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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Probability.InvariantIsing.Core.Variational
namespace OAI
/-! The upper-tail order used in the finite-spectrum upper comparison. -/
noncomputable section
open MeasureTheory Set
namespace InvariantIsing
def pathTail (p : OverlapPath) (s : ℝ) : ℝ := ∫ u in s..1, p u
lemma posPart_integral_eq_interval (p : OverlapPath) (r : ℝ) :
(∫ u, max (p u - r) 0 ∂pathMeasure) = ∫ u in 0..1, max (p u - r) 0 := by
rw [intervalIntegral.integral_of_le zero_le_one, integral_Ioc_eq_integral_Ioo]
rfl
lemma pathTail_sub_le_posPart (p : OverlapPath) (r : ℝ) {s : ℝ}
(hs : s ∈ Icc (0 : ℝ) 1) :
pathTail p s - (1 - s) * r ≤ ∫ u, max (p u - r) 0 ∂pathMeasure := by
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 hsum := intervalIntegral.integral_add_adjacent_intervals
(hg.intervalIntegrable (μ := volume) (a := 0) (b := s))
(hg.intervalIntegrable (μ := volume) (a := s) (b := 1))
have hlo : 0 ≤ ∫ u in 0..s, g u :=
intervalIntegral.integral_nonneg_of_forall hs.1 (fun _ => le_max_right _ _)
have hhi : (∫ u in s..1, p u - r) ≤ ∫ u in s..1, g u :=
intervalIntegral.integral_mono hs.2
(p.monotone.intervalIntegrable.sub intervalIntegrable_const) hg.intervalIntegrable
(fun _ => le_max_left _ _)
rw [intervalIntegral.integral_sub p.monotone.intervalIntegrable intervalIntegrable_const,
intervalIntegral.integral_const, smul_eq_mul] at hhi
rw [posPart_integral_eq_interval]
change (∫ u in s..1, p u) - (1 - s) * r ≤ ∫ u in 0..1, g u
linarith
/-- 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 exists_path_level_cutoff (p : OverlapPath) (r : ℝ) :
∃ s ∈ Icc (0 : ℝ) 1,
(∀ u ∈ Ioo 0 s, p u ≤ r) ∧ (∀ u ∈ Ioo s 1, r ≤ p u) := by
let A : Set ℝ := {u | u ∈ Icc (0 : ℝ) 1 ∧ r < p u} ∪ {1}
have hne : A.Nonempty := ⟨1, Or.inr rfl⟩
have hb : BddBelow A := by
refine ⟨0, ?_⟩
intro u hu
rcases hu with hu | hu
· exact hu.1.1
· rw [Set.mem_singleton_iff.mp hu]; 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)
/-- 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
/-- 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
end InvariantIsing
end
end OAI
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