| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationConcentration |
|
|
| namespace OAI |
|
|
| /-! |
| # Orienting three pairs of mixer halves |
|
|
| The orientations here describe the conditional sign experiment; they are |
| analysis variables, not extra random draws made by the algorithm. Every |
| unordered pair of halves has a good orientation by deficiency |
| superadditivity. A uniform vector of three orientations therefore succeeds |
| with probability at least one eighth. The product-space bound also permits |
| an arbitrary preceding event depending on the unordered pairs. |
| -/ |
|
|
| namespace LowSpaceSubsetSum |
|
|
| theorem finiteUniformProbability_ge_card_inv_of_exists {Ω : Type*} [Fintype Ω] |
| [Nonempty Ω] (P : Ω → Prop) (hP : ∃ ω, P ω) : |
| (Fintype.card Ω : ℝ)⁻¹ ≤ finiteUniformProbability P := by |
| classical |
| unfold finiteUniformProbability |
| rw [inv_eq_one_div] |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| have hcard : 0 < Fintype.card {ω // P ω} := by |
| apply Fintype.card_pos_iff.mpr |
| obtain ⟨ω, hω⟩ := hP |
| exact ⟨⟨ω, hω⟩⟩ |
| exact_mod_cast hcard |
|
|
| def orientedHalf {ι : Type*} (A B : Finset ι) (b : Bool) : Finset ι := |
| if b then A else B |
|
|
| theorem exists_good_half_orientation {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (a : ι → ℤ) (A B M : Finset ι) (hAB : Disjoint A B) (hM : A ∪ B = M) |
| (τ : ℝ) (hA : deficiency a A ≤ τ) (hB : deficiency a B ≤ τ) : |
| ∃ b : Bool, deficiency a (orientedHalf A B b) ≤ min τ (deficiency a M / 2) := by |
| have hsum := deficiency_superadditive a hAB |
| rw [hM] at hsum |
| by_cases ha : deficiency a A ≤ deficiency a M / 2 |
| · exact ⟨true, by simpa [orientedHalf] using le_min hA ha⟩ |
| · refine ⟨false, ?_⟩ |
| simp only [orientedHalf, Bool.false_eq_true, ↓reduceIte] |
| exact le_min hB (by linarith) |
|
|
| theorem three_half_orientations_probability {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (a : ι → ℤ) (A B M : Fin 3 → Finset ι) |
| (hAB : ∀ i, Disjoint (A i) (B i)) (hM : ∀ i, A i ∪ B i = M i) |
| (τ : ℝ) (hA : ∀ i, deficiency a (A i) ≤ τ) |
| (hB : ∀ i, deficiency a (B i) ≤ τ) : |
| (1 / 8 : ℝ) ≤ finiteUniformProbability (fun b : Fin 3 → Bool => |
| ∀ i, deficiency a (orientedHalf (A i) (B i) (b i)) ≤ |
| min τ (deficiency a (M i) / 2)) := by |
| classical |
| have hgood (i : Fin 3) := exists_good_half_orientation a (A i) (B i) (M i) |
| (hAB i) (hM i) τ (hA i) (hB i) |
| have hex : ∃ b : Fin 3 → Bool, ∀ i, |
| deficiency a (orientedHalf (A i) (B i) (b i)) ≤ min τ (deficiency a (M i) / 2) := |
| ⟨fun i => Classical.choose (hgood i), fun i => Classical.choose_spec (hgood i)⟩ |
| have h := finiteUniformProbability_ge_card_inv_of_exists _ hex |
| simpa using h |
|
|
| / |
| gives the joint lower bound without assuming that the preceding event and |
| the successful orientation event are independent. -/ |
| theorem finiteUniformProbability_joint_orientation {Ω : Type*} [Fintype Ω] |
| [Nonempty Ω] (E : Ω → Prop) (G : Ω → (Fin 3 → Bool) → Prop) |
| (hG : ∀ ω, E ω → ∃ b, G ω b) : |
| finiteUniformProbability E / 8 ≤ |
| finiteUniformProbability (fun p : Ω × (Fin 3 → Bool) => E p.1 ∧ G p.1 p.2) := by |
| classical |
| let : DecidablePred E := fun _ => Classical.propDecidable _ |
| let : DecidablePred (fun p : Ω × (Fin 3 → Bool) => E p.1 ∧ G p.1 p.2) := |
| fun _ => Classical.propDecidable _ |
| let witness (ω : {ω // E ω}) : Fin 3 → Bool := Classical.choose (hG ω.val ω.property) |
| let embed : {ω // E ω} → {p : Ω × (Fin 3 → Bool) // E p.1 ∧ G p.1 p.2} := |
| fun ω => ⟨(ω.val, witness ω), ω.property, Classical.choose_spec (hG ω.val ω.property)⟩ |
| have hinj : Function.Injective embed := by |
| intro x y h |
| apply Subtype.ext |
| exact congrArg (fun z => z.val.1) h |
| have hcard := Fintype.card_le_of_injective embed hinj |
| have hΩ : (0 : ℝ) < Fintype.card Ω := by exact_mod_cast Fintype.card_pos |
| unfold finiteUniformProbability |
| rw [Fintype.card_prod] |
| simp only [Fintype.card_fun, Fintype.card_bool, Fintype.card_fin, Nat.cast_mul] |
| norm_num |
| rw [div_div] |
| exact div_le_div_of_nonneg_right (by exact_mod_cast hcard) (by positivity) |
|
|
| end LowSpaceSubsetSum |
|
|
| end OAI |
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|
