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5bdeca1 | 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 | 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
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