (not_le_of_gt hu.2) (csInf_le hb humem) |
| · intro u hu |
| obtain ⟨v, hv, hvu⟩ := exists_lt_of_csInf_lt hne hu.1 |
| rcases hv with hv | hv |
| · exact (hv.2.trans_le (p.monotone hvu.le)).le |
| · rw [Set.mem_singleton_iff.mp hv] at hvu |
| exact False.elim ((not_lt_of_ge hu.2.le) hvu) |
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| /-- The positive-part integral is attained by a tail of the quantile. -/ |
| lemma exists_posPart_eq_pathTail (p : OverlapPath) (r : ℝ) : |
| ∃ s ∈ Icc (0 : ℝ) 1, |
| (∫ u, max (p u - r) 0 ∂pathMeasure) = pathTail p s - (1 - s) * r := by |
| obtain ⟨s, hs, hlo, hhi⟩ := exists_path_level_cutoff p r |
| let g : ℝ → ℝ := fun u => max (p u - r) 0 |
| have hg : Monotone g := fun _ _ huv => |
| max_le_max (sub_le_sub_right (p.monotone huv) r) le_rfl |
| have hlow : (∫ u in 0..s, g u) = 0 := by |
| calc |
| _ = ∫ _ in 0..s, (0 : ℝ) := by |
| apply intervalIntegral.integral_congr_Ioo_of_le hs.1 |
| intro u hu |
| exact max_eq_right (sub_nonpos.mpr (hlo u hu)) |
| _ = 0 := by simp |
| have hhigh : (∫ u in s..1, g u) = pathTail p s - (1 - s) * r := by |
| calc |
| _ = ∫ u in s..1, p u - r := by |
| apply intervalIntegral.integral_congr_Ioo_of_le hs.2 |
| intro u hu |
| exact max_eq_left (sub_nonneg.mpr (hhi u hu)) |
| _ = _ := by |
| rw [intervalIntegral.integral_sub p.monotone.intervalIntegrable intervalIntegrable_const, |
| intervalIntegral.integral_const, smul_eq_mul] |
| rfl |
| have hsum := intervalIntegral.integral_add_adjacent_intervals |
| (hg.intervalIntegrable (μ := volume) (a := 0) (b := s)) |
| (hg.intervalIntegrable (μ := volume) (a := s) (b := 1)) |
| refine ⟨s, hs, ?_⟩ |
| rw [posPart_integral_eq_interval] |
| simpa only [hlow, hhigh, zero_add] using hsum.symm |
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| /-- Manuscript `up:tail-order`: ordering every upper tail reverses the entire |
| deficit function, including at jumps and atoms of either overlap quantile. -/ |
| theorem deficit_le_of_pathTail_le (p q : OverlapPath) |
| (htail : ∀ s ∈ Icc (0 : ℝ) 1, pathTail q s ≤ pathTail p s) (r : ℝ) : |
| deficit p r ≤ deficit q r := by |
| obtain ⟨s, hs, he⟩ := exists_posPart_eq_pathTail q r |
| have hpos : (∫ u, max (q u - r) 0 ∂pathMeasure) ≤ |
| ∫ u, max (p u - r) 0 ∂pathMeasure := by |
| rw [he] |
| exact (sub_le_sub_right (htail s hs) _).trans (pathTail_sub_le_posPart p r hs) |
| unfold deficit |
| linarith |
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| end InvariantIsing |
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| end |
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| end OAI |
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