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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 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! Exact branch and sacrificed-mask law of the manuscript's concrete source rule.
The full-seed integral retains unused mask and parity coins. This is the
source-law side of the secretary reconstruction, not a prefix-mask assumption. -/
namespace MatroidProphet
open MeasureTheory Finset
lemma bitsExpectation_union_three {α : Type*} [DecidableEq α]
(p q r : α → ℝ) (V : Finset α) (f : Finset α → ℝ) :
bitsExpectation p V (fun H => bitsExpectation q V (fun D =>
bitsExpectation r V (fun C => f (H ∪ D ∪ C)))) =
bitsExpectation (fun e => p e + (1 - p e) * (q e + (1 - q e) * r e)) V f := by
simp_rw [Finset.union_assoc]
have h (H : Finset α) := bitsExpectation_union q r V (fun U => f (H ∪ U))
simp_rw [h]
exact bitsExpectation_union p (fun e => q e + (1 - q e) * r e) V f
lemma source_main_mask_rate :
(1 / 2 : ℝ) + (1 - 1 / 2) * (1 / 4 + (1 - 1 / 4) * thinningRate) =
5 / 8 + 3 * ((2 : ℝ) ^ 143)⁻¹ := by
norm_num [thinningRate]
lemma source_main_mask_rate_pos : 0 < (5 / 8 : ℝ) + 3 * ((2 : ℝ) ^ 143)⁻¹ := by
norm_num
lemma source_main_mask_rate_lt_one : (5 / 8 : ℝ) + 3 * ((2 : ℝ) ^ 143)⁻¹ < 1 := by
norm_num
lemma completeHiddenRule_mask_eq {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (r : Seed (mainSeedBits n)) :
(completeHiddenRule M hE).mask r =
if mainBranch r then (mainMasks r).H ∪ (mainMasks r).D ∪ (mainMasks r).C
else (mainMasks r).H := by
rfl
lemma integral_source_branch_mask {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (f : Bool → Finset (Fin n) → ℝ) :
(∫ r, f (mainBranch r) ((completeHiddenRule M hE).mask r) ∂sourceSeedLaw n) =
(bitsExpectation (fun _ : Fin n => (1 / 2 : ℝ)) univ (f false) +
bitsExpectation (fun _ : Fin n => (5 / 8 : ℝ) + 3 * ((2 : ℝ) ^ 143)⁻¹)
univ (f true)) / 2 := by
simp_rw [completeHiddenRule_mask_eq]
rw [integral_sourceSeedLaw (fun d b => f b (if b then d.H ∪ d.D ∪ d.C else d.H))]
simp only [fairParityExpectation, Bool.false_eq_true, ↓reduceIte]
simp only [bitsExpectation_const]
have hsplit (H D C : Finset (Fin n)) :
((f false H + f true (H ∪ D ∪ C)) / 2 +
(f false H + f true (H ∪ D ∪ C)) / 2) / 2 =
(1 / 2 : ℝ) * f false H + (1 / 2 : ℝ) * f true (H ∪ D ∪ C) := by ring
simp_rw [hsplit, bitsExpectation_add, bitsExpectation_mul_const, bitsExpectation_const]
rw [bitsExpectation_union_three]
simp_rw [source_main_mask_rate]
ring
end MatroidProphet
end OAI
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