openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /FrameResidueDirections.lean
| import OAI.NumberTheory.PiExponent.Jets.NormalFrameBasis | |
| import OAI.NumberTheory.PiExponent.Polynomials.PolynomialFrame | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent | |
| open scoped BigOperators | |
| open PiExponentApprox NormalBasisRigidity | |
| theorem residue_polynomialFrame_eq_frameBasis_dot {E : Type*} [Field E] | |
| (m : ℕ) (phi : FramePolynomial m →+* E) | |
| (hy : phi (MvPolynomial.X (0 : Fin (m + 1))) ≠ 0) | |
| (j : Fin (m + 1)) (p : FramePolynomial m) : | |
| phi (polynomialFrame m j p) = | |
| ∑ i, frameBasis m (phi (MvPolynomial.X 0)) hy j i * | |
| phi (MvPolynomial.pderiv i p) := by | |
| refine Fin.cases ?_ (fun l => ?_) j | |
| · rw [polynomialFrame_zero, logarithmicDerivation_apply, map_add, map_mul, | |
| map_sum, Fin.sum_univ_succ] | |
| simp only [frameBasis_zero_zero, frameBasis_zero_succ, one_mul] | |
| · rw [polynomialFrame_pos m l.succ (Fin.succ_ne_zero l), frameBasis_succ] | |
| simp [Pi.basisFun_apply, Pi.single_apply] | |
| end PiExponent | |
| end | |
| end OAI | |