File size: 3,564 Bytes
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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import OAI.Probability.MatroidProphet.BatchReveal
namespace OAI
namespace MatroidProphet
open Finset
variable {α β : Type*} [DecidableEq α]
theorem RevealTree.continuation_expectation (tree : RevealTree α β)
(q : α → ℝ) (V : Finset α) (fresh : tree.Fresh V)
(f : β → Finset α → ℝ) :
bitsExpectation q V (fun S => f (tree.run S) (S \ tree.queried S)) =
bitsExpectation q V (fun S => bitsExpectation q (V \ tree.queried S)
(fun T => f (tree.run S) T)) := by
induction tree generalizing V with
| leaf b => simp only [run, queried, sdiff_empty, bitsExpectation_const]
| query e no yes ihn ihy =>
rcases fresh with ⟨he, hn, hy⟩
let W := V.erase e
have heW : e ∉ W := notMem_erase e V
have hWV : insert e W = V := insert_erase he
have leftSplit := bitsExpectation_insert q W e heW
(fun S => f ((RevealTree.query e no yes).run S)
(S \ (RevealTree.query e no yes).queried S))
rw [hWV] at leftSplit
have rightSplit := bitsExpectation_insert q W e heW
(fun S => bitsExpectation q (V \ (RevealTree.query e no yes).queried S)
(fun T => f ((RevealTree.query e no yes).run S) T))
rw [hWV] at rightSplit
have hSD (S Q : Finset α) (hS : S ⊆ W) :
S \ insert e Q = S \ Q ∧ insert e S \ insert e Q = S \ Q := by
have heS : e ∉ S := fun h => heW (hS h)
constructor <;> ext x <;> by_cases hxe : x = e
· subst x; simp [heS]
· simp [hxe]
· subst x; simp [heS]
· simp [hxe]
have hnoLeft : bitsExpectation q W (fun S => f ((RevealTree.query e no yes).run S)
(S \ (RevealTree.query e no yes).queried S)) =
bitsExpectation q W (fun S => f (no.run S) (S \ no.queried S)) := by
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun h => heW (hS h)
simp only [run, queried, ite_eq_right heS, erase_eq_of_notMem heS, (hSD S _ hS).1]
have hyesLeft : bitsExpectation q W (fun S => f ((RevealTree.query e no yes).run (insert e S))
(insert e S \ (RevealTree.query e no yes).queried (insert e S))) =
bitsExpectation q W (fun S => f (yes.run S) (S \ yes.queried S)) := by
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun h => heW (hS h)
simp only [run, queried, mem_insert_self, ite_true, erase_insert heS, (hSD S _ hS).2]
have hnoRight : bitsExpectation q W (fun S =>
bitsExpectation q (V \ (RevealTree.query e no yes).queried S)
(fun T => f ((RevealTree.query e no yes).run S) T)) =
bitsExpectation q W (fun S => bitsExpectation q (W \ no.queried S)
(fun T => f (no.run S) T)) := by
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun h => heW (hS h)
simp only [run, queried, ite_eq_right heS, erase_eq_of_notMem heS,
← erase_sdiff_eq_sdiff_insert, W]
have hyesRight : bitsExpectation q W (fun S =>
bitsExpectation q (V \ (RevealTree.query e no yes).queried (insert e S))
(fun T => f ((RevealTree.query e no yes).run (insert e S)) T)) =
bitsExpectation q W (fun S => bitsExpectation q (W \ yes.queried S)
(fun T => f (yes.run S) T)) := by
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun h => heW (hS h)
simp only [run, queried, mem_insert_self, ite_true, erase_insert heS,
← erase_sdiff_eq_sdiff_insert, W]
rw [leftSplit, rightSplit, hnoLeft, hyesLeft, hnoRight, hyesRight, ihn W hn, ihy W hy]
end MatroidProphet
end OAI
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