l3lab/ntp-mathlib huggingface dataset:
theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
    (N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
  let ⟨s, hs⟩ := hN
  haveI := Classical.decEq M
  haveI := Classical.decEq R
  have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
  refine' @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ _ _ isArtinian_finsupp
  · exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
  · rw [← LinearMap.range_eq_top, eq_top_iff,
       ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)
         from Subtype.val_injective), Submodule.map_top, range_subtype,
         ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]
    

Proof state:
case refine'_2
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
N : Submodule R M
inst✝ : IsArtinianRing R
hN : FG N
s : Finset M
hs : span R ↑s = N
this✝¹ : DecidableEq M
this✝ : DecidableEq R
this : ∀ x ∈ s, x ∈ N
⊢ N ≤
    LinearMap.range (Submodule.subtype N ∘ₗ Finsupp.total (↑↑s) (↥N) R fun i => { val := ↑i, property := (_ : ↑i ∈ N) })

Next tactic:
subst N
theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
    (N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
  let ⟨s, hs⟩ := hN
  haveI := Classical.decEq M
  haveI := Classical.decEq R
  have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
  refine' @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ _ _ isArtinian_finsupp
  · exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
  · rw [← LinearMap.range_eq_top, eq_top_iff,
       ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)
         from Subtype.val_injective), Submodule.map_top, range_subtype,
         ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]
    subst N
    

Proof state:
case refine'_2
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinianRing R
s : Finset M
this✝¹ : DecidableEq M
this✝ : DecidableEq R
hN : FG (span R ↑s)
this : ∀ x ∈ s, x ∈ span R ↑s
⊢ span R ↑s ≤
    LinearMap.range
      (Submodule.subtype (span R ↑s) ∘ₗ
        Finsupp.total (↑↑s) (↥(span R ↑s)) R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) })

Next tactic:
refine span_le.2 (fun i hi => ?_)
theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
    (N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
  let ⟨s, hs⟩ := hN
  haveI := Classical.decEq M
  haveI := Classical.decEq R
  have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
  refine' @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ _ _ isArtinian_finsupp
  · exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
  · rw [← LinearMap.range_eq_top, eq_top_iff,
       ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)
         from Subtype.val_injective), Submodule.map_top, range_subtype,
         ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]
    subst N
    refine span_le.2 (fun i hi => ?_)
    

Proof state:
case refine'_2
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinianRing R
s : Finset M
this✝¹ : DecidableEq M
this✝ : DecidableEq R
hN : FG (span R ↑s)
this : ∀ x ∈ s, x ∈ span R ↑s
i : M
hi : i ∈ ↑s
⊢ i ∈
    ↑(LinearMap.range
        (Submodule.subtype (span R ↑s) ∘ₗ
          Finsupp.total (↑↑s) (↥(span R ↑s)) R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) }))

Next tactic:
use Finsupp.single ⟨i, hi⟩ 1
theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
    (N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
  let ⟨s, hs⟩ := hN
  haveI := Classical.decEq M
  haveI := Classical.decEq R
  have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
  refine' @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ _ _ isArtinian_finsupp
  · exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
  · rw [← LinearMap.range_eq_top, eq_top_iff,
       ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)
         from Subtype.val_injective), Submodule.map_top, range_subtype,
         ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]
    subst N
    refine span_le.2 (fun i hi => ?_)
    use Finsupp.single ⟨i, hi⟩ 1
    

Proof state:
case h
R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinianRing R
s : Finset M
this✝¹ : DecidableEq M
this✝ : DecidableEq R
hN : FG (span R ↑s)
this : ∀ x ∈ s, x ∈ span R ↑s
i : M
hi : i ∈ ↑s
⊢ ((Submodule.subtype (span R ↑s) ∘ₗ
        Finsupp.total (↑↑s) (↥(span R ↑s)) R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) })
      fun₀ | { val := i, property := hi } => 1) =
    i

Next tactic:
simp
theorem Function.Surjective.isArtinianRing {R} [Ring R] {S} [Ring S] {F} [RingHomClass F R S]
    {f : F} (hf : Function.Surjective f) [H : IsArtinianRing R] : IsArtinianRing S := by
  

Proof state:
R : Type u_1
inst✝² : Ring R
S : Type u_2
inst✝¹ : Ring S
F : Type u_3
inst✝ : RingHomClass F R S
f : F
hf : Surjective ⇑f
H : IsArtinianRing R
⊢ IsArtinianRing S

Next tactic:
rw [isArtinianRing_iff, isArtinian_iff_wellFounded] at H ⊢
theorem Function.Surjective.isArtinianRing {R} [Ring R] {S} [Ring S] {F} [RingHomClass F R S]
    {f : F} (hf : Function.Surjective f) [H : IsArtinianRing R] : IsArtinianRing S := by
  rw [isArtinianRing_iff, isArtinian_iff_wellFounded] at H ⊢
  

Proof state:
R : Type u_1
inst✝² : Ring R
S : Type u_2
inst✝¹ : Ring S
F : Type u_3
inst✝ : RingHomClass F R S
f : F
hf : Surjective ⇑f
H : WellFounded fun x x_1 => x < x_1
⊢ WellFounded fun x x_1 => x < x_1

Next tactic:
exact (Ideal.orderEmbeddingOfSurjective f hf).wellFounded H
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
⊢ IsNilpotent (Ideal.jacobson ⊥)

Next tactic:
let Jac := Ideal.jacobson (⊥ : Ideal R)
theorem isNilpotent_jacobson_bot : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
  let Jac := Ideal.jacobson (⊥ : Ideal R)
  

Proof state:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
Jac : Ideal R := Ideal.jacobson ⊥
⊢ IsNilpotent (Ideal.jacobson ⊥)

Next tactic:
let f : ℕ →o (Ideal R)ᵒᵈ := ⟨fun n => Jac ^ n, fun _ _ h => Ideal.pow_le_pow_right 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⟩
  

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) }
⊢ IsNilpotent (Ideal.jacobson ⊥)

Next tactic:
obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → Jac ^ n = Jac ^ m := IsArtinian.monotone_stabilizes f
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
  

Proof state:
case 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
⊢ IsNilpotent (Ideal.jacobson ⊥)

Next tactic:
refine' ⟨n, _⟩
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, _⟩
  

Proof state:
case 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
⊢ Ideal.jacobson ⊥ ^ n = 0

Next tactic:
let J : Ideal R := annihilator (Jac ^ n)
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)
  

Proof state:
case 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)
⊢ Ideal.jacobson ⊥ ^ n = 0

Next tactic:
suffices J = ⊤ by
    have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
    simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
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
    

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)
this : J = ⊤
⊢ Ideal.jacobson ⊥ ^ n = 0

Next tactic:
have hJ : J • Jac ^ n = ⊥ := annihilator_smul (Jac ^ n)
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)
    

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)
this : J = ⊤
hJ : J • Jac ^ n = ⊥
⊢ Ideal.jacobson ⊥ ^ n = 0

Next tactic:
simpa only [this, top_smul, Ideal.zero_eq_bot] using hJ
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
  

Proof state:
case 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)
⊢ J = ⊤

Next tactic:
by_contra hJ
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
  

Proof state:
case 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 = ⊤
⊢ False

Next tactic:
change J ≠ ⊤ at hJ
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
  

Proof state:
case 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 ≠ ⊤
⊢ False

Next tactic:
rcases IsArtinian.set_has_minimal { J' : Ideal R | J < J' } ⟨⊤, hJ.lt_top⟩ with
    ⟨J', hJJ' : J < J', hJ' : ∀ I, J < I → ¬I < J'⟩
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'⟩
  

Proof state:
case 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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
⊢ False

Next tactic:
rcases SetLike.exists_of_lt hJJ' with ⟨x, hxJ', hxJ⟩
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⟩
  

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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
x : R
hxJ' : x ∈ J'
hxJ : x ∉ J
⊢ False

Next tactic:
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⟩⟩
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
    

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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
x : R
hxJ' : x ∈ J'
hxJ : x ∉ J
⊢ J ⊔ Ideal.span {x} = J'

Next tactic:
apply eq_of_le_of_not_lt _ (hJ' (J ⊔ Ideal.span {x}) _)
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}) _)
    · 

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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
x : R
hxJ' : x ∈ J'
hxJ : x ∉ J
⊢ J ⊔ Ideal.span {x} ≤ J'

Next tactic:
exact sup_le hJJ'.le (span_le.2 (singleton_subset_iff.2 hxJ'))
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'))
    · 

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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
x : R
hxJ' : x ∈ J'
hxJ : x ∉ J
⊢ J < J ⊔ Ideal.span {x}

Next tactic:
rw [SetLike.lt_iff_le_and_exists]
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]
      

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 ≠ ⊤
J' : Submodule R R
hJJ' : J < J'
hJ' : ∀ (I : Ideal R), J < I → ¬I < J'
x : R
hxJ' : x ∈ J'
hxJ : x ∉ J
⊢ J ≤ J ⊔ Ideal.span {x} ∧ ∃ x_1 ∈ J ⊔ Ideal.span {x}, x_1 ∉ J

Next tactic:
exact ⟨le_sup_left, ⟨x, mem_sup_right (mem_span_singleton_self x), hxJ⟩⟩
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⟩⟩
  

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}
⊢ False

Next tactic:
have : J ⊔ Jac • Ideal.span {x} ≤ J ⊔ Ideal.span {x} :=
    sup_le_sup_left (smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _