Full ring finite A3Row Form exists one le norm

import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.GCD
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_FiniteA3IntegralRows_Definitions

section
open scoped NumberField
open Module

namespace D1Bound5

variable {R S : Type*} [CommRing R] [CommRing S]
















theorem _root_.D1Bound5.full_ring_finiteA3RowForm_exists_one_le_norm
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    (x y c z sX sY : 𝓞 K) (hx : x ≠ 0) (hy : y ≠ 0) (hc : c ≠ 0) (hxy : x ≠ y) (hz : z ≠ 0) :
    ∃ i : Fin 3, 1 ≤ ‖finiteA3RowForm (phi (x : K)) (phi (y : K))
      (phi (c : K)) (phi (z : K)) (phi (sX : K)) (phi (sY : K)) i‖ := by sorry

end D1Bound5
end