l3lab/ntp-mathlib huggingface dataset:
/-- An integral domain is a DVR iff it's a PID with a unique non-zero prime ideal. -/
theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
    DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P := by
  constructor
  · intro RDVR
    rcases id RDVR with ⟨Rlocal⟩
    constructor
    assumption
    use LocalRing.maximalIdeal R
    constructor
    exact ⟨Rlocal, inferInstance⟩
    · rintro Q ⟨hQ1, hQ2⟩
      obtain ⟨q, rfl⟩ := (IsPrincipalIdealRing.principal Q).1
      have hq : q ≠ 0 := by
        rintro rfl
        apply hQ1
        simp
      erw [span_singleton_prime hq] at hQ2
      replace hQ2 := hQ2.irreducible
      rw [irreducible_iff_uniformizer] at hQ2
      exact hQ2.symm
  · rintro ⟨RPID, Punique⟩
    haveI : LocalRing R := LocalRing.of_unique_nonzero_prime Punique
    refine' { not_a_field' := _ }
    rcases Punique with ⟨P, ⟨hP1, hP2⟩, _⟩
    have hPM : P ≤ maximalIdeal R := le_maximalIdeal hP2.1
    intro h
    rw [h, le_bot_iff] at hPM
    

Proof state:
case mpr.intro.intro.intro.intro
R✝ : Type u
inst✝⁴ : CommRing R✝
inst✝³ : IsDomain R✝
inst✝² : DiscreteValuationRing R✝
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
this : LocalRing R
P : Ideal R
right✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P
hP1 : P ≠ ⊥
hP2 : IsPrime P
hPM : P = ⊥
h : maximalIdeal R = ⊥
⊢ False

Next tactic:
exact hP1 hPM
theorem associated_of_irreducible {a b : R} (ha : Irreducible a) (hb : Irreducible b) :
    Associated a b := by
  

Proof state:
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
a b : R
ha : Irreducible a
hb : Irreducible b
⊢ Associated a b

Next tactic:
rw [irreducible_iff_uniformizer] at ha hb
theorem associated_of_irreducible {a b : R} (ha : Irreducible a) (hb : Irreducible b) :
    Associated a b := by
  rw [irreducible_iff_uniformizer] at ha hb
  

Proof state:
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
a b : R
ha : maximalIdeal R = span {a}
hb : maximalIdeal R = span {b}
⊢ Associated a b

Next tactic:
rw [← span_singleton_eq_span_singleton, ← ha, hb]
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  

Proof state:
R : Type u_1
inst✝ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
p q : R
hp : Irreducible p
hq : Irreducible q
⊢ Associated p q

Next tactic:
rcases hR with ⟨ϖ, hϖ, hR⟩
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  

Proof state:
case intro.intro
R : Type u_1
inst✝ : CommRing R
p q : R
hp : Irreducible p
hq : Irreducible q
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
⊢ Associated p q

Next tactic:
suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    

Proof state:
R : Type u_1
inst✝ : CommRing R
p q : R
hp : Irreducible p
hq : Irreducible q
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
this : ∀ {p : R}, Irreducible p → Associated p ϖ
⊢ Associated p q

Next tactic:
apply Associated.trans (this hp) (this hq).symm
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  

Proof state:
case intro.intro
R : Type u_1
inst✝ : CommRing R
p q : R
hp : Irreducible p
hq : Irreducible q
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
⊢ ∀ {p : R}, Irreducible p → Associated p ϖ

Next tactic:
clear hp hq p q
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  

Proof state:
case intro.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
⊢ ∀ {p : R}, Irreducible p → Associated p ϖ

Next tactic:
intro p hp
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  

Proof state:
case intro.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
⊢ Associated p ϖ

Next tactic:
obtain ⟨n, hn⟩ := hR hp.ne_zero
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  

Proof state:
case intro.intro.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
⊢ Associated p ϖ

Next tactic:
have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  

Proof state:
case intro.intro.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
⊢ Associated p ϖ

Next tactic:
rcases lt_trichotomy n 1 with (H | rfl | H)
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · 

Proof state:
case intro.intro.intro.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : n < 1
⊢ Associated p ϖ

Next tactic:
obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      

Proof state:
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : n < 1
⊢ n = 0

Next tactic:
clear hn this
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      

Proof state:
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
H : n < 1
⊢ n = 0

Next tactic:
revert H n
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      

Proof state:
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
⊢ ∀ n < 1, n = 0

Next tactic:
exact by decide
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by 

Proof state:
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
⊢ ∀ n < 1, n = 0

Next tactic:
decide
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    

Proof state:
case intro.intro.intro.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
hn : Associated (ϖ ^ 0) p
this : Irreducible (ϖ ^ 0)
H : 0 < 1
⊢ Associated p ϖ

Next tactic:
simp [not_irreducible_one, pow_zero] at this
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · 

Proof state:
case intro.intro.intro.inr.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
hn : Associated (ϖ ^ 1) p
this : Irreducible (ϖ ^ 1)
⊢ Associated p ϖ

Next tactic:
simpa only [pow_one] using hn.symm
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · 

Proof state:
case intro.intro.intro.inr.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ n) p
this : Irreducible (ϖ ^ n)
H : 1 < n
⊢ Associated p ϖ

Next tactic:
obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
    

Proof state:
case intro.intro.intro.inr.inr.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ ^ (1 + n + 1))
H : 1 < 1 + n + 1
⊢ Associated p ϖ

Next tactic:
rw [pow_succ] at this
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
    rw [pow_succ] at this
    

Proof state:
case intro.intro.intro.inr.inr.intro
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
⊢ Associated p ϖ

Next tactic:
rcases this.isUnit_or_isUnit rfl with (H0 | H0)
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
    rw [pow_succ] at this
    rcases this.isUnit_or_isUnit rfl with (H0 | H0)
    · 

Proof state:
case intro.intro.intro.inr.inr.intro.inl
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit ϖ
⊢ Associated p ϖ

Next tactic:
exact (hϖ.not_unit H0).elim
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
    rw [pow_succ] at this
    rcases this.isUnit_or_isUnit rfl with (H0 | H0)
    · exact (hϖ.not_unit H0).elim
    · 

Proof state:
case intro.intro.intro.inr.inr.intro.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit (ϖ ^ (1 + n))
⊢ Associated p ϖ

Next tactic:
rw [add_comm, pow_succ] at H0
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q := by
  rcases hR with ⟨ϖ, hϖ, hR⟩
  suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
    apply Associated.trans (this hp) (this hq).symm
  clear hp hq p q
  intro p hp
  obtain ⟨n, hn⟩ := hR hp.ne_zero
  have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
  rcases lt_trichotomy n 1 with (H | rfl | H)
  · obtain rfl : n = 0 := by
      clear hn this
      revert H n
      exact by decide
    simp [not_irreducible_one, pow_zero] at this
  · simpa only [pow_one] using hn.symm
  · obtain ⟨n, rfl⟩ : ∃ k, n = 1 + k + 1 := Nat.exists_eq_add_of_lt H
    rw [pow_succ] at this
    rcases this.isUnit_or_isUnit rfl with (H0 | H0)
    · exact (hϖ.not_unit H0).elim
    · rw [add_comm, pow_succ] at H0
      

Proof state:
case intro.intro.intro.inr.inr.intro.inr
R : Type u_1
inst✝ : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x
p : R
hp : Irreducible p
n : ℕ
hn : Associated (ϖ ^ (1 + n + 1)) p
this : Irreducible (ϖ * ϖ ^ (1 + n))
H : 1 < 1 + n + 1
H0 : IsUnit (ϖ * ϖ ^ n)
⊢ Associated p ϖ

Next tactic:
exact (hϖ.not_unit (isUnit_of_mul_isUnit_left H0)).elim
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) x

Next tactic:
use Multiset.replicate (Classical.choose (spec.2 hx)) p
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    use Multiset.replicate (Classical.choose (spec.2 hx)) p
    

Proof state:
case h
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
⊢ (∀ b ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p, Prime b) ∧
    Associated (Multiset.prod (Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p)) x

Next tactic:
constructor
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    use Multiset.replicate (Classical.choose (spec.2 hx)) p
    constructor
    · 

Proof state:
case h.left
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
⊢ ∀ b ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p, Prime b

Next tactic:
intro q hq
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    use Multiset.replicate (Classical.choose (spec.2 hx)) p
    constructor
    · intro q hq
      

Proof state:
case h.left
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p
⊢ Prime q

Next tactic:
have hpq := Multiset.eq_of_mem_replicate hq
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    use Multiset.replicate (Classical.choose (spec.2 hx)) p
    constructor
    · intro q hq
      have hpq := Multiset.eq_of_mem_replicate hq
      

Proof state:
case h.left
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p
hpq : q = p
⊢ Prime q

Next tactic:
rw [hpq]
/-- An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization`. -/
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
  let p := Classical.choose hR
  let spec := Classical.choose_spec hR
  UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
    use Multiset.replicate (Classical.choose (spec.2 hx)) p
    constructor
    · intro q hq
      have hpq := Multiset.eq_of_mem_replicate hq
      rw [hpq]
      

Proof state:
case h.left
R : Type u_1
inst✝¹ : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝ : IsDomain R
p : R := choose hR
spec : Irreducible (choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (choose hR ^ n) x := choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p
hpq : q = p
⊢ Prime p

Next tactic:
refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩