awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R} (hb : b ≠ 0), a ^ n ∣ b ↔ replicate n ((normalize : (a : R) → R) a) ≤ normalizedFactors b
c : R
hb : ¬a = 0 ∧ ¬a ^ n * c = 0
⊢ a ^ n ∣ a ^ n * c

Tactic:
apply Dvd.intro _ rfl

Premises used: Dvd.intro, rfl
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R} (hb : b ≠ 0), a ^ n ∣ b ↔ replicate n ((normalize : (a : R) → R) a) ≤ normalizedFactors b
b : R
hb : b ≠ 0
succ :
  (∃ (u : Multiset R),
      normalizedFactors b =
        HAdd.hAdd (α := Multiset R) (replicate (n + 1) ((normalize : (a : R) → R) a) : Multiset R) u) →
    a ^ (n + 1) ∣ b
⊢ replicate (n + 1) ((normalize : (a : R) → R) a) ≤ normalizedFactors b → a ^ (n + 1) ∣ b

Tactic:
rw [Multiset.le_iff_exists_add]

Premises used: Multiset.le_iff_exists_add
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R} (hb : b ≠ 0), a ^ n ∣ b ↔ replicate n ((normalize : (a : R) → R) a) ≤ normalizedFactors b
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = HAdd.hAdd (α := Multiset R) (replicate (n + 1) ((normalize : (a : R) → R) a) : Multiset R) u
succ : a ^ (n + 1) ∣ HPow.hPow (α := R) ((normalize : (a : R) → R) a : R) (n + 1) * u.prod
⊢ a ^ (n + 1) ∣ b

Tactic:
rw [← (prod_normalizedFactors hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate]

Premises used: Multiset.prod_replicate, UniqueFactorizationMonoid.prod_normalizedFactors, Multiset.prod_add, Associated.dvd_iff_dvd_right
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R} (hb : b ≠ 0), a ^ n ∣ b ↔ replicate n ((normalize : (a : R) → R) a) ≤ normalizedFactors b
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = HAdd.hAdd (α := Multiset R) (replicate (n + 1) ((normalize : (a : R) → R) a) : Multiset R) u
⊢ a ^ (n + 1) ∣ HPow.hPow (α := R) ((normalize : (a : R) → R) a : R) (n + 1) * u.prod

Tactic:
(Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)

Premises used: Associated.dvd, Dvd.intro, Multiset.prod, Associated.pow_pow, rfl, Dvd.dvd.trans, associated_normalize
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R} (hb : b ≠ 0), a ^ n ∣ b ↔ replicate n ((normalize : (a : R) → R) a) ≤ normalizedFactors b
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = HAdd.hAdd (α := Multiset R) (replicate (n + 1) ((normalize : (a : R) → R) a) : Multiset R) u
⊢ a ^ (n + 1) ∣ HPow.hPow (α := R) ((normalize : (a : R) → R) a : R) (n + 1) * u.prod

Tactic:
exact (Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)

Premises used: Associated.dvd, Dvd.intro, Multiset.prod, Associated.pow_pow, rfl, Dvd.dvd.trans, associated_normalize
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
a₁ : emultiplicity a b ≤ (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)
a₂ : (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞) ≤ emultiplicity a b
⊢ emultiplicity a b = (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)

Tactic:
apply le_antisymm

Premises used: le_antisymm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
a₁ : emultiplicity a b < HAdd.hAdd (α := ℕ∞) (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞) 1
⊢ emultiplicity a b ≤ (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)

Tactic:
apply Order.le_of_lt_add_one

Premises used: Order.le_of_lt_add_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
a₁ :
  ¬count ((normalize : (a : R) → R) a) (normalizedFactors b) + 1 ≤
      count ((normalize : (a : R) → R) a) (normalizedFactors b)
⊢ emultiplicity a b < HAdd.hAdd (α := ℕ∞) (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞) 1

Tactic:
rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge,
      le_emultiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]

Premises used: lt_iff_not_ge, Nat.cast_add, UniqueFactorizationMonoid.le_emultiplicity_iff_replicate_le_normalizedFactors, Multiset.le_count_iff_replicate_le, Nat.cast_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ ¬count ((normalize : (a : R) → R) a) (normalizedFactors b) + 1 ≤
      count ((normalize : (a : R) → R) a) (normalizedFactors b)

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
a₁ :
  replicate (count ((normalize : (a : R) → R) a) (normalizedFactors b)) ((normalize : (a : R) → R) a) ≤
    normalizedFactors b
a₂ a₃ :
  count ((normalize : (a : R) → R) a) (normalizedFactors b) ≤ count ((normalize : (a : R) → R) a) (normalizedFactors b)
⊢ (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞) ≤ emultiplicity a b

Tactic:
rw [le_emultiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]

Premises used: UniqueFactorizationMonoid.le_emultiplicity_iff_replicate_le_normalizedFactors, Multiset.le_count_iff_replicate_le
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ emultiplicity a b = (↑(count ((normalize : (a : R) → R) a) (normalizedFactors b)) : ℕ∞)

Tactic:
have := emultiplicity_eq_count_normalizedFactors ha hb
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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)
rwa₁ rwa₂ rwa₃ rwa₄ :
  ∀ (this : multiplicity a b = count ((normalize : (a : R) → R) a) (normalizedFactors b)),
    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

Premises used: FiniteMultiplicity.emultiplicity_eq_multiplicity, ENat.coe_inj, finiteMultiplicity_of_emultiplicity_eq_natCast
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
neg : ∀ (hx0 : ¬x = 0), count p (normalizedFactors x) = n
⊢ count p (normalizedFactors x) = n

Tactic:
by_cases hx0 : x = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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

Tactic:
simp [hx0] at hlt
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
neg : Eq (α := ℕ∞) (↑(count p (normalizedFactors x)) : ℕ∞) (↑n : ℕ∞)
⊢ count p (normalizedFactors x) = n

Tactic:
apply Nat.cast_injective (R := ℕ∞)

Premises used: Nat.cast_injective
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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 ((normalize : (a : R) → R) p) (normalizedFactors x)) : ℕ∞) = emultiplicity p x

Tactic:
(emultiplicity_eq_count_normalizedFactors hp hx0).symm

Premises used: UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors, Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
h₁ : (↑n : ℕ∞) = emultiplicity p x
⊢ Eq (α := ℕ∞) (↑(count p (normalizedFactors x)) : ℕ∞) (↑n : ℕ∞)

Tactic:
convert! (emultiplicity_eq_count_normalizedFactors hp hx0).symm

Premises used: UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors, Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ p = (normalize : (a : R) → R) p

Tactic:
hnorm.symm

Premises used: Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ p = (normalize : (a : R) → R) p

Tactic:
exact hnorm.symm

Premises used: Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ (↑n : ℕ∞) = emultiplicity p x

Tactic:
(emultiplicity_eq_coe.mpr ⟨hle, hlt⟩).symm

Premises used: Iff.mpr, And.intro, emultiplicity_eq_coe, Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ (↑n : ℕ∞) = emultiplicity p x

Tactic:
exact (emultiplicity_eq_coe.mpr ⟨hle, hlt⟩).symm

Premises used: Iff.mpr, And.intro, emultiplicity_eq_coe, Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
inr : ∀ (hp : Irreducible p), count p (normalizedFactors x) = n
⊢ count p (normalizedFactors x) = n

Tactic:
rcases hp with (rfl | hp)

Premises used: rfl
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
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
⊢ count 0 (normalizedFactors x) = n

Tactic:
cases n
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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)
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:
count_eq_zero.2 (zero_notMem_normalizedFactors _)

Premises used: Iff.mpr, Multiset.count_eq_zero, UniqueFactorizationMonoid.zero_notMem_normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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)
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:
exact count_eq_zero.2 (zero_notMem_normalizedFactors _)

Premises used: Iff.mpr, Multiset.count_eq_zero, UniqueFactorizationMonoid.zero_notMem_normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ 0 ∣ x

Tactic:
rw [zero_pow (Nat.succ_ne_zero _)] at hle hlt

Premises used: Nat.succ_ne_zero, zero_pow
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ ¬0 ∣ x

Tactic:
rw [zero_pow (Nat.succ_ne_zero _)] at hle hlt

Premises used: Nat.succ_ne_zero, zero_pow
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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 : 0 ∣ x
hlt : ¬0 ∣ x
⊢ count 0 (normalizedFactors x) = n + 1

Tactic:
absurd hle hlt

Premises used: absurd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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 : 0 ∣ x
hlt : ¬0 ∣ x
⊢ count 0 (normalizedFactors x) = n + 1

Tactic:
exact absurd hle hlt

Premises used: absurd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hp : Irreducible p
⊢ count p (normalizedFactors x) = n

Tactic:
count_normalizedFactors_eq hp hnorm hle hlt

Premises used: UniqueFactorizationMonoid.count_normalizedFactors_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
p x : R
hnorm : (normalize : (a : R) → R) p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hp : Irreducible p
⊢ count p (normalizedFactors x) = n

Tactic:
exact count_normalizedFactors_eq hp hnorm hle hlt

Premises used: UniqueFactorizationMonoid.count_normalizedFactors_eq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
rw : (normalizedFactors x).prod ~ᵤ x
⊢ ∏ᶠ (p : R), p ^ count p (normalizedFactors x) ~ᵤ x

Tactic:
rw [← Multiset.prod_map_eq_finprod, Multiset.map_id']

Premises used: Multiset.prod_map_eq_finprod, Multiset.map_id'
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ (normalizedFactors x).prod ~ᵤ x

Tactic:
prod_normalizedFactors hx

Premises used: UniqueFactorizationMonoid.prod_normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ (normalizedFactors x).prod ~ᵤ x

Tactic:
exact prod_normalizedFactors hx

Premises used: UniqueFactorizationMonoid.prod_normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
inst_2 : NormalizationMonoid R
inst_3 : DecidableEq R
inst_4 : Subsingleton.{u_2 + 1} Rˣ
x : R
hx : x ≠ 0
⊢ ∏ᶠ (p : R), p ^ count p (normalizedFactors x) = x

Tactic:
associated_iff_eq.mp <| associated_finprod_pow_count hx

Premises used: UniqueFactorizationMonoid.associated_finprod_pow_count, associated_iff_eq, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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

Tactic:
⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩

Premises used: emultiplicity_le_emultiplicity_of_dvd_right, Iff.intro
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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

Tactic:
refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
neg : ∀ (hb : ¬b = 0), a ∣ b
⊢ a ∣ b

Tactic:
by_cases hb : b = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ a ∣ b

Tactic:
simp_all
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
neg : ∀ (a_1 : R), Multiset.count a_1 (normalizedFactors a) ≤ Multiset.count a_1 (normalizedFactors b)
⊢ a ∣ b

Tactic:
rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]

Premises used: Multiset.le_iff_count, UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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)
neg : ∀ (hq : q ∉ normalizedFactors a), Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)
⊢ Multiset.count q (normalizedFactors a) ≤ Multiset.count q (normalizedFactors b)

Tactic:
by_cases hq : q ∈ normalizedFactors a

Premises used: UniqueFactorizationMonoid.normalizedFactors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ Prime q

Tactic:
have hqprime : Prime q := prime_of_normalized_factor q hq
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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
⊢ emultiplicity q a = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors a)) : ℕ∞)

Tactic:
have h1 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible ha
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
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∞)
⊢ emultiplicity q b = (↑(Multiset.count ((normalize : (a : R) → R) q) (normalizedFactors b)) : ℕ∞)

Tactic:
have h2 := emultiplicity_eq_count_normalizedFactors hqprime.irreducible hb