First Pair Table third rows
ProvedD1Bound5.firstPairTable_third_rowsdiophantine-tuplesnumber-theory
Auxiliary result firstPairTable_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 import Definitions.Def_D1Bound5_SmallPairClassification_Definitions
Formal statement
section
namespace D1Bound5
open QuadInt
theorem _root_.D1Bound5.firstPairTable_third_rows {tau nu : ℤ}
(hmodel : (tau,nu) ∈ smallPairModels) (a b s : QuadInt tau nu)
(hrow : firstPairCoordinates a b ∈ firstPairTable tau nu)
(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 166-184, SHA-256 d4fb0e36db302c645e881c0f79ec075a8e198e085e82153313649478e5dda480. Project: https://github.com/tomislav-math/d1-bound5-lean