| import OAI.Probability.MatroidProphet.Reverse.BirthSafety |
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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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| lemma reverseCost_add_steps (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (cost : ℤ → (ℕ → Set α) → (ℕ → Set α) → ℝ) |
| (m r : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) : |
| reverseCost M hE κ D G n cost (m+r) k S C = |
| reverseCost M hE κ D G n cost m k S C + |
| reverseCost M hE κ D G n cost r (k-m) (reverseFinal M hE κ D G n m k S C) C := by |
| induction m generalizing k S with |
| | zero => simp only [reverseCost, reverseFinal, Nat.cast_zero, sub_zero, zero_add] |
| | succ m ih => |
| rw [Nat.succ_add, reverseCost, reverseCost, reverseFinal, ih] |
| have ht : k - 1 - (m:ℤ) = k - ((m+1:ℕ):ℤ) := by omega |
| rw [ht] |
| ring |
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| lemma reverseCost_eq_sum (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ) |
| (cost : ℤ → (ℕ → Set α) → (ℕ → Set α) → ℝ) |
| (m : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) : |
| reverseCost M hE κ D G n cost m k S C = |
| ∑ i ∈ range m, cost (k-i) (reverseFinal M hE κ D G n i k S C) |
| ((reverseStepTree M hE κ (k-i) D (reverseFinal M hE κ D G n i k S C) G n).run C) := by |
| induction m with |
| | zero => simp only [reverseCost, range_zero, sum_empty] |
| | succ m ih => |
| rw [reverseCost_add_steps M hE κ D G n cost m 1, ih, sum_range_succ] |
| simp only [reverseCost, add_zero] |
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| noncomputable def pathExitSquare (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) (m : ℕ) |
| (C : Finset α) : ℝ := |
| if d ∈ nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h (-(m:ℤ)) then |
| (reverseHazard M hE κ D G q d h (-(m:ℤ)-1) |
| (fun j => nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (-(m:ℤ))))^2 else 0 |
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| lemma reverse_square_eq_path_sum (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) |
| (d : α) (h : ℕ) (hh : h ≤ n) (m : ℕ) : |
| reverseMeanCost M hE κ D G n q (squareHazardCost M hE κ D G q d h) m (-1) (fun _ => Set.univ) = |
| ∑ i ∈ range m, bitsExpectation q univ (pathExitSquare M hE κ D G q d h i) := by |
| unfold reverseMeanCost |
| rw [← bitsExpectation_sum] |
| apply bitsExpectation_congr |
| intro C hC |
| rw [reverseCost_eq_sum] |
| apply sum_congr rfl |
| intro i hi |
| have hS : ∀ j ≤ n, (fun _ => Set.univ) j = |
| nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) j (-1+1) := by |
| intro j hj |
| exact (nominalPath_nonnegative M hE κ D _ j (by omega)).symm |
| have hf (j : ℕ) (hj : j ≤ n) : reverseFinal M hE κ D G n i (-1) (fun _ => Set.univ) C j = |
| nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) j (-(i:ℤ)) := by |
| simpa only [neg_add_cancel, zero_sub] using reverseFinal_path M hE κ D G n hG i (-1) _ C hS hj |
| have ht : (-1:ℤ) - i = -(i:ℤ)-1 := by omega |
| simp only [squareHazardCost, pathExitSquare, ht, hf h hh] |
| rw [reverseHazard_state_congr M hE κ D _ _ G q _ h d (fun j hj => hf j (by omega))] |
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| theorem path_squared_exit_mass (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) |
| (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) : |
| (((2:ℝ)^12)⁻¹) / 2 ≤ |
| ∑ i ∈ range (pathHorizon h+1), bitsExpectation q univ (pathExitSquare M hE κ D G q d h i) := by |
| rw [← reverse_square_eq_path_sum M hE κ D G h hG q d h le_rfl] |
| exact reverse_squared_exit_mass M hE κ D G h hG q hq0 hq1 d hd h le_rfl |
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| end MatroidProphet |
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| end OAI |
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