-- Module: Mathlib.Algebra.Lie.Abelian
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
s : Set L
h : ∀ x ∈ s, ∀ y ∈ s, ⁅x, y⁆ = 0
x_1 x_2 : ↥(lieSpan R L s)
x : L
hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x
y : L
hy : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) y
refine_2 :
  ∀ (w : L) (hw : w ∈ s),
    Bracket.bracket (L := ↥(lieSpan R L s))
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) w ⋯)
        (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy) =
      0
refine_2₁ :
  Bracket.bracket (L := ↥(lieSpan R L s))
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) 0 ⋯)
      (Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy) =
    0
refine_2₂ :
  ∀ (u v : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hy_1 : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) v)
    (hu :
      ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx,
          Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
        0)
    (hv :
      ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) v hy_1,
          Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
        0),
    ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) (u + v) ⋯,
        Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
      0
refine_2₃ :
  ∀ (t : R) (u : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hu :
      ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx,
          Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
        0),
    ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) (t • u) ⋯,
        Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
      0
refine_2₄ :
  ∀ (u v : L) (hx : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) u)
    (hy_1 : Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) v)
    (hu :
      ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) u hx,
          Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
        0)
    (hv :
      ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) v hy_1,
          Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
        0),
    ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) ⁅u, v⁆ ⋯,
        Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
      0
⊢ ⁅Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) x hx,
      Subtype.mk (p := fun (x : L) ↦ Membership.mem (γ := LieSubalgebra R L) (lieSpan R L s) x) y hy⁆ =
    0

Tactic:
induction hx using lieSpan_induction with
    | mem w hw =>
      induction hy using lieSpan_induction with
      | mem u hu => simpa [Subtype.ext_iff] using h w hw u hu
      | zero => simp [Subtype.ext_iff]
      | add u v _ _ hu hv =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero, lie_add] at hu hv ⊢
        simp [hu, hv]
      | smul t u _ hu =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu
        simp [Subtype.ext_iff, hu]
      | lie u v _ _ hu hv =>
        simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu hv ⊢
        rw [leibniz_lie]
        simp [hu, hv]
    | zero => simp [Subtype.ext_iff]
    | add u v _ _ hu hv =>
      simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero, add_lie] at hu hv ⊢
      simp [hu, hv]
    | smul t u _ hu =>
      simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu
      simp [Subtype.ext_iff, hu]
    | lie u v _ _ hu hv =>
      simp only [Subtype.ext_iff, coe_bracket, ZeroMemClass.coe_zero] at hu hv ⊢
      simp [hu, hv]

Premises used: leibniz_lie, Subtype.ext_iff, LieSubalgebra.coe_bracket, ZeroMemClass.coe_zero, lie_add, add_lie