Eisenstein degenerate pair

Proved

D1Bound5.eisenstein_degenerate_pair

by tp · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Auxiliary result eisenstein_degenerate_pair 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 Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Fin.Tuple.Sort
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_DegeneratePairClassification_Definitions

Formal statement

section
namespace D1Bound5
open QuadInt















theorem _root_.D1Bound5.eisenstein_degenerate_pair (a b : QuadInt 1 1) (hne : a ≠ b)
    (hprod : a*b+1=0) : RationalUnitPair a b ∨
      EisensteinCanonicalPair a b ∨ EisensteinCanonicalPair (-a) (-b) := by sorry

end D1Bound5
end

Source

D1Bound5/DegeneratePairClassification.lean, lines 101-106, SHA-256 61da8c7aa871a81d9d56989ed28825c79c1cef687d7de61a56038260054cdd37. Project: https://github.com/tomislav-math/d1-bound5-lean

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