l3lab/ntp-mathlib huggingface dataset:
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ m₁ : M
hm₀ : m₀ ∈ weightSpace M 0
hm₁ : m₁ ∈ posFittingComp R L M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
this : ∀ (x : L), ∀ m ∈ posFittingCompOf R M x, (B m₀) m = 0
⊢ ∀ (y z : M), (B m₀) y = 0 → (B m₀) z = 0 → (B m₀) (y + z) = 0

Next tactic:
aesop
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  

Proof state:
case this
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ m₁ : M
hm₀ : m₀ ∈ weightSpace M 0
hm₁ : m₁ ∈ posFittingComp R L M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
⊢ ∀ (x : L), ∀ m ∈ posFittingCompOf R M x, (B m₀) m = 0

Next tactic:
clear hm₁ m₁
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  clear hm₁ m₁; 

Proof state:
case this
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ : M
hm₀ : m₀ ∈ weightSpace M 0
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
⊢ ∀ (x : L), ∀ m ∈ posFittingCompOf R M x, (B m₀) m = 0

Next tactic:
intro x m₁ hm₁
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  clear hm₁ m₁; intro x m₁ hm₁
  

Proof state:
case this
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ : M
hm₀ : m₀ ∈ weightSpace M 0
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
x : L
m₁ : M
hm₁ : m₁ ∈ posFittingCompOf R M x
⊢ (B m₀) m₁ = 0

Next tactic:
simp only [mem_weightSpace, Pi.zero_apply, zero_smul, sub_zero] at hm₀
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  clear hm₁ m₁; intro x m₁ hm₁
  simp only [mem_weightSpace, Pi.zero_apply, zero_smul, sub_zero] at hm₀
  

Proof state:
case this
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ : M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
x : L
m₁ : M
hm₁ : m₁ ∈ posFittingCompOf R M x
hm₀ : ∀ (x : L), ∃ k, (φ x ^ k) m₀ = 0
⊢ (B m₀) m₁ = 0

Next tactic:
obtain ⟨k, hk⟩ := hm₀ x
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  clear hm₁ m₁; intro x m₁ hm₁
  simp only [mem_weightSpace, Pi.zero_apply, zero_smul, sub_zero] at hm₀
  obtain ⟨k, hk⟩ := hm₀ x
  

Proof state:
case this.intro
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ : M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
x : L
m₁ : M
hm₁ : m₁ ∈ posFittingCompOf R M x
hm₀ : ∀ (x : L), ∃ k, (φ x ^ k) m₀ = 0
k : ℕ
hk : (φ x ^ k) m₀ = 0
⊢ (B m₀) m₁ = 0

Next tactic:
obtain ⟨m, rfl⟩ := (mem_posFittingCompOf R x m₁).mp hm₁ k
/-- Given a bilinear form `B` on a representation `M` of a nilpotent Lie algebra `L`, if `B` is
invariant (in the sense that the action of `L` is skew-adjoint wrt `B`) then components of the
Fitting decomposition of `M` are orthogonal wrt `B`. -/
lemma eq_zero_of_mem_weightSpace_mem_posFitting [LieAlgebra.IsNilpotent R L]
    {B : M →ₗ[R] M →ₗ[R] R} (hB : ∀ (x : L) (m n : M), B ⁅x, m⁆ n = - B m ⁅x, n⁆)
    {m₀ m₁ : M} (hm₀ : m₀ ∈ weightSpace M (0 : L → R)) (hm₁ : m₁ ∈ posFittingComp R L M) :
    B m₀ m₁ = 0 := by
  replace hB : ∀ x (k : ℕ) m n, B m ((φ x ^ k) n) = (- 1 : R) ^ k • B ((φ x ^ k) m) n := by
    intro x k
    induction k with
    | zero => simp
    | succ k ih =>
    intro m n
    replace hB : ∀ m, B m (φ x n) = (- 1 : R) • B (φ x m) n := by simp [hB]
    have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ' (-1 : R), smul_eq_mul]
    conv_lhs => rw [pow_succ', LinearMap.mul_eq_comp, LinearMap.comp_apply, ih, hB,
      ← (φ x).comp_apply, ← LinearMap.mul_eq_comp, ← pow_succ, ← smul_assoc, this]
  suffices : ∀ (x : L) m, m ∈ posFittingCompOf R M x → B m₀ m = 0
  · apply LieSubmodule.iSup_induction _ hm₁ this (map_zero _)
    aesop
  clear hm₁ m₁; intro x m₁ hm₁
  simp only [mem_weightSpace, Pi.zero_apply, zero_smul, sub_zero] at hm₀
  obtain ⟨k, hk⟩ := hm₀ x
  obtain ⟨m, rfl⟩ := (mem_posFittingCompOf R x m₁).mp hm₁ k
  

Proof state:
case this.intro.intro
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁴ : CommRing R
inst✝¹³ : LieRing L
inst✝¹² : LieAlgebra R L
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : LieRingModule L M
inst✝⁸ : LieModule R L M
inst✝⁷ : Module.Free R M
inst✝⁶ : Module.Finite R M
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : Module K M
inst✝² : LieModule K L M
inst✝¹ : FiniteDimensional K M
inst✝ : LieAlgebra.IsNilpotent R L
B : M →ₗ[R] M →ₗ[R] R
m₀ : M
hB : ∀ (x : L) (k : ℕ) (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
x : L
hm₀ : ∀ (x : L), ∃ k, (φ x ^ k) m₀ = 0
k : ℕ
hk : (φ x ^ k) m₀ = 0
m : M
hm₁ : (φ x ^ k) m ∈ posFittingCompOf R M x
⊢ (B m₀) ((φ x ^ k) m) = 0

Next tactic:
simp [hB, hk]
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L k
⊢ (trace R M) (φ x) = 0

Next tactic:
replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L 1
⊢ (trace R M) (φ x) = 0

Next tactic:
replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L 1
⊢ x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}

Next tactic:
rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ x ∈ ⊤, ∃ n ∈ lowerCentralSeries R L L 0, ⁅x, n⁆ = m}
⊢ x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}

Next tactic:
simpa using hx
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
⊢ (trace R M) (φ x) = 0

Next tactic:
refine Submodule.span_induction (p := fun x ↦ trace R _ (toEndomorphism R L M x) = 0) hx
    (fun y ⟨u, v, huv⟩ ↦ ?_) ?_ (fun u v hu hv ↦ ?_) (fun t u hu ↦ ?_)
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
  refine Submodule.span_induction (p := fun x ↦ trace R _ (toEndomorphism R L M x) = 0) hx
    (fun y ⟨u, v, huv⟩ ↦ ?_) ?_ (fun u v hu hv ↦ ?_) (fun t u hu ↦ ?_)
  · 

Proof state:
case refine_1
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
y : L
x✝ : y ∈ {m | ∃ u v, ⁅u, v⁆ = m}
u v : L
huv : ⁅u, v⁆ = y
⊢ (fun x => (trace R M) (φ x) = 0) y

Next tactic:
simp_rw [← huv, LieHom.map_lie, Ring.lie_def, map_sub, LinearMap.trace_mul_comm, sub_self]
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
  refine Submodule.span_induction (p := fun x ↦ trace R _ (toEndomorphism R L M x) = 0) hx
    (fun y ⟨u, v, huv⟩ ↦ ?_) ?_ (fun u v hu hv ↦ ?_) (fun t u hu ↦ ?_)
  · simp_rw [← huv, LieHom.map_lie, Ring.lie_def, map_sub, LinearMap.trace_mul_comm, sub_self]
  · 

Proof state:
case refine_2
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
⊢ (fun x => (trace R M) (φ x) = 0) 0

Next tactic:
simp
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
  refine Submodule.span_induction (p := fun x ↦ trace R _ (toEndomorphism R L M x) = 0) hx
    (fun y ⟨u, v, huv⟩ ↦ ?_) ?_ (fun u v hu hv ↦ ?_) (fun t u hu ↦ ?_)
  · simp_rw [← huv, LieHom.map_lie, Ring.lie_def, map_sub, LinearMap.trace_mul_comm, sub_self]
  · simp
  · 

Proof state:
case refine_3
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
u v : L
hu : (fun x => (trace R M) (φ x) = 0) u
hv : (fun x => (trace R M) (φ x) = 0) v
⊢ (fun x => (trace R M) (φ x) = 0) (u + v)

Next tactic:
simp [hu, hv]
lemma trace_toEndomorphism_eq_zero_of_mem_lcs
    {k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
    trace R _ (toEndomorphism R L M x) = 0 := by
  replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx
  replace hx : x ∈ Submodule.span R {m | ∃ u v : L, ⁅u, v⁆ = m} := by
    rw [lowerCentralSeries_succ, ← LieSubmodule.mem_coeSubmodule,
      LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx
    simpa using hx
  refine Submodule.span_induction (p := fun x ↦ trace R _ (toEndomorphism R L M x) = 0) hx
    (fun y ⟨u, v, huv⟩ ↦ ?_) ?_ (fun u v hu hv ↦ ?_) (fun t u hu ↦ ?_)
  · simp_rw [← huv, LieHom.map_lie, Ring.lie_def, map_sub, LinearMap.trace_mul_comm, sub_self]
  · simp
  · simp [hu, hv]
  · 

Proof state:
case refine_4
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m}
t : R
u : L
hu : (fun x => (trace R M) (φ x) = 0) u
⊢ (fun x => (trace R M) (φ x) = 0) (t • u)

Next tactic:
simp [hu]