awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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
inr : ∀ (H0 : IsUnit ϖ), Associated p ϖ
inr₁ : ∀ (H0 : IsUnit (ϖ ^ (1 + n))), Associated p ϖ
⊢ Associated p ϖ

Tactic:
rcases this.isUnit_or_isUnit rfl with (H0 | H0)

Premises used: Irreducible.isUnit_or_isUnit, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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

Tactic:
(hϖ.not_isUnit H0).elim

Premises used: False.elim, Irreducible.not_isUnit
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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

Tactic:
exact (hϖ.not_isUnit H0).elim

Premises used: False.elim, Irreducible.not_isUnit
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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))
⊢ IsUnit (ϖ * ϖ ^ n)

Tactic:
rw [add_comm, pow_succ'] at H0

Premises used: add_comm, pow_succ'
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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

Tactic:
(hϖ.not_isUnit (isUnit_of_mul_isUnit_left H0)).elim

Premises used: False.elim, Irreducible.not_isUnit, isUnit_of_mul_isUnit_left
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
ϖ : R
hϖ : Irreducible ϖ
hR : ∀ {x : R}, x ≠ 0 → ∃ (n : 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

Tactic:
exact (hϖ.not_isUnit (isUnit_of_mul_isUnit_left H0)).elim

Premises used: False.elim, Irreducible.not_isUnit, isUnit_of_mul_isUnit_left
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
h :
  (∀
      b ∈
        Multiset.replicate
          (Classical.choose (p := fun (x_1 : ℕ) ↦
            Associated
              (Classical.choose (p := fun (x : R) ↦
                  Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
                x_1)
              x)
            ⋯)
          p,
      Prime b) ∧
    Associated
      (Multiset.replicate
          (Classical.choose (p := fun (x_1 : ℕ) ↦
            Associated
              (Classical.choose (p := fun (x : R) ↦
                  Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
                x_1)
              x)
            ⋯)
          p).prod
      x
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod x

Tactic:
use Multiset.replicate (Classical.choose (spec.2 hx)) p

Premises used: And.right, Classical.choose, Multiset.replicate
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
h :
  ∀
    b ∈
      Multiset.replicate
        (Classical.choose (p := fun (x_1 : ℕ) ↦
          Associated
            (Classical.choose (p := fun (x : R) ↦
                Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
              x_1)
            x)
          ⋯)
        p,
    Prime b
h₁ :
  Associated
    (Multiset.replicate
        (Classical.choose (p := fun (x_1 : ℕ) ↦
          Associated
            (Classical.choose (p := fun (x : R) ↦
                Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
              x_1)
            x)
          ⋯)
        p).prod
    x
⊢ (∀
      b ∈
        Multiset.replicate
          (Classical.choose (p := fun (x_1 : ℕ) ↦
            Associated
              (Classical.choose (p := fun (x : R) ↦
                  Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
                x_1)
              x)
            ⋯)
          p,
      Prime b) ∧
    Associated
      (Multiset.replicate
          (Classical.choose (p := fun (x_1 : ℕ) ↦
            Associated
              (Classical.choose (p := fun (x : R) ↦
                  Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
                x_1)
              x)
            ⋯)
          p).prod
      x

Tactic:
constructor
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
⊢ q = p

Tactic:
have hpq := Multiset.eq_of_mem_replicate hq
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
h : Prime p
⊢ Prime q

Tactic:
rw [hpq]
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
⊢ Prime p

Tactic:
⟨spec.1.ne_zero, spec.1.not_isUnit, ?_⟩

Premises used: Prime, Irreducible.not_isUnit, And.left, And.intro, Irreducible.ne_zero
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
h : ∀ (a b : R), p ∣ a * b → p ∣ a ∨ p ∣ b
⊢ Prime p

Tactic:
refine ⟨spec.1.ne_zero, spec.1.not_isUnit, ?_⟩
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
a b : R
h : p ∣ a * b
pos : ∀ (ha : a = 0), p ∣ a ∨ p ∣ b
neg : ∀ (ha : ¬a = 0), p ∣ a ∨ p ∣ b
⊢ p ∣ a ∨ p ∣ b

Tactic:
by_cases ha : a = 0
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
a b : R
h : p ∣ a * b
ha : a = 0
pos : p ∣ 0 ∨ p ∣ b
⊢ p ∣ a ∨ p ∣ b

Tactic:
rw [ha]
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
a b : R
h : p ∣ a * b
ha : a = 0
⊢ p ∣ 0 ∨ p ∣ b

Tactic:
simp only [true_or, dvd_zero]

Premises used: dvd_zero, true_or
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
a b : R
h : p ∣ a * b
ha : ¬a = 0
neg :
  ∀ (m : ℕ) (u : Rˣ)
    (h :
      p ∣
        Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
                hR ^
              m *
            (↑u : R) *
          b)
    (ha :
      ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
                hR ^
              m *
            (↑u : R) =
          0),
    p ∣
        Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
              hR ^
            m *
          (↑u : R) ∨
      p ∣ b
⊢ p ∣ a ∨ p ∣ b

Tactic:
obtain ⟨m, u, rfl⟩ := spec.2 ha

Premises used: And.right, rfl
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h :
  p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) *
      b
ha :
  ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) =
      0
⊢ p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
        m *
      b

Tactic:
rw [mul_assoc, mul_left_comm, Units.dvd_mul_left] at h

Premises used: mul_assoc, Units.dvd_mul_left, mul_left_comm
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h :
  p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) *
      b
ha :
  ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) =
      0
h₁ :
  p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
        m *
      b
⊢ ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) =
      0

Tactic:
rw [mul_assoc, mul_left_comm, Units.dvd_mul_left] at h

Premises used: mul_assoc, Units.dvd_mul_left, mul_left_comm
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h :
  p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
        m *
      b
ha :
  ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) =
      0
neg :
  p ∣
      Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
          hR ^
        m ∨
    p ∣ b
⊢ p ∣
      Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
            hR ^
          m *
        (↑u : R) ∨
    p ∣ b

Tactic:
rw [Units.dvd_mul_right]

Premises used: Units.dvd_mul_right
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsCancelMulZero R
hR : HasUnitMulPowIrreducibleFactorization R
p : R :=
  Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR
spec : Irreducible
    (Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
      hR) ∧
  ∀ {x : R},
    x ≠ 0 →
      ∃ (n : ℕ),
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
            n)
          x :=
  ⋯
x : R
hx : x ≠ 0
q : R
hq :
  q ∈
    Multiset.replicate
      (Classical.choose (p := fun (x_1 : ℕ) ↦
        Associated
          (Classical.choose (p := fun (x : R) ↦
              Irreducible x ∧ ∀ {x_2 : R}, x_2 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_2) hR ^
            x_1)
          x)
        ⋯)
      p
hpq : q = p
b : R
m : ℕ
u : Rˣ
h :
  p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
        m *
      b
ha :
  ¬Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
          m *
        (↑u : R) =
      0
pos :
  ∀ (hm : m = 0),
    p ∣
        Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
            hR ^
          m ∨
      p ∣ b
neg :
  ∀ (hm : ¬m = 0),
    p ∣
        Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
            hR ^
          m ∨
      p ∣ b
⊢ p ∣
      Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
          hR ^
        m ∨
    p ∣ b

Tactic:
by_cases hm : m = 0