| 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 |
|
|
