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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitRandomGate
import OAI.NumberTheory.TwoPoint.Circuits.CircuitSamplingGrid
namespace OAI
/-! The sampled AND polynomial agrees with the Boolean gate whenever the
samplers isolate a false child; all-true inputs are always exact. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
def sampledCoordinates {k : ℕ} (x : BooleanCube k) : Finset (Fin k) :=
univ.filter (fun i => x i = true)
def falseCoordinates {k : ℕ} (b : Fin k → Bool) : Finset (Fin k) :=
univ.filter (fun i => b i = false)
noncomputable def sampledAndPolynomial {k s : ℕ}
(x : GateSamplingChoices k k s) (u : Fin k → ℝ) : ℝ :=
∏ a, ∏ i, andGateFactor (sampledCoordinates (x a i)) u
noncomputable def boolAndValue {k : ℕ} (b : Fin k → Bool) : ℝ :=
if ∀ i, b i = true then 1 else 0
lemma sampledAndPolynomial_degree {n k s : ℕ} (x : GateSamplingChoices k k s)
(P : Fin k → BooleanCube n → ℝ) (d : ℕ) (hP : ∀ i, WalshDegreeLE (P i) d) :
WalshDegreeLE (fun y => sampledAndPolynomial x (fun i => P i y))
(d * s * (Nat.log 2 k + 3)) := by
have hinner (a : Fin (Nat.log 2 k + 3)) :
WalshDegreeLE (fun y => ∏ i : Fin s,
andGateFactor (sampledCoordinates (x a i)) (fun j => P j y)) (d * s) := by
have hh := WalshDegreeLE.prod univ
(fun i y => andGateFactor (sampledCoordinates (x a i)) (fun j => P j y)) d
(fun i _ => WalshDegreeLE.andGateFactor _ P d (fun j _ => hP j))
simpa only [card_univ, Fintype.card_fin] using hh
have hh := WalshDegreeLE.prod univ
(fun a y => ∏ i : Fin s,
andGateFactor (sampledCoordinates (x a i)) (fun j => P j y)) (d * s)
(fun a _ => hinner a)
change WalshDegreeLE (fun y => ∏ a, ∏ i : Fin s,
andGateFactor (sampledCoordinates (x a i)) (fun j => P j y)) _
simpa only [card_univ, Fintype.card_fin] using hh
lemma isolatesOne_card {k : ℕ} {S : Finset (Fin k)} {x : BooleanCube k}
(h : isolatesOne S x) : (S.filter (fun i => x i = true)).card = 1 := by
obtain ⟨i, hi, hx⟩ := h
have heq : S.filter (fun j => x j = true) = {i} := by
ext j
simp only [mem_filter, mem_singleton]
constructor
· rintro ⟨hj, htrue⟩
have hh := hx j hj
rw [htrue] at hh
simpa [singleBitPattern] using hh.symm
· intro hj
subst j
exact ⟨hi, by simpa [singleBitPattern] using hx i hi⟩
rw [heq, card_singleton]
lemma sampled_false_card {k : ℕ} (b : Fin k → Bool) (x : BooleanCube k)
(h : isolatesOne (falseCoordinates b) x) :
((sampledCoordinates x).filter (fun i => b i = false)).card = 1 := by
have heq : (sampledCoordinates x).filter (fun i => b i = false) =
(falseCoordinates b).filter (fun i => x i = true) := by
ext i
simp only [sampledCoordinates, falseCoordinates, mem_filter, mem_univ, true_and]
tauto
rw [heq]
exact isolatesOne_card h
lemma sampledAndPolynomial_all_true {k s : ℕ} (x : GateSamplingChoices k k s)
(b : Fin k → Bool) (hb : ∀ i, b i = true) :
sampledAndPolynomial x (fun i => if b i then 1 else 0) = 1 := by
unfold sampledAndPolynomial
have hh (a : Fin (Nat.log 2 k + 3)) (i : Fin s) :
andGateFactor (sampledCoordinates (x a i)) (fun j => if b j then 1 else 0) = 1 :=
andGateFactor_all_true _ b (fun j _ => hb j)
simp only [hh, prod_const_one]
lemma sampledAndPolynomial_isolated {k s : ℕ} (x : GateSamplingChoices k k s)
(b : Fin k → Bool) (h : ∃ a i, isolatesOne (falseCoordinates b) (x a i)) :
sampledAndPolynomial x (fun i => if b i then 1 else 0) = 0 := by
obtain ⟨a, i, hi⟩ := h
unfold sampledAndPolynomial
apply prod_eq_zero (mem_univ a)
apply prod_eq_zero (mem_univ i)
exact andGateFactor_one_false _ b (sampled_false_card b (x a i) hi)
lemma sampledAndPolynomial_failure {k s : ℕ} (x : GateSamplingChoices k k s)
(b : Fin k → Bool)
(h : sampledAndPolynomial x (fun i => if b i then 1 else 0) ≠ boolAndValue b) :
∀ a i, ¬isolatesOne (falseCoordinates b) (x a i) := by
have hn : ¬∀ i, b i = true := by
intro hb
apply h
rw [sampledAndPolynomial_all_true x b hb]
exact (ite_eq_left hb).symm
intro a i hi
apply h
rw [sampledAndPolynomial_isolated x b ⟨a, i, hi⟩]
exact (ite_eq_right hn).symm
end TwoPointCorrelations
end OAI
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