/-- In characteristic zero if there is no torsion, the correspondence between roots and coroots is
unique.

Formally, the point is that the hypothesis `hc` depends only on the range of the coroot mappings. -/
@[ext]
protected lemma ext [CharZero R] [NoZeroSMulDivisors R M]
    {P₁ P₂ : RootPairing ι R M N}
    (he : P₁.toLin = P₂.toLin)
    (hr : P₁.root = P₂.root)
    (hc : range P₁.coroot = range P₂.coroot) :
    P₁ = P₂ := by
  suffices P₁.coroot = P₂.coroot by cases' P₁ with p₁; cases' P₂ with p₂; 

Proof state:
case mk.mk
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁷ : _root_.Finite ι
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
i : ι
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
p₁ : PerfectPairing R M N
root✝¹ : ι ↪ M
coroot✝¹ : ι ↪ N
root_coroot_two✝¹ : ∀ (i : ι), (p₁.toLin (root✝¹ i)) (coroot✝¹ i) = 2
mapsTo_preReflection_root✝¹ :
  ∀ (i : ι),
    MapsTo (⇑(preReflection (root✝¹ i) ((LinearMap.flip p₁.toLin) (coroot✝¹ i)))) (range ⇑root✝¹) (range ⇑root✝¹)
mapsTo_preReflection_coroot✝¹ :
  ∀ (i : ι), MapsTo (⇑(preReflection (coroot✝¹ i) (p₁.toLin (root✝¹ i)))) (range ⇑coroot✝¹) (range ⇑coroot✝¹)
p₂ : PerfectPairing R M N
root✝ : ι ↪ M
coroot✝ : ι ↪ N
root_coroot_two✝ : ∀ (i : ι), (p₂.toLin (root✝ i)) (coroot✝ i) = 2
mapsTo_preReflection_root✝ :
  ∀ (i : ι), MapsTo (⇑(preReflection (root✝ i) ((LinearMap.flip p₂.toLin) (coroot✝ i)))) (range ⇑root✝) (range ⇑root✝)
mapsTo_preReflection_coroot✝ :
  ∀ (i : ι), MapsTo (⇑(preReflection (coroot✝ i) (p₂.toLin (root✝ i)))) (range ⇑coroot✝) (range ⇑coroot✝)
he :
  { toPerfectPairing := p₁, root := root✝¹, coroot := coroot✝¹, root_coroot_two := root_coroot_two✝¹,
          mapsTo_preReflection_root := mapsTo_preReflection_root✝¹,
          mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝¹ }.toPerfectPairing.toLin =
    { toPerfectPairing := p₂, root := root✝, coroot := coroot✝, root_coroot_two := root_coroot_two✝,
          mapsTo_preReflection_root := mapsTo_preReflection_root✝,
          mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝ }.toPerfectPairing.toLin
hr :
  { toPerfectPairing := p₁, root := root✝¹, coroot := coroot✝¹, root_coroot_two := root_coroot_two✝¹,
        mapsTo_preReflection_root := mapsTo_preReflection_root✝¹,
        mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝¹ }.root =
    { toPerfectPairing := p₂, root := root✝, coroot := coroot✝, root_coroot_two := root_coroot_two✝,
        mapsTo_preReflection_root := mapsTo_preReflection_root✝,
        mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝ }.root
hc :
  range
      ⇑{ toPerfectPairing := p₁, root := root✝¹, coroot := coroot✝¹, root_coroot_two := root_coroot_two✝¹,
            mapsTo_preReflection_root := mapsTo_preReflection_root✝¹,
            mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝¹ }.coroot =
    range
      ⇑{ toPerfectPairing := p₂, root := root✝, coroot := coroot✝, root_coroot_two := root_coroot_two✝,
            mapsTo_preReflection_root := mapsTo_preReflection_root✝,
            mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝ }.coroot
this :
  { toPerfectPairing := p₁, root := root✝¹, coroot := coroot✝¹, root_coroot_two := root_coroot_two✝¹,
        mapsTo_preReflection_root := mapsTo_preReflection_root✝¹,
        mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝¹ }.coroot =
    { toPerfectPairing := p₂, root := root✝, coroot := coroot✝, root_coroot_two := root_coroot_two✝,
        mapsTo_preReflection_root := mapsTo_preReflection_root✝,
        mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝ }.coroot
⊢ { toPerfectPairing := p₁, root := root✝¹, coroot := coroot✝¹, root_coroot_two := root_coroot_two✝¹,
      mapsTo_preReflection_root := mapsTo_preReflection_root✝¹,
      mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝¹ } =
    { toPerfectPairing := p₂, root := root✝, coroot := coroot✝, root_coroot_two := root_coroot_two✝,
      mapsTo_preReflection_root := mapsTo_preReflection_root✝,
      mapsTo_preReflection_coroot := mapsTo_preReflection_coroot✝ }

Next tactic:
cases p₁