| 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 |
|
|
