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a2ee19a | 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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
namespace MatroidProphet.SourceContracts
open Finset Set MainAlgorithm
/-- Positive rounding does not remove a positive weight or create one at zero. -/
theorem roundedWeight_pos_iff {B x : ℝ} (hB : 0 < B) :
0 < roundedWeight B x ↔ 0 < x := by
unfold roundedWeight roundedLevel
by_cases hx : 0 < x
· rw [ite_eq_left hx, levelWeight]
exact iff_of_true (zpow_pos hB _) hx
· rw [ite_eq_right hx, levelWeight]
exact iff_of_false (lt_irrefl 0) hx
/-- The actual candidate test is exactly `eq:candidates`, with higher rounded
levels taking priority and the original fixed label order breaking ties. -/
theorem mem_candidateLabels_iff {n : ℕ} (M : Matroid (Fin n))
(w : Weights n) (B : ℝ) (H : Finset (Fin n)) (e : Fin n) :
e ∈ candidateLabels M (fun f => roundedLevel B (w f)) H ↔
0 < w e ∧ e ∉ H ∧
e ∉ M.closure {f | f ∈ H ∧
higher f (roundedLevel B (w f)) e (roundedLevel B (w e))} := by
classical
have hpos : roundedLevel B (w e) ≠ none ↔ 0 < w e := by
simp [roundedLevel]
constructor
· intro hcandidate
obtain ⟨hlevel, houtside, hclosure⟩ := (Finset.mem_filter.mp hcandidate).2
exact ⟨hpos.mp hlevel, houtside, hclosure⟩
· rintro ⟨hpositive, houtside, hclosure⟩
exact Finset.mem_filter.mpr
⟨Finset.mem_univ _, hpos.mpr hpositive, houtside, hclosure⟩
/-- The same test in the source's literal `u_e > 0` notation. -/
theorem mem_candidateLabels_iff_rounded {n : ℕ} (M : Matroid (Fin n))
(w : Weights n) {B : ℝ} (hB : 0 < B) (H : Finset (Fin n)) (e : Fin n) :
e ∈ candidateLabels M (fun f => roundedLevel B (w f)) H ↔
0 < roundedWeight B (w e) ∧ e ∉ H ∧
e ∉ M.closure {f | f ∈ H ∧
higher f (roundedLevel B (w f)) e (roundedLevel B (w e))} := by
rw [roundedWeight_pos_iff hB]
exact mem_candidateLabels_iff M w B H e
/-- Candidates cannot be loops, including when loops have positive weights. -/
theorem candidateLabels_nonloop {n : ℕ} (M : Matroid (Fin n))
(a : Fin n → Option ℤ) (H : Finset (Fin n)) (e : Fin n)
(he : e ∈ candidateLabels M a H) : e ∉ M.closure ∅ := by
classical
exact candidate_not_mem_closure_empty M ⟨H, ∅, ∅, ∅, false⟩ a e (a e)
(Finset.mem_filter.mp he).2
/-- Both deterministic assertions and both expectation assertions in `lem:filter`.
The assumptions include no positive-rank, nonempty-ground, or no-ties condition. -/
theorem candidate_mass {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (_hw : ∀ e, 0 ≤ w e) {B : ℝ} (hB : 1 < B) :
(∀ H : Finset (Fin n),
M.Indep ((positiveGreedy M (fun e => roundedLevel B (w e)) \ H :
Finset (Fin n)) : Set (Fin n)) ∧
positiveGreedy M (fun e => roundedLevel B (w e)) \ H ⊆
candidateLabels M (fun e => roundedLevel B (w e)) H) ∧
bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ
(fun H => ∑ e ∈ positiveGreedy M (fun e => roundedLevel B (w e)) \ H,
roundedWeight B (w e)) = optimum M (fun e => roundedWeight B (w e)) / 2 ∧
bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ
(fun H => ∑ e ∈ candidateLabels M (fun e => roundedLevel B (w e)) H,
roundedWeight B (w e)) ≤ optimum M (fun e => roundedWeight B (w e)) := by
refine ⟨?_, expectation_positiveGreedy_survivors M hE w hB,
expectation_candidateLabels_le_optimum M hE w B⟩
intro H
exact ⟨positiveGreedy_survivors_indep M hE _ H,
positiveGreedy_survivors_subset M _ H⟩
/-- The two-sided optimum rounding estimate `eq:rounding`, including zero weights. -/
theorem optimum_rounding {n : ℕ} (M : Matroid (Fin n))
(w : Weights n) (hw : ∀ e, 0 ≤ w e) {B : ℝ} (hB : 1 < B) :
optimum M w / B ≤ optimum M (fun e => roundedWeight B (w e)) ∧
optimum M (fun e => roundedWeight B (w e)) ≤ optimum M w := by
constructor
· apply (div_le_iff₀ (zero_lt_one.trans hB)).2
simpa only [mul_comm] using optimum_rounding_bound M B hB w hw
· exact optimum_mono M _ w (fun e => roundedWeight_le hB (hw e))
end MatroidProphet.SourceContracts
end OAI
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