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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 | import OAI.Probability.MatroidProphet.ProductRestriction
import OAI.Probability.MatroidProphet.AdaptiveResampling
namespace OAI
namespace MatroidProphet
open Finset
variable {α β : Type*} [DecidableEq α] [DecidableEq β]
lemma bitsExpectation_comm (p : α → ℝ) (q : β → ℝ) (U : Finset α) (V : Finset β)
(f : Finset α → Finset β → ℝ) :
bitsExpectation p U (fun S => bitsExpectation q V (f S)) =
bitsExpectation q V (fun T => bitsExpectation p U (fun S => f S T)) := by
simp only [bitsExpectation, mul_sum]
rw [sum_comm]
apply sum_congr rfl
intro T hT
apply sum_congr rfl
intro S hS
ring
lemma bitsExpectation_disjoint_union (q : α → ℝ) (U V : Finset α) (hUV : Disjoint U V)
(f : Finset α → ℝ) :
bitsExpectation q (U ∪ V) f =
bitsExpectation q U (fun S => bitsExpectation q V (fun T => f (S ∪ T))) := by
induction U using Finset.induction_on generalizing f with
| empty => simp only [empty_union, bitsExpectation_empty]
| @insert e U he ih =>
have heV : e ∉ V := fun hx => (Finset.disjoint_left.mp hUV (mem_insert_self e U) hx)
have hU : Disjoint U V := hUV.mono_left (subset_insert e U)
rw [insert_union, bitsExpectation_insert q (U ∪ V) e (by simp [he, heV]),
bitsExpectation_insert q U e he, ih hU f, ih hU (fun S => f (insert e S))]
simp only [insert_union]
lemma bitsExpectation_split (q : α → ℝ) (V U : Finset α) (hU : U ⊆ V)
(f : Finset α → ℝ) :
bitsExpectation q V f =
bitsExpectation q (V \ U) (fun S => bitsExpectation q U (fun T => f (S ∪ T))) := by
have heq : (V \ U) ∪ U = V := sdiff_union_of_subset hU
rw [← bitsExpectation_disjoint_union q (V \ U) U sdiff_disjoint f, heq]
end MatroidProphet
end OAI
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