-- Mathlib/Algebra/Lie/Weights/RootSystem.lean
lemma rootSpace_neg_nsmul_add_chainTop_of_lt (hα : α.IsNonZero) {n : ℕ} (hn : chainLength α β < n) :
    rootSpace H (- (n • α) + chainTop α β) = ⊥

Proof state:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
n : ℕ
hn : chainLength α β < n
e : ¬rootSpace H (-(n • ⇑α) + ⇑(chainTop (⇑α) β)) = ⊥
W : Weight K (↥H) L := { toFun := -(n • ⇑α) + ⇑(chainTop (⇑α) β), genWeightSpace_ne_bot' := e }
hW : ⇑W = -(n • ⇑α) + ⇑(chainTop (⇑α) β)
H₁ : 1 + n + chainTopCoeff (⇑(-α)) W ≤ chainLength (-α) W
⊢ False

Tactic:
have H₂ : ((1 + n + chainTopCoeff (-α) W) • α + chainTop (-α) W : H → K) =
    (chainTopCoeff α β + 1) • α + β := by
  simp only [Weight.coe_neg, ← Nat.cast_smul_eq_nsmul ℤ, Nat.cast_add, Nat.cast_one, coe_chainTop,
    smul_neg, ← neg_smul, hW, ← add_assoc, ← add_smul, ← sub_eq_add_neg]
  congr 2
  ring

Result:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
n : ℕ
hn : chainLength α β < n
e : ¬rootSpace H (-(n • ⇑α) + ⇑(chainTop (⇑α) β)) = ⊥
W : Weight K (↥H) L := { toFun := -(n • ⇑α) + ⇑(chainTop (⇑α) β), genWeightSpace_ne_bot' := e }
hW : ⇑W = -(n • ⇑α) + ⇑(chainTop (⇑α) β)
H₁ : 1 + n + chainTopCoeff (⇑(-α)) W ≤ chainLength (-α) W
H₂ : (1 + n + chainTopCoeff (⇑(-α)) W) • ⇑α + ⇑(chainTop (⇑(-α)) W) = (chainTopCoeff (⇑α) β + 1) • ⇑α + ⇑β
⊢ False

Proof state:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
n : ℕ
hn : chainLength α β < n
e : ¬rootSpace H (-(n • ⇑α) + ⇑(chainTop (⇑α) β)) = ⊥
W : Weight K (↥H) L := { toFun := -(n • ⇑α) + ⇑(chainTop (⇑α) β), genWeightSpace_ne_bot' := e }
hW : ⇑W = -(n • ⇑α) + ⇑(chainTop (⇑α) β)
H₁ : 1 + n + chainTopCoeff (⇑(-α)) W ≤ chainLength (-α) W
H₂ : (1 + n + chainTopCoeff (⇑(-α)) W) • ⇑α + ⇑(chainTop (⇑(-α)) W) = (chainTopCoeff (⇑α) β + 1) • ⇑α + ⇑β
⊢ False

Tactic:
have := rootSpace_neg_nsmul_add_chainTop_of_le (-α) W H₁

Result:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
hα : α.IsNonZero
n : ℕ
hn : chainLength α β < n
e : ¬rootSpace H (-(n • ⇑α) + ⇑(chainTop (⇑α) β)) = ⊥
W : Weight K (↥H) L := { toFun := -(n • ⇑α) + ⇑(chainTop (⇑α) β), genWeightSpace_ne_bot' := e }
hW : ⇑W = -(n • ⇑α) + ⇑(chainTop (⇑α) β)
H₁ : 1 + n + chainTopCoeff (⇑(-α)) W ≤ chainLength (-α) W
H₂ : (1 + n + chainTopCoeff (⇑(-α)) W) • ⇑α + ⇑(chainTop (⇑(-α)) W) = (chainTopCoeff (⇑α) β + 1) • ⇑α + ⇑β
this : rootSpace H (-((1 + n + chainTopCoeff (⇑(-α)) W) • ⇑(-α)) + ⇑(chainTop (⇑(-α)) W)) ≠ ⊥
⊢ False