File size: 3,727 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Probability.MatroidProphet.Maximum.Basic
namespace OAI
namespace MatroidProphet.Maximum
open Finset
variable {n : ℕ}
lemma expectation_mem (q : Fin n → ℝ) (V : Finset (Fin n))
(e : Fin n) (he : e ∈ V) (c : ℝ) :
bitsExpectation q V (fun H => if e ∈ H then c else 0) = q e * c := by
have hno : bitsExpectation q (V.erase e) (fun H => if e ∈ H then c else 0) = 0 := by
calc
_ = bitsExpectation q (V.erase e) (fun _ => 0) := by
apply bitsExpectation_congr
intro H hH
have heH : e ∉ H := fun hh => (Finset.mem_erase.mp (hH hh)).1 rfl
simp [heH]
_ = 0 := bitsExpectation_const q _ 0
have hyes : bitsExpectation q (V.erase e) (fun H => if e ∈ insert e H then c else 0) = c := by
simp only [Finset.mem_insert_self, ite_true]
exact bitsExpectation_const q _ c
calc
_ = bitsExpectation q (insert e (V.erase e)) (fun H => if e ∈ H then c else 0) := by
rw [Finset.insert_erase he]
_ = q e * c := by
rw [bitsExpectation_insert q _ e (by simp), hno, hyes]
ring
lemma expectation_top_two_event (V : Finset (Fin n)) (g h : Fin n)
(hg : g ∈ V) (hh : h ∈ V) (hgh : g ≠ h) (c : ℝ) :
bitsExpectation (fun _ => (1 / 2 : ℝ)) V
(fun H => if g ∈ H then 0 else if h ∈ H then c else 0) = c / 4 := by
let q : Fin n → ℝ := fun _ => 1 / 2
have hno : bitsExpectation q (V.erase g)
(fun H => if g ∈ H then 0 else if h ∈ H then c else 0) = (1 / 2 : ℝ) * c := by
calc
_ = bitsExpectation q (V.erase g) (fun H => if h ∈ H then c else 0) := by
apply bitsExpectation_congr
intro H hH
have hgH : g ∉ H := fun hm => (Finset.mem_erase.mp (hH hm)).1 rfl
simp [hgH]
_ = _ := expectation_mem q (V.erase g) h (Finset.mem_erase.mpr ⟨hgh.symm, hh⟩) c
have hyes : bitsExpectation q (V.erase g)
(fun H => if g ∈ insert g H then 0 else if h ∈ insert g H then c else 0) = 0 := by
simp only [Finset.mem_insert_self, ite_true]
exact bitsExpectation_const q _ 0
change bitsExpectation q V _ = _
conv_lhs => rw [← Finset.insert_erase hg]
rw [bitsExpectation_insert q _ g (by simp), hno, hyes]
dsimp [q]
ring
theorem maximum_expectation_two (V : Finset (Fin n)) (g h : Fin n)
(hg : g ∈ V) (hh : h ∈ V) (hgh : g ≠ h)
(F : Finset (Fin n) → ℝ) (c : ℝ)
(hF : ∀ H ⊆ V, 0 ≤ F H)
(hsuccess : ∀ H ⊆ V, g ∉ H → h ∈ H → c ≤ F H) :
c / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by
rw [← expectation_top_two_event V g h hg hh hgh c]
apply bitsExpectation_mono _ (fun _ => by norm_num) (fun _ => by norm_num)
intro H hH
split_ifs with hgH hhH
· exact hF H hH
· exact hsuccess H hH hgH hhH
· exact hF H hH
theorem maximum_expectation_one (V : Finset (Fin n)) (g : Fin n) (hg : g ∈ V)
(F : Finset (Fin n) → ℝ) (c : ℝ) (hc : 0 ≤ c)
(hF : ∀ H ⊆ V, 0 ≤ F H)
(hsuccess : ∀ H ⊆ V, g ∉ H → c ≤ F H) :
c / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by
have hhalf : c / 2 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) V F := by
have hevent := Candidates.expectation_not_mem (fun _ => (1 / 2 : ℝ)) V g hg c
have hevent' : bitsExpectation (fun _ => (1 / 2 : ℝ)) V
(fun H => if g ∉ H then c else 0) = c / 2 := by
rw [hevent]
ring
rw [← hevent']
apply bitsExpectation_mono _ (fun _ => by norm_num) (fun _ => by norm_num)
intro H hH
by_cases hgH : g ∈ H
· simpa only [hgH, not_true_eq_false, ite_false] using hF H hH
· simpa only [hgH, not_false_eq_true, ite_true] using hsuccess H hH hgH
linarith
end MatroidProphet.Maximum
end OAI
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