-- Module: Mathlib.Algebra.Lie.Weights.RootSystem
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : IsKilling K L
inst_5 : FiniteDimensional K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
α_1 β_1 α : Weight K (↥H) L
hα : Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) α
β : Weight K (↥H) L
hβ : Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) β
u : K
hu : u • Weight.toLinear K (↥H) L α = Weight.toLinear K (↥H) L β
inl :
  ∀ (h : β = -α),
    Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
          Dual.{u_1, u_2} K ↥H) ∨
      Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        (-(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
              (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
                Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β
                hβ) :
          Dual.{u_1, u_2} K ↥H)
inr :
  ∀ (h : β = α),
    Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
          Dual.{u_1, u_2} K ↥H) ∨
      Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        (-(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
              (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
                Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β
                hβ) :
          Dual.{u_1, u_2} K ↥H)
⊢ Eq (α := Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
        Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
        Dual.{u_1, u_2} K ↥H) ∨
    Eq (α := Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
        Dual.{u_1, u_2} K ↥H)
      (-(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
        Dual.{u_1, u_2} K ↥H)

Tactic:
obtain (h | h) := eq_neg_or_eq_of_eq_smul α β (by simpa using hβ) u
      (by ext x; exact DFunLike.congr_fun hu.symm x)

Premises used: LieAlgebra.IsKilling.eq_neg_or_eq_of_eq_smul