First Pair Table geometry normalize

import Lean.Elab.Tactic.Omega
import Lean.Util.CollectAxioms
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharZero.Infinite
import Mathlib.Algebra.Group.Int.Units
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.PartialFractions
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Enumerative.Catalan.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Fin.Tuple.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Data.List.Range
import Mathlib.Data.Real.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.Tactic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_FirstPairData_Definitions
import Definitions.Def_D1Bound5_FirstPairGeometryData_Definitions_001
import Definitions.Def_D1Bound5_FirstPairGeometryData_Definitions_002
import Definitions.Def_D1Bound5_OneSmallFiniteCheck_Definitions
import Definitions.Def_D1Bound5_OneSmallRows_Definitions
import Definitions.Def_D1Bound5_SmallExtensionSymmetry_Definitions
import Definitions.Def_D1Bound5_SmallPairClassification_Definitions

section
namespace D1Bound5
open QuadInt




theorem _root_.D1Bound5.firstPairTable_geometry_normalize {tau nu : ℤ}
    (hmodel : (tau,nu) ∈ smallPairModels) (a b : QuadInt tau nu)
    (hrow : firstPairCoordinates a b ∈ firstPairTable tau nu) :
    ∃ g ∈ firstPairGeometryRows, g.tau=tau ∧ g.nu=nu ∧
      ∃ sgn conj : Bool,
        firstPairCoordinates (smallExtensionSymmetry sgn conj a)
          (smallExtensionSymmetry sgn conj b) = (g.a,g.b) := by sorry

end D1Bound5
end