Frontier fifteen rank One upper

Proved

D1Bound5.frontier_fifteen_rankOne_upper

by tp · Oct 24, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Auxiliary result frontier_fifteen_rankOne_upper in the proof of the bound five for Diophantine D(1) sets in imaginary quadratic integer rings. The formal statement specifies its hypotheses and conclusion; a mathematical prose description remains to be supplied.

Preamble

import Lean.Util.CollectAxioms
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.Real.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_FrontierFifteenCofactors_Definitions

Formal statement

section
open scoped Matrix.Norms.L2Operator BigOperators
open Matrix

namespace D1Bound5





theorem _root_.D1Bound5.frontier_fifteen_rankOne_upper
    (F : Matrix (Fin 4) (Fin 4) ℂ) (w : Fin 4 → ℂ)
    (hF : F.det ≠ 0) (hdet : ‖F.det‖ ≤ 1)
    (hG : ∀ i j, ‖F.adjugate i j‖ ≤ frontierFifteenG i j) :
    ‖(F+Matrix.vecMulVec w w).det‖ ≤
      (252/1000 : ℝ)*‖w 0‖^2+(256/1000)*‖w 1‖^2+(277/1000)*‖w 2‖^2+(282/1000)*‖w 3‖^2+
      (284/1000)*‖w 0‖*‖w 1‖+(284/1000)*‖w 0‖*‖w 2‖+(288/1000)*‖w 0‖*‖w 3‖+
      (286/1000)*‖w 1‖*‖w 2‖+(296/1000)*‖w 1‖*‖w 3‖+(362/1000)*‖w 2‖*‖w 3‖+1 := by sorry

end D1Bound5
end

Source

D1Bound5/FrontierFifteenNumeric.lean, lines 120-156, SHA-256 3122c98662e60a3fcf8105b6ce6ac200558a87cd3502de98958615f5f2e0f062. Project: https://github.com/tomislav-math/d1-bound5-lean

View graph