l3lab/ntp-mathlib huggingface dataset:
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · 

Proof state:
case intro
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 • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ IsUnit (1 + -x * y)

Next tactic:
rw [isUnit_iff_exists_inv]
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    

Proof state:
case intro
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 • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ ∃ b, (1 + -x * y) * b = 1

Next tactic:
use L
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    

Proof state:
case h
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 • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ (1 + -x * y) * L = 1

Next tactic:
rw [← sub_eq_zero, neg_mul]
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    

Proof state:
case h
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 • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ (1 + -(x * y)) * L - 1 = 0

Next tactic:
apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ ∀ (n : ℕ), (1 + -(x * y)) * L - 1 ≡ 0 [SMOD I ^ n • ⊤]

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 _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
n : ℕ
⊢ (1 + -(x * y)) * L - 1 ≡ 0 [SMOD I ^ n • ⊤]

Next tactic:
specialize hL 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 _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
n : ℕ
hL : f n ≡ L [SMOD I ^ n • ⊤]
⊢ (1 + -(x * y)) * L - 1 ≡ 0 [SMOD I ^ n • ⊤]

Next tactic:
rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
n : ℕ
hL : f n - L ∈ I ^ n
⊢ (1 + -(x * y)) * L - 1 - 0 ∈ I ^ n

Next tactic:
rw [sub_zero]
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
n : ℕ
hL : f n - L ∈ I ^ n
⊢ (1 + -(x * y)) * L - 1 ∈ I ^ n

Next tactic:
suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      ring
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ 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✝ : 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 • ⊤]
L : R
n : ℕ
hL : f n - L ∈ I ^ n
this : (1 - x * y) * f n - 1 ∈ I ^ n
⊢ (1 + -(x * y)) * L - 1 ∈ I ^ n

Next tactic:
convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      

Proof state:
case h.e'_4
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 • ⊤]
L : R
n : ℕ
hL : f n - L ∈ I ^ n
this : (1 - x * y) * f n - 1 ∈ I ^ n
⊢ (1 + -(x * y)) * L - 1 = (1 - x * y) * f n - 1 - (1 + -(x * y)) * (f n - L)

Next tactic:
ring
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      ring
    

Proof state:
case h.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
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
n : ℕ
hL : f n - L ∈ I ^ n
⊢ (1 - x * y) * f n - 1 ∈ I ^ n

Next tactic:
cases 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 _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      ring
    cases n
    · 

Proof state:
case h.a.zero
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 • ⊤]
L : R
hL : f Nat.zero - L ∈ I ^ Nat.zero
⊢ (1 - x * y) * f Nat.zero - 1 ∈ I ^ Nat.zero

Next tactic:
simp only [Ideal.one_eq_top, pow_zero, Nat.zero_eq, mem_top]
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      ring
    cases n
    · simp only [Ideal.one_eq_top, pow_zero, Nat.zero_eq, mem_top]
    · 

Proof state:
case h.a.succ
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 • ⊤]
L : R
n✝ : ℕ
hL : f (Nat.succ n✝) - L ∈ I ^ Nat.succ n✝
⊢ (1 - x * y) * f (Nat.succ n✝) - 1 ∈ I ^ Nat.succ n✝

Next tactic:
rw [← neg_sub _ (1 : R), neg_mul, mul_geom_sum, neg_sub, sub_sub, add_comm, ← sub_sub,
        sub_self, zero_sub, @neg_mem_iff, mul_pow]
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)
  obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
  · rw [isUnit_iff_exists_inv]
    use L
    rw [← sub_eq_zero, neg_mul]
    apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
    intro n
    specialize hL n
    rw [SModEq.sub_mem, Algebra.id.smul_eq_mul, Ideal.mul_top] at hL ⊢
    rw [sub_zero]
    suffices (1 - x * y) * f n - 1 ∈ I ^ n by
      convert Ideal.sub_mem _ this (Ideal.mul_mem_left _ (1 + -(x * y)) hL) using 1
      ring
    cases n
    · simp only [Ideal.one_eq_top, pow_zero, Nat.zero_eq, mem_top]
    · rw [← neg_sub _ (1 : R), neg_mul, mul_geom_sum, neg_sub, sub_sub, add_comm, ← sub_sub,
        sub_self, zero_sub, @neg_mem_iff, mul_pow]
      

Proof state:
case h.a.succ
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 • ⊤]
L : R
n✝ : ℕ
hL : f (Nat.succ n✝) - L ∈ I ^ Nat.succ n✝
⊢ x ^ Nat.succ n✝ * y ^ Nat.succ n✝ ∈ I ^ Nat.succ n✝

Next tactic:
exact Ideal.mul_mem_right _ (I ^ _) (Ideal.pow_mem_pow hx _)
/-- The image of `canonicalEmbedding` lives in the `ℝ`-submodule of the `x ∈ ((K →+* ℂ) → ℂ)` such
that `conj x_φ = x_(conj φ)` for all `∀ φ : K →+* ℂ`. -/
theorem conj_apply {x : ((K →+* ℂ) → ℂ)} (φ : K →+* ℂ)
    (hx : x ∈ Submodule.span ℝ (Set.range (canonicalEmbedding K))) :
    conj (x φ) = x (ComplexEmbedding.conjugate φ) := by
  

Proof state:
K : Type u_1
inst✝ : Field K
x : (K →+* ℂ) → ℂ
φ : K →+* ℂ
hx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
⊢ (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)

Next tactic:
refine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)
/-- The image of `canonicalEmbedding` lives in the `ℝ`-submodule of the `x ∈ ((K →+* ℂ) → ℂ)` such
that `conj x_φ = x_(conj φ)` for all `∀ φ : K →+* ℂ`. -/
theorem conj_apply {x : ((K →+* ℂ) → ℂ)} (φ : K →+* ℂ)
    (hx : x ∈ Submodule.span ℝ (Set.range (canonicalEmbedding K))) :
    conj (x φ) = x (ComplexEmbedding.conjugate φ) := by
  refine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)
  · 

Proof state:
case refine_1
K : Type u_1
inst✝ : Field K
x : (K →+* ℂ) → ℂ
φ : K →+* ℂ
hx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
⊢ ∀ x ∈ Set.range ⇑(canonicalEmbedding K), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)

Next tactic:
rintro _ ⟨x, rfl⟩
/-- The image of `canonicalEmbedding` lives in the `ℝ`-submodule of the `x ∈ ((K →+* ℂ) → ℂ)` such
that `conj x_φ = x_(conj φ)` for all `∀ φ : K →+* ℂ`. -/
theorem conj_apply {x : ((K →+* ℂ) → ℂ)} (φ : K →+* ℂ)
    (hx : x ∈ Submodule.span ℝ (Set.range (canonicalEmbedding K))) :
    conj (x φ) = x (ComplexEmbedding.conjugate φ) := by
  refine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)
  · rintro _ ⟨x, rfl⟩
    

Proof state:
case refine_1.intro
K : Type u_1
inst✝ : Field K
x✝ : (K →+* ℂ) → ℂ
φ : K →+* ℂ
hx : x✝ ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
x : K
⊢ (starRingEnd ℂ) ((canonicalEmbedding K) x φ) = (canonicalEmbedding K) x (ComplexEmbedding.conjugate φ)

Next tactic:
rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq]
/-- The image of `canonicalEmbedding` lives in the `ℝ`-submodule of the `x ∈ ((K →+* ℂ) → ℂ)` such
that `conj x_φ = x_(conj φ)` for all `∀ φ : K →+* ℂ`. -/
theorem conj_apply {x : ((K →+* ℂ) → ℂ)} (φ : K →+* ℂ)
    (hx : x ∈ Submodule.span ℝ (Set.range (canonicalEmbedding K))) :
    conj (x φ) = x (ComplexEmbedding.conjugate φ) := by
  refine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)
  · rintro _ ⟨x, rfl⟩
    rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq]
  · 

Proof state:
case refine_2
K : Type u_1
inst✝ : Field K
x : (K →+* ℂ) → ℂ
φ : K →+* ℂ
hx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
⊢ (starRingEnd ℂ) (OfNat.ofNat 0 φ) = OfNat.ofNat 0 (ComplexEmbedding.conjugate φ)

Next tactic:
rw [Pi.zero_apply, Pi.zero_apply, map_zero]