Full ring three separated norm inequality

import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Int.GCD
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Mathlib.Tactic.ComputeDegree
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_CoherentIntegralForms_Definitions
import Definitions.Def_D1Bound5_CoherentPolynomialWeights_Definitions
import Definitions.Def_D1Bound5_CoherentQuotientForms_Definitions
import Definitions.Def_D1Bound5_IsD1

section
open Finset
open scoped NumberField
open Module

namespace D1Bound5


theorem _root_.D1Bound5.full_ring_three_separated_norm_inequality
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    {Tup : Finset (𝓞 K)} (hTup : IsD1 Tup) {p q : 𝓞 K}
    (hp : p ∈ Tup) (hq : q ∈ Tup) (hpq : p ≠ q)
    (a : Fin 3 → 𝓞 K) (hinj : Function.Injective a)
    (ha : ∀ i, a i ∈ Tup) (hap : ∀ i, a i ≠ p) (haq : ∀ i, a i ≠ q)
    (hAB : ‖phi (a 0 : K)‖ ≤ ‖phi (a 1 : K)‖)
    (hBS : ‖phi (a 1 : K)‖ ≤ ‖phi (a 2 : K)‖)
    (hP : 0 < ‖phi (p : K)‖) (hAP : 1000 ≤ ‖phi (a 0 : K)‖*‖phi (p : K)‖)
    (hQ : 4*‖phi (p : K)‖ ≤ ‖phi (q : K)‖)
    (hPQ : 10000 < ‖phi (p : K)‖*‖phi (q : K)‖)
    (eta : ℂ) (heta : eta^2 = phi (p : K)*phi (q : K))
    (hL : coherentQuotientForm (fun i => phi (a i : K))
      (coherentVandermonde3 (fun i => phi (a i : K)))
        (phi (p : K)) (phi (q : K)) eta ≠ 0) :
    1 < 5*‖phi (a 1 : K)‖*‖phi (a 2 : K)‖^2/
          ‖coherentTerminalRoot (phi (p : K)) (phi (q : K)) eta‖ +
      3*‖phi (a 2 : K)‖*‖coherentTerminalRoot (phi (p : K)) (phi (q : K)) eta‖/
        (4*‖phi (a 0 : K)‖*‖phi (p : K)‖^3) := by sorry



end D1Bound5
end