| import OAI.Probability.MatroidProphet.Residual.Probability |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Finset |
| variable {α : Type*} [DecidableEq α] |
|
|
| lemma bitsExpectation_union (p q : α → ℝ) (V : Finset α) (f : Finset α → ℝ) : |
| bitsExpectation p V (fun D => bitsExpectation q V (fun C => f (D ∪ C))) = |
| bitsExpectation (fun e => p e + (1 - p e) * q e) V f := by |
| induction V using Finset.induction_on generalizing f with |
| | empty => simp |
| | @insert e V he ih => |
| rw [bitsExpectation_insert p V e he, |
| bitsExpectation_insert (fun e => p e + (1 - p e) * q e) V e he] |
| have hno : bitsExpectation p V (fun D => bitsExpectation q (insert e V) (fun C => f (D ∪ C))) = |
| (1 - q e) * bitsExpectation p V (fun D => bitsExpectation q V (fun C => f (D ∪ C))) + |
| q e * bitsExpectation p V (fun D => bitsExpectation q V (fun C => f (insert e (D ∪ C)))) := by |
| simp_rw [bitsExpectation_insert q V e he, union_insert] |
| rw [bitsExpectation_add, bitsExpectation_mul_const, bitsExpectation_mul_const] |
| have hyes : bitsExpectation p V (fun D => bitsExpectation q (insert e V) (fun C => f (insert e D ∪ C))) = |
| bitsExpectation p V (fun D => bitsExpectation q V (fun C => f (insert e (D ∪ C)))) := by |
| apply bitsExpectation_congr |
| intro D hD |
| rw [bitsExpectation_insert q V e he] |
| simp only [insert_union, union_insert, insert_idem] |
| ring |
| rw [hno, hyes, ih f, ih (fun S => f (insert e S))] |
| ring |
|
|
| end MatroidProphet |
|
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| end OAI |
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|
