openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /AdmissibleMatrixInterpolation.lean
| import OAI.NumberTheory.PiExponent.Ampleness.AdmissibleBlowupGeometry | |
| import OAI.NumberTheory.PiExponent.Approximation.DeterminantContradiction | |
| import OAI.NumberTheory.PiExponent.Approximation.FormalMatrixSurjectivity | |
| import OAI.NumberTheory.PiExponent.Approximation.WeightedSliceDegree | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.AdmissibleMatrixInterpolation | |
| open Filter | |
| open DeterminantContradiction AdmissibleBlowupGeometry | |
| variable {ν : ℝ} (d : FixedData ν) | |
| theorem degreeWeights_eq_matrix (i : Fin (d.m+1)) : | |
| (d.curveDegreeWeights i : ℝ) = | |
| InterpolationMatrix.columnWeights d.w0 | |
| (MatrixArithmetic.logWeights (finiteDenominators d)) i := by | |
| refine Fin.cases ?_ (fun j => ?_) i | |
| · rfl | |
| · simp only [AdmissibleParameters.curveDegreeWeights_succ, | |
| d.cast_rationalWeight, InterpolationMatrix.columnWeights, Fin.cases_succ, | |
| logWeights_eq] | |
| theorem jetWeights_eq_matrix (i : Fin (d.m+1)) : | |
| (d.curveJetWeights i : ℝ) = | |
| InterpolationMatrix.rowWeights d.v0 d.base.theta | |
| (MatrixArithmetic.logWeights (finiteDenominators d)) i := by | |
| refine Fin.cases ?_ (fun j => ?_) i | |
| · rfl | |
| · simp only [AdmissibleParameters.curveJetWeights_succ, Rat.cast_div, | |
| d.cast_rationalWeight, InterpolationMatrix.rowWeights, Fin.cases_succ, | |
| logWeights_eq] | |
| theorem actual_truncation_weight_bound (i : Fin d.m) : | |
| d.curveJetWeights i.succ ≤ | |
| (MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℚ) * | |
| d.curveJetWeights 0 := by | |
| have hw : 0 < MatrixArithmetic.logWeights (finiteDenominators d) i := by | |
| rw [logWeights_eq] | |
| exact zero_lt_one.trans_le (d.x_one_le _) | |
| have hF : 1 / (d.base.theta : ℝ) < d.F0 := | |
| (div_lt_div_of_pos_right (by norm_num : (1 : ℝ) < 2) d.base.theta_pos).trans d.F0_large | |
| have hceil := Nat.le_ceil | |
| (d.F0 * MatrixArithmetic.logWeights (finiteDenominators d) i / (d.v0 : ℝ)) | |
| have hbound : MatrixArithmetic.logWeights (finiteDenominators d) i / (d.base.theta : ℝ) ≤ | |
| (MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℝ) * | |
| (d.v0 : ℝ) := by | |
| calc | |
| _ = (1 / (d.base.theta : ℝ)) * | |
| MatrixArithmetic.logWeights (finiteDenominators d) i := by ring | |
| _ ≤ d.F0 * MatrixArithmetic.logWeights (finiteDenominators d) i := | |
| mul_le_mul_of_nonneg_right hF.le hw.le | |
| _ ≤ _ := (div_le_iff₀ d.v0_pos).mp hceil | |
| have hcast : (d.curveJetWeights i.succ : ℝ) ≤ | |
| (MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0 i : ℝ) * | |
| (d.curveJetWeights 0 : ℝ) := by | |
| simpa only [AdmissibleParameters.curveJetWeights_zero, | |
| AdmissibleParameters.curveJetWeights_succ, Rat.cast_div, | |
| d.cast_rationalWeight, logWeights_eq] using hbound | |
| exact_mod_cast hcast | |
| theorem matrix_degree_of_supportBound (H : ℝ) (P : PiExponentApprox.FramePolynomial d.m) | |
| (hP : WeightedSliceDegree.SupportBound (fun i => (d.curveDegreeWeights i : ℝ)) H P) : | |
| PiExponentApprox.HasWeightedDegreeLE | |
| (InterpolationMatrix.columnWeights d.w0 | |
| (MatrixArithmetic.logWeights (finiteDenominators d))) H P := by | |
| intro a ha | |
| have h := hP a ha | |
| have hw : (fun i => (d.curveDegreeWeights i : ℝ)) = | |
| InterpolationMatrix.columnWeights d.w0 | |
| (MatrixArithmetic.logWeights (finiteDenominators d)) := | |
| funext (degreeWeights_eq_matrix d) | |
| rw [hw] at h | |
| simpa only [PiExponentApprox.monomialWeight, Finsupp.weight_eq_sum, nsmul_eq_mul] using h | |
| theorem actualMatrix_surjective_of_formalPackets {α : Type*} (H : ℚ) | |
| (P : α → PiExponentApprox.FramePolynomial d.m) | |
| (hdegree : ∀ a, WeightedSliceDegree.SupportBound | |
| (fun i => (d.curveDegreeWeights i : ℝ)) (H : ℝ) (P a)) | |
| (hpacket : Function.Surjective (fun a => fun j : Fin d.K => | |
| JetGeometry.rationalCoefficientPacket d.curveJetWeights H | |
| (FormalLogJet.formalJet (d.curveCenters j) (P a)))) : | |
| Function.Surjective (actualMatrix d (H : ℝ)).mulVecLin := by | |
| apply FormalMatrixBridge.truncatedLogMatrix_surjective_of_formalLog_packets | |
| d.K d.w0 d.v0 d.base.theta | |
| (MatrixArithmetic.logWeights (finiteDenominators d)) H | |
| d.w0_pos (fun i => ?_) d.curveJetWeights d.curveJetWeights_pos | |
| (jetWeights_eq_matrix d) | |
| (MatrixArithmetic.rationalCenters (finiteNumerators d) (finiteDenominators d)) | |
| (MatrixArithmetic.truncationOrders (finiteDenominators d) d.F0 d.v0) | |
| (actual_truncation_weight_bound d) P | |
| (fun a => matrix_degree_of_supportBound d _ _ (hdegree a)) hpacket | |
| rw [logWeights_eq] | |
| exact zero_lt_one.trans_le (d.x_one_le _) | |
| def WeightedPolynomials (n : ℕ) := | |
| {P : PiExponentApprox.FramePolynomial d.m // | |
| WeightedSliceDegree.SupportBound (fun i => (d.curveDegreeWeights i : ℝ)) | |
| ((n : ℝ) * (scale d).radius) P} | |
| def packetMap (n : ℕ) (P : WeightedPolynomials d n) := | |
| fun j : Fin d.K => JetGeometry.rationalCoefficientPacket d.curveJetWeights | |
| ((n : ℚ) * (scale d).radius) (FormalLogJet.formalJet (d.curveCenters j) P.val) | |
| theorem packetMap_surjective_of_family {α : Type*} (n : ℕ) | |
| (P : α → PiExponentApprox.FramePolynomial d.m) | |
| (hdegree : ∀ a, WeightedSliceDegree.SupportBound | |
| (fun i => (d.curveDegreeWeights i : ℝ)) ((n : ℝ) * (scale d).radius) (P a)) | |
| (hpacket : Function.Surjective (fun a => fun j : Fin d.K => | |
| JetGeometry.rationalCoefficientPacket d.curveJetWeights | |
| ((n : ℚ) * (scale d).radius) (FormalLogJet.formalJet (d.curveCenters j) (P a)))) : | |
| Function.Surjective (packetMap d n) := by | |
| intro y | |
| obtain ⟨a,ha⟩ := hpacket y | |
| exact ⟨⟨P a,hdegree a⟩,ha⟩ | |
| theorem actualMatrix_surjective_of_packetMap (n : ℕ) | |
| (hpacket : Function.Surjective (packetMap d n)) : | |
| Function.Surjective (actualMatrix d ((n : ℝ) * (scale d).radius)).mulVecLin := by | |
| have h := actualMatrix_surjective_of_formalPackets d | |
| ((n : ℚ) * (scale d).radius) (fun P : WeightedPolynomials d n => P.val) | |
| (fun P => by simpa only [Rat.cast_mul, Rat.cast_natCast] using P.property) hpacket | |
| have hc : (((n : ℚ) * (scale d).radius : ℚ) : ℝ) = | |
| (n : ℝ) * (scale d).radius := by push_cast; rfl | |
| exact hc ▸ h | |
| theorem cofinal_actualMatrix_of_eventual_packets | |
| (hpacket : ∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n)) | |
| (L : ℝ) : ∃ H : ℝ, L ≤ H ∧ Function.Surjective (actualMatrix d H).mulVecLin := by | |
| obtain ⟨N,hN⟩ := eventually_atTop.mp hpacket | |
| let n := max N (Nat.ceil (L / ((scale d).radius : ℝ))) | |
| have hR : (0 : ℝ) < (scale d).radius := by exact_mod_cast (scale d).radius_pos | |
| have hL : L ≤ (n : ℝ) * (scale d).radius := by | |
| apply (div_le_iff₀ hR).mp | |
| exact (Nat.le_ceil _).trans (by exact_mod_cast (le_max_right N | |
| (Nat.ceil (L / ((scale d).radius : ℝ))))) | |
| exact ⟨(n : ℝ) * (scale d).radius, hL, | |
| actualMatrix_surjective_of_packetMap d n (hN n (le_max_left _ _))⟩ | |
| theorem globalInterpolation_of_eventual_packets | |
| (hpacket : ∀ ν : ℝ, 2 < ν → ∀ d : FixedData ν, | |
| ∀ᶠ n : ℕ in atTop, Function.Surjective (packetMap d n)) : | |
| GlobalInterpolationStatement := by | |
| intro ν hν d L | |
| exact cofinal_actualMatrix_of_eventual_packets d (hpacket ν hν d) L | |
| end PiExponent.AdmissibleMatrixInterpolation | |
| end | |
| end OAI | |