l3lab/ntp-mathlib huggingface dataset:
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  

Proof state:
case succ.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
⊢ (f x * ↑x) ^ (n + 1) ∈ ↑(span ((fun x => x ^ Nat.succ n) '' s))

Next tactic:
have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    exact subset_span ⟨x, x.prop, rfl⟩
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
⊢ span {↑x ^ (n + 1)} ≤ span ((fun x => x ^ (n + 1)) '' s)

Next tactic:
rw [span_le, Set.singleton_subset_iff]
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
⊢ ↑x ^ (n + 1) ∈ ↑(span ((fun x => x ^ (n + 1)) '' s))

Next tactic:
exact subset_span ⟨x, x.prop, rfl⟩
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    exact subset_span ⟨x, x.prop, rfl⟩
  

Proof state:
case succ.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this✝ : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
this : span {↑x ^ (n + 1)} ≤ span ((fun x => x ^ (n + 1)) '' s)
⊢ (f x * ↑x) ^ (n + 1) ∈ ↑(span ((fun x => x ^ Nat.succ n) '' s))

Next tactic:
refine' this _
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    exact subset_span ⟨x, x.prop, rfl⟩
  refine' this _
  

Proof state:
case succ.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this✝ : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
this : span {↑x ^ (n + 1)} ≤ span ((fun x => x ^ (n + 1)) '' s)
⊢ (f x * ↑x) ^ (n + 1) ∈ span {↑x ^ (n + 1)}

Next tactic:
rw [mul_pow, mem_span_singleton]
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
    span ((fun (x : α) => x ^ n) '' s) = ⊤ := by
  rw [eq_top_iff_one]
  cases' n with n
  · obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s
    · rw [Set.image_empty, hs]
      trivial
    · exact subset_span ⟨_, hx, pow_zero _⟩
  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
  rcases hs with ⟨f, hf⟩
  have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
  have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
  rw [hf, one_pow] at this
  refine' span_le.mpr _ this
  rintro _ hx
  simp_rw [Set.mem_image] at hx
  rcases hx with ⟨x, _, rfl⟩
  have : span ({(x:α) ^ (n + 1)} : Set α) ≤ span ((fun x : α => x ^ (n + 1)) '' s) := by
    rw [span_le, Set.singleton_subset_iff]
    exact subset_span ⟨x, x.prop, rfl⟩
  refine' this _
  rw [mul_pow, mem_span_singleton]
  

Proof state:
case succ.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
s : Set α
n : ℕ
f : ↑s →₀ α
hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1
hf : ∑ a in f.support, f a * ↑a = 1
this✝ : 1 ∈ span ((fun i => (f i * ↑i) ^ (n + 1)) '' ↑f.support)
x : ↑s
left✝ : x ∈ ↑f.support
this : span {↑x ^ (n + 1)} ≤ span ((fun x => x ^ (n + 1)) '' s)
⊢ ↑x ^ (n + 1) ∣ f x ^ (n + 1) * ↑x ^ (n + 1)

Next tactic:
exact ⟨f x ^ (n + 1), mul_comm _ _⟩
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
⊢ I ≠ ⊤

Next tactic:
rintro rfl
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑⊤ ↑S
maximally_disjoint : ∀ (J : Ideal α), ⊤ < J → ¬Disjoint ↑J ↑S
⊢ False

Next tactic:
have : 1 ∈ (S : Set α) := S.one_mem
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑⊤ ↑S
maximally_disjoint : ∀ (J : Ideal α), ⊤ < J → ¬Disjoint ↑J ↑S
this : 1 ∈ ↑S
⊢ False

Next tactic:
aesop
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
⊢ x ∈ I ∨ y ∈ I

Next tactic:
by_contra! rid
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
⊢ False

Next tactic:
have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
hx : ¬Disjoint ↑(I ⊔ span {x}) ↑S
⊢ False

Next tactic:
have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
hx : ¬Disjoint ↑(I ⊔ span {x}) ↑S
hy : ¬Disjoint ↑(I ⊔ span {y}) ↑S
⊢ False

Next tactic:
simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
    

Proof state:
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
hx : ∃ x_1, (∃ y ∈ I, ∃ z, x ∣ z ∧ y + z = x_1) ∧ x_1 ∈ S
hy : ∃ x, (∃ y_1 ∈ I, ∃ z, y ∣ z ∧ y_1 + z = x) ∧ x ∈ S
⊢ False

Next tactic:
obtain ⟨s₁, ⟨i₁, hi₁, ⟨_, ⟨r₁, rfl⟩, hr₁⟩⟩, hs₁⟩ := hx
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
    obtain ⟨s₁, ⟨i₁, hi₁, ⟨_, ⟨r₁, rfl⟩, hr₁⟩⟩, hs₁⟩ := hx
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
hy : ∃ x, (∃ y_1 ∈ I, ∃ z, y ∣ z ∧ y_1 + z = x) ∧ x ∈ S
s₁ : α
hs₁ : s₁ ∈ S
i₁ : α
hi₁ : i₁ ∈ I
r₁ : α
hr₁ : i₁ + x * r₁ = s₁
⊢ False

Next tactic:
obtain ⟨s₂, ⟨i₂, hi₂, ⟨_, ⟨r₂, rfl⟩, hr₂⟩⟩, hs₂⟩ := hy
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
    obtain ⟨s₁, ⟨i₁, hi₁, ⟨_, ⟨r₁, rfl⟩, hr₁⟩⟩, hs₁⟩ := hx
    obtain ⟨s₂, ⟨i₂, hi₂, ⟨_, ⟨r₂, rfl⟩, hr₂⟩⟩, hs₂⟩ := hy
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
s₁ : α
hs₁ : s₁ ∈ S
i₁ : α
hi₁ : i₁ ∈ I
r₁ : α
hr₁ : i₁ + x * r₁ = s₁
s₂ : α
hs₂ : s₂ ∈ S
i₂ : α
hi₂ : i₂ ∈ I
r₂ : α
hr₂ : i₂ + y * r₂ = s₂
⊢ False

Next tactic:
refine disjoint.ne_of_mem
      (I.add_mem (I.mul_mem_left (i₁ + x * r₁) hi₂) <| I.add_mem (I.mul_mem_right (y * r₂) hi₁) <|
        I.mul_mem_right (r₁ * r₂) hxy)
      (S.mul_mem hs₁ hs₂) ?_
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
    obtain ⟨s₁, ⟨i₁, hi₁, ⟨_, ⟨r₁, rfl⟩, hr₁⟩⟩, hs₁⟩ := hx
    obtain ⟨s₂, ⟨i₂, hi₂, ⟨_, ⟨r₂, rfl⟩, hr₂⟩⟩, hs₂⟩ := hy
    refine disjoint.ne_of_mem
      (I.add_mem (I.mul_mem_left (i₁ + x * r₁) hi₂) <| I.add_mem (I.mul_mem_right (y * r₂) hi₁) <|
        I.mul_mem_right (r₁ * r₂) hxy)
      (S.mul_mem hs₁ hs₂) ?_
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
s₁ : α
hs₁ : s₁ ∈ S
i₁ : α
hi₁ : i₁ ∈ I
r₁ : α
hr₁ : i₁ + x * r₁ = s₁
s₂ : α
hs₂ : s₂ ∈ S
i₂ : α
hi₂ : i₂ ∈ I
r₂ : α
hr₂ : i₂ + y * r₂ = s₂
⊢ (i₁ + x * r₁) * i₂ + (i₁ * (y * r₂) + x * y * (r₁ * r₂)) = s₁ * s₂

Next tactic:
rw [← hr₁, ← hr₂]
lemma isPrime_of_maximally_disjoint (I : Ideal α)
    (S : Submonoid α)
    (disjoint : Disjoint (I : Set α) S)
    (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬ Disjoint (J : Set α) S) :
    I.IsPrime where
  ne_top' := by
    rintro rfl
    have : 1 ∈ (S : Set α) := S.one_mem
    aesop
  mem_or_mem' {x y} hxy := by
    by_contra! rid
    have hx := maximally_disjoint (I ⊔ span {x}) (Submodule.lt_sup_iff_not_mem.mpr rid.1)
    have hy := maximally_disjoint (I ⊔ span {y}) (Submodule.lt_sup_iff_not_mem.mpr rid.2)
    simp only [Set.not_disjoint_iff, mem_inter_iff, SetLike.mem_coe, Submodule.mem_sup,
      mem_span_singleton] at hx hy
    obtain ⟨s₁, ⟨i₁, hi₁, ⟨_, ⟨r₁, rfl⟩, hr₁⟩⟩, hs₁⟩ := hx
    obtain ⟨s₂, ⟨i₂, hi₂, ⟨_, ⟨r₂, rfl⟩, hr₂⟩⟩, hs₂⟩ := hy
    refine disjoint.ne_of_mem
      (I.add_mem (I.mul_mem_left (i₁ + x * r₁) hi₂) <| I.add_mem (I.mul_mem_right (y * r₂) hi₁) <|
        I.mul_mem_right (r₁ * r₂) hxy)
      (S.mul_mem hs₁ hs₂) ?_
    rw [← hr₁, ← hr₂]
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u
β : Type v
a b : α
inst✝ : CommSemiring α
I✝ I : Ideal α
S : Submonoid α
disjoint : Disjoint ↑I ↑S
maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S
x y : α
hxy : x * y ∈ I
rid : x ∉ I ∧ y ∉ I
s₁ : α
hs₁ : s₁ ∈ S
i₁ : α
hi₁ : i₁ ∈ I
r₁ : α
hr₁ : i₁ + x * r₁ = s₁
s₂ : α
hs₂ : s₂ ∈ S
i₂ : α
hi₂ : i₂ ∈ I
r₂ : α
hr₂ : i₂ + y * r₂ = s₂
⊢ (i₁ + x * r₁) * i₂ + (i₁ * (y * r₂) + x * y * (r₁ * r₂)) = (i₁ + x * r₁) * (i₂ + y * r₂)

Next tactic:
ring
@[simp]
theorem span_singleton_neg (x : α) : (span {-x} : Ideal α) = span {x} := by
  

Proof state:
α : Type u
β : Type v
inst✝ : Ring α
I : Ideal α
a b x : α
⊢ span {-x} = span {x}

Next tactic:
ext
@[simp]
theorem span_singleton_neg (x : α) : (span {-x} : Ideal α) = span {x} := by
  ext
  

Proof state:
case h
α : Type u
β : Type v
inst✝ : Ring α
I : Ideal α
a b x x✝ : α
⊢ x✝ ∈ span {-x} ↔ x✝ ∈ span {x}

Next tactic:
simp only [mem_span_singleton']
@[simp]
theorem span_singleton_neg (x : α) : (span {-x} : Ideal α) = span {x} := by
  ext
  simp only [mem_span_singleton']
  

Proof state:
case h
α : Type u
β : Type v
inst✝ : Ring α
I : Ideal α
a b x x✝ : α
⊢ (∃ a, a * -x = x✝) ↔ ∃ a, a * x = x✝

Next tactic:
exact ⟨fun ⟨y, h⟩ => ⟨-y, h ▸ neg_mul_comm y x⟩, fun ⟨y, h⟩ => ⟨-y, h ▸ neg_mul_neg y x⟩⟩
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
⊢ I = ⊥ ∨ I = ⊤

Next tactic:
rw [or_iff_not_imp_right]
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
⊢ ¬I = ⊤ → I = ⊥

Next tactic:
change _ ≠ _ → _
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
⊢ I ≠ ⊤ → I = ⊥

Next tactic:
rw [Ideal.ne_top_iff_one]
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
⊢ 1 ∉ I → I = ⊥

Next tactic:
intro h1
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  intro h1
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
⊢ I = ⊥

Next tactic:
rw [eq_bot_iff]
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  intro h1
  rw [eq_bot_iff]
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
⊢ I ≤ ⊥

Next tactic:
intro r hr
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  intro h1
  rw [eq_bot_iff]
  intro r hr
  

Proof state:
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
r : K
hr : r ∈ I
⊢ r ∈ ⊥

Next tactic:
by_cases H : r = 0
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  intro h1
  rw [eq_bot_iff]
  intro r hr
  by_cases H : r = 0; · 

Proof state:
case pos
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
r : K
hr : r ∈ I
H : r = 0
⊢ r ∈ ⊥

Next tactic:
simpa
/-- All ideals in a division (semi)ring are trivial. -/
theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by
  rw [or_iff_not_imp_right]
  change _ ≠ _ → _
  rw [Ideal.ne_top_iff_one]
  intro h1
  rw [eq_bot_iff]
  intro r hr
  by_cases H : r = 0; · simpa
  

Proof state:
case neg
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
r : K
hr : r ∈ I
H : ¬r = 0
⊢ r ∈ ⊥

Next tactic:
simpa [H, h1] using I.mul_mem_left r⁻¹ hr
variable (K) in
/-- A bijection between between (left) ideals of a division ring and `{0, 1}`, sending `⊥` to `0`
and `⊤` to `1`. -/
def equivFinTwo [DecidableEq (Ideal K)] : Ideal K ≃ Fin 2 where
  toFun := fun I ↦ if I = ⊥ then 0 else 1
  invFun := ![⊥, ⊤]
  left_inv := fun I ↦ by 

Proof state:
α : Type u
β : Type v
K : Type u
inst✝¹ : DivisionSemiring K
I✝ : Ideal K
inst✝ : DecidableEq (Ideal K)
I : Ideal K
⊢ Matrix.vecCons ⊥ ![⊤] ((fun I => if I = ⊥ then 0 else 1) I) = I

Next tactic:
rcases eq_bot_or_top I with rfl | rfl
variable (K) in
/-- A bijection between between (left) ideals of a division ring and `{0, 1}`, sending `⊥` to `0`
and `⊤` to `1`. -/
def equivFinTwo [DecidableEq (Ideal K)] : Ideal K ≃ Fin 2 where
  toFun := fun I ↦ if I = ⊥ then 0 else 1
  invFun := ![⊥, ⊤]
  left_inv := fun I ↦ by rcases eq_bot_or_top I with rfl | rfl <;> 

Proof state:
case inl
α : Type u
β : Type v
K : Type u
inst✝¹ : DivisionSemiring K
I : Ideal K
inst✝ : DecidableEq (Ideal K)
⊢ Matrix.vecCons ⊥ ![⊤] ((fun I => if I = ⊥ then 0 else 1) ⊥) = ⊥

Next tactic:
simp
variable (K) in
/-- A bijection between between (left) ideals of a division ring and `{0, 1}`, sending `⊥` to `0`
and `⊤` to `1`. -/
def equivFinTwo [DecidableEq (Ideal K)] : Ideal K ≃ Fin 2 where
  toFun := fun I ↦ if I = ⊥ then 0 else 1
  invFun := ![⊥, ⊤]
  left_inv := fun I ↦ by rcases eq_bot_or_top I with rfl | rfl <;> 

Proof state:
case inr
α : Type u
β : Type v
K : Type u
inst✝¹ : DivisionSemiring K
I : Ideal K
inst✝ : DecidableEq (Ideal K)
⊢ Matrix.vecCons ⊥ ![⊤] ((fun I => if I = ⊥ then 0 else 1) ⊤) = ⊤

Next tactic:
simp