Pair Closure complete full ring

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.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_PairExtensionCore_Definitions

section
open scoped NumberField
open Module

namespace D1Bound5
open QuadInt


theorem _root_.D1Bound5.pairClosure_complete_full_ring
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    {tau nu : ℤ} (hd : 0 < discriminant tau nu)
    (e : QuadInt tau nu ≃+* 𝓞 K) (a b : QuadInt tau nu)
    (ha : a ≠ 0) (hb : b ≠ 0) (hab : a*b+1 ≠ 0)
    (hroot : ∃ r0, r0*r0 = a*b+1)
    (hNa : a.norm ≤ 8) (hNb : b.norm ≤ 8)
    (bound : ℤ) (vertices : List (QuadInt tau nu))
    (hseeds : ∀ z, PairExtension a b z → z.norm ≤ 1089 → z ∈ vertices)
    (hclosed : ∀ z ∈ vertices, ∀ w,
      (w = pairMutationPlus a b z ∨ w = pairMutationMinus a b z) →
      w.norm ≤ bound → w ∈ vertices)
    (z : QuadInt tau nu) (hz : PairExtension a b z) (hNz : z.norm ≤ bound) :
    z ∈ vertices := by sorry

end D1Bound5
end