Pair Mutation eq supplied roots

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Definitions.Def_D1Bound5_Coordinates_Definitions
import Definitions.Def_D1Bound5_DescentAlgebra_Definitions
import Definitions.Def_D1Bound5_PairExtensionCore_Definitions

section
namespace D1Bound5
open QuadInt














theorem _root_.D1Bound5.pairMutation_eq_supplied_roots {tau nu : ℤ}
    (hd : 0 < discriminant tau nu) (a b z r0 r s : QuadInt tau nu)
    (h0 : r0 * r0 = a * b + 1) (hr : r * r = a * z + 1)
    (hs : s * s = b * z + 1) {w : QuadInt tau nu}
    (hw : w = pairMutationPlus a b z ∨ w = pairMutationMinus a b z) :
    w = mutate a b z r0 r s ∨ w = mutate a b z r0 r (-s) := by sorry







end D1Bound5
end