Pair Cached Independent Bool sound

Proved

D1Bound5.pairCachedIndependentBool_sound

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

diophantine-tuplesnumber-theory

Auxiliary result pairCachedIndependentBool_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_PairBipartiteCheck_Definitions
import Definitions.Def_D1Bound5_PairCachedResidues_Definitions

Formal statement

section
namespace D1Bound5





















theorem _root_.D1Bound5.pairCachedIndependentBool_sound {tau nu : ℤ} (sieve : List (ℕ × ℕ))
    (cached : List (PairResidueVertex tau nu))
    (hvalid : ∀ x ∈ cached, x.signature = pairCoordinateSignature sieve x.vertex)
    (hcheck : pairCachedIndependentBool sieve cached = true) :
    pairIndependentBool sieve (cached.map PairResidueVertex.vertex) = true := by sorry





end D1Bound5
end

Source

D1Bound5/PairCachedResidues.lean, lines 94-123, SHA-256 8d30539028d401e948637a33f705cb33707d8ad4e1fe939eafd5ed7c690a9100. Project: https://github.com/tomislav-math/d1-bound5-lean

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