| import OAI.NumberTheory.TwoPoint.Circuits.CircuitDyadicSampling |
| import OAI.NumberTheory.TwoPoint.Circuits.CircuitProbabilisticMethod |
| import OAI.NumberTheory.TwoPoint.Bounds.IndependentSampling |
|
|
| namespace OAI |
|
|
| /-! Independent repetition amplifies the single-hit experiment. These finite |
| identities are the random-subset step of the circuit polynomial construction. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| def isolatesOne {n : ℕ} (S : Finset (Fin n)) (x : BooleanCube n) : Prop := |
| ∃ i ∈ S, ∀ j ∈ S, x j = singleBitPattern i j |
|
|
| lemma singleHitScore_eq_indicator {n : ℕ} (S : Finset (Fin n)) (x : BooleanCube n) : |
| singleHitScore S x = if isolatesOne S x then 1 else 0 := by |
| by_cases hx : isolatesOne S x |
| · rw [ite_eq_left hx] |
| obtain ⟨i, hi, hxi⟩ := hx |
| unfold singleHitScore |
| rw [sum_eq_single i] |
| · exact ite_eq_left hxi |
| · intro j hj hji |
| unfold cubePattern |
| apply ite_eq_right |
| intro hxj |
| have hh := (hxi i hi).symm.trans (hxj i hi) |
| simp [singleBitPattern, Ne.symm hji] at hh |
| · exact fun h => (h hi).elim |
| · rw [ite_eq_right hx] |
| unfold singleHitScore |
| apply sum_eq_zero |
| intro i hi |
| unfold cubePattern |
| exact ite_eq_right (fun h => hx ⟨i, hi, h⟩) |
|
|
| theorem bernoulliCubeLaw_isolatesOne (n : ℕ) (p : ℝ) (hp : 0 ≤ p) (hp1 : p ≤ 1) |
| (S : Finset (Fin n)) : |
| (bernoulliCubeLaw n p hp hp1).probability (isolatesOne S) = |
| (S.card : ℝ) * p * (1 - p) ^ (S.card - 1) := by |
| unfold FiniteLaw.probability |
| have heq : (fun x => if isolatesOne S x then (1 : ℝ) else 0) = singleHitScore S := |
| funext (fun x => (singleHitScore_eq_indicator S x).symm) |
| rw [heq] |
| exact bernoulliCubeLaw_singleHitScore n p hp hp1 S |
|
|
| lemma FiniteLaw.probability_complement {α : Type*} [Fintype α] |
| (μ : FiniteLaw α) (E : α → Prop) : |
| μ.probability (fun x => ¬E x) = 1 - μ.probability E := by |
| unfold FiniteLaw.probability FiniteLaw.average |
| calc |
| _ = ∑ x, (μ.weight x - μ.weight x * (if E x then (1 : ℝ) else 0)) := by |
| apply sum_congr rfl |
| intro x _ |
| by_cases hx : E x <;> simp [hx] |
| _ = _ := by rw [sum_sub_distrib, μ.total] |
|
|
| theorem FiniteLaw.independent_failure {α : Type*} [Fintype α] |
| (μ : FiniteLaw α) (E : α → Prop) (s : ℕ) : |
| (FiniteLaw.independent (fun _ : Fin s => μ)).probability |
| (fun x => ∀ i, ¬E (x i)) = (1 - μ.probability E) ^ s := by |
| calc |
| _ = ∏ _i : Fin s, μ.probability (fun x => ¬E x) := |
| FiniteLaw.independent_probability_all (fun _ : Fin s => μ) (fun _ x => ¬E x) |
| _ = _ := by |
| simp only [μ.probability_complement, prod_const, card_univ, Fintype.card_fin] |
|
|
| theorem FiniteLaw.independent_failure_le {α : Type*} [Fintype α] |
| (μ : FiniteLaw α) (E : α → Prop) (s : ℕ) |
| (hE : (1 / 8 : ℝ) ≤ μ.probability E) : |
| (FiniteLaw.independent (fun _ : Fin s => μ)).probability |
| (fun x => ∀ i, ¬E (x i)) ≤ (7 / 8 : ℝ) ^ s := by |
| rw [μ.independent_failure] |
| apply pow_le_pow_left₀ |
| · linarith [μ.probability_le_one E] |
| · linarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
