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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Probability.MatroidSecretary.Secretary.MaskModel
import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! Exact branch/sacrificed-mask marginal of the complete source law.
This marginal is taken after integrating all source coins, including test,
parity and unused coins. It is not a product-law assertion conditional on the
entire secretary seed.
Pinned source: `sections/secretary.tex`, lines 41--52 and 68--78,
SHA256 `3554fe0f7296782edf63cc7f1305c377e7a7b9d8f8948b9510ea4ad02a702d63`.
These are supporting obligations of `cor:secretary`, not the complete endpoint.
-/
namespace MatroidProphet.Secretary
open MeasureTheory Finset
lemma branchRate_pos (j : Bool) : 0 < branchRate j := by
cases j <;> norm_num [branchRate]
lemma branchRate_lt_one (j : Bool) : branchRate j < 1 := by
cases j <;> norm_num [branchRate]
lemma branchRate_main :
branchRate true = 1 / 2 + (1 - 1 / 2) * (1 / 4 + (1 - 1 / 4) * thinningRate) := by
norm_num [branchRate, thinningRate]
lemma sourceMask_eq_completeHiddenRule {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (r : Seed (mainSeedBits n)) :
sourceMask r = (completeHiddenRule M hE).mask r := by
rfl
lemma triple_mask_expectation {n : ℕ} (f : Finset (Fin n) → ℝ) :
bitsExpectation (fun _ : Fin n => (1 / 2 : ℝ)) univ (fun H =>
bitsExpectation (fun _ : Fin n => (1 / 4 : ℝ)) univ (fun D =>
bitsExpectation (fun _ : Fin n => thinningRate) univ (fun C => f (H ∪ D ∪ C)))) =
bitsExpectation (fun _ : Fin n => branchRate true) univ f := by
have hDC (H : Finset (Fin n)) :
bitsExpectation (fun _ : Fin n => (1 / 4 : ℝ)) univ (fun D =>
bitsExpectation (fun _ : Fin n => thinningRate) univ (fun C => f (H ∪ D ∪ C))) =
bitsExpectation (fun _ : Fin n => 1 / 4 + (1 - 1 / 4) * thinningRate) univ
(fun S => f (H ∪ S)) := by
simpa only [Finset.union_assoc] using
bitsExpectation_union (fun _ : Fin n => (1 / 4 : ℝ))
(fun _ : Fin n => thinningRate) univ (fun S => f (H ∪ S))
simp_rw [hDC]
rw [bitsExpectation_union]
congr 1
funext e
exact branchRate_main.symm
/-- Joint branch/mask law. All unused source coins are integrated, not discarded
from the information available to the adversary. -/
theorem integral_branch_sourceMask {n : ℕ} (j : Bool) (f : Finset (Fin n) → ℝ) :
(∫ r, (if mainBranch r = j then f (sourceMask r) else 0) ∂sourceSeedLaw n) =
(1 / 2 : ℝ) * bitsExpectation (fun _ : Fin n => branchRate j) univ f := by
have h := integral_sourceSeedLaw (fun d b =>
if b = j then f (if b then d.H ∪ d.D ∪ d.C else d.H) else 0)
change (∫ r, (if mainBranch r = j then
f (if mainBranch r then (mainMasks r).H ∪ (mainMasks r).D ∪ (mainMasks r).C
else (mainMasks r).H) else 0) ∂sourceSeedLaw n) = _
rw [h]
cases j
· simp only [fairParityExpectation, Bool.false_eq_true, Bool.true_eq_false,
↓reduceIte, add_zero]
simp only [bitsExpectation_const]
simp only [div_eq_mul_inv, bitsExpectation_mul_right, bitsExpectation_add]
simp only [branchRate, Bool.false_eq_true, ↓reduceIte]
simp only [div_eq_mul_inv, one_mul]
ring
· simp only [fairParityExpectation, Bool.false_eq_true, ↓reduceIte, zero_add]
simp only [bitsExpectation_const]
simp only [div_eq_mul_inv, bitsExpectation_mul_right, bitsExpectation_add]
have ht := triple_mask_expectation f
simp only [div_eq_mul_inv, one_mul] at ht ⊢
rw [ht]
ring
/-- Every branch/mask fiber has its explicit positive Bernoulli product mass. -/
lemma integral_sourceMask_atom {n : ℕ} (j : Bool) (P : Finset (Fin n)) :
(∫ r, (if mainBranch r = j ∧ sourceMask r = P then (1 : ℝ) else 0)
∂sourceSeedLaw n) =
(1 / 2 : ℝ) * bitsWeight (fun _ : Fin n => branchRate j) univ P := by
classical
simpa [bitsExpectation, ite_and] using
integral_branch_sourceMask j (fun S => if S = P then (1 : ℝ) else 0)
lemma bitsWeight_branchRate_pos {n : ℕ} (j : Bool) (P : Finset (Fin n)) :
0 < bitsWeight (fun _ : Fin n => branchRate j) univ P := by
unfold bitsWeight
apply Finset.prod_pos
intro e he
split_ifs
· exact branchRate_pos j
· exact sub_pos.mpr (branchRate_lt_one j)
lemma sourceMask_atom_real {n : ℕ} (j : Bool) (P : Finset (Fin n)) :
(sourceSeedLaw n).real {r | mainBranch r = j ∧ sourceMask r = P} =
(1 / 2 : ℝ) * bitsWeight (fun _ : Fin n => branchRate j) univ P := by
classical
rw [← integral_indicator_one (Set.to_countable _).measurableSet]
simpa only [Set.indicator_apply, Set.mem_ofPred_eq, Pi.one_apply] using
integral_sourceMask_atom j P
lemma sourceMask_atom {n : ℕ} (j : Bool) (P : Finset (Fin n)) :
sourceSeedLaw n {r | mainBranch r = j ∧ sourceMask r = P} =
ENNReal.ofReal ((1 / 2 : ℝ) * bitsWeight (fun _ : Fin n => branchRate j) univ P) := by
rw [← sourceMask_atom_real, ofReal_measureReal]
lemma sourceMask_atom_pos {n : ℕ} (j : Bool) (P : Finset (Fin n)) :
0 < sourceSeedLaw n {r | mainBranch r = j ∧ sourceMask r = P} := by
rw [sourceMask_atom, ENNReal.ofReal_pos]
exact mul_pos (by norm_num) (bitsWeight_branchRate_pos j P)
/-- The same atom identity stated directly for the source algorithm's actual
branch-tagged observation; this is the finite conditional-kernel interface. -/
theorem completeHiddenRule_joint_atom {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (j : Bool) (P : Finset (Fin n)) :
sourceSeedLaw n {r | (mainBranch r, (completeHiddenRule M hE).mask r) = (j, P)} =
ENNReal.ofReal ((1 / 2 : ℝ) * bitsWeight (fun _ : Fin n => branchRate j) univ P) := by
simpa only [Prod.mk.injEq, ← sourceMask_eq_completeHiddenRule M hE] using
sourceMask_atom j P
theorem completeHiddenRule_joint_atom_pos {n : ℕ} (M : Matroid (Fin n))
(hE : M.E = Set.univ) (j : Bool) (P : Finset (Fin n)) :
0 < sourceSeedLaw n {r | (mainBranch r, (completeHiddenRule M hE).mask r) = (j, P)} := by
simpa only [Prod.mk.injEq, ← sourceMask_eq_completeHiddenRule M hE] using
sourceMask_atom_pos j P
end MatroidProphet.Secretary
end OAI
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