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9dbc64e | 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 28 29 30 31 32 33 34 35 36 37 38 39 40 | 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
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