| import OAI.NumberTheory.TwoPoint.Circuits.CircuitEncoding |
|
|
| namespace OAI |
|
|
| /-! Relabel, complement and conjoin the residue circuits used in a |
| closed-word expansion without increasing their internal depth. -/ |
|
|
| namespace TwoPointCorrelations.AC0Circuit |
|
|
| open Finset |
|
|
| def negate {n : ℕ} : AC0Circuit n → AC0Circuit n |
| | .literal i b => .literal i (!b) |
| | .andGate c => .orGate (fun i => negate (c i)) |
| | .orGate c => .andGate (fun i => negate (c i)) |
|
|
| lemma negate_eval {n : ℕ} (c : AC0Circuit n) (x : BooleanCube n) : |
| c.negate.eval x = true ↔ c.eval x ≠ true := by |
| induction c with |
| | literal i b => cases b <;> cases hx : x i <;> simp [negate, eval, hx] |
| | andGate c ih => simp only [negate, eval, ne_eq, decide_eq_true_eq, ih, not_forall] |
| | orGate c ih => simp only [negate, eval, ne_eq, decide_eq_true_eq, ih, not_exists] |
|
|
| lemma negate_depth {n : ℕ} (c : AC0Circuit n) : c.negate.depth = c.depth := by |
| induction c with |
| | literal i b => rfl |
| | andGate c ih => simp only [negate, depth, ih] |
| | orGate c ih => simp only [negate, depth, ih] |
|
|
| lemma negate_size {n : ℕ} (c : AC0Circuit n) : c.negate.size = c.size := by |
| induction c with |
| | literal i b => rfl |
| | andGate c ih => simp only [negate, size, ih] |
| | orGate c ih => simp only [negate, size, ih] |
|
|
| def relabel {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) : AC0Circuit m := |
| substitute (fun i b => .literal (f i) b) c |
|
|
| lemma relabel_eval {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) (x : BooleanCube m) : |
| (relabel f c).eval x = c.eval (fun i => x (f i)) := |
| substitute_eval _ (fun x i => x (f i)) (fun _ _ _ => rfl) c x |
|
|
| lemma relabel_depth {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) : |
| (relabel f c).depth ≤ c.depth := by |
| simpa only [relabel, Nat.add_zero] using substitute_depth (fun i b => literal (f i) b) |
| (d := 0) (fun _ _ => Nat.le_refl 0) c |
|
|
| lemma relabel_size {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) : |
| (relabel f c).size ≤ c.size := by |
| simpa only [relabel, Nat.one_mul] using substitute_size (fun i b => literal (f i) b) |
| (K := 1) (by omega) (fun _ _ => Nat.le_refl 1) c |
|
|
| noncomputable def conjunction {ι : Type*} [Fintype ι] {n : ℕ} |
| (c : ι → AC0Circuit n) : AC0Circuit n := |
| .andGate (fun i : Fin (Fintype.card ι) => c ((Fintype.equivFin ι).symm i)) |
|
|
| lemma conjunction_eval {ι : Type*} [Fintype ι] {n : ℕ} |
| (c : ι → AC0Circuit n) (x : BooleanCube n) : |
| (conjunction c).eval x = true ↔ ∀ i, (c i).eval x = true := by |
| simp only [conjunction, eval, decide_eq_true_eq] |
| constructor |
| · intro h i |
| simpa only [Equiv.symm_apply_apply] using h ((Fintype.equivFin ι) i) |
| · intro h i |
| exact h _ |
|
|
| lemma conjunction_depth {ι : Type*} [Fintype ι] {n d : ℕ} |
| (c : ι → AC0Circuit n) (hc : ∀ i, (c i).depth ≤ d) : |
| (conjunction c).depth ≤ d + 1 := by |
| simp only [conjunction, depth] |
| have hh : (univ.sup fun i : Fin (Fintype.card ι) => |
| (c ((Fintype.equivFin ι).symm i)).depth) ≤ d := |
| Finset.sup_le (fun i _ => hc _) |
| omega |
|
|
| lemma conjunction_size {ι : Type*} [Fintype ι] {n : ℕ} |
| (c : ι → AC0Circuit n) : (conjunction c).size = 1 + ∑ i, (c i).size := by |
| simp only [conjunction, size] |
| congr 1 |
| exact (Fintype.equivFin ι).symm.sum_comp (fun i => (c i).size) |
|
|
| end TwoPointCorrelations.AC0Circuit |
|
|
| end OAI |
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|
