l3lab/ntp-mathlib huggingface dataset:
theorem isClosed_nonneg {E} [NormedLatticeAddCommGroup E] : IsClosed { x : E | 0 ≤ x } := by
  suffices { x : E | 0 ≤ x } = NegPart.neg ⁻¹' {(0 : E)} by
    rw [this]
    exact IsClosed.preimage continuous_neg' isClosed_singleton
  ext1 x
  

Proof state:
case h
α : Type u_1
inst✝¹ : NormedLatticeAddCommGroup α
E : Type u_2
inst✝ : NormedLatticeAddCommGroup E
x : E
⊢ x ∈ {x | 0 ≤ x} ↔ x ∈ NegPart.neg ⁻¹' {0}

Next tactic:
simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq,
    @neg_eq_zero_iff E _ _ (OrderedAddCommGroup.to_covariantClass_left_le E)]
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } := by
  

Proof state:
α : Type u_1
inst✝³ : NormedLatticeAddCommGroup α
G : Type u_2
inst✝² : OrderedAddCommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
h : IsClosed {x | 0 ≤ x}
⊢ IsClosed {p | p.1 ≤ p.2}

Next tactic:
have : { p : G × G | p.fst ≤ p.snd } = (fun p : G × G => p.snd - p.fst) ⁻¹' { x : G | 0 ≤ x } :=
    by ext1 p; simp only [sub_nonneg, Set.preimage_setOf_eq]
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } := by
  have : { p : G × G | p.fst ≤ p.snd } = (fun p : G × G => p.snd - p.fst) ⁻¹' { x : G | 0 ≤ x } :=
    by 

Proof state:
α : Type u_1
inst✝³ : NormedLatticeAddCommGroup α
G : Type u_2
inst✝² : OrderedAddCommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
h : IsClosed {x | 0 ≤ x}
⊢ {p | p.1 ≤ p.2} = (fun p => p.2 - p.1) ⁻¹' {x | 0 ≤ x}

Next tactic:
ext1 p
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } := by
  have : { p : G × G | p.fst ≤ p.snd } = (fun p : G × G => p.snd - p.fst) ⁻¹' { x : G | 0 ≤ x } :=
    by ext1 p; 

Proof state:
case h
α : Type u_1
inst✝³ : NormedLatticeAddCommGroup α
G : Type u_2
inst✝² : OrderedAddCommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
h : IsClosed {x | 0 ≤ x}
p : G × G
⊢ p ∈ {p | p.1 ≤ p.2} ↔ p ∈ (fun p => p.2 - p.1) ⁻¹' {x | 0 ≤ x}

Next tactic:
simp only [sub_nonneg, Set.preimage_setOf_eq]
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } := by
  have : { p : G × G | p.fst ≤ p.snd } = (fun p : G × G => p.snd - p.fst) ⁻¹' { x : G | 0 ≤ x } :=
    by ext1 p; simp only [sub_nonneg, Set.preimage_setOf_eq]
  

Proof state:
α : Type u_1
inst✝³ : NormedLatticeAddCommGroup α
G : Type u_2
inst✝² : OrderedAddCommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
h : IsClosed {x | 0 ≤ x}
this : {p | p.1 ≤ p.2} = (fun p => p.2 - p.1) ⁻¹' {x | 0 ≤ x}
⊢ IsClosed {p | p.1 ≤ p.2}

Next tactic:
rw [this]
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } := by
  have : { p : G × G | p.fst ≤ p.snd } = (fun p : G × G => p.snd - p.fst) ⁻¹' { x : G | 0 ≤ x } :=
    by ext1 p; simp only [sub_nonneg, Set.preimage_setOf_eq]
  rw [this]
  

Proof state:
α : Type u_1
inst✝³ : NormedLatticeAddCommGroup α
G : Type u_2
inst✝² : OrderedAddCommGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
h : IsClosed {x | 0 ≤ x}
this : {p | p.1 ≤ p.2} = (fun p => p.2 - p.1) ⁻¹' {x | 0 ≤ x}
⊢ IsClosed ((fun p => p.2 - p.1) ⁻¹' {x | 0 ≤ x})

Next tactic:
exact IsClosed.preimage (continuous_snd.sub continuous_fst) h
@[simp]
theorem normalizer_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
    H.toLieSubmodule.normalizer = H.toLieSubmodule := by
  

Proof state:
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H✝ H : LieSubalgebra R L
inst✝ : IsCartanSubalgebra H
⊢ LieSubmodule.normalizer (toLieSubmodule H) = toLieSubmodule H

Next tactic:
rw [← LieSubmodule.coe_toSubmodule_eq_iff, coe_normalizer_eq_normalizer,
    IsCartanSubalgebra.self_normalizing, coe_toLieSubmodule]
@[simp]
theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
    H.toLieSubmodule.ucs k = H.toLieSubmodule := by
  

Proof state:
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H✝ H : LieSubalgebra R L
inst✝ : IsCartanSubalgebra H
k : ℕ
⊢ LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H

Next tactic:
induction' k with k ih
@[simp]
theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
    H.toLieSubmodule.ucs k = H.toLieSubmodule := by
  induction' k with k ih
  · 

Proof state:
case zero
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H✝ H : LieSubalgebra R L
inst✝ : IsCartanSubalgebra H
⊢ LieSubmodule.ucs Nat.zero (toLieSubmodule H) = toLieSubmodule H

Next tactic:
simp
@[simp]
theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
    H.toLieSubmodule.ucs k = H.toLieSubmodule := by
  induction' k with k ih
  · simp
  · 

Proof state:
case succ
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H✝ H : LieSubalgebra R L
inst✝ : IsCartanSubalgebra H
k : ℕ
ih : LieSubmodule.ucs k (toLieSubmodule H) = toLieSubmodule H
⊢ LieSubmodule.ucs (Nat.succ k) (toLieSubmodule H) = toLieSubmodule H

Next tactic:
simp [ih]
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
⊢ IsCartanSubalgebra H ↔ LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
constructor
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · 

Proof state:
case mp
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
⊢ IsCartanSubalgebra H → LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
intro h
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    

Proof state:
case mp
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
⊢ LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by 

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
⊢ LieAlgebra.IsNilpotent R ↥H

Next tactic:
infer_instance
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    

Proof state:
case mp
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
h₁ : LieAlgebra.IsNilpotent R ↥H
⊢ LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    

Proof state:
case mp.intro
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
h₁ : LieAlgebra.IsNilpotent R ↥H
k : ℕ
hk : toLieSubmodule H ≤ LieSubmodule.ucs k ⊥
⊢ LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    

Proof state:
case mp.intro
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
h₁ : LieAlgebra.IsNilpotent R ↥H
k : ℕ
hk : toLieSubmodule H = LieSubmodule.ucs k ⊥
⊢ LieSubmodule.IsUcsLimit (toLieSubmodule H)

Next tactic:
refine' ⟨k, fun l hl => _⟩
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    

Proof state:
case mp.intro
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
h : IsCartanSubalgebra H
h₁ : LieAlgebra.IsNilpotent R ↥H
k : ℕ
hk : toLieSubmodule H = LieSubmodule.ucs k ⊥
l : ℕ
hl : k ≤ l
⊢ LieSubmodule.ucs l ⊥ = toLieSubmodule H

Next tactic:
rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · 

Proof state:
case mpr
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
⊢ LieSubmodule.IsUcsLimit (toLieSubmodule H) → IsCartanSubalgebra H

Next tactic:
rintro ⟨k, hk⟩
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    

Proof state:
case mpr.intro
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ IsCartanSubalgebra H

Next tactic:
exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          use k
          rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]
        self_normalizing := by
          have hk' := hk (k + 1) k.le_succ
          rw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk'
          rw [← LieSubalgebra.coe_to_submodule_eq_iff, ← LieSubalgebra.coe_normalizer_eq_normalizer,
            hk', LieSubalgebra.coe_toLieSubmodule] }
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ LieAlgebra.IsNilpotent R ↥H

Next tactic:
dsimp only [LieAlgebra.IsNilpotent]
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ LieModule.IsNilpotent R ↥H ↥H

Next tactic:
erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ ∃ k, LieSubmodule.lcs k (toLieSubmodule H) = ⊥

Next tactic:
use k
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          use k
          

Proof state:
case h
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ LieSubmodule.lcs k (toLieSubmodule H) = ⊥

Next tactic:
rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          use k
          rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]
        self_normalizing := by
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
⊢ normalizer H = H

Next tactic:
have hk' := hk (k + 1) k.le_succ
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          use k
          rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]
        self_normalizing := by
          have hk' := hk (k + 1) k.le_succ
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
hk' : LieSubmodule.ucs (k + 1) ⊥ = toLieSubmodule H
⊢ normalizer H = H

Next tactic:
rw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk'
theorem isCartanSubalgebra_iff_isUcsLimit : H.IsCartanSubalgebra ↔ H.toLieSubmodule.IsUcsLimit := by
  constructor
  · intro h
    have h₁ : LieAlgebra.IsNilpotent R H := by infer_instance
    obtain ⟨k, hk⟩ := H.toLieSubmodule.isNilpotent_iff_exists_self_le_ucs.mp h₁
    replace hk : H.toLieSubmodule = LieSubmodule.ucs k ⊥ :=
      le_antisymm hk
        (LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
    refine' ⟨k, fun l hl => _⟩
    rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
      LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
  · rintro ⟨k, hk⟩
    exact
      { nilpotent := by
          dsimp only [LieAlgebra.IsNilpotent]
          erw [H.toLieSubmodule.isNilpotent_iff_exists_lcs_eq_bot]
          use k
          rw [_root_.eq_bot_iff, LieSubmodule.lcs_le_iff, hk k (le_refl k)]
        self_normalizing := by
          have hk' := hk (k + 1) k.le_succ
          rw [LieSubmodule.ucs_succ, hk k (le_refl k)] at hk'
          

Proof state:
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
k : ℕ
hk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = toLieSubmodule H
hk' : LieSubmodule.normalizer (toLieSubmodule H) = toLieSubmodule H
⊢ normalizer H = H

Next tactic:
rw [← LieSubalgebra.coe_to_submodule_eq_iff, ← LieSubalgebra.coe_normalizer_eq_normalizer,
            hk', LieSubalgebra.coe_toLieSubmodule]
@[simp]
theorem LieIdeal.normalizer_eq_top {R : Type u} {L : Type v} [CommRing R] [LieRing L]
    [LieAlgebra R L] (I : LieIdeal R L) : (I : LieSubalgebra R L).normalizer = ⊤ := by
  

Proof state:
R✝ : Type u
L✝ : Type v
inst✝⁵ : CommRing R✝
inst✝⁴ : LieRing L✝
inst✝³ : LieAlgebra R✝ L✝
H : LieSubalgebra R✝ L✝
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
⊢ LieSubalgebra.normalizer (lieIdealSubalgebra R L I) = ⊤

Next tactic:
ext x
@[simp]
theorem LieIdeal.normalizer_eq_top {R : Type u} {L : Type v} [CommRing R] [LieRing L]
    [LieAlgebra R L] (I : LieIdeal R L) : (I : LieSubalgebra R L).normalizer = ⊤ := by
  ext x
  

Proof state:
case h
R✝ : Type u
L✝ : Type v
inst✝⁵ : CommRing R✝
inst✝⁴ : LieRing L✝
inst✝³ : LieAlgebra R✝ L✝
H : LieSubalgebra R✝ L✝
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
x : L
⊢ x ∈ LieSubalgebra.normalizer (lieIdealSubalgebra R L I) ↔ x ∈ ⊤

Next tactic:
simpa only [LieSubalgebra.mem_normalizer_iff, LieSubalgebra.mem_top, iff_true_iff] using
    fun y hy => I.lie_mem hy
/-- A nilpotent Lie algebra is its own Cartan subalgebra. -/
instance LieAlgebra.top_isCartanSubalgebra_of_nilpotent [LieAlgebra.IsNilpotent R L] :
    LieSubalgebra.IsCartanSubalgebra (⊤ : LieSubalgebra R L) where
  nilpotent := inferInstance
  self_normalizing := by 

Proof state:
R : Type u
L : Type v
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
inst✝ : IsNilpotent R L
⊢ LieSubalgebra.normalizer ⊤ = ⊤

Next tactic:
rw [← top_coe_lieSubalgebra, normalizer_eq_top, top_coe_lieSubalgebra]
instance : SetLike (ConvexBody V) V where
  coe := ConvexBody.carrier
  coe_injective' K L h := by
    

Proof state:
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K L : ConvexBody V
h : K.carrier = L.carrier
⊢ K = L

Next tactic:
cases K
instance : SetLike (ConvexBody V) V where
  coe := ConvexBody.carrier
  coe_injective' K L h := by
    cases K
    

Proof state:
case mk
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
L : ConvexBody V
carrier✝ : Set V
convex'✝ : Convex ℝ carrier✝
isCompact'✝ : IsCompact carrier✝
nonempty'✝ : Set.Nonempty carrier✝
h : { carrier := carrier✝, convex' := convex'✝, isCompact' := isCompact'✝, nonempty' := nonempty'✝ }.carrier = L.carrier
⊢ { carrier := carrier✝, convex' := convex'✝, isCompact' := isCompact'✝, nonempty' := nonempty'✝ } = L

Next tactic:
cases L
instance : SetLike (ConvexBody V) V where
  coe := ConvexBody.carrier
  coe_injective' K L h := by
    cases K
    cases L
    

Proof state:
case mk.mk
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
carrier✝¹ : Set V
convex'✝¹ : Convex ℝ carrier✝¹
isCompact'✝¹ : IsCompact carrier✝¹
nonempty'✝¹ : Set.Nonempty carrier✝¹
carrier✝ : Set V
convex'✝ : Convex ℝ carrier✝
isCompact'✝ : IsCompact carrier✝
nonempty'✝ : Set.Nonempty carrier✝
h :
  { carrier := carrier✝¹, convex' := convex'✝¹, isCompact' := isCompact'✝¹, nonempty' := nonempty'✝¹ }.carrier =
    { carrier := carrier✝, convex' := convex'✝, isCompact' := isCompact'✝, nonempty' := nonempty'✝ }.carrier
⊢ { carrier := carrier✝¹, convex' := convex'✝¹, isCompact' := isCompact'✝¹, nonempty' := nonempty'✝¹ } =
    { carrier := carrier✝, convex' := convex'✝, isCompact' := isCompact'✝, nonempty' := nonempty'✝ }

Next tactic:
congr
/-- A convex body that is symmetric contains `0`. -/
theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K := by
  

Proof state:
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
⊢ 0 ∈ K

Next tactic:
obtain ⟨x, hx⟩ := K.nonempty
/-- A convex body that is symmetric contains `0`. -/
theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K := by
  obtain ⟨x, hx⟩ := K.nonempty
  

Proof state:
case intro
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
x : V
hx : x ∈ ↑K
⊢ 0 ∈ K

Next tactic:
rw [show 0 = (1/2 : ℝ) • x + (1/2 : ℝ) • (- x) by field_simp]