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ae00908 | 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 | 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
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