| import OAI.Computability.DepthThree.LanguageParser |
| import OAI.Computability.DepthThree.AlgebraPolynomial |
| import OAI.Computability.DepthThree.Hash |
| import Mathlib.Algebra.Polynomial.Div |
|
|
| namespace OAI |
|
|
| namespace DepthThreeLowerBound |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace ParsedInput |
|
|
| def hashPolynomial (q : ParsedInput) : Polynomial (ZMod 2) := |
| BinaryAlgebra.pack fun i => |
| BinaryAlgebra.bitValue (BinaryHash.hashBool q.hashSeed q.data i) |
|
|
| def coefficientPolynomial (q : ParsedInput) (j : Fin q.coefficientCount) : |
| Polynomial (ZMod 2) := |
| BinaryAlgebra.pack fun i => BinaryAlgebra.bitValue (q.coefficients j i) |
|
|
| def modulus (q : ParsedInput) : Polynomial (ZMod 2) := |
| BinaryAlgebra.inputPolynomialBits q.polynomial |
|
|
| theorem modulus_monic (q : ParsedInput) : q.modulus.Monic := |
| BinaryAlgebra.inputPolynomialBits_monic q.polynomial |
|
|
| theorem modulus_degree (q : ParsedInput) : |
| q.modulus.degree = (q.ringDegree : WithBot ℕ) := |
| BinaryAlgebra.inputPolynomialBits_degree q.polynomial |
|
|
| def evaluationPolynomial (q : ParsedInput) : Polynomial (ZMod 2) := |
| ∑ j : Fin q.coefficientCount, |
| q.coefficientPolynomial j * q.hashPolynomial ^ j.val |
|
|
| def evaluationRemainder (q : ParsedInput) : Polynomial (ZMod 2) := |
| q.evaluationPolynomial %ₘ q.modulus |
|
|
| theorem evaluationRemainder_degree_lt (q : ParsedInput) : |
| q.evaluationRemainder.degree < (q.ringDegree : WithBot ℕ) := by |
| rw [evaluationRemainder, ← q.modulus_degree] |
| exact Polynomial.degree_modByMonic_lt _ q.modulus_monic |
|
|
| def evaluate (q : ParsedInput) : Bool := |
| decide (q.evaluationRemainder.coeff 0 = 0) |
|
|
| theorem evaluate_eq_true_iff (q : ParsedInput) : |
| q.evaluate = true ↔ q.evaluationRemainder.coeff 0 = 0 := by |
| simp only [evaluate, decide_eq_true_eq] |
|
|
| end ParsedInput |
|
|
| def language (w : List Bool) : Bool := |
| match parseInput w with |
| | none => false |
| | some q => q.evaluate |
|
|
| theorem language_of_not_fits {w : List Bool} (h : ¬InputFits w) : |
| language w = false := by |
| simp only [language, parseInput_of_not_fits h] |
|
|
| theorem language_of_fits {w : List Bool} (h : InputFits w) : |
| language w = (decodeInput w).evaluate := by |
| simp only [language, parseInput_of_fits h] |
|
|
| theorem language_eq_true_iff (w : List Bool) : |
| language w = true ↔ |
| InputFits w ∧ (decodeInput w).evaluationRemainder.coeff 0 = 0 := by |
| by_cases h : InputFits w |
| · rw [language_of_fits h, ParsedInput.evaluate_eq_true_iff] |
| simp only [h, true_and] |
| · rw [language_of_not_fits h] |
| simp only [Bool.false_eq_true, h, false_and] |
|
|
| end |
|
|
| end DepthThreeLowerBound |
|
|
| end OAI |
|
|
