| import OAI.Probability.MatroidProphet.SafeSamples |
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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 safeEndpointParityCount (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) : ℝ := |
| ∑ i ∈ range (pathHorizon h-1), if safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T ∧ |
| (-(i:ℤ)) % 2 = (if parity then 1 else 0) then 1 else 0 |
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| noncomputable def safeSampleCount (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) : ℝ := |
| ∑ d ∈ I, safeEndpointParityCount M hE κ D G h d C T parity |
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| omit [DecidableEq α] in |
| lemma safeSampleCount_parity_split (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) : |
| safeSampleCount M hE κ D G h I C T false + safeSampleCount M hE κ D G h I C T true = |
| safeCountBeforeParity M hE κ D G h I C T := by |
| unfold safeSampleCount safeCountBeforeParity |
| rw [← sum_add_distrib] |
| apply sum_congr rfl |
| intro d hd |
| unfold safeEndpointParityCount safeEndpointCount |
| rw [← sum_add_distrib] |
| apply sum_congr rfl |
| intro i hi |
| have hm0 := Int.emod_nonneg (-(i:ℤ)) (by norm_num : (2:ℤ) ≠ 0) |
| have hm1 := Int.emod_lt_of_pos (-(i:ℤ)) (by norm_num : (0:ℤ) < 2) |
| have hm : (-(i:ℤ)) % 2 = 0 ∨ (-(i:ℤ)) % 2 = 1 := by omega |
| by_cases hs : safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T |
| · rcases hm with hm | hm <;> simp [hs, hm] |
| · simp [hs] |
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| noncomputable def fairParityExpectation (f : Bool → ℝ) : ℝ := (f false + f true) / 2 |
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| omit [DecidableEq α] in |
| lemma safeSampleCount_fair_parity (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) : |
| fairParityExpectation (safeSampleCount M hE κ D G h I C T) = |
| safeCountBeforeParity M hE κ D G h I C T / 2 := by |
| unfold fairParityExpectation |
| rw [safeSampleCount_parity_split] |
|
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| theorem safeSampleCount_lower (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (hκ : 0 < κ) (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) (I : Finset α) (hI : M.Indep (I : Set α)) : |
| (((2:ℝ)^12)⁻¹) / 4 * I.card - (D h).ncard / (κ:ℝ) ≤ |
| bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| fairParityExpectation (safeSampleCount M hE κ D G h I C T))) := by |
| have hpre := safeCountBeforeParity_lower M hE κ hκ D G hG q hq0 hq1 h I hI |
| have heq : bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| fairParityExpectation (safeSampleCount M hE κ D G h I C T))) = |
| (bitsExpectation q univ (fun C => bitsExpectation q univ (safeCountBeforeParity M hE κ D G h I C))) / 2 := by |
| simp only [safeSampleCount_fair_parity, div_eq_mul_inv] |
| rw [show bitsExpectation q univ (fun C => bitsExpectation q univ (fun T => |
| safeCountBeforeParity M hE κ D G h I C T * (2:ℝ)⁻¹)) = |
| bitsExpectation q univ (fun C => bitsExpectation q univ (safeCountBeforeParity M hE κ D G h I C) * (2:ℝ)⁻¹) from by |
| apply bitsExpectation_congr; intro C hC; exact bitsExpectation_mul_right q univ _ _] |
| exact bitsExpectation_mul_right q univ _ _ |
| rw [heq] |
| have hnonneg : 0 ≤ (D h).ncard / (κ:ℝ) := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _) |
| linarith |
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| omit [Fintype α] [DecidableEq α] in |
| lemma sum_unique_indicators {ι : Type*} (J : Finset ι) (p : ι → Prop) [DecidablePred p] |
| (huniq : ∀ i ∈ J, ∀ j ∈ J, p i → p j → i = j) : |
| (∑ i ∈ J, if p i then (1:ℝ) else 0) = if ∃ i ∈ J, p i then 1 else 0 := by |
| classical |
| by_cases hex : ∃ i ∈ J, p i |
| · obtain ⟨i, hi, hp⟩ := hex |
| rw [ite_eq_left ⟨i, hi, hp⟩] |
| calc |
| _ = if p i then (1:ℝ) else 0 := by |
| apply sum_eq_single i |
| · intro j hj hji |
| exact ite_eq_right (fun hjp => hji (huniq j hj i hi hjp hp)) |
| · intro hnot |
| exact (hnot hi).elim |
| _ = 1 := ite_eq_left hp |
| · rw [ite_eq_right hex] |
| apply sum_eq_zero |
| intro i hi |
| exact ite_eq_right (fun hp => hex ⟨i, hi, hp⟩) |
|
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| noncomputable def IsSafeSample (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) : Prop := |
| let b := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d |
| activation h + 2 ≤ b ∧ b ≤ 0 ∧ b % 2 = (if parity then 1 else 0) ∧ |
| d ∉ densityExpansion M hE κ (D h) |
| (guardedPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h (b-2) ∪ |
| lowerCompetition M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) |
| (fun i => (T : Set α) ∩ (G i : Set α)) b h) |
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| omit [DecidableEq α] in |
| lemma isSafeSample_iff_endpoint (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (d : α) (C T : Finset α) (parity : Bool) : |
| IsSafeSample M hE κ D G h d C T parity ↔ ∃ i ∈ range (pathHorizon h-1), |
| safeBirthEvent M hE κ D G h d (-(i:ℤ)) C T ∧ (-(i:ℤ)) % 2 = (if parity then 1 else 0) := by |
| let b := nominalBirth M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h d |
| have ht := activation_add_horizon h |
| have hH := pathHorizon_pos h |
| constructor |
| · rintro ⟨hlo, hhi, hp, hs⟩ |
| have hi0 : 0 ≤ -b := by omega |
| have hieq : -(((-b).toNat:ℕ):ℤ) = b := by rw [Int.toNat_of_nonneg hi0]; omega |
| refine ⟨(-b).toNat, mem_range.mpr ?_, ?_, ?_⟩ |
| · have hiNat := Int.toNat_of_nonneg hi0 |
| change activation h + 2 ≤ b at hlo |
| omega |
| · rw [hieq] |
| exact ⟨rfl, hs⟩ |
| · simpa only [hieq] using hp |
| · rintro ⟨i, hi, hs, hp⟩ |
| have hbirth := hs.1 |
| have hbound := mem_range.mp hi |
| change _ ∧ _ ∧ _ ∧ _ |
| rw [hbirth] |
| exact ⟨by omega, by omega, hp, hs.2⟩ |
|
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| noncomputable def safeSampleSet (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) : Finset α := |
| I.filter (fun d => IsSafeSample M hE κ D G h d C T parity) |
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| omit [DecidableEq α] in |
|
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| lemma safeSampleCount_eq_card (M : Matroid α) (hE : M.E = Set.univ) |
| (κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (h : ℕ) (I C T : Finset α) (parity : Bool) : |
| safeSampleCount M hE κ D G h I C T parity = (safeSampleSet M hE κ D G h I C T parity).card := by |
| unfold safeSampleCount safeSampleSet |
| have hpoint (d : α) : safeEndpointParityCount M hE κ D G h d C T parity = |
| if IsSafeSample M hE κ D G h d C T parity then 1 else 0 := by |
| unfold safeEndpointParityCount |
| rw [sum_unique_indicators] |
| · simp only [← isSafeSample_iff_endpoint] |
| · intro i hi j hj hsi hsj |
| have hbi := hsi.1.1 |
| have hbj := hsj.1.1 |
| omega |
| simp_rw [hpoint] |
| simp |
|
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| end MatroidProphet |
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| end OAI |
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