| import OAI.Computability.DepthThree.AlgebraPolynomial |
| import OAI.Computability.DepthThree.IrreduciblePolynomial |
|
|
| namespace OAI |
|
|
| namespace DepthThreeLowerBound |
| namespace BinaryAlgebra |
|
|
| def polynomialBits (P : Polynomial F2) (r : ℕ) : Fin r → Bool := |
| fun i => fieldBit (P.coeff i.val) |
|
|
| theorem inputPolynomialBits_polynomialBits (P : Polynomial F2) (hP : P.Monic) |
| {r : ℕ} (hdegree : P.natDegree = r) : |
| inputPolynomialBits (polynomialBits P r) = P := by |
| ext j |
| rcases lt_trichotomy j r with hj | rfl | hj |
| · simpa only [polynomialBits, bitValue_fieldBit] using |
| inputPolynomialBits_coeff_lt (polynomialBits P r) ⟨j, hj⟩ |
| · rw [inputPolynomialBits_coeff_leading] |
| simpa only [hdegree] using hP.coeff_natDegree.symm |
| · have hleft : (inputPolynomialBits (polynomialBits P r)).coeff j = 0 := |
| inputPolynomial_coeff_of_gt _ hj |
| rw [hleft] |
| symm |
| apply Polynomial.coeff_eq_zero_of_natDegree_lt |
| simpa only [hdegree] using hj |
|
|
| end BinaryAlgebra |
|
|
| theorem exists_irreducible_inputPolynomialBits (r : ℕ) (hr : 0 < r) : |
| ∃ p : Fin r → Bool, Irreducible (BinaryAlgebra.inputPolynomialBits p) := by |
| obtain ⟨P, hP, hI, hdegree⟩ := exists_monic_irreducible_natDegree_two r hr |
| refine ⟨BinaryAlgebra.polynomialBits P r, ?_⟩ |
| rw [BinaryAlgebra.inputPolynomialBits_polynomialBits P hP hdegree] |
| exact hI |
|
|
| end DepthThreeLowerBound |
|
|
| end OAI |
|
|
