Pair Graph Prepared3 — indexed coverage checked

Proved

D1Bound5.PairGraphPrepared3.indexed_coverage_checked

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

diophantine-tuplesnumber-theory

Auxiliary result PairGraphPrepared3.indexed_coverage_checked 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_PairClosureWitness_Definitions
import Definitions.Def_D1Bound5_SmallPairGraphInputs3_Definitions_001
import Definitions.Def_D1Bound5_SmallPairGraphInputs3_Definitions_002
import Definitions.Def_D1Bound5_SmallPairGraphWitnessData3_Definitions_001
import Definitions.Def_D1Bound5_SmallPairGraphWitnessData3_Definitions_003

Formal statement

section
namespace D1Bound5.PairGraphPrepared3
set_option autoImplicit false
set_option maxRecDepth 100000
set_option maxHeartbeats 0

theorem _root_.D1Bound5.PairGraphPrepared3.indexed_coverage_checked :
    decide (anchorRoot * anchorRoot = smallPairA_3 * smallPairB_3 + 1 ∧
      witnesses.map PairRootWitness.vertex = smallPairClosure_3) = true := by sorry

end D1Bound5.PairGraphPrepared3
end

Source

D1Bound5/SmallPairGraphClosure3Coverage.lean, lines 9-29, SHA-256 e0c07ec5b4d30ca59dc77923c28914f9390a1a13800575fa2c7a3d856db4b3c0. Project: https://github.com/tomislav-math/d1-bound5-lean

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