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Is D1 — card le four of one three negative

Proved
D1Bound5.IsD1.card_le_four_of_one_three_negative

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

diophantine-tuplesnumber-theory

Complete negative-entry branch of the manuscript's {1,3} lemma, over all imaginary quadratic full rings. The extra entry may be -1.

Preamble
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Int.GCD
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_IsD1

Formal statement
section
open scoped NumberField
open Module

namespace D1Bound5




























theorem _root_.D1Bound5.IsD1.card_le_four_of_one_three_negative
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) {A : Finset (𝓞 K)} (hA : IsD1 A)
    (h1 : (1 : 𝓞 K) ∈ A) (h3 : (3 : 𝓞 K) ∈ A)
    (hn : ∃ n : ℤ, n<0 ∧ (n : 𝓞 K) ∈ A) : A.card ≤ 4 := by sorry



end D1Bound5
end
Source
D1Bound5/OneThreeNegative.lean, lines 179-239, SHA-256 608590903d9651b83081c7c7b14962c497bd29944ae7411568eb23eee5734f08. Project: https://github.com/tomislav-math/d1-bound5-lean

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