| 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 |
|
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| end OAI |
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