Pair Independent Bool sound

Proved

D1Bound5.pairIndependentBool_sound

by tp · Oct 24, 2026 · Mathlib 0df444a (Lean v4.33.1)

diophantine-tuplesnumber-theory

Auxiliary result pairIndependentBool_sound 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.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_FastSquareTest_Definitions
import Definitions.Def_D1Bound5_PairBipartiteCheck_Definitions

Formal statement

section
namespace D1Bound5
open QuadInt







theorem _root_.D1Bound5.pairIndependentBool_sound {tau nu : ℤ} (sieve : List (ℕ × ℕ))
    (vertices : List (QuadInt tau nu)) (hcheck : pairIndependentBool sieve vertices = true)
    (x y : QuadInt tau nu) (hx : x ∈ vertices) (hy : y ∈ vertices) (hxy : x ≠ y) :
    hasSquareRootFast sieve (x*y+1) = false := by sorry





end D1Bound5
end

Source

D1Bound5/PairBipartiteCheck.lean, lines 37-57, SHA-256 d1497b46c9dcbae577d7172e6c4401388a1797e68d33ba795d4e6a6a05dd4c0c. Project: https://github.com/tomislav-math/d1-bound5-lean

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