File size: 4,455 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 | import OAI.Probability.MatroidProphet.Reverse.LowHazard
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
attribute [local instance] Classical.propDecidable
noncomputable def exitHazardCost (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (q : α → ℝ) (d : α) (h : ℕ)
(k : ℤ) (S _ : ℕ → Set α) : ℝ :=
if d ∈ S h then reverseHazard M hE κ D G q d h k S else 0
noncomputable def squareHazardCost (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (q : α → ℝ) (d : α) (h : ℕ)
(k : ℤ) (S _ : ℕ → Set α) : ℝ :=
if d ∈ S h then (reverseHazard M hE κ D G q d h k S)^2 else 0
lemma exitHazardCost_mean_eq_exit (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(d : α) (h : ℕ) (hh : h ≤ n) (m : ℕ) (k : ℤ) (S : ℕ → Set α)
(closed : ReverseClosed M hE κ D k S) :
reverseMeanCost M hE κ D G n q (exitHazardCost M hE κ D G q d h) m k S =
reverseMeanCost M hE κ D G n q (exitCost d h) m k S := by
apply reverseMeanCost_eq_of_step M hE κ D G n hG q hq0 hq1
· intro l R hR
by_cases hd : d ∈ R h
· simp only [exitHazardCost, hd, ite_true, bitsExpectation_const, exitCost, true_and]
exact reverseHazard_eq_step M hE κ D G n hG q d h hh l R
· simp only [exitHazardCost, hd, ite_false, bitsExpectation_const, exitCost, false_and]
· exact closed
theorem reverse_squared_exit_mass (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(d : α) (hd : d ∉ M.closure ∅) (h : ℕ) (hh : h ≤ n) :
(((2:ℝ)^12)⁻¹) / 2 ≤
reverseMeanCost M hE κ D G n q (squareHazardCost M hE κ D G q d h)
(pathHorizon h+1) (-1) (fun _ => Set.univ) := by
let η : ℝ := ((2:ℝ)^12)⁻¹
have hη : 0 ≤ η := by dsimp only [η]; positivity
have hpoint (k : ℤ) (S S' : ℕ → Set α) :
η * exitHazardCost M hE κ D G q d h k S S' ≤
squareHazardCost M hE κ D G q d h k S S' + η * lowHazardCost M hE κ D G q d h η k S S' := by
by_cases hs : d ∈ S h
· by_cases hp : reverseHazard M hE κ D G q d h k S < η
· simp only [exitHazardCost, squareHazardCost, lowHazardCost, hs, hp, true_and, ite_true]
nlinarith [sq_nonneg (reverseHazard M hE κ D G q d h k S)]
· have hp' : η ≤ reverseHazard M hE κ D G q d h k S := le_of_not_gt hp
have hp0 := reverseHazard_nonneg M hE κ D G q hq0 hq1 d h k S
simp only [exitHazardCost, squareHazardCost, lowHazardCost, hs, hp, true_and, ite_true, ite_false, mul_zero, add_zero]
nlinarith
· simp [exitHazardCost, squareHazardCost, lowHazardCost, hs]
have hmean :
η * reverseMeanCost M hE κ D G n q (exitHazardCost M hE κ D G q d h) (pathHorizon h+1) (-1) (fun _ => Set.univ) ≤
reverseMeanCost M hE κ D G n q (squareHazardCost M hE κ D G q d h) (pathHorizon h+1) (-1) (fun _ => Set.univ) +
η * reverseMeanCost M hE κ D G n q (lowHazardCost M hE κ D G q d h η) (pathHorizon h+1) (-1) (fun _ => Set.univ) := by
rw [← reverseMeanCost_mul, ← reverseMeanCost_mul, ← reverseMeanCost_add]
unfold reverseMeanCost
apply bitsExpectation_mono q hq0 hq1
intro C hC
exact reverseCost_mono M hE κ D G n _ _ hpoint _ _ _ C
rw [exitHazardCost_mean_eq_exit M hE κ D G n hG q hq0 hq1 d h hh _ _ _
(reverseClosed_univ M hE κ D (-1)), reverse_exit_total_one M hE κ D G n hG q d hd h hh,
mul_one] at hmean
have hlow := reverse_low_hazard_lt_half M hE κ D G n hG q hq0 hq1 d h hh
(pathHorizon h+1) (-1) (fun _ => Set.univ) (reverseClosed_univ M hE κ D (-1))
change η / 2 ≤ reverseMeanCost M hE κ D G n q _ _ _ _
change reverseMeanCost M hE κ D G n q (lowHazardCost M hE κ D G q d h η) _ _ _ < _ at hlow
have hmul := mul_le_mul_of_nonneg_left hlow.le hη
have hsub := (sub_le_iff_le_add).mpr (hmean.trans (add_le_add le_rfl hmul))
calc
η / 2 = η - η * (1 / 2) := by ring
_ ≤ _ := hsub
end MatroidProphet
end OAI
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