l3lab/ntp-mathlib huggingface dataset:
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · 

Proof state:
case mp
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ (∃ a b, a * I + b * J = 1) → I ⊔ J = ⊤

Next tactic:
rintro ⟨x, y, hxy⟩
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    

Proof state:
case mp.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ I ⊔ J = ⊤

Next tactic:
rw [eq_top_iff_one]
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    

Proof state:
case mp.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 1 ∈ I ⊔ J

Next tactic:
apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
    

Proof state:
case mp.intro.intro.a
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 1 ∈ x * I + y * J

Next tactic:
rw [hxy]
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
    rw [hxy]
    

Proof state:
case mp.intro.intro.a
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L x y : Ideal R
hxy : x * I + y * J = 1
⊢ 1 ∈ 1

Next tactic:
simp only [one_eq_top, Submodule.mem_top]
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
    rw [hxy]
    simp only [one_eq_top, Submodule.mem_top]
  · 

Proof state:
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ I ⊔ J = ⊤ → ∃ a b, a * I + b * J = 1

Next tactic:
intro h
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
    rw [hxy]
    simp only [one_eq_top, Submodule.mem_top]
  · intro h
    

Proof state:
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
⊢ ∃ a b, a * I + b * J = 1

Next tactic:
refine' ⟨1, 1, _⟩
theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
  rw [IsCoprime, codisjoint_iff]
  constructor
  · rintro ⟨x, y, hxy⟩
    rw [eq_top_iff_one]
    apply (show x * I + y * J ≤ I ⊔ J from
      sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
    rw [hxy]
    simp only [one_eq_top, Submodule.mem_top]
  · intro h
    refine' ⟨1, 1, _⟩
    

Proof state:
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
h : I ⊔ J = ⊤
⊢ 1 * I + 1 * J = 1

Next tactic:
simpa only [one_eq_top, top_mul, Submodule.add_eq_sup]
theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ I + J = 1

Next tactic:
rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top]
theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1

Next tactic:
rw [← add_eq_one_iff, isCoprime_iff_add]
theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ IsCoprime I J ↔ I ⊔ J = ⊤

Next tactic:
rw [isCoprime_iff_codisjoint, codisjoint_iff]
open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
    ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤]

Next tactic:
rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
      ← isCoprime_iff_sup_eq]
open List in
theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
    ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by
  rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
      ← isCoprime_iff_sup_eq]
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ TFAE [IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J]

Next tactic:
simp
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ IsCoprime (span {x}) (span {y}) ↔ IsCoprime x y

Next tactic:
simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ (∃ a b, y ∣ b ∧ a * x + b = 1) ↔ IsCoprime x y

Next tactic:
constructor
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
  constructor
  · 

Proof state:
case mp
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ (∃ a b, y ∣ b ∧ a * x + b = 1) → IsCoprime x y

Next tactic:
rintro ⟨a, _, ⟨b, rfl⟩, e⟩
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
  constructor
  · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; 

Proof state:
case mp.intro.intro.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y a b : R
e : a * x + y * b = 1
⊢ IsCoprime x y

Next tactic:
exact ⟨a, b, mul_comm b y ▸ e⟩
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
  constructor
  · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩
  · 

Proof state:
case mpr
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y : R
⊢ IsCoprime x y → ∃ a b, y ∣ b ∧ a * x + b = 1

Next tactic:
rintro ⟨a, b, e⟩
theorem isCoprime_span_singleton_iff (x y : R) :
    IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
  simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
    mem_span_singleton]
  constructor
  · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩
  · rintro ⟨a, b, e⟩; 

Proof state:
case mpr.intro.intro
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
x y a b : R
e : a * x + b * y = 1
⊢ ∃ a b, y ∣ b ∧ a * x + b = 1

Next tactic:
exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, IsCoprime I (J j)
⊢ IsCoprime I (⨅ j ∈ s, J j)

Next tactic:
classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1+K)*I + K*J i  := by ring
        _ ≤ I + K ⊓ J i      := add_le_add mul_le_left mul_le_inf
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, IsCoprime I (J j)
⊢ IsCoprime I (⨅ j ∈ s, J j)

Next tactic:
simp_rw [isCoprime_iff_add] at *
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
s : Finset ι
hf : ∀ j ∈ s, I + J j = 1
⊢ I + ⨅ j ∈ s, J j = 1

Next tactic:
induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1+K)*I + K*J i  := by ring
        _ ≤ I + K ⊓ J i      := add_le_add mul_le_left mul_le_inf
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  

Proof state:
case empty
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
hf : ∀ j ∈ ∅, I + J j = 1
⊢ I + ⨅ j ∈ ∅, J j = 1

Next tactic:
| empty =>
      simp
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      

Proof state:
case empty
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
hf : ∀ j ∈ ∅, I + J j = 1
⊢ I + ⨅ j ∈ ∅, J j = 1

Next tactic:
simp
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  

Proof state:
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ I + ⨅ j ∈ insert i s, J j = 1

Next tactic:
| @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1+K)*I + K*J i  := by ring
        _ ≤ I + K ⊓ J i      := add_le_add mul_le_left mul_le_inf
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      

Proof state:
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ I + ⨅ j ∈ insert i s, J j = 1

Next tactic:
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      

Proof state:
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hs : (∀ j ∈ s, I + J j = 1) → I + ⨅ j ∈ s, J j = 1
hf : ∀ j ∈ insert i s, I + J j = 1
⊢ 1 ≤ I + (⨅ x ∈ s, J x) ⊓ J i

Next tactic:
set K := ⨅ j ∈ s, J j
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      

Proof state:
case insert
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ 1 ≤ I + K ⊓ J i

Next tactic:
calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1+K)*I + K*J i  := by ring
        _ ≤ I + K ⊓ J i      := add_le_add mul_le_left mul_le_inf
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ I + K = I + K * (I + J i)

Next tactic:
rw [hf i (Finset.mem_insert_self i s), mul_one]
theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K            := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K*(I + J i)  := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1+K)*I + K*J i  := by 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J✝ K✝ L : Ideal R
J : ι → Ideal R
i : ι
s : Finset ι
a✝ : i ∉ s
hf : ∀ j ∈ insert i s, I + J j = 1
K : Ideal R := ⨅ j ∈ s, J j
hs : (∀ j ∈ s, I + J j = 1) → I + K = 1
⊢ I + K * (I + J i) = (1 + K) * I + K * J i

Next tactic:
ring
/-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/
def radical (I : Ideal R) : Ideal R where
  carrier := { r | ∃ n : ℕ, r ^ n ∈ I }
  zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩
  add_mem' :=
  fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ =>
    ⟨m + n,
      (add_pow x y (m + n)).symm ▸ I.sum_mem <|
        show
          ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I
          from fun c _ =>
          Or.casesOn (le_total c m) (fun hcm =>
              I.mul_mem_right _ <|
                I.mul_mem_left _ <|
                  Nat.add_comm n m ▸
                    (add_tsub_assoc_of_le hcm n).symm ▸
                      (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc =>
               I.mul_mem_right _ <|
                I.mul_mem_right _ <|
                  add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩
-- Porting note: Below gives weird errors without `by exact`
  smul_mem' {r s} := by 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r s : R
⊢ s ∈
      {
            toAddSubsemigroup :=
              { carrier := {r | ∃ n, r ^ n ∈ I},
                add_mem' :=
                  (_ :
                    ∀ {x y : R}, x ∈ {r | ∃ n, r ^ n ∈ I} → y ∈ {r | ∃ n, r ^ n ∈ I} → x + y ∈ {r | ∃ n, r ^ n ∈ I}) },
            zero_mem' := (_ : ∃ n, 0 ^ n ∈ I) }.toAddSubsemigroup.carrier →
    r • s ∈
      {
            toAddSubsemigroup :=
              { carrier := {r | ∃ n, r ^ n ∈ I},
                add_mem' :=
                  (_ :
                    ∀ {x y : R}, x ∈ {r | ∃ n, r ^ n ∈ I} → y ∈ {r | ∃ n, r ^ n ∈ I} → x + y ∈ {r | ∃ n, r ^ n ∈ I}) },
            zero_mem' := (_ : ∃ n, 0 ^ n ∈ I) }.toAddSubsemigroup.carrier

Next tactic:
exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩
/-- An ideal is radical iff it is equal to its radical. -/
theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical I = I ↔ IsRadical I

Next tactic:
rw [le_antisymm_iff, and_iff_left le_radical, IsRadical]
theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical (I * J) = radical I ⊓ radical J

Next tactic:
refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
    ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
  refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
    ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
⊢ radical (I * J) ≤ radical I ⊓ radical J

Next tactic:
have := radical_mono <| @mul_le_inf _ _ I J
theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
  refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
    ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
  have := radical_mono <| @mul_le_inf _ _ I J
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
this : radical (I * J) ≤ radical (I ⊓ J)
⊢ radical (I * J) ≤ radical I ⊓ radical J

Next tactic:
simp_rw [radical_inf I J] at this
theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by
  refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
    ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
  have := radical_mono <| @mul_le_inf _ _ I J
  simp_rw [radical_inf I J] at this
  

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
this : radical (I * J) ≤ radical I ⊓ radical J
⊢ radical (I * J) ≤ radical I ⊓ radical J

Next tactic:
assumption
theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
  le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
    by_contradiction fun hri ↦
      let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
        zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
          (fun c hc hcc y hyc =>
            ⟨sSup c, fun ⟨n, hrnc⟩ =>
              let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc
              hc hyc ⟨n, hrny⟩,
              fun z => le_sSup⟩)
          I hri
      have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm =>
        by_contradiction fun hrmx =>
          hxm <|
            hm (m ⊔ span {x}) hrmx le_sup_left ▸
              (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _)
      have : IsPrime m :=
        ⟨by 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
m : Ideal R
hrm : r ∉ radical m
him : I ≤ m
hm : ∀ z ∈ {K | r ∉ radical K}, m ≤ z → z = m
this : ∀ x ∉ m, r ∈ radical (m ⊔ span {x})
⊢ m ≠ ⊤

Next tactic:
rintro rfl
theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
  le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
    by_contradiction fun hri ↦
      let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
        zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
          (fun c hc hcc y hyc =>
            ⟨sSup c, fun ⟨n, hrnc⟩ =>
              let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc
              hc hyc ⟨n, hrny⟩,
              fun z => le_sSup⟩)
          I hri
      have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm =>
        by_contradiction fun hrmx =>
          hxm <|
            hm (m ⊔ span {x}) hrmx le_sup_left ▸
              (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _)
      have : IsPrime m :=
        ⟨by rintro rfl; 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
hrm : r ∉ radical ⊤
him : I ≤ ⊤
hm : ∀ z ∈ {K | r ∉ radical K}, ⊤ ≤ z → z = ⊤
this : ∀ x ∉ ⊤, r ∈ radical (⊤ ⊔ span {x})
⊢ False

Next tactic:
rw [radical_top] at hrm
theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
  le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
    by_contradiction fun hri ↦
      let ⟨m, (hrm : r ∉ radical m), him, hm⟩ :=
        zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K }
          (fun c hc hcc y hyc =>
            ⟨sSup c, fun ⟨n, hrnc⟩ =>
              let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc
              hc hyc ⟨n, hrny⟩,
              fun z => le_sSup⟩)
          I hri
      have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm =>
        by_contradiction fun hrmx =>
          hxm <|
            hm (m ⊔ span {x}) hrmx le_sup_left ▸
              (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _)
      have : IsPrime m :=
        ⟨by rintro rfl; rw [radical_top] at hrm; 

Proof state:
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L I : Ideal R
r : R
hr : r ∈ sInf {J | I ≤ J ∧ IsPrime J}
hri : r ∉ radical I
hrm : r ∉ ⊤
him : I ≤ ⊤
hm : ∀ z ∈ {K | r ∉ radical K}, ⊤ ≤ z → z = ⊤
this : ∀ x ∉ ⊤, r ∈ radical (⊤ ⊔ span {x})
⊢ False

Next tactic:
exact hrm trivial