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First Pair Table 0 2 third rows

Proved
D1Bound5.firstPairTable_0_2_third_rows

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

diophantine-tuplesnumber-theory

Auxiliary result firstPairTable_0_2_third_rows 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.Group.Int.Units
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.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
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.Data.List.Range
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_FirstPairData_Definitions
import Definitions.Def_D1Bound5_OneSmallFiniteCheck_Definitions
import Definitions.Def_D1Bound5_OneSmallRows_Definitions
import Definitions.Def_D1Bound5_OneSmallSymmetry_Definitions
import Definitions.Def_D1Bound5_PairExtensionCore_Definitions

Formal statement
section
namespace D1Bound5
open QuadInt



theorem _root_.D1Bound5.firstPairTable_0_2_third_rows (a b s : QuadInt 0 2)
    (hrow : firstPairCoordinates a b ∈ firstPairTable 0 2)
    (hbound : s.norm ≤ 224) (horder : b.norm ≤ s.norm)
    (ha : s ≠ a) (hb : s ≠ b) (hpair : PairExtension a b s) :
    OneSmallNormalizedThirdRows a b s := by sorry



















end D1Bound5
end
Source
D1Bound5/OneSmallOrbitRows.lean, lines 27-40, SHA-256 d4fb0e36db302c645e881c0f79ec075a8e198e085e82153313649478e5dda480. Project: https://github.com/tomislav-math/d1-bound5-lean

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