| import Mathlib | |
| namespace OAI | |
| namespace SingleFold | |
| /-- Evaluation over the integers of an integer polynomial at natural inputs. -/ | |
| noncomputable def evalNat {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) | |
| (a : Fin n → ℕ) (w : Fin m → ℕ) : ℤ := | |
| MvPolynomial.eval (Sum.elim (fun i => (a i : ℤ)) (fun j => (w j : ℤ))) P | |
| /-- Exact existence on members, no witness on nonmembers, and uniqueness of the | |
| whole witness on every input. No coordinates are hidden outside `w`. -/ | |
| def Represents {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) | |
| (S : Set (Fin n → ℕ)) : Prop := | |
| ∀ a, (a ∈ S ↔ ∃ w, evalNat P a w = 0) ∧ | |
| ∀ w v, evalNat P a w = 0 → evalNat P a v = 0 → w = v | |
| /-- Single-fold representation with positive input and witness lengths. -/ | |
| def MainStatement : Prop := | |
| ∀ (n : ℕ), 1 ≤ n → ∀ (S : Set (Fin n → ℕ)), REPred (fun a => a ∈ S) → | |
| ∃ (m : ℕ), 1 ≤ m ∧ ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P S | |
| end SingleFold | |
| end OAI | |