Pair Graph Triangle Free Bool of trace zero tau zero

Proved

D1Bound5.pairGraphTriangleFreeBool_of_trace_zero_tau_zero

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

diophantine-tuplesnumber-theory

In the trace-zero model with nu>1, checking the trace of every listed vertex replaces all pair comparisons. The original graph predicate remains unchanged, and its nonzero guards supply the algebraic endpoint's hypotheses.

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_PairGraphNoThreeTransfer_Definitions
import Definitions.Def_D1Bound5_SmallPairGraphCheck_Definitions

Formal statement

section
namespace D1Bound5
open QuadInt


theorem _root_.D1Bound5.pairGraphTriangleFreeBool_of_trace_zero_tau_zero {nu : ℤ}
    (hnu : 1 < nu) (sieve : List (ℕ × ℕ))
    (hvalid : CoordinateSquareSieveValid 0 nu sieve)
    (vertices : List (QuadInt 0 nu))
    (htrace : pairAllTraceZeroBool vertices = true) :
    pairGraphTriangleFreeBool sieve vertices = true := by sorry



end D1Bound5
end

Source

D1Bound5/PairTraceZeroGraph.lean, lines 7-25, SHA-256 0d64054a8011cced0bc7f67ef2726805eba0ee9f00ed409f60ae1163ef294ee0. Project: https://github.com/tomislav-math/d1-bound5-lean

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