First pair geometry bounds

import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Mathlib.Tactic.GCongr
import Definitions.Def_D1Bound5_CoherentPolynomialWeights_Definitions
import Definitions.Def_D1Bound5_FirstPairGeometry_Definitions

section
namespace D1Bound5






theorem _root_.D1Bound5.first_pair_geometry_bounds (a b s : ℂ) (A1 B1 d1 : ℝ)
    (hb : b ≠ 0) (hBS : ‖b‖ ≤ ‖s‖) (hS15 : 15 ≤ ‖s‖)
    (hA1 : ‖a‖ ≤ A1) (hB1 : ‖b‖ ≤ B1)
    (hd1 : ‖a-b‖/‖b‖ ≤ d1) :
    let Q := ‖coherentVandermonde3 (![a,b,s] : Fin 3 → ℂ)‖/(‖b‖*‖s‖^2)
    Q ≤ d1*(1+A1/15)*(1+B1/15) ∧
      Q^2*firstPairPhi b s ≤
        (‖a-b‖^2/‖b‖^2)*(1+A1/15)^2*(1+B1/15)^8 := by sorry

end D1Bound5
end