l3lab/ntp-mathlib huggingface dataset:
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      

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
⊢ ∀ (a b : R), p ∣ a * b → p ∣ a ∨ p ∣ b

Next tactic:
intro a b h
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      

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
a b : R
h : p ∣ a * b
⊢ p ∣ a ∨ p ∣ b

Next tactic:
by_cases ha : a = 0
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · 

Proof state:
case pos
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
a b : R
h : p ∣ a * b
ha : a = 0
⊢ p ∣ a ∨ p ∣ b

Next tactic:
rw [ha]
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        

Proof state:
case pos
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
a b : R
h : p ∣ a * b
ha : a = 0
⊢ p ∣ 0 ∨ p ∣ b

Next tactic:
simp only [true_or_iff, dvd_zero]
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      

Proof state:
case neg
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
a b : R
h : p ∣ a * b
ha : ¬a = 0
⊢ p ∣ a ∨ p ∣ b

Next tactic:
obtain ⟨m, u, rfl⟩ := spec.2 ha
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      

Proof state:
case neg.intro.intro
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
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * ↑u * b
ha : ¬choose hR ^ m * ↑u = 0
⊢ p ∣ choose hR ^ m * ↑u ∨ p ∣ b

Next tactic:
rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      

Proof state:
case neg.intro.intro
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
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * b
ha : ¬choose hR ^ m * ↑u = 0
⊢ p ∣ choose hR ^ m * ↑u ∨ p ∣ b

Next tactic:
rw [IsUnit.dvd_mul_right (Units.isUnit _)]
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      

Proof state:
case neg.intro.intro
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
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * b
ha : ¬choose hR ^ m * ↑u = 0
⊢ p ∣ choose hR ^ m ∨ p ∣ b

Next tactic:
by_cases hm : m = 0
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · 

Proof state:
case pos
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
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * b
ha : ¬choose hR ^ m * ↑u = 0
hm : m = 0
⊢ p ∣ choose hR ^ m ∨ p ∣ b

Next tactic:
simp only [hm, one_mul, pow_zero] at h ⊢
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        

Proof state:
case pos
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
b : R
m : ℕ
u : Rˣ
ha : ¬choose hR ^ m * ↑u = 0
hm : m = 0
h : choose hR ∣ b
⊢ choose hR ∣ 1 ∨ choose hR ∣ b

Next tactic:
right
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        

Proof state:
case pos.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
q : R
hq : q ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p
hpq : q = p
b : R
m : ℕ
u : Rˣ
ha : ¬choose hR ^ m * ↑u = 0
hm : m = 0
h : choose hR ∣ b
⊢ choose hR ∣ b

Next tactic:
exact h
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      

Proof state:
case neg
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
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * b
ha : ¬choose hR ^ m * ↑u = 0
hm : ¬m = 0
⊢ p ∣ choose hR ^ m ∨ p ∣ b

Next tactic:
left
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      left
      

Proof state:
case neg.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
q : R
hq : q ∈ Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h : p ∣ choose hR ^ m * b
ha : ¬choose hR ^ m * ↑u = 0
hm : ¬m = 0
⊢ p ∣ choose hR ^ m

Next tactic:
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      left
      obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
      

Proof state:
case neg.h.intro
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
b : R
u : Rˣ
m : ℕ
h : p ∣ choose hR ^ Nat.succ m * b
ha : ¬choose hR ^ Nat.succ m * ↑u = 0
hm : ¬Nat.succ m = 0
⊢ p ∣ choose hR ^ Nat.succ m

Next tactic:
rw [pow_succ]
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      left
      obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
      rw [pow_succ]
      

Proof state:
case neg.h.intro
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
b : R
u : Rˣ
m : ℕ
h : p ∣ choose hR ^ Nat.succ m * b
ha : ¬choose hR ^ Nat.succ m * ↑u = 0
hm : ¬Nat.succ m = 0
⊢ p ∣ choose hR * choose hR ^ m

Next tactic:
apply dvd_mul_of_dvd_left dvd_rfl _
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      left
      obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
      rw [pow_succ]
      apply dvd_mul_of_dvd_left dvd_rfl _
    · 

Proof state:
case h.right
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
⊢ Associated (Multiset.prod (Multiset.replicate (choose (_ : ∃ n, Associated (choose hR ^ n) x)) p)) x

Next tactic:
rw [Multiset.prod_replicate]
/-- 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]
      refine' ⟨spec.1.ne_zero, spec.1.not_unit, _⟩
      intro a b h
      by_cases ha : a = 0
      · rw [ha]
        simp only [true_or_iff, dvd_zero]
      obtain ⟨m, u, rfl⟩ := spec.2 ha
      rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h
      rw [IsUnit.dvd_mul_right (Units.isUnit _)]
      by_cases hm : m = 0
      · simp only [hm, one_mul, pow_zero] at h ⊢
        right
        exact h
      left
      obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
      rw [pow_succ]
      apply dvd_mul_of_dvd_left dvd_rfl _
    · rw [Multiset.prod_replicate]
      

Proof state:
case h.right
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
⊢ Associated (p ^ choose (_ : ∃ n, Associated (choose hR ^ n) x)) x

Next tactic:
exact Classical.choose_spec (spec.2 hx)