| import OAI.NumberTheory.PiExponent.Analysis.FormalLogTruncation |
| import OAI.NumberTheory.PiExponent.Jets.AuxiliaryJetMap |
| import OAI.NumberTheory.PiExponent.Polynomials.NestedPolynomialSwap |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PiExponent.FormalMatrixBridge |
|
|
| open MvPowerSeries |
| open scoped BigOperators |
|
|
| def flatten (m : ℕ) : |
| MvPolynomial (Fin m) (Polynomial ℂ) →ₐ[ℂ] MvPowerSeries (Fin (m+1)) ℂ := |
| (MvPolynomial.coeToMvPowerSeries.algHom ℂ).comp |
| ((MvPolynomial.finSuccEquiv ℂ m).symm.toAlgHom.comp |
| (nestedPolynomialSwap ℂ m).toAlgHom) |
|
|
| @[simp] theorem coeff_flatten {m : ℕ} |
| (p : MvPolynomial (Fin m) (Polynomial ℂ)) (d : Fin (m+1) →₀ ℕ) : |
| coeff d (flatten m p) = (p.coeff d.tail).coeff (d 0) := by |
| change ((MvPolynomial.finSuccEquiv ℂ m).symm (nestedPolynomialSwap ℂ m p)).coeff d = _ |
| rw [← Finsupp.cons_tail d, ← MvPolynomial.finSuccEquiv_coeff_coeff, |
| AlgEquiv.apply_symm_apply] |
| exact coeff_nestedPolynomialSwap p (d 0) d.tail |
|
|
| @[simp] theorem flatten_C {m : ℕ} (p : Polynomial ℂ) : |
| flatten m (MvPolynomial.C p) = FormalLogJet.liftSeries m (p : PowerSeries ℂ) := by |
| classical |
| ext d |
| rw [coeff_flatten, FormalLogJet.coeff_liftSeries] |
| simp only [MvPolynomial.coeff_C, Polynomial.coeff_coe] |
| split_ifs <;> simp_all |
|
|
| @[simp] theorem flatten_X {m : ℕ} (i : Fin m) : |
| flatten m (MvPolynomial.X i) = X i.succ := by |
| classical |
| ext d |
| rw [coeff_flatten, MvPolynomial.coeff_X, coeff_X] |
| have hd : d = Finsupp.single i.succ 1 ↔ |
| d.tail = Finsupp.single i 1 ∧ d 0 = 0 := by |
| conv_lhs => rw [← Finsupp.cons_tail d] |
| exact Finsupp.cons_eq_single_succ_iff i |
| simp only [hd] |
| by_cases ht : Finsupp.single i 1 = d.tail |
| · rw [ite_eq_left ht] |
| have ht' : d.tail = Finsupp.single i 1 := ht.symm |
| simp only [ht', true_and, Polynomial.coeff_one] |
| · rw [ite_eq_right ht, Polynomial.coeff_zero] |
| have ht' : ¬ d.tail = Finsupp.single i 1 := Ne.symm ht |
| simp only [ht', false_and, ite_false] |
|
|
| theorem flatten_literalJetSubstitution {m : ℕ} |
| (r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ) |
| (p : PiExponentApprox.FramePolynomial m) : |
| flatten m (literalJetSubstitution r |
| (fun i => InterpolationMatrix.truncatedLog (T i)) j p) = |
| FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T p := by |
| have hh : (flatten m).comp |
| (literalJetSubstitution r (fun i => InterpolationMatrix.truncatedLog (T i)) j) = |
| FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T := by |
| ext i |
| cases i using Fin.cases with |
| | zero => |
| simp [literalJetSubstitution, FormalLogTruncation.truncatedFormalJet] |
| | succ i => |
| simp [literalJetSubstitution, FormalLogTruncation.truncatedFormalJet, |
| InterpolationMatrix.truncatedLog, add_assoc, add_left_comm, add_comm] |
| exact DFunLike.congr_fun hh p |
|
|
| theorem coeff_truncatedFormalJet {m : ℕ} |
| (r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ) |
| (p : PiExponentApprox.FramePolynomial m) (d : Fin (m+1) →₀ ℕ) : |
| coeff d (FormalLogTruncation.truncatedFormalJet |
| (fun i => (j : ℂ) * r i) T p) = |
| ((literalJetSubstitution r (fun i => InterpolationMatrix.truncatedLog (T i)) j p).coeff |
| d.tail).coeff (d 0) := by |
| rw [← flatten_literalJetSubstitution, coeff_flatten] |
|
|
| @[simp] theorem exponentVector_tail {m : ℕ} (a : Fin (m+1) → ℕ) : |
| (InterpolationMatrix.exponentVector a).tail = |
| InterpolationMatrix.exponentVector (fun i => a i.succ) := by |
| ext i |
| simp [Finsupp.tail_apply] |
|
|
| theorem entry_eq_coeff_truncatedFormalJet {m : ℕ} |
| (r : Fin m → ℂ) (T : Fin m → ℕ) (j : ℕ) |
| (b e : Fin (m+1) → ℕ) : |
| InterpolationMatrix.entry r (fun i => InterpolationMatrix.truncatedLog (T i)) |
| j (b 0) (fun i => b i.succ) (e 0) (fun i => e i.succ) = |
| coeff (InterpolationMatrix.exponentVector b) |
| (FormalLogTruncation.truncatedFormalJet (fun i => (j : ℂ) * r i) T |
| (MvPolynomial.monomial (InterpolationMatrix.exponentVector e) 1)) := by |
| rw [coeff_truncatedFormalJet, literalJetSubstitution_monomial] |
| simp [InterpolationMatrix.entry] |
|
|
| theorem truncatedLogMatrix_mulVec_eq_coeff {m : ℕ} |
| (K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℝ) |
| (r : Fin m → ℂ) (T : Fin m → ℕ) |
| (x : InterpolationMatrix.Column w0 w H → ℂ) |
| (ρ : InterpolationMatrix.Row K v0 θ w H) : |
| (InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w H r T).mulVecLin x ρ = |
| coeff (InterpolationMatrix.exponentVector ρ.2.val) |
| (FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T |
| (polynomialOfCoefficients |
| (realWeightedSimplex (InterpolationMatrix.columnWeights w0 w) H) x)) := by |
| rw [coeff_truncatedFormalJet, literalJetSubstitution_polynomialOfCoefficients] |
| simpa only [exponentVector_tail, InterpolationMatrix.exponentVector_apply, |
| InterpolationMatrix.linearEvaluation, InterpolationMatrix.truncatedLogMatrix] using |
| InterpolationMatrix.linearEvaluation_eq_coeff K w0 v0 θ w H r |
| (fun i => InterpolationMatrix.truncatedLog (T i)) x ρ |
|
|
| end PiExponent.FormalMatrixBridge |
|
|
| end |
|
|
| end OAI |
|
|
