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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 | import OAI.Probability.InvariantIsing.Core.TailOrder
namespace OAI
/-! Contact inequalities at trial-partition endpoints imply the full upper
tail order. Monotone quantile tails lie above their interval chords. -/
noncomputable section
open MeasureTheory Set
open scoped BigOperators
namespace InvariantIsing
lemma pathTail_chord (p : OverlapPath) {a s b : ℝ} (has : a ≤ s) (hsb : s ≤ b) :
(b - s) * pathTail p a + (s - a) * pathTail p b ≤ (b - a) * pathTail p s := by
have hleft : (∫ u in a..s, p u) ≤ ∫ _u in a..s, p s :=
intervalIntegral.integral_mono_on has p.monotone.intervalIntegrable
intervalIntegrable_const (fun u hu => p.monotone hu.2)
have hright : (∫ _u in s..b, p s) ≤ ∫ u in s..b, p u :=
intervalIntegral.integral_mono_on hsb intervalIntegrable_const
p.monotone.intervalIntegrable (fun u hu => p.monotone hu.1)
rw [intervalIntegral.integral_const, smul_eq_mul] at hleft hright
have h1 := intervalIntegral.integral_add_adjacent_intervals
(p.monotone.intervalIntegrable (μ := volume) (a := a) (b := s))
(p.monotone.intervalIntegrable (μ := volume) (a := s) (b := 1))
have h2 := intervalIntegral.integral_add_adjacent_intervals
(p.monotone.intervalIntegrable (μ := volume) (a := s) (b := b))
(p.monotone.intervalIntegrable (μ := volume) (a := b) (b := 1))
have hL := mul_le_mul_of_nonneg_left hleft (sub_nonneg.mpr hsb)
have hR := mul_le_mul_of_nonneg_left hright (sub_nonneg.mpr has)
unfold pathTail
nlinarith
lemma pathTail_chord_eq_of_const (q : OverlapPath) {a s b c : ℝ}
(has : a ≤ s) (hsb : s ≤ b) (hc : ∀ u ∈ Ioo a b, q u = c) :
(b - s) * pathTail q a + (s - a) * pathTail q b = (b - a) * pathTail q s := by
have hleft : (∫ u in a..s, q u) = (s - a) * c := by
calc
_ = ∫ _u in a..s, c := intervalIntegral.integral_congr_Ioo_of_le has
(fun u hu => hc u ⟨hu.1, hu.2.trans_le hsb⟩)
_ = _ := by simp only [intervalIntegral.integral_const, smul_eq_mul]
have hright : (∫ u in s..b, q u) = (b - s) * c := by
calc
_ = ∫ _u in s..b, c := intervalIntegral.integral_congr_Ioo_of_le hsb
(fun u hu => hc u ⟨has.trans_lt hu.1, hu.2⟩)
_ = _ := by simp only [intervalIntegral.integral_const, smul_eq_mul]
have h1 := intervalIntegral.integral_add_adjacent_intervals
(q.monotone.intervalIntegrable (μ := volume) (a := a) (b := s))
(q.monotone.intervalIntegrable (μ := volume) (a := s) (b := 1))
have h2 := intervalIntegral.integral_add_adjacent_intervals
(q.monotone.intervalIntegrable (μ := volume) (a := s) (b := b))
(q.monotone.intervalIntegrable (μ := volume) (a := b) (b := 1))
rw [hleft] at h1
rw [hright] at h2
unfold pathTail
nlinarith
lemma pathTail_le_of_interval_endpoints (p q : OverlapPath) {a s b c : ℝ}
(hab : a < b) (has : a ≤ s) (hsb : s ≤ b) (hc : ∀ u ∈ Ioo a b, q u = c)
(ha : pathTail q a ≤ pathTail p a) (hb : pathTail q b ≤ pathTail p b) :
pathTail q s ≤ pathTail p s := by
have hp := pathTail_chord p has hsb
have hq := pathTail_chord_eq_of_const q has hsb hc
have h1 := mul_le_mul_of_nonneg_left ha (sub_nonneg.mpr hsb)
have h2 := mul_le_mul_of_nonneg_left hb (sub_nonneg.mpr has)
nlinarith
lemma finite_cut_bracket {n : ℕ} (cut : Fin (n + 2) → ℝ)
(hfirst : cut 0 = 0) (hlast : cut (Fin.last (n + 1)) = 1)
{s : ℝ} (hs : s ∈ Icc (0 : ℝ) 1) :
∃ i : Fin (n + 1), cut i.castSucc ≤ s ∧ s ≤ cut i.succ := by
classical
let S := Finset.univ.filter (fun i : Fin (n + 1) => cut i.castSucc ≤ s)
have hS : S.Nonempty := ⟨0, by simp only [S, Finset.mem_filter, Finset.mem_univ,
true_and, Fin.castSucc_zero, hfirst]; exact hs.1⟩
let i := S.max' hS
have hiS : i ∈ S := Finset.max'_mem _ _
have hi : cut i.castSucc ≤ s := (Finset.mem_filter.mp hiS).2
refine ⟨i, hi, ?_⟩
by_cases hin : i.val = n
· have he : i.succ = Fin.last (n + 1) := by ext; simp only [Fin.val_succ, Fin.val_last]; omega
rw [he, hlast]
exact hs.2
· by_contra hsb
let j : Fin (n + 1) := ⟨i.val + 1, by omega⟩
have he : j.castSucc = i.succ := by ext; rfl
have hjS : j ∈ S := Finset.mem_filter.mpr ⟨Finset.mem_univ _, by
rw [he]; exact (lt_of_not_ge hsb).le⟩
have hji : j ≤ i := Finset.le_max' S j hjS
change j.val ≤ i.val at hji
dsimp only [j] at hji
omega
theorem pathTail_order_of_finite_partition (p q : OverlapPath) {n : ℕ}
(cut : Fin (n + 2) → ℝ) (hcut : StrictMono cut)
(hfirst : cut 0 = 0) (hlast : cut (Fin.last (n + 1)) = 1)
(v : Fin (n + 1) → ℝ)
(hq : ∀ i u, u ∈ Ioo (cut i.castSucc) (cut i.succ) → q u = v i)
(hend : ∀ i, pathTail q (cut i) ≤ pathTail p (cut i)) :
∀ s ∈ Icc (0 : ℝ) 1, pathTail q s ≤ pathTail p s := by
intro s hs
obtain ⟨i, has, hsb⟩ := finite_cut_bracket cut hfirst hlast hs
exact pathTail_le_of_interval_endpoints p q
(hcut (by exact_mod_cast (show i.val < i.val + 1 by omega))) has hsb (hq i)
(hend i.castSucc) (hend i.succ)
end InvariantIsing
end
end OAI
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