l3lab/ntp-mathlib huggingface dataset:
/-- The completion of a module with respect to an ideal. This is not necessarily Hausdorff.
In fact, this is only complete if the ideal is finitely generated. -/
def adicCompletion : Submodule R (∀ n : ℕ, M ⧸ (I ^ n • ⊤ : Submodule R M)) where
  carrier := { f | ∀ {m n} (h : m ≤ n), liftQ _ (mkQ _) (by
      rw [ker_mkQ]
      exact smul_mono (Ideal.pow_le_pow_right h) le_rfl)
    (f n) = f m }
  zero_mem' hmn := by rw [Pi.zero_apply, Pi.zero_apply, LinearMap.map_zero]
  add_mem' hf hg m n hmn := by
    rw [Pi.add_apply, Pi.add_apply, LinearMap.map_add, hf hmn, hg hmn]
  smul_mem' c f hf m n hmn := by 

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
c : R
f : (n : ℕ) → M ⧸ I ^ n • ⊤
hf :
  f ∈
    {
          toAddSubsemigroup :=
            {
              carrier :=
                {f |
                  ∀ {m n : ℕ} (h : m ≤ n),
                    (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤)))) (f n) =
                      f m},
              add_mem' :=
                (_ :
                  ∀ {a b : (n : ℕ) → M ⧸ I ^ n • ⊤},
                    a ∈
                        {f |
                          ∀ {m n : ℕ} (h : m ≤ n),
                            (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤))))
                                (f n) =
                              f m} →
                      b ∈
                          {f |
                            ∀ {m n : ℕ} (h : m ≤ n),
                              (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤))))
                                  (f n) =
                                f m} →
                        ∀ (m n : ℕ) (hmn : m ≤ n),
                          (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤))))
                              ((a + b) n) =
                            (a + b) m) },
          zero_mem' :=
            (_ :
              ∀ {m n : ℕ} (hmn : m ≤ n),
                (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤))))
                    (OfNat.ofNat 0 n) =
                  OfNat.ofNat 0 m) }.toAddSubsemigroup.carrier
m n : ℕ
hmn : m ≤ n
⊢ (liftQ (I ^ n • ⊤) (mkQ (I ^ m • ⊤)) (_ : I ^ n • ⊤ ≤ LinearMap.ker (mkQ (I ^ m • ⊤)))) ((c • f) n) = (c • f) m

Next tactic:
rw [Pi.smul_apply, Pi.smul_apply, LinearMap.map_smul, hf hmn]
instance bot : IsHausdorff (⊥ : Ideal R) M :=
  ⟨fun x hx => by 

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x : M
hx : ∀ (n : ℕ), x ≡ 0 [SMOD ⊥ ^ n • ⊤]
⊢ x = 0

Next tactic:
simpa only [pow_one ⊥, bot_smul, SModEq.bot] using hx 1
protected theorem subsingleton (h : IsHausdorff (⊤ : Ideal R) M) : Subsingleton M :=
  ⟨fun x y => eq_of_sub_eq_zero <| h.haus (x - y) fun n => by
    

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : IsHausdorff ⊤ M
x y : M
n : ℕ
⊢ x - y ≡ 0 [SMOD ⊤ ^ n • ⊤]

Next tactic:
rw [Ideal.top_pow, top_smul]
protected theorem subsingleton (h : IsHausdorff (⊤ : Ideal R) M) : Subsingleton M :=
  ⟨fun x y => eq_of_sub_eq_zero <| h.haus (x - y) fun n => by
    rw [Ideal.top_pow, top_smul]
    

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : IsHausdorff ⊤ M
x y : M
n : ℕ
⊢ x - y ≡ 0 [SMOD ⊤]

Next tactic:
exact SModEq.top
instance : IsHausdorff I (Hausdorffification I M) :=
  ⟨fun x => Quotient.inductionOn' x fun x hx =>
    (Quotient.mk_eq_zero _).2 <| (mem_iInf _).2 fun n => by
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝ : Hausdorffification I M
x : M
hx : ∀ (n : ℕ), Quotient.mk'' x ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
⊢ x ∈ I ^ n • ⊤

Next tactic:
have := comap_map_mkQ (⨅ n : ℕ, I ^ n • ⊤ : Submodule R M) (I ^ n • ⊤)
instance : IsHausdorff I (Hausdorffification I M) :=
  ⟨fun x => Quotient.inductionOn' x fun x hx =>
    (Quotient.mk_eq_zero _).2 <| (mem_iInf _).2 fun n => by
      have := comap_map_mkQ (⨅ n : ℕ, I ^ n • ⊤ : Submodule R M) (I ^ n • ⊤)
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝ : Hausdorffification I M
x : M
hx : ∀ (n : ℕ), Quotient.mk'' x ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
this : comap (mkQ (⨅ n, I ^ n • ⊤)) (map (mkQ (⨅ n, I ^ n • ⊤)) (I ^ n • ⊤)) = (⨅ n, I ^ n • ⊤) ⊔ I ^ n • ⊤
⊢ x ∈ I ^ n • ⊤

Next tactic:
simp only [sup_of_le_right (iInf_le (fun n => (I ^ n • ⊤ : Submodule R M)) n)] at this
instance : IsHausdorff I (Hausdorffification I M) :=
  ⟨fun x => Quotient.inductionOn' x fun x hx =>
    (Quotient.mk_eq_zero _).2 <| (mem_iInf _).2 fun n => by
      have := comap_map_mkQ (⨅ n : ℕ, I ^ n • ⊤ : Submodule R M) (I ^ n • ⊤)
      simp only [sup_of_le_right (iInf_le (fun n => (I ^ n • ⊤ : Submodule R M)) n)] at this
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝ : Hausdorffification I M
x : M
hx : ∀ (n : ℕ), Quotient.mk'' x ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
this : comap (mkQ (⨅ n, I ^ n • ⊤)) (map (mkQ (⨅ n, I ^ n • ⊤)) (I ^ n • ⊤)) = I ^ n • ⊤
⊢ x ∈ I ^ n • ⊤

Next tactic:
rw [← this, map_smul'', mem_comap, Submodule.map_top, range_mkQ, ← SModEq.zero]
instance : IsHausdorff I (Hausdorffification I M) :=
  ⟨fun x => Quotient.inductionOn' x fun x hx =>
    (Quotient.mk_eq_zero _).2 <| (mem_iInf _).2 fun n => by
      have := comap_map_mkQ (⨅ n : ℕ, I ^ n • ⊤ : Submodule R M) (I ^ n • ⊤)
      simp only [sup_of_le_right (iInf_le (fun n => (I ^ n • ⊤ : Submodule R M)) n)] at this
      rw [← this, map_smul'', mem_comap, Submodule.map_top, range_mkQ, ← SModEq.zero]
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝ : Hausdorffification I M
x : M
hx : ∀ (n : ℕ), Quotient.mk'' x ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
this : comap (mkQ (⨅ n, I ^ n • ⊤)) (map (mkQ (⨅ n, I ^ n • ⊤)) (I ^ n • ⊤)) = I ^ n • ⊤
⊢ (mkQ (⨅ n, I ^ n • ⊤)) x ≡ 0 [SMOD I ^ n • ⊤]

Next tactic:
exact hx n
/-- Universal property of Hausdorffification: any linear map to a Hausdorff module extends to a
unique map from the Hausdorffification. -/
def lift (f : M →ₗ[R] N) : Hausdorffification I M →ₗ[R] N :=
  liftQ _ f <| map_le_iff_le_comap.1 <| h.iInf_pow_smul ▸ le_iInf fun n =>
    le_trans (map_mono <| iInf_le _ n) <| by
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : IsHausdorff I N
f : M →ₗ[R] N
n : ℕ
⊢ map f (I ^ n • ⊤) ≤ I ^ n • ⊤

Next tactic:
rw [map_smul'']
/-- Universal property of Hausdorffification: any linear map to a Hausdorff module extends to a
unique map from the Hausdorffification. -/
def lift (f : M →ₗ[R] N) : Hausdorffification I M →ₗ[R] N :=
  liftQ _ f <| map_le_iff_le_comap.1 <| h.iInf_pow_smul ▸ le_iInf fun n =>
    le_trans (map_mono <| iInf_le _ n) <| by
      rw [map_smul'']
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : IsHausdorff I N
f : M →ₗ[R] N
n : ℕ
⊢ I ^ n • map f ⊤ ≤ I ^ n • ⊤

Next tactic:
exact smul_mono le_rfl le_top
/-- Uniqueness of lift. -/
theorem lift_eq (f : M →ₗ[R] N) (g : Hausdorffification I M →ₗ[R] N) (hg : g.comp (of I M) = f) :
    g = lift I f :=
  LinearMap.ext fun x => induction_on x fun x => by 

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : IsHausdorff I N
f : M →ₗ[R] N
g : Hausdorffification I M →ₗ[R] N
hg : g ∘ₗ of I M = f
x✝ : Hausdorffification I M
x : M
⊢ g ((of I M) x) = (lift I f) ((of I M) x)

Next tactic:
rw [lift_of, ← hg, LinearMap.comp_apply]
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
⊢ IsPrecomplete ⊥ M

Next tactic:
refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊥ ^ m • ⊤]
n : ℕ
⊢ f n ≡ f 1 [SMOD ⊥ ^ n • ⊤]

Next tactic:
cases' n with n
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  cases' n with n
  · 

Proof state:
case zero
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊥ ^ m • ⊤]
⊢ f Nat.zero ≡ f 1 [SMOD ⊥ ^ Nat.zero • ⊤]

Next tactic:
rw [pow_zero, Ideal.one_eq_top, top_smul]
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  cases' n with n
  · rw [pow_zero, Ideal.one_eq_top, top_smul]
    

Proof state:
case zero
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊥ ^ m • ⊤]
⊢ f Nat.zero ≡ f 1 [SMOD ⊤]

Next tactic:
exact SModEq.top
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  cases' n with n
  · rw [pow_zero, Ideal.one_eq_top, top_smul]
    exact SModEq.top
  

Proof state:
case succ
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊥ ^ m • ⊤]
n : ℕ
⊢ f (Nat.succ n) ≡ f 1 [SMOD ⊥ ^ Nat.succ n • ⊤]

Next tactic:
specialize hf (Nat.le_add_left 1 n)
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  cases' n with n
  · rw [pow_zero, Ideal.one_eq_top, top_smul]
    exact SModEq.top
  specialize hf (Nat.le_add_left 1 n)
  

Proof state:
case succ
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
n : ℕ
hf : f 1 ≡ f (n + 1) [SMOD ⊥ ^ 1 • ⊤]
⊢ f (Nat.succ n) ≡ f 1 [SMOD ⊥ ^ Nat.succ n • ⊤]

Next tactic:
rw [pow_one, bot_smul, SModEq.bot] at hf
instance bot : IsPrecomplete (⊥ : Ideal R) M := by
  refine' ⟨fun f hf => ⟨f 1, fun n => _⟩⟩
  cases' n with n
  · rw [pow_zero, Ideal.one_eq_top, top_smul]
    exact SModEq.top
  specialize hf (Nat.le_add_left 1 n)
  rw [pow_one, bot_smul, SModEq.bot] at hf; 

Proof state:
case succ
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
n : ℕ
hf : f 1 = f (n + 1)
⊢ f (Nat.succ n) ≡ f 1 [SMOD ⊥ ^ Nat.succ n • ⊤]

Next tactic:
rw [hf]
instance top : IsPrecomplete (⊤ : Ideal R) M :=
  ⟨fun f _ =>
    ⟨0, fun n => by
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
x✝ : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊤ ^ m • ⊤]
n : ℕ
⊢ f n ≡ 0 [SMOD ⊤ ^ n • ⊤]

Next tactic:
rw [Ideal.top_pow, top_smul]
instance top : IsPrecomplete (⊤ : Ideal R) M :=
  ⟨fun f _ =>
    ⟨0, fun n => by
      rw [Ideal.top_pow, top_smul]
      

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : ℕ → M
x✝ : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊤ ^ m • ⊤]
n : ℕ
⊢ f n ≡ 0 [SMOD ⊤]

Next tactic:
exact SModEq.top
instance (priority := 100) of_subsingleton [Subsingleton M] : IsPrecomplete I M :=
  ⟨fun f _ => ⟨0, fun n => by 

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Subsingleton M
f : ℕ → M
x✝ : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
n : ℕ
⊢ f n ≡ 0 [SMOD I ^ n • ⊤]

Next tactic:
rw [Subsingleton.elim (f n) 0]
instance : IsHausdorff I (adicCompletion I M) :=
  ⟨fun x hx => ext fun n => smul_induction_on (SModEq.zero.1 <| hx n) (fun r hr x _ =>
    ((eval I M n).map_smul r x).symm ▸
      Quotient.inductionOn' (eval I M n x) fun x => SModEq.zero.2 <| smul_mem_smul hr mem_top)
    fun _ _ ih1 ih2 => by 

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x : ↥(adicCompletion I M)
hx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
x✝¹ x✝ : ↥(adicCompletion I M)
ih1 : (eval I M n) x✝¹ = (eval I M n) 0
ih2 : (eval I M n) x✝ = (eval I M n) 0
⊢ (eval I M n) (x✝¹ + x✝) = (eval I M n) 0

Next tactic:
rw [LinearMap.map_add, ih1, ih2, LinearMap.map_zero, add_zero]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
⊢ I ≤ Ideal.jacobson ⊥

Next tactic:
intro x hx
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
⊢ x ∈ Ideal.jacobson ⊥

Next tactic:
rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
⊢ ∀ (y : R), IsUnit (-x * y + 1)

Next tactic:
intro y
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
⊢ IsUnit (-x * y + 1)

Next tactic:
rw [add_comm]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
⊢ IsUnit (1 + -x * y)

Next tactic:
let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
⊢ IsUnit (1 + -x * y)

Next tactic:
have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    intro n _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
⊢ ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]

Next tactic:
intro m n h
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ f m ≡ f n [SMOD I ^ m • ⊤]

Next tactic:
simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ ∑ i in range m, (x * y) ^ i - ∑ i in range n, (x * y) ^ i ∈ I ^ m

Next tactic:
rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ ∑ x_1 in range (n - m), (x * y) ^ (m + x_1) ∈ I ^ m

Next tactic:
apply Submodule.sum_mem
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    

Proof state:
case a
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ ∀ c ∈ range (n - m), (x * y) ^ (m + c) ∈ I ^ m

Next tactic:
intro n _
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    intro n _
    

Proof state:
case a
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n✝ : ℕ
h : m ≤ n✝
n : ℕ
a✝ : n ∈ range (n✝ - m)
⊢ (x * y) ^ (m + n) ∈ I ^ m

Next tactic:
rw [mul_pow, pow_add, mul_assoc]
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    intro n _
    rw [mul_pow, pow_add, mul_assoc]
    

Proof state:
case a
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
m n✝ : ℕ
h : m ≤ n✝
n : ℕ
a✝ : n ∈ range (n✝ - m)
⊢ x ^ m * (x ^ n * y ^ (m + n)) ∈ I ^ m

Next tactic:
exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson := by
  intro x hx
  rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
  intro y
  rw [add_comm]
  let f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
  have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [Algebra.id.smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    intro n _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
  

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i in range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
⊢ IsUnit (1 + -x * y)

Next tactic:
obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf