| import Mathlib |
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| namespace OAI |
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| universe u_Ω u_T u_I u_J |
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| noncomputable section |
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| open Set MeasureTheory |
| open scoped BigOperators ENNReal |
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| namespace Problem310.RepresentativeFailure |
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| / |
| into an all-parameters bound. No measurability of the existential event is |
| needed: the outer-measure union bound is sufficient. -/ |
| theorem failure_le_card_mul {Ω : Type u_Ω} {T : Type u_T} [MeasurableSpace Ω] |
| (μ : Measure Ω) (A : Set Ω) (K : Set T) (R : Finset T) |
| (failure test : Ω → T → Prop) (q : ℝ≥0∞) |
| (htransfer : ∀ ω ∈ A, ∀ t ∈ K, failure ω t → ∃ r ∈ R, test ω r) |
| (hbound : ∀ r ∈ R, μ {ω | ω ∈ A ∧ test ω r} ≤ q) : |
| μ {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} ≤ R.card * q := by |
| classical |
| have hsub : {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} ⊆ |
| ⋃ r ∈ R, {ω | ω ∈ A ∧ test ω r} := by |
| rintro ω ⟨hω, t, ht, hf⟩ |
| obtain ⟨r, hr, htest⟩ := htransfer ω hω t ht hf |
| exact mem_iUnion.mpr ⟨r, mem_iUnion.mpr ⟨hr, hω, htest⟩⟩ |
| calc |
| μ {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} |
| ≤ μ (⋃ r ∈ R, {ω | ω ∈ A ∧ test ω r}) := measure_mono hsub |
| _ ≤ ∑ r ∈ R, μ {ω | ω ∈ A ∧ test ω r} := measure_biUnion_finset_le _ _ |
| _ ≤ ∑ _r ∈ R, q := Finset.sum_le_sum hbound |
| _ = R.card * q := by simp [nsmul_eq_mul] |
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| / |
| transfer hypothesis in the local-routing argument. -/ |
| theorem local_failure_le_card_mul {Ω : Type u_Ω} {T : Type u_T} {I : Type u_I} [MeasurableSpace Ω] |
| (μ : Measure Ω) (A : Set Ω) (K : Set T) (R : Finset T) |
| (failure : Ω → T → Prop) (localHit : Ω → I → T → Prop) (q : ℝ≥0∞) |
| (hlocal : ∀ ω ∈ A, ∀ t ∈ K, failure ω t → ∀ i, ¬ localHit ω i t) |
| (hreps : ∀ t ∈ K, ∃ r ∈ R, |
| ∀ ω ∈ A, ∀ i, localHit ω i r ↔ localHit ω i t) |
| (hbound : ∀ r ∈ R, μ {ω | ω ∈ A ∧ ∀ i, ¬ localHit ω i r} ≤ q) : |
| μ {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} ≤ R.card * q := by |
| apply failure_le_card_mul μ A K R failure (fun ω t => ∀ i, ¬ localHit ω i t) q ?_ hbound |
| intro ω hω t ht hf |
| obtain ⟨r, hr, hagree⟩ := hreps t ht |
| refine ⟨r, hr, ?_⟩ |
| intro i hi |
| exact hlocal ω hω t ht hf i ((hagree ω hω i).mp hi) |
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| / |
| joint bound `q * μ A` and an entropy budget `card R * q ≤ p` give the |
| all-scales joint bound `p * μ A`. -/ |
| theorem local_failure_le_mul_mass {Ω : Type u_Ω} {T : Type u_T} {I : Type u_I} [MeasurableSpace Ω] |
| (μ : Measure Ω) (A : Set Ω) (K : Set T) (R : Finset T) |
| (failure : Ω → T → Prop) (localHit : Ω → I → T → Prop) (p q : ℝ≥0∞) |
| (hlocal : ∀ ω ∈ A, ∀ t ∈ K, failure ω t → ∀ i, ¬ localHit ω i t) |
| (hreps : ∀ t ∈ K, ∃ r ∈ R, |
| ∀ ω ∈ A, ∀ i, localHit ω i r ↔ localHit ω i t) |
| (hbound : ∀ r ∈ R, μ {ω | ω ∈ A ∧ ∀ i, ¬ localHit ω i r} ≤ q * μ A) |
| (hbudget : R.card * q ≤ p) : |
| μ {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} ≤ p * μ A := by |
| calc |
| μ {ω | ω ∈ A ∧ ∃ t ∈ K, failure ω t} ≤ R.card * (q * μ A) := |
| local_failure_le_card_mul μ A K R failure localHit (q * μ A) hlocal hreps hbound |
| _ = (R.card * q) * μ A := (mul_assoc _ _ _).symm |
| _ ≤ p * μ A := mul_le_mul hbudget le_rfl (by positivity) (by positivity) |
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| / |
| exceptional event on which the first default does not exist. The mass bound |
| is separated out so this works for finite partitions, subpartitions, and |
| any finite-family construction with total mass at most one. -/ |
| theorem failure_le_bad_add_atom_bound {Ω : Type u_Ω} {J : Type u_J} [MeasurableSpace Ω] [Fintype J] |
| (μ : Measure Ω) (F D : Set Ω) (A : J → Set Ω) (p q : ℝ≥0∞) |
| (hcover : F ⊆ D ∪ ⋃ j, A j) |
| (hbad : μ D ≤ q) |
| (hatom : ∀ j, μ (F ∩ A j) ≤ p * μ (A j)) |
| (hmass : ∑ j, μ (A j) ≤ 1) : μ F ≤ q + p := by |
| classical |
| have hsub : F ⊆ D ∪ ⋃ j, F ∩ A j := by |
| intro ω hω |
| rcases hcover hω with hD | hA |
| · exact Or.inl hD |
| · obtain ⟨j, hj⟩ := mem_iUnion.mp hA |
| exact Or.inr (mem_iUnion.mpr ⟨j, hω, hj⟩) |
| calc |
| μ F ≤ μ (D ∪ ⋃ j, F ∩ A j) := measure_mono hsub |
| _ ≤ μ D + μ (⋃ j, F ∩ A j) := measure_union_le _ _ |
| _ ≤ q + ∑ j, μ (F ∩ A j) := |
| add_le_add hbad (measure_iUnion_fintype_le μ (fun j => F ∩ A j)) |
| _ ≤ q + ∑ j, p * μ (A j) := add_le_add le_rfl (Finset.sum_le_sum fun j _ => hatom j) |
| _ = q + p * ∑ j, μ (A j) := by rw [Finset.mul_sum] |
| _ ≤ q + p * 1 := add_le_add le_rfl (mul_le_mul le_rfl hmass (by positivity) (by positivity)) |
| _ = q + p := by rw [mul_one] |
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| / |
| finite table construction. Good exposure atoms may have different trial |
| types, representatives and fixed-scale failure probabilities. -/ |
| theorem raw_failure_le_two_budgets {Ω : Type u_Ω} {T : Type u_T} {J : Type u_J} [MeasurableSpace Ω] [Fintype J] |
| (μ : Measure Ω) (K : Set T) (D : Set Ω) (A : J → Set Ω) |
| (I : J → Type u_I) (R : J → Finset T) (failure : Ω → T → Prop) |
| (localHit : ∀ j, Ω → I j → T → Prop) (p q : ℝ≥0∞) (rate : J → ℝ≥0∞) |
| (hcover : univ ⊆ D ∪ ⋃ j, A j) (hbad : μ D ≤ q) |
| (hmass : ∑ j, μ (A j) ≤ 1) |
| (hlocal : ∀ j, ∀ ω ∈ A j, ∀ t ∈ K, failure ω t → ∀ i, ¬ localHit j ω i t) |
| (hreps : ∀ j, ∀ t ∈ K, ∃ r ∈ R j, |
| ∀ ω ∈ A j, ∀ i, localHit j ω i r ↔ localHit j ω i t) |
| (hbound : ∀ j, ∀ r ∈ R j, |
| μ {ω | ω ∈ A j ∧ ∀ i, ¬ localHit j ω i r} ≤ rate j * μ (A j)) |
| (hbudget : ∀ j, (R j).card * rate j ≤ p) : |
| μ {ω | ∃ t ∈ K, failure ω t} ≤ q + p := by |
| apply failure_le_bad_add_atom_bound μ {ω | ∃ t ∈ K, failure ω t} D A p q |
| ((subset_univ _).trans hcover) hbad ?_ hmass |
| intro j |
| have h := local_failure_le_mul_mass μ (A j) K (R j) failure (localHit j) p (rate j) |
| (hlocal j) (hreps j) (hbound j) (hbudget j) |
| simpa only [Set.ofPred_and, Set.ofPred_mem_eq, Set.inter_comm] using h |
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| end Problem310.RepresentativeFailure |
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| end |
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| end OAI |
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