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dd14c26 | 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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
namespace MatroidProphet
open Finset
lemma sum_zpow_lt_two (B : ℝ) (hB : 2 ≤ B) (s : Finset ℤ) (l : ℤ)
(hsl : ∀ i ∈ s, i ≤ l) :
(∑ i ∈ s, B ^ i) < 2 * B ^ l := by
have hB1 : 1 < B := lt_of_lt_of_le (by norm_num) hB
have hs := sum_zpow_lt_upper B hB1 s (l + 1)
(fun i hi => by have := hsl i hi; omega)
have hp : 0 < B ^ l := zpow_pos (by linarith) _
have hden : 0 < B - 1 := by linarith
have hratio : B / (B - 1) ≤ 2 := (div_le_iff₀ hden).mpr (by linarith)
rw [zpow_add_one₀ (ne_of_gt (zero_lt_one.trans hB1)), mul_div_assoc] at hs
nlinarith
lemma sum_zpow_lt_two_of_pos_bound (B : ℝ) (hB : 2 ≤ B) (levels : Finset ℤ)
(m : ℝ) (hm : 0 < m) (hbound : ∀ i ∈ levels, B ^ i ≤ m) :
(∑ i ∈ levels, B ^ i) < 2 * m := by
classical
by_cases hn : levels.Nonempty
· have h := sum_zpow_lt_two B hB levels (levels.max' hn)
(fun i hi => levels.le_max' i hi)
exact h.trans_le (mul_le_mul_of_nonneg_left
(hbound _ (levels.max'_mem hn)) (by norm_num))
· have he : levels = ∅ := Finset.not_nonempty_iff_eq_empty.mp hn
simp only [he, Finset.sum_empty]
positivity
lemma small_group_mass_strict (B κ m : ℝ) (hB : 2 ≤ B) (hκ : 0 < κ) (hm : 0 < m)
(levels : Finset ℤ) (Y : ℤ → ℝ) (hY : ∀ i ∈ levels, Y i ≤ κ)
(hbound : ∀ i ∈ levels, B ^ i ≤ m) :
(∑ i ∈ levels, B ^ i * Y i) < 2 * κ * m := by
have hB0 : 0 < B := lt_of_lt_of_le (by norm_num) hB
calc
_ ≤ ∑ i ∈ levels, B ^ i * κ :=
Finset.sum_le_sum fun i hi => mul_le_mul_of_nonneg_left (hY i hi) (zpow_pos hB0 i).le
_ = κ * ∑ i ∈ levels, B ^ i := by rw [← Finset.sum_mul]; ring
_ < κ * (2 * m) := mul_lt_mul_of_pos_left
(sum_zpow_lt_two_of_pos_bound B hB levels m hm hbound) hκ
_ = _ := by ring
lemma maximumRounded_pos_of_optimum_pos {n : ℕ} (M : Matroid (Fin n)) (w : Weights n)
(hW : 0 < optimum M (fun e => roundedWeight weightBase (w e))) :
0 < maximumRounded M w := by
classical
by_contra! hm
have hopt : optimum M (fun e => roundedWeight weightBase (w e)) ≤ 0 := by
apply optimum_le_of_independent_bound M _ 0 le_rfl
intro I hI
apply Finset.sum_nonpos
intro e he
have hei : M.Indep ({e} : Set (Fin n)) := hI.subset (by
intro f hf
have hfe : f = e := Set.mem_singleton_iff.mp hf
change f ∈ I
rwa [hfe])
exact (roundedWeight_le_maximumRounded M w e hei).trans hm
exact (not_lt_of_ge hopt) hW
namespace MainAlgorithm
/-- Exact strict version of the discarded-group bound in the positive-W case. -/
lemma small_comparison_mass_strict {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (w : Weights n) (H : Finset (Fin n))
(hW : 0 < optimum M (fun e => roundedWeight weightBase (w e))) :
comparisonMass M (fun e => roundedLevel weightBase (w e)) H <
analyzedComparisonMass M (fun e => roundedLevel weightBase (w e)) H +
2 * densityThreshold * maximumRounded M w := by
classical
let a : Fin n → Option ℤ := fun e => roundedLevel weightBase (w e)
let Y := positiveGreedy M a \ H
let U := candidateLabels M a H
let L := trueLevels M a H
let A := analyzedLevels M a H
have hAU : A ⊆ L := Finset.filter_subset _ _
have hsmall : ∀ i ∈ L \ A, ((Y.filter (fun e => a e = some i)).card : ℝ) ≤ densityThreshold := by
intro i hi
obtain ⟨hiL, hiA⟩ := Finset.mem_sdiff.mp hi
have hn : ¬densityThreshold ≤ ((U.filter (fun e => a e = some i)).card : ℝ) := by
intro hh
exact hiA (Finset.mem_filter.mpr ⟨hiL, hh⟩)
have hc : (Y.filter (fun e => a e = some i)).card ≤ (U.filter (fun e => a e = some i)).card :=
Finset.card_le_card (Finset.filter_subset_filter _ (positiveGreedy_survivors_subset M a H))
have hcR : ((Y.filter (fun e => a e = some i)).card : ℝ) ≤
((U.filter (fun e => a e = some i)).card : ℝ) := by exact_mod_cast hc
exact hcR.trans (le_of_lt (lt_of_not_ge hn))
have hbound := small_group_mass_strict weightBase densityThreshold (maximumRounded M w)
(by norm_num [weightBase]) constants_positive.2.1 (maximumRounded_pos_of_optimum_pos M w hW)
(L \ A) (fun i => ((Y.filter (fun e => a e = some i)).card : ℝ)) hsmall
(fun i hi => candidate_level_le_maximumRounded M hE w H i (Finset.mem_sdiff.mp hi).1)
have hpartition := Finset.sum_sdiff hAU
(f := fun i => weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ))
rw [comparisonMass_partition]
change (∑ i ∈ L, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) <
(∑ i ∈ A, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) + _
linarith
end MainAlgorithm
end MatroidProphet
end OAI
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