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namespace OAI
namespace MatroidProphet
open Finset
variable {α β : Type*} [DecidableEq α]
lemma bitsExpectation_mul_left (q : α → ℝ) (V : Finset α) (c : ℝ) (f : Finset α → ℝ) :
bitsExpectation q V (fun S => c * f S) = c * bitsExpectation q V f := by
simp only [bitsExpectation, mul_sum]
apply sum_congr rfl
intro S hS
ring
lemma bitsExpectation_mul_right (q : α → ℝ) (V : Finset α) (c : ℝ) (f : Finset α → ℝ) :
bitsExpectation q V (fun S => f S * c) = bitsExpectation q V f * c := by
simp only [bitsExpectation, sum_mul]
apply sum_congr rfl
intro S hS
ring
inductive RevealTree (α β : Type*) where
| leaf (value : β)
| query (e : α) (no yes : RevealTree α β)
def RevealTree.Fresh : RevealTree α β → Finset α → Prop
| .leaf _, _ => True
| .query e no yes, V => e ∈ V ∧ no.Fresh (V.erase e) ∧ yes.Fresh (V.erase e)
def RevealTree.run : RevealTree α β → Finset α → β
| .leaf value, _ => value
| .query e no yes, S => if e ∈ S then yes.run (S.erase e) else no.run (S.erase e)
def RevealTree.queried : RevealTree α β → Finset α → Finset α
| .leaf _, _ => ∅
| .query e no yes, S => insert e (if e ∈ S then yes.queried (S.erase e) else no.queried (S.erase e))
def RevealTree.hybrid : RevealTree α β → Finset α → Finset α → Finset α
| .leaf _, S, _ => S
| .query e no yes, S, T =>
let H := if e ∈ S then yes.hybrid (S.erase e) (T.erase e)
else no.hybrid (S.erase e) (T.erase e)
if e ∈ T then insert e H else H
lemma RevealTree.queried_subset (tree : RevealTree α β) (V S : Finset α) (fresh : tree.Fresh V) :
tree.queried S ⊆ V := by
induction tree generalizing V S with
| leaf value => exact empty_subset V
| query e no yes ihn ihy =>
rcases fresh with ⟨he, hn, hy⟩
apply insert_subset he
by_cases heS : e ∈ S
· simpa [queried, heS] using (ihy (V.erase e) (S.erase e) hy).trans (erase_subset e V)
· simpa [queried, heS] using (ihn (V.erase e) (S.erase e) hn).trans (erase_subset e V)
lemma RevealTree.run_query_no (e : α) (no yes : RevealTree α β) (S : Finset α) (he : e ∉ S) :
(RevealTree.query e no yes).run S = no.run S := by simp [run, he]
lemma RevealTree.run_query_yes (e : α) (no yes : RevealTree α β) (S : Finset α) (he : e ∉ S) :
(RevealTree.query e no yes).run (insert e S) = yes.run S := by simp [run, he]
lemma RevealTree.hybrid_query (e : α) (no yes : RevealTree α β) (S T : Finset α)
(heS : e ∉ S) (heT : e ∉ T) :
(RevealTree.query e no yes).hybrid S T = no.hybrid S T ∧
(RevealTree.query e no yes).hybrid S (insert e T) = insert e (no.hybrid S T) ∧
(RevealTree.query e no yes).hybrid (insert e S) T = yes.hybrid S T ∧
(RevealTree.query e no yes).hybrid (insert e S) (insert e T) = insert e (yes.hybrid S T) := by
simp [hybrid, heS, heT]
lemma RevealTree.mean_run_query (q : α → ℝ) (V : Finset α) (e : α)
(no yes : RevealTree α β) (φ : β → ℝ) (he : e ∈ V) :
bitsExpectation q V (fun S => φ ((RevealTree.query e no yes).run S)) =
(1-q e) * bitsExpectation q (V.erase e) (fun S => φ (no.run S)) +
q e * bitsExpectation q (V.erase e) (fun S => φ (yes.run S)) := by
have hs := bitsExpectation_insert q (V.erase e) e (notMem_erase e V)
(fun S => φ ((RevealTree.query e no yes).run S))
rw [insert_erase he] at hs
rw [hs]
congr 1
· congr 1
apply bitsExpectation_congr
intro S hS
rw [run_query_no _ _ _ _ (fun heS => notMem_erase e V (hS heS))]
· congr 1
apply bitsExpectation_congr
intro S hS
rw [run_query_yes _ _ _ _ (fun heS => notMem_erase e V (hS heS))]
theorem RevealTree.hybrid_independent (tree : RevealTree α β)
(q : α → ℝ) (V : Finset α) (fresh : tree.Fresh V)
(φ : β → ℝ) (ψ : Finset α → ℝ) :
bitsExpectation q V (fun S => bitsExpectation q V (fun T => φ (tree.run S) * ψ (tree.hybrid S T))) =
bitsExpectation q V (fun S => φ (tree.run S)) * bitsExpectation q V ψ := by
induction tree generalizing V ψ with
| leaf value =>
simp only [run, hybrid, bitsExpectation_const]
exact bitsExpectation_mul_left q V (φ value) ψ
| 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 hsplit (S : Finset α) := bitsExpectation_insert q W e heW
(fun T => φ ((RevealTree.query e no yes).run S) * ψ ((RevealTree.query e no yes).hybrid S T))
have hNo : bitsExpectation q W (fun S => bitsExpectation q V
(fun T => φ ((RevealTree.query e no yes).run S) * ψ ((RevealTree.query e no yes).hybrid S T))) =
(1-q e) * bitsExpectation q W (fun S => bitsExpectation q W
(fun T => φ (no.run S) * ψ (no.hybrid S T))) +
q e * bitsExpectation q W (fun S => bitsExpectation q W
(fun T => φ (no.run S) * ψ (insert e (no.hybrid S T)))) := by
rw [← bitsExpectation_mul_left, ← bitsExpectation_mul_left, ← bitsExpectation_add]
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun hs => heW (hS hs)
have h := hsplit S
rw [hWV] at h
rw [h]
congr 1
· congr 1
apply bitsExpectation_congr
intro T hT
have heT : e ∉ T := fun ht => heW (hT ht)
rw [run_query_no _ _ _ _ heS, (hybrid_query e no yes S T heS heT).1]
· congr 1
apply bitsExpectation_congr
intro T hT
have heT : e ∉ T := fun ht => heW (hT ht)
rw [run_query_no _ _ _ _ heS, (hybrid_query e no yes S T heS heT).2.1]
have hYes : bitsExpectation q W (fun S => bitsExpectation q V
(fun T => φ ((RevealTree.query e no yes).run (insert e S)) *
ψ ((RevealTree.query e no yes).hybrid (insert e S) T))) =
(1-q e) * bitsExpectation q W (fun S => bitsExpectation q W
(fun T => φ (yes.run S) * ψ (yes.hybrid S T))) +
q e * bitsExpectation q W (fun S => bitsExpectation q W
(fun T => φ (yes.run S) * ψ (insert e (yes.hybrid S T)))) := by
rw [← bitsExpectation_mul_left, ← bitsExpectation_mul_left, ← bitsExpectation_add]
apply bitsExpectation_congr
intro S hS
have heS : e ∉ S := fun hs => heW (hS hs)
have h := hsplit (insert e S)
rw [hWV] at h
rw [h]
congr 1
· congr 1
apply bitsExpectation_congr
intro T hT
have heT : e ∉ T := fun ht => heW (hT ht)
rw [run_query_yes _ _ _ _ heS, (hybrid_query e no yes S T heS heT).2.2.1]
· congr 1
apply bitsExpectation_congr
intro T hT
have heT : e ∉ T := fun ht => heW (hT ht)
rw [run_query_yes _ _ _ _ heS, (hybrid_query e no yes S T heS heT).2.2.2]
have hOuter := bitsExpectation_insert q W e heW
(fun S => bitsExpectation q V (fun T => φ ((RevealTree.query e no yes).run S) *
ψ ((RevealTree.query e no yes).hybrid S T)))
rw [hWV, hNo, hYes] at hOuter
rw [hOuter, ihn W hn ψ, ihn W hn (fun S => ψ (insert e S)),
ihy W hy ψ, ihy W hy (fun S => ψ (insert e S)), mean_run_query q V e no yes φ he]
have hPsi := bitsExpectation_insert q W e heW ψ
rw [hWV] at hPsi
rw [hPsi]
ring
lemma RevealTree.hybrid_eq_resample (tree : RevealTree α β) (V S T : Finset α)
(fresh : tree.Fresh V) (hS : S ⊆ V) (hT : T ⊆ V) :
tree.hybrid S T = (S \ tree.queried S) ∪ (T ∩ tree.queried S) := by
induction tree generalizing V S T with
| leaf value => simp [hybrid, queried]
| query e no yes ihn ihy =>
rcases fresh with ⟨he, hn, hy⟩
have hSerase : S.erase e ⊆ V.erase e := erase_subset_erase e hS
have hTerase : T.erase e ⊆ V.erase e := erase_subset_erase e hT
by_cases heS : e ∈ S
· have hq := yes.queried_subset (V.erase e) (S.erase e) hy
have heq : e ∉ yes.queried (S.erase e) := fun h => notMem_erase e V (hq h)
have hi := ihy (V.erase e) (S.erase e) (T.erase e) hy hSerase hTerase
simp only [hybrid, queried, ite_eq_left heS, hi]
by_cases heT : e ∈ T
· rw [ite_eq_left heT]
ext x
by_cases hxe : x = e
· subst x; simp [heS, heT, heq]
· simp [hxe]
· rw [ite_eq_right heT]
ext x
by_cases hxe : x = e
· subst x; simp [heS, heT, heq]
· simp [hxe]
· have hq := no.queried_subset (V.erase e) (S.erase e) hn
have heq : e ∉ no.queried (S.erase e) := fun h => notMem_erase e V (hq h)
have hi := ihn (V.erase e) (S.erase e) (T.erase e) hn hSerase hTerase
simp only [hybrid, queried, ite_eq_right heS, hi]
by_cases heT : e ∈ T
· rw [ite_eq_left heT]
ext x
by_cases hxe : x = e
· subst x; simp [heS, heT]
· simp [hxe]
· rw [ite_eq_right heT]
ext x
by_cases hxe : x = e
· subst x; simp [heS, heT]
· simp [hxe]
lemma RevealTree.double_event (tree : RevealTree α β) (q : α → ℝ) (V : Finset α)
(fresh : tree.Fresh V) (event : β → Prop) [DecidablePred event] :
bitsExpectation q V (fun S => bitsExpectation q V (fun T =>
(if event (tree.run S) then 1 else 0) *
(if event (tree.run (tree.hybrid S T)) then 1 else 0))) =
(bitsExpectation q V (fun S => if event (tree.run S) then 1 else 0)) ^ 2 := by
rw [tree.hybrid_independent q V fresh (fun b => if event b then 1 else 0)
(fun S => if event (tree.run S) then 1 else 0)]
ring
end MatroidProphet
end OAI
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