awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
this : emultiplicity a b = (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)
rwa :
  ∀
    (this :
      Eq (α := ℕ∞) (↑(multiplicity a b) : ℕ∞) (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)),
    multiplicity a b = count ((normalize : (a : R) → R) a) (normalizedFactors b)
⊢ multiplicity a b = count ((normalize : (a : R) → R) a) (normalizedFactors b)
have rwa₁ : ∀ (this : multiplicity a b = count ((normalize : (a : R) → R) a) (normalizedFactors b)),
  multiplicity a b = count ((normalize : (a : R) → R) a) (normalizedFactors b)
Tactic: rwa [(finiteMultiplicity_of_emultiplicity_eq_natCast this).emultiplicity_eq_multiplicity,
    ENat.coe_inj] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
⊢ count p (normalizedFactors x) = n
have pos : ∀ (hx0 : x = 0), count p (normalizedFactors x) = n
Tactic: by_cases hx0 : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
pos : ∀ (hx0 : x = 0), count p (normalizedFactors x) = n
⊢ count p (normalizedFactors x) = n
have neg : ∀ (hx0 : ¬x = 0), count p (normalizedFactors x) = n
Tactic: by_cases hx0 : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ count p (normalizedFactors x) = n
have neg : Eq (α := ℕ∞) (↑(count p (normalizedFactors x)) : ℕ∞) (↑n : ℕ∞)
Tactic: apply Nat.cast_injective (R := ℕ∞)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
⊢ Eq (α := ℕ∞) (↑(count p (normalizedFactors x)) : ℕ∞) (↑n : ℕ∞)
have h : p = (normalize : (a : R) → R) p
Tactic: convert! (emultiplicity_eq_count_normalizedFactors hp hx0).symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hx0 : ¬x = 0
h : p = (normalize : (a : R) → R) p
⊢ Eq (α := ℕ∞) (↑(count p (normalizedFactors x)) : ℕ∞) (↑n : ℕ∞)
have h₁ : (↑n : ℕ∞) = emultiplicity p x
Tactic: convert! (emultiplicity_eq_count_normalizedFactors hp hx0).symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : p = 0 ∨ Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
⊢ count p (normalizedFactors x) = n
have inl : ∀ (hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)) (hle : HPow.hPow (α := R) 0 n ∣ x)
  (hlt : ¬HPow.hPow (α := R) 0 (n + 1) ∣ x), count 0 (normalizedFactors x) = n
Tactic: rcases hp with (rfl | hp)
Cleared hypotheses: p, hp, hnorm, hle, hlt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hp : p = 0 ∨ Irreducible p
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
inl :
  ∀ (hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)) (hle : HPow.hPow (α := R) 0 n ∣ x)
    (hlt : ¬HPow.hPow (α := R) 0 (n + 1) ∣ x), count 0 (normalizedFactors x) = n
⊢ count p (normalizedFactors x) = n
have inr : ∀ (hp : Irreducible p), count p (normalizedFactors x) = n
Tactic: rcases hp with (rfl | hp)
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
x : R
n : ℕ
hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)
hle : HPow.hPow (α := R) 0 n ∣ x
hlt : ¬HPow.hPow (α := R) 0 (n + 1) ∣ x
⊢ count 0 (normalizedFactors x) = n
have inl : ∀ (hle : HPow.hPow (α := R) (β := ℕ) 0 0 ∣ x) (hlt : ¬HPow.hPow (α := R) 0 (HAdd.hAdd (α := ℕ) 0 1) ∣ x),
  count 0 (normalizedFactors x) = 0
Tactic: cases n
Cleared hypotheses: n, hle, hlt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
x : R
n : ℕ
hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)
hle : HPow.hPow (α := R) 0 n ∣ x
hlt : ¬HPow.hPow (α := R) 0 (n + 1) ∣ x
inl :
  ∀ (hle : HPow.hPow (α := R) (β := ℕ) 0 0 ∣ x) (hlt : ¬HPow.hPow (α := R) 0 (HAdd.hAdd (α := ℕ) 0 1) ∣ x),
    count 0 (normalizedFactors x) = 0
⊢ count 0 (normalizedFactors x) = n
have inl₁ : ∀ (n : ℕ) (hle : HPow.hPow (α := R) 0 (n + 1) ∣ x) (hlt : ¬HPow.hPow (α := R) 0 (n + 1 + 1) ∣ x),
  count 0 (normalizedFactors x) = n + 1
Tactic: cases n
Cleared hypotheses: n, hle, hlt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
x : R
hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)
n : ℕ
hle : HPow.hPow (α := R) 0 (n + 1) ∣ x
hlt : ¬HPow.hPow (α := R) 0 (n + 1 + 1) ∣ x
⊢ count 0 (normalizedFactors x) = n + 1
have hle₁ : 0 ∣ x
Tactic: rw [zero_pow (Nat.succ_ne_zero _)] at hle hlt
Cleared hypotheses: hle, hlt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
x : R
hnorm : Eq.{u_2 + 1} (α := R) ((normalize : (a : R) → R) 0 : R) (0 : R)
n : ℕ
hle : HPow.hPow (α := R) 0 (n + 1) ∣ x
hlt : ¬HPow.hPow (α := R) 0 (n + 1 + 1) ∣ x
hle₁ : 0 ∣ x
⊢ count 0 (normalizedFactors x) = n + 1
have hlt₁ : ¬0 ∣ x
Tactic: rw [zero_pow (Nat.succ_ne_zero _)] at hle hlt
Cleared hypotheses: hle, hlt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
x : R
hx : x ≠ 0
⊢ ∏ᶠ (p : R), p ^ count p (normalizedFactors x) ~ᵤ x
have rw : (normalizedFactors x).prod ~ᵤ x
Tactic: rw [← Multiset.prod_map_eq_finprod, Multiset.map_id']
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
Tactic: refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
⊢ a ∣ b
have pos : ∀ (hb : b = 0), a ∣ b
Tactic: by_cases hb : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
pos : ∀ (hb : b = 0), a ∣ b
⊢ a ∣ b
have neg : ∀ (hb : ¬b = 0), a ∣ b
Tactic: by_cases hb : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
⊢ a ∣ b
have neg : ∀ (a_1 : R), Multiset.count a_1 (normalizedFactors a) ≤ Multiset.count a_1 (normalizedFactors b)
Tactic: rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have pos : ∀ (hq : q ∈ normalizedFactors a), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
Tactic: by_cases hq : q ∈ normalizedFactors a
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
pos : ∀ (hq : q ∈ normalizedFactors a), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have neg : ∀ (hq : q ∉ normalizedFactors a), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
Tactic: by_cases hq : q ∈ normalizedFactors a
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
hq : q ∈ normalizedFactors a
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have hqprime : Prime q
Tactic: have hqprime : Prime q := prime_of_normalized_factor q hq
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have h1 : emultiplicity q a = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors a)) : ℕ∞)
Tactic: have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
h1 : emultiplicity q a = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors a)) : ℕ∞)
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have h2 : emultiplicity q b = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors b)) : ℕ∞)
Tactic: have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
h1 : emultiplicity q a = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors a)) : ℕ∞)
h2 : emultiplicity q b = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors b)) : ℕ∞)
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have h1₁ : emultiplicity q a = (↑(Multiset.count q (normalizedFactors a)) : ℕ∞)
Tactic: rw [normalize_normalized_factor q hq] at h1 h2
Cleared hypotheses: h1, h2
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
hb : ¬b = 0
this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
q : R
hq : q ∈ normalizedFactors a
hqprime : Prime q
h1 : emultiplicity q a = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors a)) : ℕ∞)
h2 : emultiplicity q b = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors b)) : ℕ∞)
h1₁ : emultiplicity q a = (↑(Multiset.count q (normalizedFactors a)) : ℕ∞)
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
have h2₁ : emultiplicity q b = (↑(Multiset.count q (normalizedFactors b)) : ℕ∞)
Tactic: rw [normalize_normalized_factor q hq] at h1 h2
Cleared hypotheses: h1, h2
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have pos : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : b = 0), a ∣ b
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
pos : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : b = 0), a ∣ b
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have neg : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
pos : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : b = 0), a ∣ b
neg neg₁ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
neg₂ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have neg₃ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
  let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
  ∀ (a_1 : R), Multiset.count a_1 (normalizedFactors a) ≤ Multiset.count a_1 (normalizedFactors b)
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
pos : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : b = 0), a ∣ b
neg neg₁ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
neg₂ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
neg₃ neg₄ :
  ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
    let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
    ∀ (q : R), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have pos₁ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
  let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
  ∀ (q : R) (hq : q ∈ normalizedFactors a),
    Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
ha : a ≠ 0
refine : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b), a ∣ b
pos : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : b = 0), a ∣ b
neg neg₁ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
neg₂ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0), a ∣ b
neg₃ neg₄ :
  ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
    let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
    ∀ (q : R), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
pos₁ :
  ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
    let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
    ∀ (q : R) (hq : q ∈ normalizedFactors a),
      Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
have neg₅ : ∀ (h : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b) (hb : ¬b = 0),
  let this : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid;
  ∀ (q : R) (hq : q ∉ normalizedFactors a),
    Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
Tactic: classical
  refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
  by_cases hb : b = 0
  · simp_all
  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
  rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
  intro q
  by_cases hq : q ∈ normalizedFactors a
  · have hqprime : Prime q := prime_of_normalized_factor q hq
    have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
    have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb
    rw [normalize_normalized_factor q hq] at h1 h2
    simpa [h1, h2] using h q hqprime
  · simp [Multiset.count_eq_zero_of_notMem hq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
⊢ a ^ n ∣ b ^ n ↔ a ∣ b
have pos : ∀ (ha : a = 0), a ^ n ∣ b ^ n ↔ a ∣ b
Tactic: by_cases ha : a = 0