awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
s : Set (↥H → R)
hs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ s, χ₁ + χ₂ ∈ s
x : L
m : M
hx : Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ χ ∈ s, rootSpace H χ) x
hm : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) m
zero :
  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ)
    (Bracket.bracket.{u_2, u_3} (L := L) 0 m)
⊢ Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x, m⁆
have add : ∀ (x y : L),
  Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ i ∈ s, rootSpace H i) x →
    Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ i ∈ s, rootSpace H i) y →
      ∀ (hu : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x, m⁆)
        (hv : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅y, m⁆),
        Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x + y, m⁆
Tactic: induction hx using LieSubmodule.iSup_induction' with
  | zero => simp
  | add _ _ _ _ hu hv => rw [add_lie]; exact add_mem hu hv
  | mem χ₁ u hu =>
    by_cases hχ₁ : χ₁ ∈ s; swap
    · simp_all
    replace hu : u ∈ rootSpace H χ₁ := by simpa [hχ₁] using hu
    induction hm using LieSubmodule.iSup_induction' with
    | zero => simp
    | add _ _ _ _ hv hw => rw [lie_add]; exact add_mem hv hw
    | mem χ₂ v hv =>
      by_cases hχ₂ : χ₂ ∈ s; swap
      · simp_all
      apply LieSubmodule.mem_iSup_of_mem (χ₁ + χ₂)
      simp_all [lie_mem_genWeightSpace_of_mem_genWeightSpace]
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
s : Set (↥H → R)
hs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ s, χ₁ + χ₂ ∈ s
x : L
m : M
hx : Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ χ ∈ s, rootSpace H χ) x
hm : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) m
zero :
  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ)
    (Bracket.bracket.{u_2, u_3} (L := L) 0 m)
add :
  ∀ (x y : L),
    Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ i ∈ s, rootSpace H i) x →
      Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ i ∈ s, rootSpace H i) y →
        ∀ (hu : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x, m⁆)
          (hv : Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅y, m⁆),
          Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x + y, m⁆
⊢ Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅x, m⁆
have mem : ∀ (χ₁ : ↥H → R) (u : L) (hu : Membership.mem (γ := LieSubmodule R (↥H) L) (⨆ (_ : χ₁ ∈ s), rootSpace H χ₁) u),
  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (⨆ χ ∈ s, genWeightSpace M χ) ⁅u, m⁆
Tactic: induction hx using LieSubmodule.iSup_induction' with
  | zero => simp
  | add _ _ _ _ hu hv => rw [add_lie]; exact add_mem hu hv
  | mem χ₁ u hu =>
    by_cases hχ₁ : χ₁ ∈ s; swap
    · simp_all
    replace hu : u ∈ rootSpace H χ₁ := by simpa [hχ₁] using hu
    induction hm using LieSubmodule.iSup_induction' with
    | zero => simp
    | add _ _ _ _ hv hw => rw [lie_add]; exact add_mem hv hw
    | mem χ₂ v hv =>
      by_cases hχ₂ : χ₂ ∈ s; swap
      · simp_all
      apply LieSubmodule.mem_iSup_of_mem (χ₁ + χ₂)
      simp_all [lie_mem_genWeightSpace_of_mem_genWeightSpace]
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
x y : ↥(rootSpace H χ₁)
⊢ {
      toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
        Subtype.mk (p := fun (x : M) ↦
          Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
          ⁅(↑(x + y) : L), (↑m : M)⁆ ⋯,
      map_add' := ⋯, map_smul' := ⋯ } =
    HAdd.hAdd (α := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))
      {
        toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
          Subtype.mk (p := fun (x : M) ↦
            Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑x : L), (↑m : M)⁆ ⋯,
        map_add' := ⋯, map_smul' := ⋯ : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
      {
        toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
          Subtype.mk (p := fun (x : M) ↦
            Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑y : L), (↑m : M)⁆ ⋯,
        map_add' := ⋯, map_smul' := ⋯ : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
have h : ∀ (m : ↥(genWeightSpace M χ₂)),
  Eq.{u_3 + 1} (α := M)
    (↑(({
              toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                Subtype.mk (p := fun (x : M) ↦
                  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                  ⁅(↑(x + y) : L), (↑m : M)⁆ sorry,
              map_add' := sorry, map_smul' := sorry } :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
    (↑((HAdd.hAdd (α := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))
              {
                toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                  Subtype.mk (p := fun (x : M) ↦
                    Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                    ⁅(↑x : L), (↑m : M)⁆ sorry,
                map_add' := sorry, map_smul' := sorry : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
              {
                toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                  Subtype.mk (p := fun (x : M) ↦
                    Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                    ⁅(↑y : L), (↑m : M)⁆ sorry,
                map_add' := sorry, map_smul' := sorry : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) } :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
Tactic: ext m
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
t : R
x : ↥(rootSpace H χ₁)
⊢ {
      toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
        Subtype.mk (p := fun (x : M) ↦
          Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
          ⁅(↑(t • x) : L), (↑m : M)⁆ ⋯,
      map_add' := ⋯, map_smul' := ⋯ } =
    HSMul.hSMul (α := R) (β := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) ((RingHom.id R : (a : R) → R) t : R)
      {
        toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
          Subtype.mk (p := fun (x : M) ↦
            Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑x : L), (↑m : M)⁆ ⋯,
        map_add' := ⋯, map_smul' := ⋯ : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
have simp : {
    toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
      Subtype.mk (p := fun (x : M) ↦
        Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑(t • x) : L), (↑m : M)⁆
        sorry,
    map_add' := sorry, map_smul' := sorry } =
  HSMul.hSMul (β := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) t
    {
      toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
        Subtype.mk (p := fun (x : M) ↦
          Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑x : L), (↑m : M)⁆ sorry,
      map_add' := sorry, map_smul' := sorry : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
Tactic: simp only [RingHom.id_apply]
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
t : R
x : ↥(rootSpace H χ₁)
⊢ {
      toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
        Subtype.mk (p := fun (x : M) ↦
          Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
          ⁅(↑(t • x) : L), (↑m : M)⁆ ⋯,
      map_add' := ⋯, map_smul' := ⋯ } =
    HSMul.hSMul (β := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) t
      {
        toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
          Subtype.mk (p := fun (x : M) ↦
            Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x) ⁅(↑x : L), (↑m : M)⁆ ⋯,
        map_add' := ⋯, map_smul' := ⋯ : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) }
have h : ∀ (m : ↥(genWeightSpace M χ₂)),
  Eq.{u_3 + 1} (α := M)
    (↑(({
              toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                Subtype.mk (p := fun (x : M) ↦
                  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                  ⁅(↑(t • x) : L), (↑m : M)⁆ sorry,
              map_add' := sorry, map_smul' := sorry } :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
    (↑((HSMul.hSMul (β := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) t
              {
                toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                  Subtype.mk (p := fun (x : M) ↦
                    Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                    ⁅(↑x : L), (↑m : M)⁆ sorry,
                map_add' := sorry, map_smul' := sorry : ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃) } :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
Tactic: ext m
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
x : ↥H
y : ↥(rootSpace H χ₁)
⊢ Eq (α := ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))
    (AddHom.toFun
        (LinearMap.toAddHom
            (rootSpaceWeightSpaceProductAux R L H M hχ :
              LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
          AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
        ⁅x, y⁆ :
      ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))
    (⁅x,
        AddHom.toFun
          (LinearMap.toAddHom
              (rootSpaceWeightSpaceProductAux R L H M hχ :
                LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                  (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
            AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
          y⁆ :
      ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))
have h : ∀ (m : ↥(genWeightSpace M χ₂)),
  Eq.{u_3 + 1} (α := M)
    (↑(((AddHom.toFun
                (LinearMap.toAddHom
                    (rootSpaceWeightSpaceProductAux R L H M hχ :
                      LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                        (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
                  AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
                ⁅x, y⁆ :
              ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
    (↑(((⁅x,
                AddHom.toFun
                  (LinearMap.toAddHom
                      (rootSpaceWeightSpaceProductAux R L H M hχ :
                        LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                          (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
                    AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
                  y⁆ :
              ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
Tactic: ext m
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
x : ↥H
y : ↥(rootSpace H χ₁)
m : ↥(genWeightSpace M χ₂)
⊢ Eq.{u_3 + 1} (α := M)
    (↑(((AddHom.toFun
                (LinearMap.toAddHom
                    (rootSpaceWeightSpaceProductAux R L H M hχ :
                      LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                        (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
                  AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
                ⁅x, y⁆ :
              ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
    (↑(((⁅x,
                AddHom.toFun
                  (LinearMap.toAddHom
                      (rootSpaceWeightSpaceProductAux R L H M hχ :
                        LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) (↥(rootSpace H χ₁))
                          (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃))) :
                    AddHom.{u_2, u_3} (↥(rootSpace H χ₁)) (↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)))
                  y⁆ :
              ↥(genWeightSpace M χ₂) →ₗ[R] ↥(genWeightSpace M χ₃)) :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
have h : Eq.{u_3 + 1} (α := M)
  (↑(({
            toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
              Subtype.mk (p := fun (x : M) ↦
                Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                ⁅(↑⁅x, y⁆ : L), (↑m : M)⁆ sorry,
            map_add' := sorry, map_smul' := sorry } :
          (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
        m) :
    M)
  (↑((⁅x,
            {
              toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                Subtype.mk (p := fun (x : M) ↦
                  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                  ⁅(↑y : L), (↑m : M)⁆ sorry,
              map_add' := sorry, map_smul' := sorry }⁆ :
          (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
        m) :
    M)
Tactic: simp only [rootSpaceWeightSpaceProductAux]
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
χ₁ χ₂ χ₃ : ↥H → R
hχ : χ₁ + χ₂ = χ₃
x : ↥H
y : ↥(rootSpace H χ₁)
m : ↥(genWeightSpace M χ₂)
⊢ Eq.{u_3 + 1} (α := M)
    (↑(({
              toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                Subtype.mk (p := fun (x : M) ↦
                  Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                  ⁅(↑⁅x, y⁆ : L), (↑m : M)⁆ ⋯,
              map_add' := ⋯, map_smul' := ⋯ } :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
    (↑((⁅x,
              {
                toFun := fun (m : ↥(genWeightSpace M χ₂)) ↦
                  Subtype.mk (p := fun (x : M) ↦
                    Membership.mem (γ := LieSubmodule.{u_1, u_2, u_3} R (↥H) M) (genWeightSpace M χ₃) x)
                    ⁅(↑y : L), (↑m : M)⁆ ⋯,
                map_add' := ⋯, map_smul' := ⋯ }⁆ :
            (a : ↥(genWeightSpace M χ₂)) → ↥(genWeightSpace M χ₃))
          m) :
      M)
have h : ⁅⁅(↑x : L), (↑y : L)⁆, (↑m : M)⁆ = ⁅(↑x : L), ⁅(↑y : L), (↑m : M)⁆⁆ - ⁅(↑y : L), ⁅(↑x : L), (↑m : M)⁆⁆
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
x y : L
hx : x ∈ (↑(rootSpace H 0) : Submodule R L).carrier
hy : y ∈ (↑(rootSpace H 0) : Submodule R L).carrier
xy : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0) :=
  Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx ⊗ₜ[R] Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy
this :
  (↑((rootSpaceProduct R L H 0 0 0 ⋯ : (a : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0)) → ↥(rootSpace H 0)) xy) : L) ∈
    rootSpace H 0
⊢ ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
have rwa : ∀
  (this :
    ⁅(↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx) : L),
        (↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy) : L)⁆ ∈
      rootSpace H 0),
  ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
Tactic: rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
x y : L
hx : x ∈ (↑(rootSpace H 0) : Submodule R L).carrier
hy : y ∈ (↑(rootSpace H 0) : Submodule R L).carrier
xy : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0) :=
  Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx ⊗ₜ[R] Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy
this :
  (↑((rootSpaceProduct R L H 0 0 0 ⋯ : (a : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0)) → ↥(rootSpace H 0)) xy) : L) ∈
    rootSpace H 0
rwa :
  ∀
    (this :
      ⁅(↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx) : L),
          (↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy) : L)⁆ ∈
        rootSpace H 0),
    ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
⊢ ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
have rwa₁ : ∀ (this : ⁅x, (↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy) : L)⁆ ∈ rootSpace H 0),
  ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
Tactic: rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
x y : L
hx : x ∈ (↑(rootSpace H 0) : Submodule R L).carrier
hy : y ∈ (↑(rootSpace H 0) : Submodule R L).carrier
xy : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0) :=
  Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx ⊗ₜ[R] Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy
this :
  (↑((rootSpaceProduct R L H 0 0 0 ⋯ : (a : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0)) → ↥(rootSpace H 0)) xy) : L) ∈
    rootSpace H 0
rwa :
  ∀
    (this :
      ⁅(↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx) : L),
          (↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy) : L)⁆ ∈
        rootSpace H 0),
    ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
rwa₁ :
  ∀ (this : ⁅x, (↑(Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy) : L)⁆ ∈ rootSpace H 0),
    ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
⊢ ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
have rwa₂ : ∀ (this : ⁅x, y⁆ ∈ rootSpace H 0), ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
Tactic: rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
M : Type u_3
inst_4 : AddCommGroup M
inst_5 : Module R M
inst_6 : LieRingModule L M
inst_7 : LieModule R L M
x y : L
hx : x ∈ (↑(rootSpace H 0) : Submodule R L).carrier
hy : y ∈ (↑(rootSpace H 0) : Submodule R L).carrier
xy : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0) :=
  Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) x hx ⊗ₜ[R] Subtype.mk (p := fun (x : L) ↦ x ∈ rootSpace H 0) y hy
⊢ ⁅x, y⁆ ∈ (↑(rootSpace H 0) : Submodule R L).carrier
have suffices : (↑((rootSpaceProduct R L H 0 0 0 sorry : (a : ↥(rootSpace H 0) ⊗[R] ↥(rootSpace H 0)) → ↥(rootSpace H 0)) xy) : L) ∈
  rootSpace H 0
Tactic: suffices (rootSpaceProduct R L H 0 0 0 (add_zero 0) xy : L) ∈ rootSpace H 0 by
        rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (hypothesis)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
x : L
hx : x ∈ H.toLieSubmodule
⊢ x ∈ rootSpace H 0
have hx₁ : x ∈ H
Tactic: simp only [LieSubalgebra.mem_toLieSubmodule] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
x : L
hx : x ∈ H
⊢ x ∈ rootSpace H 0
have simp : ∀ (x_1 : ↥H),
  ∃ (k : ℕ),
    Eq (α := L)
      ((HPow.hPow (α := Module.End R L) ((toEnd R (↥H) L : (a : ↥H) → Module.End R L) x_1 : Module.End R L) k :
            (a : L) → L)
          x :
        L)
      (0 : L)
Tactic: simp only [mem_genWeightSpace, Pi.zero_apply, sub_zero, zero_smul]
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.Cartan (goal)
Goal:
R : Type u_1
L : Type u_2
inst : CommRing R
inst_1 : LieRing L
inst_2 : LieAlgebra R L
H : LieSubalgebra R L
inst_3 : LieRing.IsNilpotent.{u_2} ↥H
x : L
hx : x ∈ H
y : ↥H
⊢ ∃ (k : ℕ),
    Eq (α := L)
      ((HPow.hPow (α := Module.End R L) ((toEnd R (↥H) L : (a : ↥H) → Module.End R L) y : Module.End R L) k :
            (a : L) → L)
          x :
        L)
      (0 : L)
have obtain : ∀ (k : ℕ) (hk : Eq.{u_2 + 1} (α := LieSubmodule R ↥H ↥H) (lowerCentralSeries R (↥H) (↥H) k) ⊥),
  ∃ (k : ℕ),
    Eq (α := L)
      ((HPow.hPow (α := Module.End R L) ((toEnd R (↥H) L : (a : ↥H) → Module.End R L) y : Module.End R L) k :
            (a : L) → L)
          x :
        L)
      (0 : L)
Tactic: obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R H H