l3lab/ntp-mathlib huggingface dataset:
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    

Proof state:
case intro.intro.intro.cons
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ b ∈ fs, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))

Next tactic:
rcases eq_or_ne fs ∅ with (rfl | hf')
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · 

Proof state:
case intro.intro.intro.cons.inl
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fh : ∅ ≠ ∅ → (∀ b ∈ ∅, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ ∅))

Next tactic:
use 1
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      

Proof state:
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fh : ∅ ≠ ∅ → (∀ b ∈ ∅, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
⊢ Associated (x ^ 1) (Multiset.prod (fa ::ₘ ∅))

Next tactic:
rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      

Proof state:
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fh : ∅ ≠ ∅ → (∀ b ∈ ∅, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod ∅)
hf₂ : fa ::ₘ ∅ ≠ ∅
this : ∀ b ∈ fa ::ₘ ∅, Associated x b
⊢ Associated x fa

Next tactic:
exact this _ (Multiset.mem_cons_self _ _)
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · 

Proof state:
case intro.intro.intro.cons.inr
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ b ∈ fs, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))

Next tactic:
obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      

Proof state:
case intro.intro.intro.cons.inr.intro
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ b ∈ fs, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ ∃ n, Associated (x ^ n) (Multiset.prod (fa ::ₘ fs))

Next tactic:
use n + 1
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      

Proof state:
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ b ∈ fs, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ Associated (x ^ (n + 1)) (Multiset.prod (fa ::ₘ fs))

Next tactic:
rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      

Proof state:
case h
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this✝ : x ≠ 0
hx' : Irreducible x
fa : R
fs : Multiset R
fh : fs ≠ ∅ → (∀ b ∈ fs, Associated x b) → ∃ n, Associated (x ^ n) (Multiset.prod fs)
hf₂ : fa ::ₘ fs ≠ ∅
this : ∀ b ∈ fa ::ₘ fs, Associated x b
hf' : fs ≠ ∅
n : ℕ
hn : Associated (x ^ n) (Multiset.prod fs)
⊢ Associated (x * x ^ n) (fa * Multiset.prod fs)

Next tactic:
exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
  

Proof state:
case neg
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ n, I = maximalIdeal R ^ n

Next tactic:
have : ∃ n : ℕ, x ^ n ∈ I := by
    obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      by_contra! h; apply hI; rw [eq_bot_iff]; exact h
    obtain ⟨n, u, rfl⟩ := H' r hr₂ (le_maximalIdeal hI' hr₁)
    use n
    rwa [← I.unit_mul_mem_iff_mem u.isUnit, mul_comm]
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
  have : ∃ n : ℕ, x ^ n ∈ I := by
    

Proof state:
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ n, x ^ n ∈ I

Next tactic:
obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      by_contra! h; apply hI; rw [eq_bot_iff]; exact h
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
  have : ∃ n : ℕ, x ^ n ∈ I := by
    obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      

Proof state:
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
⊢ ∃ r ∈ I, r ≠ 0

Next tactic:
by_contra! h
theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
    (h : ¬IsField R) (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
    ∃ n : ℕ, I = maximalIdeal R ^ n := by
  classical
  obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
  by_cases hI' : I = ⊤
  · use 0; rw [pow_zero, hI', Ideal.one_eq_top]
  have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
    (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton
  have : x ≠ 0 := by
    rintro rfl
    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h
    simp [hx]
  have hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx
  have H' : ∀ r : R, r ≠ 0 → r ∈ nonunits R → ∃ n : ℕ, Associated (x ^ n) r := by
    intro r hr₁ hr₂
    obtain ⟨f, hf₁, rfl, hf₂⟩ := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr₁).mp hr₂
    have : ∀ b ∈ f, Associated x b := by
      intro b hb
      exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1)
    clear hr₁ hr₂ hf₁
    induction' f using Multiset.induction with fa fs fh
    · exact (hf₂ rfl).elim
    rcases eq_or_ne fs ∅ with (rfl | hf')
    · use 1
      rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]
      exact this _ (Multiset.mem_cons_self _ _)
    · obtain ⟨n, hn⟩ := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)
      use n + 1
      rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]
      exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn
  have : ∃ n : ℕ, x ^ n ∈ I := by
    obtain ⟨r, hr₁, hr₂⟩ : ∃ r : R, r ∈ I ∧ r ≠ 0 := by
      by_contra! h; 

Proof state:
R : Type u_1
inst✝⁶ : CommRing R
K : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : IsDomain R
h✝ : ¬IsField R
I : Ideal R
hI : I ≠ ⊥
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
H' : ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r
h : ∀ r ∈ I, r = 0
⊢ False

Next tactic:
apply hI