awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
hm : m = 0
pos : ∀ (h : p ∣ b), p ∣ 1 ∨ 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:
simp only [hm, one_mul, pow_zero] at h ⊢

Premises used: pow_zero, one_mul
-- 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
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
hm : m = 0
h : p ∣ b
pos : p ∣ b
⊢ p ∣ 1 ∨ p ∣ b

Tactic:
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
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
hm : m = 0
h : p ∣ b
⊢ p ∣ b

Tactic:
h
-- 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
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
hm : m = 0
h : p ∣ b
⊢ p ∣ b

Tactic:
exact h
-- 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
hm : ¬m = 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 ∣
      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:
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
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
hm : ¬m = 0
neg :
  ∀ (m : ℕ)
    (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.succ *
          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.succ *
            (↑u : R) =
          0)
    (hm : ¬m.succ = 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.succ
⊢ p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
      m

Tactic:
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm

Premises used: Nat.exists_eq_succ_of_ne_zero, 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
u : Rˣ
m : ℕ
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.succ *
      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.succ *
        (↑u : R) =
      0
hm : ¬m.succ = 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 *
      Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
          hR ^
        m
⊢ p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR ^
      m.succ

Tactic:
rw [pow_succ']

Premises used: pow_succ'
-- 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
u : Rˣ
m : ℕ
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.succ *
      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.succ *
        (↑u : R) =
      0
hm : ¬m.succ = 0
⊢ p ∣
    Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1) hR *
      Classical.choose (p := fun (x : R) ↦ Irreducible x ∧ ∀ {x_1 : R}, x_1 ≠ 0 → ∃ (n : ℕ), Associated (x ^ n) x_1)
          hR ^
        m

Tactic:
apply dvd_mul_of_dvd_left dvd_rfl _

Premises used: dvd_mul_of_dvd_left, dvd_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
h :
  Associated
    (p ^
      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)
        ⋯)
    x
⊢ 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:
rw [Multiset.prod_replicate]

Premises used: Multiset.prod_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
⊢ Associated
    (p ^
      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)
        ⋯)
    x

Tactic:
Classical.choose_spec (spec.2 hx)

Premises used: Classical.choose_spec, And.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
⊢ Associated
    (p ^
      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)
        ⋯)
    x

Tactic:
exact Classical.choose_spec (spec.2 hx)

Premises used: Classical.choose_spec, And.right
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₁ : ∃ (p : R), Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
obtain : ∀ (p : R) (hp : Irreducible p), HasUnitMulPowIrreducibleFactorization R
⊢ HasUnitMulPowIrreducibleFactorization R

Tactic:
obtain ⟨p, hp⟩ := h₁
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
⊢ HasUnitMulPowIrreducibleFactorization R

Tactic:
⟨p, hp, ?_⟩

Premises used: IsDiscreteValuationRing.HasUnitMulPowIrreducibleFactorization, And.intro, Exists.intro
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
refine : ∀ {x : R}, x ≠ 0 → ∃ (n : ℕ), Associated (p ^ n) x
⊢ HasUnitMulPowIrreducibleFactorization R

Tactic:
refine ⟨p, hp, ?_⟩
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
obtain : ∀ (fx : Multiset R) (hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x), ∃ (n : ℕ), Associated (p ^ n) x
⊢ ∃ (n : ℕ), Associated (p ^ n) x

Tactic:
obtain ⟨fx, hfx⟩ := WfDvdMonoid.exists_factors x hx

Premises used: WfDvdMonoid.exists_factors
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ ∃ (n : ℕ), Associated (p ^ n) x

Tactic:
⟨Multiset.card fx, ?_⟩

Premises used: Exists.intro, Multiset.card
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
refine : Associated (p ^ fx.card) x
⊢ ∃ (n : ℕ), Associated (p ^ n) x

Tactic:
refine ⟨Multiset.card fx, ?_⟩
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
⊢ Associated fx.prod x

Tactic:
have H := hfx.2
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associated fx.prod x
rw :
  ∀ (H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)),
    Eq (α := Associates R) (Associates.mk (p ^ fx.card)) (Associates.mk x)
⊢ Associated (p ^ fx.card) x

Tactic:
rw [← Associates.mk_eq_mk_iff_associated] at H ⊢

Premises used: Associates.mk_eq_mk_iff_associated
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
rw :
  Eq (α := Associates R) (Multiset.prod (Multiset.replicate fx.card (Associates.mk p)))
    (Multiset.prod (Multiset.map Associates.mk fx))
⊢ Eq (α := Associates R) (Associates.mk (p ^ fx.card)) (Associates.mk x)

Tactic:
rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]

Premises used: Multiset.prod_replicate, Associates.mk_pow, Associates.prod_mk
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
e_a : Eq (α := Multiset (Associates R)) (Multiset.replicate fx.card (Associates.mk p)) (Multiset.map Associates.mk fx)
⊢ Eq (α := Associates R) (Multiset.prod (Multiset.replicate fx.card (Associates.mk p)))
    (Multiset.prod (Multiset.map Associates.mk fx))

Tactic:
congr 1
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
e_a : Eq (α := Multiset (Associates R)) (Multiset.map Associates.mk fx) (Multiset.replicate fx.card (Associates.mk p))
⊢ Eq (α := Multiset (Associates R)) (Multiset.replicate fx.card (Associates.mk p)) (Multiset.map Associates.mk fx)

Tactic:
symm
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
e_a :
  Multiset.card (α := Associates R) (Multiset.map Associates.mk fx) = fx.card ∧
    ∀ (b : Associates R),
      Membership.mem (γ := Multiset (Associates R)) (Multiset.map Associates.mk fx) b → b = Associates.mk p
⊢ Eq (α := Multiset (Associates R)) (Multiset.map Associates.mk fx) (Multiset.replicate fx.card (Associates.mk p))

Tactic:
rw [Multiset.eq_replicate]

Premises used: Multiset.eq_replicate
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
e_a : ∀ (b : Associates R), ∀ x ∈ fx, Associates.mk x = b → b = Associates.mk p
⊢ Multiset.card (α := Associates R) (Multiset.map Associates.mk fx) = fx.card ∧
    ∀ (b : Associates R),
      Membership.mem (γ := Multiset (Associates R)) (Multiset.map Associates.mk fx) b → b = Associates.mk p

Tactic:
simp only [true_and, and_imp, Multiset.card_map, Multiset.mem_map, exists_imp]

Premises used: and_imp, Multiset.mem_map, exists_imp, true_and, Multiset.card_map
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
q : R
hq : q ∈ fx
e_a : Associated q p
⊢ Eq (α := Associates R) (Associates.mk q) (Associates.mk p)

Tactic:
rw [Associates.mk_eq_mk_iff_associated]

Premises used: Associates.mk_eq_mk_iff_associated
-- Module: Mathlib.RingTheory.DiscreteValuationRing.Basic
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Eq (α := Associates R) (Associates.mk fx.prod) (Associates.mk x)
q : R
hq : q ∈ fx
⊢ Associated q p

Tactic:
apply h₂ (hfx.1 _ hq) hp

Premises used: And.left