l3lab/ntp-mathlib huggingface dataset:
theorem eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n := by
  obtain ⟨n, hn⟩ := associated_pow_irreducible hx hirr
  obtain ⟨u, rfl⟩ := hn.symm
  

Proof state:
case intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
n : ℕ
u : Rˣ
hx : ϖ ^ n * ↑u ≠ 0
hn : Associated (ϖ ^ n * ↑u) (ϖ ^ n)
⊢ ∃ n_1 u_1, ϖ ^ n * ↑u = ↑u_1 * ϖ ^ n_1

Next tactic:
use n, u
theorem eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n := by
  obtain ⟨n, hn⟩ := associated_pow_irreducible hx hirr
  obtain ⟨u, rfl⟩ := hn.symm
  use n, u
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
n : ℕ
u : Rˣ
hx : ϖ ^ n * ↑u ≠ 0
hn : Associated (ϖ ^ n * ↑u) (ϖ ^ n)
⊢ ϖ ^ n * ↑u = ↑u * ϖ ^ n

Next tactic:
apply mul_comm
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
⊢ ∃ n, s = span {ϖ ^ n}

Next tactic:
have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
⊢ generator s ≠ 0

Next tactic:
rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
⊢ ¬s = ⊥

Next tactic:
assumption
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
⊢ ∃ n, s = span {ϖ ^ n}

Next tactic:
rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
  rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
  

Proof state:
case intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * ↑u = ϖ ^ n
⊢ ∃ n, s = span {ϖ ^ n}

Next tactic:
use n
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
  rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
  use n
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * ↑u = ϖ ^ n
⊢ s = span {ϖ ^ n}

Next tactic:
have : span _ = _ := Ideal.span_singleton_generator s
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
  rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
  use n
  have : span _ = _ := Ideal.span_singleton_generator s
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * ↑u = ϖ ^ n
this : span {generator s} = s
⊢ s = span {ϖ ^ n}

Next tactic:
rw [← this, ← hnu, span_singleton_eq_span_singleton]
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by
  have gen_ne_zero : generator s ≠ 0 := by
    rw [Ne.def, ← eq_bot_iff_generator_eq_zero]
    assumption
  rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
  use n
  have : span _ = _ := Ideal.span_singleton_generator s
  rw [← this, ← hnu, span_singleton_eq_span_singleton]
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
s : Ideal R
hs : s ≠ ⊥
ϖ : R
hirr : Irreducible ϖ
gen_ne_zero : generator s ≠ 0
n : ℕ
u : Rˣ
hnu : generator s * ↑u = ϖ ^ n
this : span {generator s} = s
⊢ Associated (generator s) (generator s * ↑u)

Next tactic:
use u
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
⊢ m = n

Next tactic:
have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
⊢ Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))

Next tactic:
rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
⊢ ∃ u, p ^ m * ↑u = q ^ n

Next tactic:
refine' ⟨u * v⁻¹, _⟩
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
⊢ p ^ m * ↑(u * v⁻¹) = q ^ n

Next tactic:
simp only [Units.val_mul]
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
⊢ p ^ m * (↑u * ↑v⁻¹) = q ^ n

Next tactic:
rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ m = n

Next tactic:
have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      assumption
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ ?m.288573

Next tactic:
refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    

Proof state:
case refine'_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ ∀ x ∈ Multiset.replicate m p, Irreducible x
case refine'_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ ∀ x ∈ Multiset.replicate n q, Irreducible x

Next tactic:
all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      assumption
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      

Proof state:
case refine'_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ ∀ x ∈ Multiset.replicate m p, Irreducible x

Next tactic:
intro x hx
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      

Proof state:
case refine'_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
x : R
hx : x ∈ Multiset.replicate m p
⊢ Irreducible x

Next tactic:
obtain rfl := Multiset.eq_of_mem_replicate hx
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      

Proof state:
case refine'_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
q : R
hq : Irreducible q
u v : Rˣ
m n : ℕ
x : R
hp : Irreducible x
h : ↑u * x ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m x)) (Multiset.prod (Multiset.replicate n q))
hx : x ∈ Multiset.replicate m x
⊢ Irreducible x

Next tactic:
assumption
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      

Proof state:
case refine'_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
⊢ ∀ x ∈ Multiset.replicate n q, Irreducible x

Next tactic:
intro x hx
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      

Proof state:
case refine'_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
x : R
hx : x ∈ Multiset.replicate n q
⊢ Irreducible x

Next tactic:
obtain rfl := Multiset.eq_of_mem_replicate hx
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      

Proof state:
case refine'_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p : R
hp : Irreducible p
u v : Rˣ
m n : ℕ
x : R
hq : Irreducible x
h : ↑u * p ^ m = ↑v * x ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n x))
hx : x ∈ Multiset.replicate n x
⊢ Irreducible x

Next tactic:
assumption
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n := by
  have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
    rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
    refine' ⟨u * v⁻¹, _⟩
    simp only [Units.val_mul]
    rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
  have := by
    refine' Multiset.card_eq_card_of_rel (UniqueFactorizationMonoid.factors_unique _ _ key)
    all_goals
      intro x hx
      obtain rfl := Multiset.eq_of_mem_replicate hx
      assumption
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
p q : R
hp : Irreducible p
hq : Irreducible q
u v : Rˣ
m n : ℕ
h : ↑u * p ^ m = ↑v * q ^ n
key : Associated (Multiset.prod (Multiset.replicate m p)) (Multiset.prod (Multiset.replicate n q))
this : Multiset.card (Multiset.replicate m p) = Multiset.card (Multiset.replicate n q)
⊢ m = n

Next tactic:
simpa only [Multiset.card_replicate]
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m n : ℕ
h : ↑u * ϖ ^ m = ↑v * ϖ ^ n
⊢ u = v

Next tactic:
obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ↑u * ϖ ^ m = ↑v * ϖ ^ m
⊢ u = v

Next tactic:
rw [← sub_eq_zero] at h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  rw [← sub_eq_zero] at h
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ↑u * ϖ ^ m - ↑v * ϖ ^ m = 0
⊢ u = v

Next tactic:
rw [← sub_mul, mul_eq_zero] at h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  rw [← sub_eq_zero] at h
  rw [← sub_mul, mul_eq_zero] at h
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ↑u - ↑v = 0 ∨ ϖ ^ m = 0
⊢ u = v

Next tactic:
cases' h with h h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  rw [← sub_eq_zero] at h
  rw [← sub_mul, mul_eq_zero] at h
  cases' h with h h
  · 

Proof state:
case inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ↑u - ↑v = 0
⊢ u = v

Next tactic:
rw [sub_eq_zero] at h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  rw [← sub_eq_zero] at h
  rw [← sub_mul, mul_eq_zero] at h
  cases' h with h h
  · rw [sub_eq_zero] at h
    

Proof state:
case inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ↑u = ↑v
⊢ u = v

Next tactic:
exact mod_cast h
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by
  obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
  rw [← sub_eq_zero] at h
  rw [← sub_mul, mul_eq_zero] at h
  cases' h with h h
  · rw [sub_eq_zero] at h
    exact mod_cast h
  · 

Proof state:
case inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hirr : Irreducible ϖ
u v : Rˣ
m : ℕ
h : ϖ ^ m = 0
⊢ u = v

Next tactic:
apply (hirr.ne_zero (pow_eq_zero h)).elim
theorem addVal_def (r : R) (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) (hr : r = u * ϖ ^ n) :
    addVal R r = n := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
r : R
u : Rˣ
ϖ : R
hϖ : Irreducible ϖ
n : ℕ
hr : r = ↑u * ϖ ^ n
⊢ (addVal R) r = ↑n

Next tactic:
rw [addVal, addValuation_apply, hr, eq_of_associated_left
      (associated_of_irreducible R hϖ (Classical.choose_spec (exists_prime R)).irreducible),
    eq_of_associated_right (Associated.symm ⟨u, mul_comm _ _⟩),
    multiplicity_pow_self_of_prime (PrincipalIdealRing.irreducible_iff_prime.1 hϖ)]
@[simp]
theorem addVal_uniformizer {ϖ : R} (hϖ : Irreducible ϖ) : addVal R ϖ = 1 := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
hϖ : Irreducible ϖ
⊢ (addVal R) ϖ = 1

Next tactic:
simpa only [one_mul, eq_self_iff_true, Units.val_one, pow_one, forall_true_left, Nat.cast_one]
    using addVal_def ϖ 1 hϖ 1
nonrec theorem _root_.Irreducible.addVal_pow {ϖ : R} (h : Irreducible ϖ) (n : ℕ) :
    addVal R (ϖ ^ n) = n := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
ϖ : R
h : Irreducible ϖ
n : ℕ
⊢ (addVal R) (ϖ ^ n) = ↑n

Next tactic:
rw [addVal_pow, addVal_uniformizer h, nsmul_one]
theorem addVal_eq_top_iff {a : R} : addVal R a = ⊤ ↔ a = 0 := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
a : R
⊢ (addVal R) a = ⊤ ↔ a = 0

Next tactic:
have hi := (Classical.choose_spec (exists_prime R)).irreducible
theorem addVal_eq_top_iff {a : R} : addVal R a = ⊤ ↔ a = 0 := by
  have hi := (Classical.choose_spec (exists_prime R)).irreducible
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
a : R
hi : Irreducible (choose (_ : ∃ ϖ, Prime ϖ))
⊢ (addVal R) a = ⊤ ↔ a = 0

Next tactic:
constructor
theorem addVal_eq_top_iff {a : R} : addVal R a = ⊤ ↔ a = 0 := by
  have hi := (Classical.choose_spec (exists_prime R)).irreducible
  constructor
  · 

Proof state:
case mp
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DiscreteValuationRing R
a : R
hi : Irreducible (choose (_ : ∃ ϖ, Prime ϖ))
⊢ (addVal R) a = ⊤ → a = 0

Next tactic:
contrapose