l3lab/ntp-mathlib huggingface dataset:
/-- The increment partition has energy greater than the original one by a known fixed amount. -/
theorem energy_increment [Nonempty α] (hP : P.IsEquipartition) (hP₇ : 7 ≤ P.parts.card)
    (hε : ↑100 < ↑4 ^ P.parts.card * ε ^ 5) (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
    (hPG : ¬P.IsUniform G ε) (hε₁ : ε ≤ 1) :
    ↑(P.energy G) + ε ^ 5 / 4 ≤ (increment hP G ε).energy G := by
  rw [coe_energy]
  

Proof state:
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP✝ : IsEquipartition P
G : SimpleGraph α
ε : ℝ
inst✝ : Nonempty α
hP : IsEquipartition P
hP₇ : 7 ≤ Finset.card P.parts
hε : 100 < 4 ^ Finset.card P.parts * ε ^ 5
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPG : ¬Finpartition.IsUniform P G ε
hε₁ : ε ≤ 1
⊢ (∑ uv in offDiag P.parts, ↑(edgeDensity G uv.1 uv.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 + ε ^ 5 / 4 ≤
    ↑(energy (increment hP G ε) G)

Next tactic:
have h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) hε₁ hP₇ hPα hε.le hPG
/-- The increment partition has energy greater than the original one by a known fixed amount. -/
theorem energy_increment [Nonempty α] (hP : P.IsEquipartition) (hP₇ : 7 ≤ P.parts.card)
    (hε : ↑100 < ↑4 ^ P.parts.card * ε ^ 5) (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
    (hPG : ¬P.IsUniform G ε) (hε₁ : ε ≤ 1) :
    ↑(P.energy G) + ε ^ 5 / 4 ≤ (increment hP G ε).energy G := by
  rw [coe_energy]
  have h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) hε₁ hP₇ hPα hε.le hPG
  

Proof state:
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP✝ : IsEquipartition P
G : SimpleGraph α
ε : ℝ
inst✝ : Nonempty α
hP : IsEquipartition P
hP₇ : 7 ≤ Finset.card P.parts
hε : 100 < 4 ^ Finset.card P.parts * ε ^ 5
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPG : ¬Finpartition.IsUniform P G ε
hε₁ : ε ≤ 1
h :
  (∑ x in offDiag P.parts, ↑(edgeDensity G x.1 x.2) ^ 2 + ↑(Finset.card P.parts) ^ 2 * (ε ^ 5 / 4)) /
      ↑(Finset.card P.parts) ^ 2 ≤
    ∑ x in attach (offDiag P.parts),
      ↑(SzemerediRegularity.pairContrib G ε hP x) / ↑(Finset.card (increment hP G ε).parts) ^ 2
⊢ (∑ uv in offDiag P.parts, ↑(edgeDensity G uv.1 uv.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 + ε ^ 5 / 4 ≤
    ↑(energy (increment hP G ε) G)

Next tactic:
rw [add_div, mul_div_cancel_left] at h
/-- The increment partition has energy greater than the original one by a known fixed amount. -/
theorem energy_increment [Nonempty α] (hP : P.IsEquipartition) (hP₇ : 7 ≤ P.parts.card)
    (hε : ↑100 < ↑4 ^ P.parts.card * ε ^ 5) (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
    (hPG : ¬P.IsUniform G ε) (hε₁ : ε ≤ 1) :
    ↑(P.energy G) + ε ^ 5 / 4 ≤ (increment hP G ε).energy G := by
  rw [coe_energy]
  have h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) hε₁ hP₇ hPα hε.le hPG
  rw [add_div, mul_div_cancel_left] at h
  

Proof state:
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP✝ : IsEquipartition P
G : SimpleGraph α
ε : ℝ
inst✝ : Nonempty α
hP : IsEquipartition P
hP₇ : 7 ≤ Finset.card P.parts
hε : 100 < 4 ^ Finset.card P.parts * ε ^ 5
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPG : ¬Finpartition.IsUniform P G ε
hε₁ : ε ≤ 1
h :
  (∑ x in offDiag P.parts, ↑(edgeDensity G x.1 x.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 + ε ^ 5 / 4 ≤
    ∑ x in attach (offDiag P.parts),
      ↑(SzemerediRegularity.pairContrib G ε hP x) / ↑(Finset.card (increment hP G ε).parts) ^ 2
⊢ (∑ uv in offDiag P.parts, ↑(edgeDensity G uv.1 uv.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 + ε ^ 5 / 4 ≤
    ↑(energy (increment hP G ε) G)
case ha
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP✝ : IsEquipartition P
G : SimpleGraph α
ε : ℝ
inst✝ : Nonempty α
hP : IsEquipartition P
hP₇ : 7 ≤ Finset.card P.parts
hε : 100 < 4 ^ Finset.card P.parts * ε ^ 5
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPG : ¬Finpartition.IsUniform P G ε
hε₁ : ε ≤ 1
h :
  (∑ x in offDiag P.parts, ↑(edgeDensity G x.1 x.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 +
      ↑(Finset.card P.parts) ^ 2 * (ε ^ 5 / 4) / ↑(Finset.card P.parts) ^ 2 ≤
    ∑ x in attach (offDiag P.parts),
      ↑(SzemerediRegularity.pairContrib G ε hP x) / ↑(Finset.card (increment hP G ε).parts) ^ 2
⊢ ↑(Finset.card P.parts) ^ 2 ≠ 0

Next tactic:
exact h.trans (mod_cast offDiag_pairs_le_increment_energy)
/-- The increment partition has energy greater than the original one by a known fixed amount. -/
theorem energy_increment [Nonempty α] (hP : P.IsEquipartition) (hP₇ : 7 ≤ P.parts.card)
    (hε : ↑100 < ↑4 ^ P.parts.card * ε ^ 5) (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
    (hPG : ¬P.IsUniform G ε) (hε₁ : ε ≤ 1) :
    ↑(P.energy G) + ε ^ 5 / 4 ≤ (increment hP G ε).energy G := by
  rw [coe_energy]
  have h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) hε₁ hP₇ hPα hε.le hPG
  rw [add_div, mul_div_cancel_left] at h
  exact h.trans (mod_cast offDiag_pairs_le_increment_energy)
  

Proof state:
case ha
α : Type u_1
inst✝¹ : Fintype α
P : Finpartition univ
hP✝ : IsEquipartition P
G : SimpleGraph α
ε : ℝ
inst✝ : Nonempty α
hP : IsEquipartition P
hP₇ : 7 ≤ Finset.card P.parts
hε : 100 < 4 ^ Finset.card P.parts * ε ^ 5
hPα : Finset.card P.parts * 16 ^ Finset.card P.parts ≤ Fintype.card α
hPG : ¬Finpartition.IsUniform P G ε
hε₁ : ε ≤ 1
h :
  (∑ x in offDiag P.parts, ↑(edgeDensity G x.1 x.2) ^ 2) / ↑(Finset.card P.parts) ^ 2 +
      ↑(Finset.card P.parts) ^ 2 * (ε ^ 5 / 4) / ↑(Finset.card P.parts) ^ 2 ≤
    ∑ x in attach (offDiag P.parts),
      ↑(SzemerediRegularity.pairContrib G ε hP x) / ↑(Finset.card (increment hP G ε).parts) ^ 2
⊢ ↑(Finset.card P.parts) ^ 2 ≠ 0

Next tactic:
positivity
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  

Proof state:
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
⊢ ⁅x, x⁆ = 0

Next tactic:
simp only [bracket_def]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  

Proof state:
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) x) x = 0

Next tactic:
refine' x.induction_on _ _ _
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · 

Proof state:
case refine'_1
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) 0) 0 = 0

Next tactic:
simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · 

Proof state:
case refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
⊢ ∀ (x : A) (y : L), ((LieAlgebra.ExtendScalars.bracket' R A L L) (x ⊗ₜ[R] y)) (x ⊗ₜ[R] y) = 0

Next tactic:
intro a l
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    

Proof state:
case refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
a : A
l : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a ⊗ₜ[R] l)) (a ⊗ₜ[R] l) = 0

Next tactic:
simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · 

Proof state:
case refine'_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x : A ⊗[R] L
⊢ ∀ (x y : A ⊗[R] L),
    ((LieAlgebra.ExtendScalars.bracket' R A L L) x) x = 0 →
      ((LieAlgebra.ExtendScalars.bracket' R A L L) y) y = 0 →
        ((LieAlgebra.ExtendScalars.bracket' R A L L) (x + y)) (x + y) = 0

Next tactic:
intro z₁ z₂ h₁ h₂
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    

Proof state:
case refine'_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (z₁ + z₂)) (z₁ + z₂) = 0

Next tactic:
suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      

Proof state:
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
this : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₂ + ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₁ = 0
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (z₁ + z₂)) (z₁ + z₂) = 0

Next tactic:
rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    

Proof state:
case refine'_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₂ + ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₁ = 0

Next tactic:
refine' z₁.induction_on _ _ _
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · 

Proof state:
case refine'_3.refine'_1
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) 0) z₂ + ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) 0 = 0

Next tactic:
simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · 

Proof state:
case refine'_3.refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
⊢ ∀ (x : A) (y : L),
    ((LieAlgebra.ExtendScalars.bracket' R A L L) (x ⊗ₜ[R] y)) z₂ +
        ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) (x ⊗ₜ[R] y) =
      0

Next tactic:
intro a₁ l₁
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; 

Proof state:
case refine'_3.refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) z₂ +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) (a₁ ⊗ₜ[R] l₁) =
    0

Next tactic:
refine' z₂.induction_on _ _ _
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; refine' z₂.induction_on _ _ _
      · 

Proof state:
case refine'_3.refine'_2.refine'_1
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) 0 +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) 0) (a₁ ⊗ₜ[R] l₁) =
    0

Next tactic:
simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; refine' z₂.induction_on _ _ _
      · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
      · 

Proof state:
case refine'_3.refine'_2.refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
⊢ ∀ (x : A) (y : L),
    ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (x ⊗ₜ[R] y) +
        ((LieAlgebra.ExtendScalars.bracket' R A L L) (x ⊗ₜ[R] y)) (a₁ ⊗ₜ[R] l₁) =
      0

Next tactic:
intro a₂ l₂
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; refine' z₂.induction_on _ _ _
      · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
      · intro a₂ l₂
        

Proof state:
case refine'_3.refine'_2.refine'_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
a₂ : A
l₂ : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (a₂ ⊗ₜ[R] l₂) +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₂ ⊗ₜ[R] l₂)) (a₁ ⊗ₜ[R] l₁) =
    0

Next tactic:
simp only [← lie_skew l₂ l₁, mul_comm a₁ a₂, TensorProduct.tmul_neg, bracket'_tmul,
          add_right_neg]
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; refine' z₂.induction_on _ _ _
      · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
      · intro a₂ l₂
        simp only [← lie_skew l₂ l₁, mul_comm a₁ a₂, TensorProduct.tmul_neg, bracket'_tmul,
          add_right_neg]
      · 

Proof state:
case refine'_3.refine'_2.refine'_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
⊢ ∀ (x y : A ⊗[R] L),
    ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) x +
          ((LieAlgebra.ExtendScalars.bracket' R A L L) x) (a₁ ⊗ₜ[R] l₁) =
        0 →
      ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y +
            ((LieAlgebra.ExtendScalars.bracket' R A L L) y) (a₁ ⊗ₜ[R] l₁) =
          0 →
        ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (x + y) +
            ((LieAlgebra.ExtendScalars.bracket' R A L L) (x + y)) (a₁ ⊗ₜ[R] l₁) =
          0

Next tactic:
intro y₁ y₂ hy₁ hy₂
private theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 := by
  simp only [bracket_def]
  refine' x.induction_on _ _ _
  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
  · intro a l
    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
  · intro z₁ z₂ h₁ h₂
    suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,
        zero_add, add_zero, add_comm, this]
    refine' z₁.induction_on _ _ _
    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
    · intro a₁ l₁; refine' z₂.induction_on _ _ _
      · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]
      · intro a₂ l₂
        simp only [← lie_skew l₂ l₁, mul_comm a₁ a₂, TensorProduct.tmul_neg, bracket'_tmul,
          add_right_neg]
      · intro y₁ y₂ hy₁ hy₂
        

Proof state:
case refine'_3.refine'_2.refine'_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x z₁ z₂ : A ⊗[R] L
h₁ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₁) z₁ = 0
h₂ : ((LieAlgebra.ExtendScalars.bracket' R A L L) z₂) z₂ = 0
a₁ : A
l₁ : L
y₁ y₂ : A ⊗[R] L
hy₁ :
  ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y₁ +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) y₁) (a₁ ⊗ₜ[R] l₁) =
    0
hy₂ :
  ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y₂ +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) y₂) (a₁ ⊗ₜ[R] l₁) =
    0
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (y₁ + y₂) +
      ((LieAlgebra.ExtendScalars.bracket' R A L L) (y₁ + y₂)) (a₁ ⊗ₜ[R] l₁) =
    0

Next tactic:
simp only [hy₁, hy₂, add_add_add_comm, add_zero, LinearMap.add_apply, LinearMap.map_add]