| import OAI.Algebra.FormulaHitting.Evaluation |
| import OAI.Algebra.FormulaHitting.Series |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace NCHitting |
| namespace Formula |
|
|
| variable {F : Type*} {n : ℕ} [Field F] |
|
|
| def series (f : Formula F n) : PowerSeries F := |
| f.polynomial.sum (fun w a => PowerSeries.C a * wordSeries w) |
|
|
| theorem series_eq_smul_sum (f : Formula F n) : |
| f.series = f.polynomial.sum (fun w a => a • wordSeries w) := by |
| simp only [series, PowerSeries.smul_eq_C_mul] |
|
|
| theorem series_eq_sum_of_cutoff (f : Formula F n) (d : ℕ) |
| (hcut : ∀ w, d ≤ w.length → f.coeff w = 0) : |
| f.series = ∑ k ∈ Finset.range d, ∑ w ∈ wordsOfLength n k, |
| PowerSeries.C (f.coeff w) * wordSeries w := by |
| exact sum_eq_sum_wordsOfLength f.polynomial hcut _ (by intro w; simp) |
|
|
| theorem series_eq_sum_wordsOfLength (f : Formula F n) {s : ℕ} (hs : f.size ≤ s) : |
| f.series = ∑ k ∈ Finset.range (s + 1), ∑ w ∈ wordsOfLength n k, |
| PowerSeries.C (f.coeff w) * wordSeries w := by |
| apply f.series_eq_sum_of_cutoff |
| intro w hw |
| exact f.coeff_eq_zero_of_length_gt w (by omega) |
|
|
| end Formula |
| end NCHitting |
|
|
| end |
|
|
| end OAI |
|
|
