openAI-math-dataset / lean /OAI /Probability /MatroidSecretary /Accounting /ConditionalExpectation.lean
| import OAI.Probability.MatroidSecretary.Accounting.Contracts | |
| import OAI.Probability.MatroidSecretary.Sampling.BernoulliConditionalFiber | |
| namespace OAI | |
| /-! | |
| # Literal conditional expectation in the order-uniform accounting lemma | |
| The canonical accounting contract averages fresh focal bits after fixing all | |
| outside bits. This corollary identifies that average with the normalized | |
| restriction of the full product distribution to the specified outside event. | |
| No order is conditioned on, and all other mask fields remain fixed. | |
| -/ | |
| namespace MatroidProphet.AccountingContracts | |
| open Set Finset MainAlgorithm | |
| /-- The fixed-outside test-thinning clause, expressed as a conditional | |
| expectation under the entire independent test-mask law. Every outside atom | |
| has positive mass at the source's interior thinning rate. -/ | |
| theorem all_orders_conditional_expectation {n : ℕ} | |
| (M : Matroid (Fin n)) (hE : M.E = Set.univ) | |
| (d : MainMasks n) (w : Fin n → Option ℤ) (h : ℕ) | |
| (A : Finset (Fin n)) (hA : Disjoint A (pathGroups M d w h)) : | |
| thinningRate * (finalRankStatistic M hE (2^100) | |
| (groupMask M d w d.D) (groupMask M d w d.C) | |
| (fun j => (A : Set (Fin n)) ∩ (pathGroups M d w j : Set (Fin n))) | |
| h (groups M d w).length (pathGroups M d w h) | |
| ((d.H ∪ d.D ∪ d.C : Finset (Fin n)) : Set (Fin n)) Set.univ | |
| (boolParity d.odd) : ℝ) ≤ | |
| bitsExpectation (fun _ : Fin n => thinningRate) univ | |
| (fun T => if T \ pathGroups M d w h = A then | |
| (listedLambda M hE {d with T := T} w h : ℝ) else 0) / | |
| bitsExpectation (fun _ : Fin n => thinningRate) univ | |
| (fun T => if T \ pathGroups M d w h = A then 1 else 0) := by | |
| have hsub : A ⊆ univ \ pathGroups M d w h := by | |
| intro e he | |
| exact Finset.mem_sdiff.mpr ⟨Finset.mem_univ e, | |
| fun heG => Finset.disjoint_left.mp hA he heG⟩ | |
| rw [bitsExpectation_condition_on_outside_const thinningRate | |
| (by norm_num [thinningRate]) (by norm_num [thinningRate]) | |
| univ (pathGroups M d w h) A (Finset.subset_univ _) hsub] | |
| exact (all_orders M hE d w h).2 A hA | |
| end MatroidProphet.AccountingContracts | |
| end OAI | |