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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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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
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