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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 91 92 93 94 95 96 | import OAI.Probability.MatroidProphet.Main
import OAI.Probability.MatroidSecretary.Transport.TransferSpecification
namespace OAI
/-!
# Source-facing sample-to-rank transfer
Source: `sections/transfer.tex`, `lem:transfer` / `eq:rank-transfer`.
The comparison set is the actual priority-greedy survivor group; its independence
and containment are proved, not supplied as extra assumptions. `mainMean` fixes
only H and averages the density, guard, test, and parity coins. `trueZ` uses zero
on unlisted outcomes, so the expectation is not conditioned on listing.
-/
namespace MatroidProphet.SourceProof
open Finset MainAlgorithm
/-- Conditional averaging depends on the fixed filter mask only. All other
fields of the initial record are overwritten by independent product samples. -/
lemma mainMean_filter_congr {n : ℕ} (d d' : MainMasks n)
(hH : d.H = d'.H) (f : MainMasks n → ℝ) : mainMean d f = mainMean d' f := by
rcases d with ⟨H, D, C, T, p⟩
rcases d' with ⟨H', D', C', T', p'⟩
change H = H' at hH
subst H'
rfl
/-- The unlisted branch is included in the product average with zero statistic. -/
lemma trueZ_unlisted {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(hi : i ∉ groups M d s) : trueZ M hE d s i = 0 := by
classical
simp only [trueZ, ite_eq_right hi]
/-- In particular, a true group missed by the discovery sample contributes zero;
no positivity or listing assumption is imposed on the individual outcomes. -/
lemma trueZ_zero_of_no_discovery {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(hmiss : d.D ∩ trueGroup M d s i = ∅) : trueZ M hE d s i = 0 := by
apply trueZ_unlisted
intro hi
obtain ⟨e, heD, heU⟩ := (mem_listed_iff M d s i).mp ((mem_groups M d s i).mp hi)
have he : e ∈ d.D ∩ trueGroup M d s i := mem_inter.mpr ⟨heD, heU⟩
rw [hmiss] at he
exact notMem_empty e he
/-- The source transfer at arbitrary rounded-level keys. This stronger helper
also covers the `none` keys representing zero/nonpositive weights. -/
theorem transfer_keys {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(s : Fin n → Option ℤ) (H : Finset (Fin n)) (i : ℤ) :
let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩
let U := trueGroup M d s i
let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i)
densityThreshold ≤ (U.card : ℝ) →
((((2 : ℝ)^16)⁻¹ * Y.card - ((2 : ℝ)^23)⁻¹ * U.card) / (2 : ℝ)^100) ≤
mainMean d (fun d' => (trueZ M hE d' s i : ℝ)) := by
classical
dsimp only
intro hn
let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩
let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i)
have hY : (Y : Set (Fin n)) ⊆ (trueGroup M d s i : Set (Fin n)) := by
intro e he
rw [trueGroup_eq_candidate_filter]
rcases mem_filter.mp he with ⟨heG, hei⟩
exact mem_filter.mpr ⟨positiveGreedy_survivors_subset M s H heG, hei⟩
have hI : M.Indep (Y : Set (Fin n)) :=
(positiveGreedy_survivors_indep M hE s H).subset
(fun _ he => (mem_filter.mp he).1)
have h := mainMean_trueZ_lower M hE d s i (Y : Set (Fin n)) hY hI hn
simpa only [Set.ncard_coe_finset, retainedFraction, densityThreshold, d, Y] using h
/-- `lem:transfer` for the manuscript's actual rounded nonnegative weights,
with literal constants gamma = 2^-16 and kappa = 2^100. -/
theorem transfer {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (_hw : ∀ e, 0 ≤ w e) (H : Finset (Fin n)) (i : ℤ) :
let s : Fin n → Option ℤ := fun e => roundedLevel weightBase (w e)
let d : MainMasks n := ⟨H, ∅, ∅, ∅, false⟩
let U := trueGroup M d s i
let Y := (positiveGreedy M s \ H).filter (fun e => s e = some i)
densityThreshold ≤ (U.card : ℝ) →
((((2 : ℝ)^16)⁻¹ * Y.card - ((2 : ℝ)^23)⁻¹ * U.card) / (2 : ℝ)^100) ≤
mainMean d (fun d' => (trueZ M hE d' s i : ℝ)) :=
transfer_keys M hE (fun e => roundedLevel weightBase (w e)) H i
/-- Thin proof alias for the independent root-target specification. -/
theorem transfer_claim : TransferClaim := by
intro n M hE w hw H i
exact transfer M hE w hw H i
end MatroidProphet.SourceProof
end OAI
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