l3lab/ntp-mathlib huggingface dataset:
theorem le_multiplicity_iff_replicate_le_normalizedFactors [DecidableEq R] {a b : R} {n : ℕ}
    (ha : Irreducible a) (hb : b ≠ 0) :
    ↑n ≤ multiplicity a b ↔ replicate n (normalize a) ≤ normalizedFactors b := by
  rw [← pow_dvd_iff_le_multiplicity]
  revert b
  induction' n with n ih; · simp
  intro b hb
  constructor
  · rintro ⟨c, rfl⟩
    rw [Ne.def, pow_succ, mul_assoc, mul_eq_zero, not_or] at hb
    rw [pow_succ, mul_assoc, normalizedFactors_mul hb.1 hb.2, replicate_succ,
      normalizedFactors_irreducible ha, singleton_add, cons_le_cons_iff, ← ih hb.2]
    apply Dvd.intro _ rfl
  · rw [Multiset.le_iff_exists_add]
    

Proof state:
case succ.mpr
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
⊢ (∃ u, normalizedFactors b = replicate (Nat.succ n) (normalize a) + u) → a ^ Nat.succ n ∣ b

Next tactic:
rintro ⟨u, hu⟩
theorem le_multiplicity_iff_replicate_le_normalizedFactors [DecidableEq R] {a b : R} {n : ℕ}
    (ha : Irreducible a) (hb : b ≠ 0) :
    ↑n ≤ multiplicity a b ↔ replicate n (normalize a) ≤ normalizedFactors b := by
  rw [← pow_dvd_iff_le_multiplicity]
  revert b
  induction' n with n ih; · simp
  intro b hb
  constructor
  · rintro ⟨c, rfl⟩
    rw [Ne.def, pow_succ, mul_assoc, mul_eq_zero, not_or] at hb
    rw [pow_succ, mul_assoc, normalizedFactors_mul hb.1 hb.2, replicate_succ,
      normalizedFactors_irreducible ha, singleton_add, cons_le_cons_iff, ← ih hb.2]
    apply Dvd.intro _ rfl
  · rw [Multiset.le_iff_exists_add]
    rintro ⟨u, hu⟩
    

Proof state:
case succ.mpr.intro
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = replicate (Nat.succ n) (normalize a) + u
⊢ a ^ Nat.succ n ∣ b

Next tactic:
rw [← (normalizedFactors_prod hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate]
theorem le_multiplicity_iff_replicate_le_normalizedFactors [DecidableEq R] {a b : R} {n : ℕ}
    (ha : Irreducible a) (hb : b ≠ 0) :
    ↑n ≤ multiplicity a b ↔ replicate n (normalize a) ≤ normalizedFactors b := by
  rw [← pow_dvd_iff_le_multiplicity]
  revert b
  induction' n with n ih; · simp
  intro b hb
  constructor
  · rintro ⟨c, rfl⟩
    rw [Ne.def, pow_succ, mul_assoc, mul_eq_zero, not_or] at hb
    rw [pow_succ, mul_assoc, normalizedFactors_mul hb.1 hb.2, replicate_succ,
      normalizedFactors_irreducible ha, singleton_add, cons_le_cons_iff, ← ih hb.2]
    apply Dvd.intro _ rfl
  · rw [Multiset.le_iff_exists_add]
    rintro ⟨u, hu⟩
    rw [← (normalizedFactors_prod hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate]
    

Proof state:
case succ.mpr.intro
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a : R
ha : Irreducible a
n : ℕ
ih : ∀ {b : R}, b ≠ 0 → (a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b)
b : R
hb : b ≠ 0
u : Multiset R
hu : normalizedFactors b = replicate (Nat.succ n) (normalize a) + u
⊢ a ^ Nat.succ n ∣ normalize a ^ Nat.succ n * prod u

Next tactic:
exact (Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)
/-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
the normalized factor occurs in the `normalizedFactors`.

See also `count_normalizedFactors_eq` which expands the definition of `multiplicity`
to produce a specification for `count (normalizedFactors _) _`..
-/
theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
    (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ multiplicity a b = ↑(count (normalize a) (normalizedFactors b))

Next tactic:
apply le_antisymm
/-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
the normalized factor occurs in the `normalizedFactors`.

See also `count_normalizedFactors_eq` which expands the definition of `multiplicity`
to produce a specification for `count (normalizedFactors _) _`..
-/
theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
    (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by
  apply le_antisymm
  · 

Proof state:
case a
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ multiplicity a b ≤ ↑(count (normalize a) (normalizedFactors b))

Next tactic:
apply PartENat.le_of_lt_add_one
/-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
the normalized factor occurs in the `normalizedFactors`.

See also `count_normalizedFactors_eq` which expands the definition of `multiplicity`
to produce a specification for `count (normalizedFactors _) _`..
-/
theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
    (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    

Proof state:
case a.h
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ multiplicity a b < ↑(count (normalize a) (normalizedFactors b)) + 1

Next tactic:
rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge, ge_iff_le,
      le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
/-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
the normalized factor occurs in the `normalizedFactors`.

See also `count_normalizedFactors_eq` which expands the definition of `multiplicity`
to produce a specification for `count (normalizedFactors _) _`..
-/
theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
    (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge, ge_iff_le,
      le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
    

Proof state:
case a.h
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ¬count (normalize a) (normalizedFactors b) + 1 ≤ count (normalize a) (normalizedFactors b)

Next tactic:
simp
/-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
the normalized factor occurs in the `normalizedFactors`.

See also `count_normalizedFactors_eq` which expands the definition of `multiplicity`
to produce a specification for `count (normalizedFactors _) _`..
-/
theorem multiplicity_eq_count_normalizedFactors [DecidableEq R] {a b : R} (ha : Irreducible a)
    (hb : b ≠ 0) : multiplicity a b = (normalizedFactors b).count (normalize a) := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    rw [← Nat.cast_one, ← Nat.cast_add, lt_iff_not_ge, ge_iff_le,
      le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
    simp
  

Proof state:
case a
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
a b : R
ha : Irreducible a
hb : b ≠ 0
⊢ ↑(count (normalize a) (normalizedFactors b)) ≤ multiplicity a b

Next tactic:
rw [le_multiplicity_iff_replicate_le_normalizedFactors ha hb, ← le_count_iff_replicate_le]
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
⊢ count p (normalizedFactors x) = n

Next tactic:
letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
⊢ count p (normalizedFactors x) = n

Next tactic:
by_cases hx0 : x = 0
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  by_cases hx0 : x = 0
  · 

Proof state:
case pos
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
hx0 : x = 0
⊢ count p (normalizedFactors x) = n

Next tactic:
simp [hx0] at hlt
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  by_cases hx0 : x = 0
  · simp [hx0] at hlt
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
hx0 : ¬x = 0
⊢ count p (normalizedFactors x) = n

Next tactic:
rw [← PartENat.natCast_inj]
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  by_cases hx0 : x = 0
  · simp [hx0] at hlt
  rw [← PartENat.natCast_inj]
  

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
hx0 : ¬x = 0
⊢ ↑(count p (normalizedFactors x)) = ↑n

Next tactic:
convert (multiplicity_eq_count_normalizedFactors hp hx0).symm
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  by_cases hx0 : x = 0
  · simp [hx0] at hlt
  rw [← PartENat.natCast_inj]
  convert (multiplicity_eq_count_normalizedFactors hp hx0).symm
  · 

Proof state:
case h.e'_2.h.e'_3.h.e'_3
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
hx0 : ¬x = 0
⊢ p = normalize p

Next tactic:
exact hnorm.symm
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq [DecidableEq R] {p x : R} (hp : Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  letI : DecidableRel ((· ∣ ·) : R → R → Prop) := fun _ _ => Classical.propDecidable _
  by_cases hx0 : x = 0
  · simp [hx0] at hlt
  rw [← PartENat.natCast_inj]
  convert (multiplicity_eq_count_normalizedFactors hp hx0).symm
  · exact hnorm.symm
  

Proof state:
case h.e'_3
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)
hx0 : ¬x = 0
⊢ ↑n = multiplicity p x

Next tactic:
exact (multiplicity.eq_coe_iff.mpr ⟨hle, hlt⟩).symm
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hp : p = 0 ∨ Irreducible p
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
⊢ count p (normalizedFactors x) = n

Next tactic:
rcases hp with (rfl | hp)
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  rcases hp with (rfl | hp)
  · 

Proof state:
case inl
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
x : R
n : ℕ
hnorm : normalize 0 = 0
hle : 0 ^ n ∣ x
hlt : ¬0 ^ (n + 1) ∣ x
⊢ count 0 (normalizedFactors x) = n

Next tactic:
cases n
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  rcases hp with (rfl | hp)
  · cases n
    · 

Proof state:
case inl.zero
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
x : R
hnorm : normalize 0 = 0
hle : 0 ^ Nat.zero ∣ x
hlt : ¬0 ^ (Nat.zero + 1) ∣ x
⊢ count 0 (normalizedFactors x) = Nat.zero

Next tactic:
exact count_eq_zero.2 (zero_not_mem_normalizedFactors _)
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  rcases hp with (rfl | hp)
  · cases n
    · exact count_eq_zero.2 (zero_not_mem_normalizedFactors _)
    · 

Proof state:
case inl.succ
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
x : R
hnorm : normalize 0 = 0
n✝ : ℕ
hle : 0 ^ Nat.succ n✝ ∣ x
hlt : ¬0 ^ (Nat.succ n✝ + 1) ∣ x
⊢ count 0 (normalizedFactors x) = Nat.succ n✝

Next tactic:
rw [zero_pow (Nat.succ_pos _)] at hle hlt
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  rcases hp with (rfl | hp)
  · cases n
    · exact count_eq_zero.2 (zero_not_mem_normalizedFactors _)
    · rw [zero_pow (Nat.succ_pos _)] at hle hlt
      

Proof state:
case inl.succ
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
x : R
hnorm : normalize 0 = 0
n✝ : ℕ
hle : 0 ∣ x
hlt : ¬0 ∣ x
⊢ count 0 (normalizedFactors x) = Nat.succ n✝

Next tactic:
exact absurd hle hlt
/-- The number of times an irreducible factor `p` appears in `normalizedFactors x` is defined by
the number of times it divides `x`. This is a slightly more general version of
`UniqueFactorizationMonoid.count_normalizedFactors_eq` that allows `p = 0`.

See also `multiplicity_eq_count_normalizedFactors` if `n` is given by `multiplicity p x`.
-/
theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 ∨ Irreducible p)
    (hnorm : normalize p = p) {n : ℕ} (hle : p ^ n ∣ x) (hlt : ¬p ^ (n + 1) ∣ x) :
    (normalizedFactors x).count p = n := by
  rcases hp with (rfl | hp)
  · cases n
    · exact count_eq_zero.2 (zero_not_mem_normalizedFactors _)
    · rw [zero_pow (Nat.succ_pos _)] at hle hlt
      exact absurd hle hlt
  · 

Proof state:
case inr
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : Nontrivial R
inst✝¹ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
inst✝ : DecidableEq R
p x : R
hnorm : normalize p = p
n : ℕ
hle : p ^ n ∣ x
hlt : ¬p ^ (n + 1) ∣ x
hp : Irreducible p
⊢ count p (normalizedFactors x) = n

Next tactic:
exact count_normalizedFactors_eq hp hnorm hle hlt
theorem max_power_factor {a₀ : R} {x : R} (h : a₀ ≠ 0) (hx : Irreducible x) :
    ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : Nontrivial R
inst✝ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
a₀ x : R
h : a₀ ≠ 0
hx : Irreducible x
⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a

Next tactic:
classical
    let n := (normalizedFactors a₀).count (normalize x)
    obtain ⟨a, ha1, ha2⟩ := @exists_eq_pow_mul_and_not_dvd R _ _ x a₀
      (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr
        -- Porting note: this was a hole that was filled at the end of the proof with `use`:
        ⟨n, multiplicity_eq_count_normalizedFactors hx h⟩))
    simp_rw [← (multiplicity_eq_count_normalizedFactors hx h).symm] at ha1
    use n, a, ha2, ha1
theorem max_power_factor {a₀ : R} {x : R} (h : a₀ ≠ 0) (hx : Irreducible x) :
    ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := by
  classical
    

Proof state:
α : Type u_1
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : Nontrivial R
inst✝ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
a₀ x : R
h : a₀ ≠ 0
hx : Irreducible x
⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a

Next tactic:
let n := (normalizedFactors a₀).count (normalize x)