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One Small Thirds 1 complete

Proved
D1Bound5.oneSmallThirds_1_complete

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

diophantine-tuplesnumber-theory

Auxiliary result oneSmallThirds_1_complete 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.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.List.Range
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_OneSmallThirdData_Definitions_001
import Definitions.Def_D1Bound5_PairExtensionCore_Definitions

Formal statement
section
namespace D1Bound5
open QuadInt
set_option maxRecDepth 32768
set_option maxHeartbeats 12000000















theorem _root_.D1Bound5.oneSmallThirds_1_complete (s : QuadInt 0 1) (hbound : s.norm ≤ 224)
    (horder : oneSmallPairB_1.norm ≤ s.norm)
    (ha : s ≠ oneSmallPairA_1) (hb : s ≠ oneSmallPairB_1)
    (hpair : PairExtension oneSmallPairA_1 oneSmallPairB_1 s) : s ∈ oneSmallThirds_1 := by sorry

















































































































































































































































































































end D1Bound5
end
Source
D1Bound5/OneSmallThirdData.lean, lines 34-40, SHA-256 1a99812615576213a57446af3223d15f39986fed5a03e5b58ab6c3210a77a7c2. Project: https://github.com/tomislav-math/d1-bound5-lean

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