| import OAI.NumberTheory.TwoPoint.Circuits.CircuitBernoulli |
|
|
| namespace OAI |
|
|
| /-! The averaging step that chooses one polynomial approximation from a |
| finite random construction, for an arbitrary input distribution. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma FiniteLaw.exists_average_le {α : Type*} [Fintype α] |
| (μ : FiniteLaw α) (f : α → ℝ) {a : ℝ} (ha : μ.average f ≤ a) : |
| ∃ x, f x ≤ a := by |
| by_contra hn |
| have hf : ∀ x, a < f x := by simpa only [not_exists, not_le] using hn |
| have hpos : ∃ x, 0 < μ.weight x := by |
| have ht : 0 < ∑ x, μ.weight x := by rw [μ.total]; norm_num |
| simpa only [sum_pos_iff_of_nonneg (fun x _ => μ.nonneg x), mem_univ, true_and] using ht |
| obtain ⟨x, hx⟩ := hpos |
| have hlt : (∑ y, μ.weight y * a) < ∑ y, μ.weight y * f y := by |
| apply sum_lt_sum |
| · intro y _ |
| exact mul_le_mul_of_nonneg_left (hf y).le (μ.nonneg y) |
| · exact ⟨x, mem_univ _, mul_lt_mul_of_pos_left (hf x) hx⟩ |
| change μ.average (fun _ => a) < μ.average f at hlt |
| rw [μ.average_const] at hlt |
| exact (not_lt_of_ge ha) hlt |
|
|
| /-- If every fixed input has small failure probability over random choices, |
| one fixed choice has small failure probability for the supplied input law. -/ |
| theorem exists_choice_of_pointwise_failure {α β : Type*} [Fintype α] [Fintype β] |
| (μ : FiniteLaw α) (ν : FiniteLaw β) (E : α → β → Prop) {ε : ℝ} |
| (hE : ∀ x, μ.probability (fun a => E a x) ≤ ε) : |
| ∃ a, ν.probability (E a) ≤ ε := by |
| apply μ.exists_average_le |
| unfold FiniteLaw.probability |
| rw [μ.average_comm ν] |
| exact (ν.average_mono hE).trans_eq (ν.average_const ε) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
