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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 | 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
end OAI
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