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8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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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