l3lab/ntp-mathlib huggingface dataset:
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
⊢ False

Next tactic:
have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
⊢ Jac * Ideal.span {x} ≤ J

Next tactic:
by_contra H
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
H : ¬Jac * Ideal.span {x} ≤ J
⊢ False

Next tactic:
refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
H : ¬Jac * Ideal.span {x} ≤ J
⊢ J < J ⊔ Jac • Ideal.span {x}

Next tactic:
exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this : Jac * Ideal.span {x} ≤ J
⊢ False

Next tactic:
have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  

Proof state:
case this
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this : Jac * Ideal.span {x} ≤ J
⊢ Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
case intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝¹ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this✝ : Jac * Ideal.span {x} ≤ J
this : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
⊢ False

Next tactic:
calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by rw [pow_succ, ← mul_assoc]
    _ ≤ J * Jac ^ n := (mul_le_mul (by rwa [mul_comm]) le_rfl)
    _ = ⊥ := by simp
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by 

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this : Jac * Ideal.span {x} ≤ J
⊢ Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n

Next tactic:
rw [pow_succ, ← mul_assoc]
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by rw [pow_succ, ← mul_assoc]
    _ ≤ J * Jac ^ n := (mul_le_mul (by 

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this : Jac * Ideal.span {x} ≤ J
⊢ Ideal.span {x} * Jac ≤ J

Next tactic:
rwa [mul_comm]
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by rw [pow_succ, ← mul_assoc]
    _ ≤ J * Jac ^ n := (mul_le_mul (by rwa [mul_comm]) le_rfl)
    _ = ⊥ := by 

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this : Jac * Ideal.span {x} ≤ J
⊢ J * Jac ^ n = ⊥

Next tactic:
simp
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by rw [pow_succ, ← mul_assoc]
    _ ≤ J * Jac ^ n := (mul_le_mul (by rwa [mul_comm]) le_rfl)
    _ = ⊥ := by simp
  

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝¹ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this✝ : Jac * Ideal.span {x} ≤ J
this : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
⊢ False

Next tactic:
refine' hxJ (mem_annihilator.2 fun y hy => (mem_bot R).1 _)
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right h⟩
  obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
  refine' ⟨n, _⟩
  let J : Ideal R := annihilator (Jac ^ n)
  suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
  by_contra hJ
  change J ≠ ⊤ at hJ
  rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
  rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
  obtain rfl : J ⊔ Ideal.span {x} = J' := by
    apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
    · exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
    · rw [SetLike.lt_iff_le_and_exists]
      exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
  have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _
  have : Jac * Ideal.span {x} ≤ J := by -- Need version 4 of Nakayama's lemma on Stacks
    by_contra H
    refine' H (smul_le_of_le_smul_of_le_jacobson_bot (fg_span_singleton _) le_rfl
      (this.eq_of_not_lt (hJ' _ _)).ge)
    exact lt_of_le_of_ne le_sup_left fun h => H <| h.symm ▸ le_sup_right
  have : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
  calc
    Ideal.span {x} * Jac ^ (n + 1) = Ideal.span {x} * Jac * Jac ^ n := by rw [pow_succ, ← mul_assoc]
    _ ≤ J * Jac ^ n := (mul_le_mul (by rwa [mul_comm]) le_rfl)
    _ = ⊥ := by simp
  refine' hxJ (mem_annihilator.2 fun y hy => (mem_bot R).1 _)
  

Proof state:
case intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := (_ : ∀ (x x_1 : ℕ), x ≤ x_1 → Jac ^ x_1 ≤ Jac ^ x) }
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m
J : Ideal R := annihilator (Jac ^ n)
hJ : J ≠ ⊤
x : R
hxJ : x ∉ J
hJJ' : J < J ⊔ Ideal.span {x}
hJ' : ∀ (I : Ideal R), J < I → ¬I < J ⊔ Ideal.span {x}
hxJ' : x ∈ J ⊔ Ideal.span {x}
this✝¹ : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x}
this✝ : Jac * Ideal.span {x} ≤ J
this : Ideal.span {x} * Jac ^ (n + 1) ≤ ⊥
y : R
hy : y ∈ Jac ^ n
⊢ x • y ∈ ⊥

Next tactic:
refine' this (mul_mem_mul (mem_span_singleton_self x) _)