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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Combinatorics.ExpectationThreshold.RoundingMasses
import OAI.Combinatorics.ExpectationThreshold.RoundingColorProducts
import OAI.Combinatorics.ExpectationThreshold.RoundingExpectation
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
namespace OAI
/-!
# The deterministic lower bound for rounding
A weighted average at most two places at least half the mass on subsets with
average at most four. On those subsets the quotient-of-powers product is at
least one, yielding the lower bound one quarter for the full product score.
This statement is pointwise in the coloring and needs no independence premise.
-/
open scoped BigOperators
namespace TalagrandThreshold
/-- Finite weighted Markov inequality in the form used by rounding. -/
theorem half_mass_of_weighted_average_le {β : Type*} (T : Finset β)
(w v : β → ℝ) (hw : ∀ i ∈ T, 0 ≤ w i) (hv : ∀ i ∈ T, 0 ≤ v i)
(hmean : (∑ i ∈ T, w i * v i) ≤ 2 * (∑ i ∈ T, w i)) :
(∑ i ∈ T, w i) / 2 ≤ ∑ i ∈ T.filter (fun i => v i ≤ 4), w i := by
classical
have hpoint (i : β) (hi : i ∈ T) :
4 * w i ≤ w i * v i + 4 * (if v i ≤ 4 then w i else 0) := by
by_cases hgood : v i ≤ 4
· simp only [ite_eq_left hgood]
have hnonneg := mul_nonneg (hw i hi) (hv i hi)
linarith
· simp only [ite_eq_right hgood, mul_zero, add_zero]
calc
4 * w i = w i * 4 := mul_comm _ _
_ ≤ w i * v i :=
mul_le_mul_of_nonneg_left (le_of_lt (lt_of_not_ge hgood)) (hw i hi)
have hsum : 4 * (∑ i ∈ T, w i) ≤
(∑ i ∈ T, w i * v i) + 4 * (∑ i ∈ T.filter (fun i => v i ≤ 4), w i) := by
calc
_ = ∑ i ∈ T, 4 * w i := Finset.mul_sum _ _ _
_ ≤ ∑ i ∈ T, (w i * v i + 4 * (if v i ≤ 4 then w i else 0)) :=
Finset.sum_le_sum hpoint
_ = _ := by
simp only [Finset.sum_add_distrib, Finset.mul_sum, Finset.sum_filter,
mul_ite, mul_zero]
linarith
variable {α : Type*} [Fintype α] [DecidableEq α]
omit [Fintype α] [DecidableEq α] in
theorem subsetAverage_nat_nonneg (a : α → ℕ) (S : Finset α) :
0 ≤ subsetAverage (fun x => (a x : ℝ)) S := by
unfold subsetAverage
exact div_nonneg (Finset.sum_nonneg (fun x _ => Nat.cast_nonneg (a x)))
(Nat.cast_nonneg S.card)
omit [Fintype α] [DecidableEq α] in
theorem sum_le_four_card_of_subsetAverage_le (a : α → ℕ) {S : Finset α}
(hS : S.Nonempty) (haverage : subsetAverage (fun x => (a x : ℝ)) S ≤ 4) :
(∑ x ∈ S, a x) ≤ 4 * S.card := by
have hcard : (0 : ℝ) < S.card := Nat.cast_pos.mpr (Finset.card_pos.mpr hS)
have hreal : (∑ x ∈ S, (a x : ℝ)) ≤ 4 * (S.card : ℝ) :=
(div_le_iff₀ hcard).mp haverage
exact_mod_cast hreal
/-- The fractional member mass and its mean color force a product-score lower
bound. Natural colors can include zero; the actual selector uses `val + 1`. -/
theorem productScore_lower_of_mean (g : Finset α → ℝ) (H : Finset α)
(a : α → ℕ) (hg : ∀ S, 0 ≤ g S) (hMH : (1 / 2 : ℝ) ≤ memberMass g H)
(hmean : (∑ x : α, (a x : ℝ) * roundingMass g H x) ≤ 2) :
(1 / 4 : ℝ) ≤ productScore g (fun x => (512 : ℝ) ^ 4 / (512 : ℝ) ^ a x) := by
classical
have hMpos : 0 < memberMass g H := by linarith
have hweighted :
(∑ S ∈ nonemptySubsets H, g S * subsetAverage (fun x => (a x : ℝ)) S) ≤
2 * memberMass g H := by
apply (div_le_iff₀ hMpos).mp
rw [← roundingMass_mean g H (fun x => (a x : ℝ))]
exact hmean
let G := (nonemptySubsets H).filter
(fun S => subsetAverage (fun x => (a x : ℝ)) S ≤ 4)
have hgood : memberMass g H / 2 ≤ ∑ S ∈ G, g S := by
simpa only [G, memberMass] using
half_mass_of_weighted_average_le (nonemptySubsets H) g
(fun S => subsetAverage (fun x => (a x : ℝ)) S)
(fun S _ => hg S) (fun S _ => subsetAverage_nat_nonneg a S) hweighted
have hsub : G ⊆ (Finset.univ.erase ∅ : Finset (Finset α)) := by
intro S hS
have hS' : S ∈ nonemptySubsets H := (Finset.mem_filter.mp hS).1
have hSne : S ≠ ∅ := (mem_nonemptySubsets.mp hS').1
exact Finset.mem_erase.mpr ⟨hSne, Finset.mem_univ S⟩
have hproduct (S : Finset α) (hS : S ∈ G) :
1 ≤ ∏ x ∈ S, (512 : ℝ) ^ 4 / (512 : ℝ) ^ a x := by
have hS' := Finset.mem_filter.mp hS
have hSne : S ≠ ∅ := (mem_nonemptySubsets.mp hS'.1).1
exact one_le_colorProduct_of_sum_le S a
(sum_le_four_card_of_subsetAverage_le a
(Finset.nonempty_iff_ne_empty.mpr hSne) hS'.2)
have hY (x : α) : 0 ≤ (512 : ℝ) ^ 4 / (512 : ℝ) ^ a x :=
div_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (by norm_num) _)
calc
(1 / 4 : ℝ) ≤ memberMass g H / 2 := by linarith
_ ≤ ∑ S ∈ G, g S := hgood
_ ≤ ∑ S ∈ G, g S * ∏ x ∈ S, (512 : ℝ) ^ 4 / (512 : ℝ) ^ a x := by
apply Finset.sum_le_sum
intro S hS
simpa only [mul_one] using mul_le_mul_of_nonneg_left (hproduct S hS) (hg S)
_ ≤ productScore g (fun x => (512 : ℝ) ^ 4 / (512 : ℝ) ^ a x) := by
unfold productScore
exact Finset.sum_le_sum_of_subset_of_nonneg hsub
(fun S _ _ => mul_nonneg (hg S) (Finset.prod_nonneg (fun x _ => hY x)))
end TalagrandThreshold
end OAI
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