| import OAI.Probability.MatroidProphet.Reverse.LowHazard |
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| namespace OAI |
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| namespace MatroidProphet |
| open Finset |
| variable {α : Type*} [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end MatroidProphet |
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| end OAI |
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