-- Mathlib/Algebra/Squarefree/Basic.lean
theorem squarefree_iff_nodup_normalizedFactors [NormalizationMonoid R] {x : R}
    (x0 : x ≠ 0) : Squarefree x ↔ Multiset.Nodup (normalizedFactors x)

Proof state:
case mp
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
by_cases hmem : a ∈ normalizedFactors x

Result:
case pos
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
⊢ Multiset.count a (normalizedFactors x) ≤ 1

case neg
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∉ normalizedFactors x
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Proof state:
case pos
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
rcases h a with (h | h)

Result:
case pos.inl
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : emultiplicity a x ≤ 1
⊢ Multiset.count a (normalizedFactors x) ≤ 1

case pos.inr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : IsUnit a
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Proof state:
case mpr.inr.intro.intro
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
a : R
hu : ¬IsUnit a
h0 : a ≠ 0
b : R
hib : Irreducible b
hdvd : b ∣ a
⊢ emultiplicity a x ≤ 1

Tactic:
apply le_trans (emultiplicity_le_emultiplicity_of_dvd_left hdvd)

Result:
case mpr.inr.intro.intro
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
a : R
hu : ¬IsUnit a
h0 : a ≠ 0
b : R
hib : Irreducible b
hdvd : b ∣ a
⊢ emultiplicity b x ≤ 1

Proof state:
case pos.inr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : IsUnit a
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
have := ha.1

Result:
case pos.inr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this✝ : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : IsUnit a
this : ¬IsUnit a
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Proof state:
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
⊢ (∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1) ↔ ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
constructor <;> intro h a

Result:
case mp
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
⊢ Multiset.count a (normalizedFactors x) ≤ 1

case mpr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
a : R
⊢ emultiplicity a x ≤ 1 ∨ IsUnit a

Proof state:
case pos.inl
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : emultiplicity a x ≤ 1
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
rw [← normalize_normalized_factor _ hmem]

Result:
case pos.inl
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h✝ : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
h : emultiplicity a x ≤ 1
⊢ Multiset.count (normalize a) (normalizedFactors x) ≤ 1

Proof state:
case mpr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
a : R
hu : ¬IsUnit a
⊢ emultiplicity a x ≤ 1

Tactic:
rcases eq_or_ne a 0 with rfl | h0

Result:
case mpr.inl
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
hu : ¬IsUnit 0
⊢ emultiplicity 0 x ≤ 1

case mpr.inr
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1
a : R
hu : ¬IsUnit a
h0 : a ≠ 0
⊢ emultiplicity a x ≤ 1

Proof state:
case pos
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Tactic:
have ha := irreducible_of_normalized_factor _ hmem

Result:
case pos
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
this : Nontrivial R
h : ∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1
a : R
hmem : a ∈ normalizedFactors x
ha : Irreducible a
⊢ Multiset.count a (normalizedFactors x) ≤ 1

Proof state:
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
⊢ Squarefree x ↔ (normalizedFactors x).Nodup

Tactic:
rw [squarefree_iff_emultiplicity_le_one, Multiset.nodup_iff_count_le_one]

Result:
R : Type u_1
inst✝² : CancelCommMonoidWithZero R
inst✝¹ : UniqueFactorizationMonoid R
inst✝ : NormalizationMonoid R
x : R
x0 : x ≠ 0
⊢ (∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1) ↔ ∀ (a : R), Multiset.count a (normalizedFactors x) ≤ 1