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e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.SelectorTruncationDefinitions
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
/-!
# The right-hand slope at a maximal admissible cutoff
A finite minimum provides an actual larger cutoff below every currently
heavy weight. Between these cutoffs the truncated excess has the exact
affine increment contributed by the heavy coordinates. Weights equal to
the original cutoff remain saturated, including when the cutoff is zero.
-/
namespace TalagrandThreshold.Selector
open scoped BigOperators
variable {α : Type*} [Fintype α] [DecidableEq α]
omit [DecidableEq α] in
/-- There is a strictly larger cutoff, at most one, that does not pass any
weight strictly above the current cutoff. -/
theorem exists_short_right_cutoff (w : α → ℝ) {ε : ℝ} (hε : ε < 1) :
∃ u : ℝ, ε < u ∧ u ≤ 1 ∧ ∀ x, ε < w x → u ≤ w x := by
classical
let S : Finset ℝ := insert 1 ((heavySet w ε).image w)
have hS : S.Nonempty := ⟨1, Finset.mem_insert_self _ _⟩
refine ⟨S.min' hS, ?_, S.min'_le 1 (Finset.mem_insert_self _ _), ?_⟩
· apply (Finset.lt_min'_iff S hS).2
intro y hy
rcases Finset.mem_insert.mp hy with rfl | hy
· exact hε
· obtain ⟨x, hx, rfl⟩ := Finset.mem_image.mp hy
exact mem_heavySet.mp hx
· intro x hx
apply S.min'_le (w x)
exact Finset.mem_insert_of_mem
(Finset.mem_image.mpr ⟨x, mem_heavySet.mpr hx, rfl⟩)
/-- On the first interval to the right of a cutoff, precisely the strictly
heavy coordinates vary. The inclusion in `A` makes their coefficient `d`. -/
theorem truncatedExcess_sub_eq_heavy_increment (w : α → ℝ) (A : Finset α)
(d ε u : ℝ) (hεu : ε ≤ u)
(hupper : ∀ x, ε < w x → u ≤ w x)
(hheavy : ∀ x, ε < w x → x ∈ A) :
truncatedExcess w A d u - truncatedExcess w A d ε =
(u - ε) * d * ((heavySet w ε).card : ℝ) := by
classical
calc
truncatedExcess w A d u - truncatedExcess w A d ε =
∑ x, if ε < w x then (u - ε) * d else 0 := by
unfold truncatedExcess
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro x _
by_cases hx : ε < w x
· simp only [min_eq_right (hupper x hx), min_eq_right hx.le,
ite_eq_left hx, ite_eq_left (hheavy x hx)]
ring
· have hxε : w x ≤ ε := le_of_not_gt hx
simp only [min_eq_left (hxε.trans hεu), min_eq_left hxε,
ite_eq_right hx, sub_self]
_ = ∑ x ∈ heavySet w ε, (u - ε) * d := by
rw [← Finset.sum_filter]
rfl
_ = (u - ε) * d * ((heavySet w ε).card : ℝ) := by
simp only [Finset.sum_const, nsmul_eq_mul]
ring
/-- The cardinality conclusion of maximal truncation. Existence of a
maximal admissible cutoff is supplied separately by compactness. -/
theorem maximal_admissible_card_bound (w : α → ℝ) (A : Finset α) (m d ε : ℝ)
(hwnonneg : ∀ x, 0 ≤ w x) (hwsum : ∑ x, w x = 1)
(hm0 : 0 ≤ m) (_hd0 : 0 < d) (_hd1 : d < 1)
(hε : Admissible w A m d ε)
(hmax : ∀ u : ℝ, Admissible w A m d u → u ≤ ε) :
d * ((heavySet w ε).card : ℝ) ≤ m := by
classical
have hwle (x : α) : w x ≤ 1 := by
calc
w x ≤ ∑ y, w y :=
Finset.single_le_sum (fun y _ => hwnonneg y) (Finset.mem_univ x)
_ = 1 := hwsum
by_cases hεone : ε = 1
· have hempty : heavySet w ε = ∅ := by
apply Finset.eq_empty_iff_forall_notMem.mpr
intro x hx
have hlt := mem_heavySet.mp hx
rw [hεone] at hlt
exact (not_lt_of_ge (hwle x)) hlt
simpa only [hempty, Finset.card_empty, Nat.cast_zero, mul_zero] using hm0
· have hεlt : ε < 1 := lt_of_le_of_ne hε.2.1 hεone
obtain ⟨u, hεu, hu1, hupper⟩ := exists_short_right_cutoff w hεlt
by_contra hbound
have hgap : m < d * ((heavySet w ε).card : ℝ) := lt_of_not_ge hbound
have haffine := truncatedExcess_sub_eq_heavy_increment w A d ε u
hεu.le hupper hε.2.2.1
have hslope : 0 ≤ (u - ε) * (d * ((heavySet w ε).card : ℝ) - m) :=
mul_nonneg (sub_nonneg.mpr hεu.le) (sub_nonneg.mpr hgap.le)
have htest : m * u ≤ truncatedExcess w A d u := by
have htestε := hε.2.2.2
nlinarith
have hu : Admissible w A m d u :=
⟨hε.1.trans hεu.le, hu1,
fun x hx => hε.2.2.1 x (hεu.trans hx), htest⟩
exact (not_lt_of_ge (hmax u hu)) hεu
end TalagrandThreshold.Selector
end OAI
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