openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /AdmissibleMatrixGeometry.lean
| import OAI.NumberTheory.PiExponent.Approximation.AdmissibleMatrixFrame | |
| import OAI.NumberTheory.PiExponent.Approximation.AdmissibleMatrixInterpolation | |
| import OAI.NumberTheory.PiExponent.Approximation.WeightedGlobalSectionBound | |
| import OAI.NumberTheory.PiExponent.Jets.AdmissibleJetPackets | |
| import OAI.NumberTheory.PiExponent.Jets.AdmissibleJetSurjectivity | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.AdmissibleMatrixInterpolation | |
| open AlgebraicGeometry CategoryTheory Filter | |
| open PiExponentSeshadri.Geometry | |
| open DeterminantContradiction AdmissibleBlowupGeometry AdmissibleJetPackets | |
| attribute [local irreducible] WeightedCompactification.lineBundle | |
| WeightedCompactification.affineChartMap | |
| AffineJetCoefficientInterface.Frame AffineJetCoefficientInterface.Sections | |
| AffineJetCoefficientInterface.coefficient | |
| variable {ν : ℝ} (d : FixedData ν) | |
| theorem eventual_packets_of_eventual_jetRestriction | |
| (hjet : ∀ᶠ n : ℕ in atTop, Function.Surjective | |
| (BlowupJetSurjectivity.jetRestriction (centerIdeal d) (hyperplane d) n)) : | |
| ∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n) := by | |
| obtain ⟨N,hN⟩ := WeightedGlobalSectionBound.eventual_admissible_supportBound d | |
| obtain ⟨e⟩ := actualFrame_exists d | |
| filter_upwards [hjet, eventually_ge_atTop N] with n hn hnN | |
| apply packetMap_surjective_of_family d n (sectionPolynomial d n e) | |
| · intro s | |
| exact hN n hnN e s | |
| · exact formalPackets_surjective_of_jetRestriction d n e hn | |
| theorem globalInterpolation_of_eventual_jetRestriction | |
| (hjet : ∀ ν : ℝ, 2 < ν → ∀ d : FixedData ν, | |
| ∀ᶠ n : ℕ in atTop, Function.Surjective | |
| (BlowupJetSurjectivity.jetRestriction (centerIdeal d) (hyperplane d) n)) : | |
| GlobalInterpolationStatement := by | |
| apply globalInterpolation_of_eventual_packets | |
| intro ν hν d | |
| exact eventual_packets_of_eventual_jetRestriction d (hjet ν hν d) | |
| theorem globalInterpolation : GlobalInterpolationStatement := by | |
| apply globalInterpolation_of_eventual_jetRestriction | |
| intro ν _ d | |
| exact AdmissibleJetSurjectivity.eventually_jetRestriction_surjective d | |
| end PiExponent.AdmissibleMatrixInterpolation | |
| end | |
| end OAI | |