| 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 |
|
|
