| import OAI.Probability.MatroidProphet.Reverse.SquareSum |
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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 pathExitSquare_nonneg (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (q : α → ℝ) (d : α) (h m : ℕ) (C : Finset α) : |
| 0 ≤ pathExitSquare M hE κ D G q d h m C := by |
| unfold pathExitSquare |
| split_ifs |
| · exact sq_nonneg _ |
| · exact le_rfl |
|
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| lemma pathExitSquare_eq_doubleExit (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) |
| (h : ℕ) (d : α) (m : ℕ) : |
| bitsExpectation q univ (pathExitSquare M hE κ D G q d h m) = |
| bitsExpectation q univ (fun C => bitsExpectation q univ |
| (reverseDoubleExit M hE κ D ((reversePrefixTree M hE κ D G h m (-1) (fun _ => Set.univ)).run C) |
| G (-(m:ℤ)-1) h d C)) := by |
| rw [reverseDoubleExit_after_prefix M hE κ D G h d hG q m (-1) _ (reverseClosed_univ M hE κ D (-1))] |
| apply bitsExpectation_congr |
| intro C hC |
| have hp (j : ℕ) (hj : j ≤ h) := reversePrefixTree_from_univ_path M hE κ D G h hG m C hj |
| rw [hp h le_rfl, reverseHazard_state_congr M hE κ D _ _ G q _ h d (fun j hj => hp j hj.le)] |
| rfl |
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| omit [DecidableEq α] in |
| lemma doubleExit_le_birth (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D S : ℕ → Set α) (G : ℕ → Finset α) (C T : Finset α) (b : ℤ) (h : ℕ) |
| (d : α) (hd : d ∉ M.closure ∅) |
| (hS : ∀ j ≤ h, S j = nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j b) : |
| reverseDoubleExit M hE κ D S G (b-1) h d C T ≤ |
| if nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d = b then 1 else 0 := by |
| unfold reverseDoubleExit |
| by_cases hs : d ∈ S h |
| · rw [ite_eq_left hs] |
| by_cases hn : d ∉ reverseNominal M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) S (b-1) h |
| · rw [ite_eq_left hn, one_mul] |
| have hbirth : nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d = b := by |
| rw [hS h le_rfl, mem_nominal_iff_birth_le M hE κ D _ h d hd b] at hs |
| rw [reverseNominal_actual_state M hE κ D S G C b h hS, |
| mem_nominal_iff_birth_le M hE κ D _ h d hd (b-1)] at hn |
| omega |
| rw [ite_eq_left hbirth] |
| split_ifs <;> norm_num |
| · simp only [ite_eq_right hn, zero_mul] |
| split_ifs <;> norm_num |
| · simp only [ite_eq_right hs] |
| split_ifs <;> norm_num |
|
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| theorem pathExitSquare_le_birth_probability (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) |
| (h : ℕ) (d : α) (hd : d ∉ M.closure ∅) (m : ℕ) : |
| bitsExpectation q univ (pathExitSquare M hE κ D G q d h m) ≤ |
| bitsExpectation q univ (fun C => |
| if nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d = -(m:ℤ) then 1 else 0) := by |
| rw [pathExitSquare_eq_doubleExit M hE κ D G hG q h d m] |
| apply bitsExpectation_mono q hq0 hq1 |
| intro C hC |
| rw [← bitsExpectation_const q univ (if nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d = -(m:ℤ) then 1 else 0)] |
| apply bitsExpectation_mono q hq0 hq1 |
| intro T hT |
| exact doubleExit_le_birth M hE κ D _ G C T (-(m:ℤ)) h d hd |
| (fun j hj => reversePrefixTree_from_univ_path M hE κ D G h hG m C hj) |
|
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| omit [DecidableEq α] in |
| lemma nominalBirth_baseline_iff (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (d : α) (hd : d ∉ M.closure ∅) : |
| nominalBirth M hE κ D C h d = activation h ↔ d ∈ densityExpansion M hE κ (D h) (M.closure ∅) := by |
| have hm := mem_nominal_iff_birth_le M hE κ D C h d hd (activation h) |
| rw [nominalPath_baseline] at hm |
| have hb := (nominalBirth_bounds M hE κ D C h d).1 |
| exact ⟨fun he => hm.mpr he.le, fun he => le_antisymm (hm.mp he) hb⟩ |
|
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| end MatroidProphet |
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| end OAI |
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