openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /FormalMatrixSurjectivity.lean
| import OAI.NumberTheory.PiExponent.Approximation.FormalMatrixBridge | |
| import OAI.NumberTheory.PiExponent.Jets.FormalJetIndices | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.FormalMatrixBridge | |
| open MvPowerSeries | |
| theorem truncatedLogMatrix_surjective_of_rational_packets {m : ℕ} {α : Type*} | |
| (K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℚ) | |
| (hw0 : 0 < w0) (hw : ∀ i, 0 < w i) | |
| (V : Fin (m+1) → ℚ) (hV : ∀ i, 0 < V i) | |
| (hrow : ∀ i, (V i : ℝ) = InterpolationMatrix.rowWeights v0 θ w i) | |
| (r : Fin m → ℂ) (T : Fin m → ℕ) | |
| (P : α → PiExponentApprox.FramePolynomial m) | |
| (hdegree : ∀ a, PiExponentApprox.HasWeightedDegreeLE | |
| (InterpolationMatrix.columnWeights w0 w) (H : ℝ) (P a)) | |
| (hpacket : Function.Surjective (fun a => fun j : Fin K => | |
| JetGeometry.rationalCoefficientPacket V H | |
| (FormalLogTruncation.truncatedFormalJet (fun i => (j.val : ℂ) * r i) T (P a)))) : | |
| Function.Surjective | |
| (InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w (H : ℝ) r T).mulVecLin := by | |
| classical | |
| have hrow' : (fun i => (V i : ℝ)) = InterpolationMatrix.rowWeights v0 θ w := funext hrow | |
| have hW : ∀ i, 0 < InterpolationMatrix.columnWeights w0 w i := by | |
| intro i | |
| exact Fin.cases hw0 hw i | |
| intro y | |
| let e := rationalJetIndexEquiv V hV H | |
| let z : Fin K → JetGeometry.RationalCoefficientPacket (R := ℂ) V H := | |
| fun j d => y (j, ⟨(e d).val, by | |
| simpa only [← hrow'] using (e d).property⟩) | |
| obtain ⟨a, ha⟩ := hpacket z | |
| let x : InterpolationMatrix.Column w0 w (H : ℝ) → ℂ := | |
| fun c => (P a).coeff (InterpolationMatrix.exponentVector c.val) | |
| refine ⟨x, ?_⟩ | |
| funext ρ | |
| rw [truncatedLogMatrix_mulVec_eq_coeff] | |
| change coeff (InterpolationMatrix.exponentVector ρ.2.val) | |
| (FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T | |
| (polynomialOfCoefficients | |
| (realWeightedSimplex (InterpolationMatrix.columnWeights w0 w) (H : ℝ)) | |
| (fun c => (P a).coeff (InterpolationMatrix.exponentVector c.val)))) = y ρ | |
| rw [polynomialOfCoefficients_of_weighted _ hW _ (P a) (hdegree a)] | |
| let b : ↥(strictWeightedSimplex (fun i => (V i : ℝ)) (H : ℝ)) := | |
| ⟨ρ.2.val, by simpa only [hrow'] using ρ.2.property⟩ | |
| have hh := congrFun (congrFun ha ρ.1) (e.symm b) | |
| change coeff (e.symm b).val | |
| (FormalLogTruncation.truncatedFormalJet (fun i => (ρ.1.val : ℂ) * r i) T (P a)) = | |
| z ρ.1 (e.symm b) at hh | |
| simpa only [e, rationalJetIndexEquiv_symm_val, b, z, Equiv.apply_symm_apply] using hh | |
| theorem truncatedLogMatrix_surjective_of_formalLog_packets {m : ℕ} {α : Type*} | |
| (K : ℕ) (w0 v0 θ : ℝ) (w : Fin m → ℝ) (H : ℚ) | |
| (hw0 : 0 < w0) (hw : ∀ i, 0 < w i) | |
| (V : Fin (m+1) → ℚ) (hV : ∀ i, 0 < V i) | |
| (hrow : ∀ i, (V i : ℝ) = InterpolationMatrix.rowWeights v0 θ w i) | |
| (r : Fin m → ℂ) (T : Fin m → ℕ) | |
| (hT : ∀ i, V i.succ ≤ (T i : ℚ) * V 0) | |
| (P : α → PiExponentApprox.FramePolynomial m) | |
| (hdegree : ∀ a, PiExponentApprox.HasWeightedDegreeLE | |
| (InterpolationMatrix.columnWeights w0 w) (H : ℝ) (P a)) | |
| (hpacket : Function.Surjective (fun a => fun j : Fin K => | |
| JetGeometry.rationalCoefficientPacket V H | |
| (FormalLogJet.formalJet (fun i => (j.val : ℂ) * r i) (P a)))) : | |
| Function.Surjective | |
| (InterpolationMatrix.truncatedLogMatrix K w0 v0 θ w (H : ℝ) r T).mulVecLin := by | |
| apply truncatedLogMatrix_surjective_of_rational_packets | |
| K w0 v0 θ w H hw0 hw V hV hrow r T P hdegree | |
| exact (FormalLogTruncation.formalLog_packets_surjective_iff_truncated | |
| V (fun i => (hV i).le) H T hT (fun j : Fin K => fun i => (j.val : ℂ) * r i) P).mp hpacket | |
| end PiExponent.FormalMatrixBridge | |
| end | |
| end OAI | |