l3lab/ntp-mathlib huggingface dataset:
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · 

Proof state:
case neg
s : Finset ℕ
m n✝ p n : ℕ
d2 : ¬2 ∣ n
⊢ MinSqFacProp n (minSqFacAux n 3)

Next tactic:
rcases Nat.eq_zero_or_pos n with n0 | n0
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · 

Proof state:
case neg.inl
s : Finset ℕ
m n✝ p n : ℕ
d2 : ¬2 ∣ n
n0 : n = 0
⊢ MinSqFacProp n (minSqFacAux n 3)

Next tactic:
subst n0
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      

Proof state:
case neg.inl
s : Finset ℕ
m n p : ℕ
d2 : ¬2 ∣ 0
⊢ MinSqFacProp 0 (minSqFacAux 0 3)

Next tactic:
cases d2 (by decide)
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d2 (by 

Proof state:
s : Finset ℕ
m n p : ℕ
d2 : ¬2 ∣ 0
⊢ 2 ∣ 0

Next tactic:
decide
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d2 (by decide)
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
⊢ MinSqFacProp n (minSqFacAux n 3)

Next tactic:
refine' minSqFacAux_has_prop _ n0 0 rfl _
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d2 (by decide)
    refine' minSqFacAux_has_prop _ n0 0 rfl _
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
⊢ ∀ (m : ℕ), Prime m → m ∣ n → 3 ≤ m

Next tactic:
refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d2 (by decide)
    refine' minSqFacAux_has_prop _ n0 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p✝ n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
p : ℕ
pp : Prime p
dp : p ∣ n
⊢ 2 ≠ p

Next tactic:
rintro rfl
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => pp.two_le⟩
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d4 (by decide)
    refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
    refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    contradiction
  · rcases Nat.eq_zero_or_pos n with n0 | n0
    · subst n0
      cases d2 (by decide)
    refine' minSqFacAux_has_prop _ n0 0 rfl _
    refine' fun p pp dp => succ_le_of_lt (lt_of_le_of_ne pp.two_le _)
    rintro rfl
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : ¬2 ∣ n
n0 : n > 0
pp : Prime 2
dp : 2 ∣ n
⊢ False

Next tactic:
contradiction
theorem minSqFac_prime {n d : ℕ} (h : n.minSqFac = some d) : Prime d := by
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
⊢ Prime d

Next tactic:
have := minSqFac_has_prop n
theorem minSqFac_prime {n d : ℕ} (h : n.minSqFac = some d) : Prime d := by
  have := minSqFac_has_prop n
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
this : MinSqFacProp n (minSqFac n)
⊢ Prime d

Next tactic:
rw [h] at this
theorem minSqFac_prime {n d : ℕ} (h : n.minSqFac = some d) : Prime d := by
  have := minSqFac_has_prop n
  rw [h] at this
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
this : MinSqFacProp n (some d)
⊢ Prime d

Next tactic:
exact this.1
theorem minSqFac_dvd {n d : ℕ} (h : n.minSqFac = some d) : d * d ∣ n := by
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
⊢ d * d ∣ n

Next tactic:
have := minSqFac_has_prop n
theorem minSqFac_dvd {n d : ℕ} (h : n.minSqFac = some d) : d * d ∣ n := by
  have := minSqFac_has_prop n
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
this : MinSqFacProp n (minSqFac n)
⊢ d * d ∣ n

Next tactic:
rw [h] at this
theorem minSqFac_dvd {n d : ℕ} (h : n.minSqFac = some d) : d * d ∣ n := by
  have := minSqFac_has_prop n
  rw [h] at this
  

Proof state:
s : Finset ℕ
m n✝ p n d : ℕ
h : minSqFac n = some d
this : MinSqFacProp n (some d)
⊢ d * d ∣ n

Next tactic:
exact this.2.1
theorem minSqFac_le_of_dvd {n d : ℕ} (h : n.minSqFac = some d) {m} (m2 : 2 ≤ m) (md : m * m ∣ n) :
    d ≤ m := by
  

Proof state:
s : Finset ℕ
m✝ n✝ p n d : ℕ
h : minSqFac n = some d
m : ℕ
m2 : 2 ≤ m
md : m * m ∣ n
⊢ d ≤ m

Next tactic:
have := minSqFac_has_prop n
theorem minSqFac_le_of_dvd {n d : ℕ} (h : n.minSqFac = some d) {m} (m2 : 2 ≤ m) (md : m * m ∣ n) :
    d ≤ m := by
  have := minSqFac_has_prop n; 

Proof state:
s : Finset ℕ
m✝ n✝ p n d : ℕ
h : minSqFac n = some d
m : ℕ
m2 : 2 ≤ m
md : m * m ∣ n
this : MinSqFacProp n (minSqFac n)
⊢ d ≤ m

Next tactic:
rw [h] at this
theorem minSqFac_le_of_dvd {n d : ℕ} (h : n.minSqFac = some d) {m} (m2 : 2 ≤ m) (md : m * m ∣ n) :
    d ≤ m := by
  have := minSqFac_has_prop n; rw [h] at this
  

Proof state:
s : Finset ℕ
m✝ n✝ p n d : ℕ
h : minSqFac n = some d
m : ℕ
m2 : 2 ≤ m
md : m * m ∣ n
this : MinSqFacProp n (some d)
⊢ d ≤ m

Next tactic:
have fd := minFac_dvd m
theorem minSqFac_le_of_dvd {n d : ℕ} (h : n.minSqFac = some d) {m} (m2 : 2 ≤ m) (md : m * m ∣ n) :
    d ≤ m := by
  have := minSqFac_has_prop n; rw [h] at this
  have fd := minFac_dvd m
  

Proof state:
s : Finset ℕ
m✝ n✝ p n d : ℕ
h : minSqFac n = some d
m : ℕ
m2 : 2 ≤ m
md : m * m ∣ n
this : MinSqFacProp n (some d)
fd : minFac m ∣ m
⊢ d ≤ m

Next tactic:
exact
    le_trans (this.2.2 _ (minFac_prime <| ne_of_gt m2) (dvd_trans (mul_dvd_mul fd fd) md))
      (minFac_le <| lt_of_lt_of_le (by decide) m2)
theorem minSqFac_le_of_dvd {n d : ℕ} (h : n.minSqFac = some d) {m} (m2 : 2 ≤ m) (md : m * m ∣ n) :
    d ≤ m := by
  have := minSqFac_has_prop n; rw [h] at this
  have fd := minFac_dvd m
  exact
    le_trans (this.2.2 _ (minFac_prime <| ne_of_gt m2) (dvd_trans (mul_dvd_mul fd fd) md))
      (minFac_le <| lt_of_lt_of_le (by 

Proof state:
s : Finset ℕ
m✝ n✝ p n d : ℕ
h : minSqFac n = some d
m : ℕ
m2 : 2 ≤ m
md : m * m ∣ n
this : MinSqFacProp n (some d)
fd : minFac m ∣ m
⊢ 0 < 2

Next tactic:
decide
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
⊢ Squarefree n ↔ minSqFac n = none

Next tactic:
have := minSqFac_has_prop n
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
⊢ Squarefree n ↔ minSqFac n = none

Next tactic:
constructor
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> 

Proof state:
case mp
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
⊢ Squarefree n → minSqFac n = none

Next tactic:
intro H
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> 

Proof state:
case mpr
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
⊢ minSqFac n = none → Squarefree n

Next tactic:
intro H
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> intro H
  · 

Proof state:
case mp
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
H : Squarefree n
⊢ minSqFac n = none

Next tactic:
cases' e : n.minSqFac with d
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> intro H
  · cases' e : n.minSqFac with d
    · 

Proof state:
case mp.none
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
H : Squarefree n
e : minSqFac n = none
⊢ none = none

Next tactic:
rfl
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> intro H
  · cases' e : n.minSqFac with d
    · rfl
    

Proof state:
case mp.some
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
H : Squarefree n
d : ℕ
e : minSqFac n = some d
⊢ some d = none

Next tactic:
rw [e] at this
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> intro H
  · cases' e : n.minSqFac with d
    · rfl
    rw [e] at this
    

Proof state:
case mp.some
s : Finset ℕ
m n✝ p n : ℕ
H : Squarefree n
d : ℕ
this : MinSqFacProp n (some d)
e : minSqFac n = some d
⊢ some d = none

Next tactic:
cases squarefree_iff_prime_squarefree.1 H _ this.1 this.2.1
theorem squarefree_iff_minSqFac {n : ℕ} : Squarefree n ↔ n.minSqFac = none := by
  have := minSqFac_has_prop n
  constructor <;> intro H
  · cases' e : n.minSqFac with d
    · rfl
    rw [e] at this
    cases squarefree_iff_prime_squarefree.1 H _ this.1 this.2.1
  · 

Proof state:
case mpr
s : Finset ℕ
m n✝ p n : ℕ
this : MinSqFacProp n (minSqFac n)
H : minSqFac n = none
⊢ Squarefree n

Next tactic:
rwa [H] at this
theorem squarefree_two : Squarefree 2 := by
  

Proof state:
s : Finset ℕ
m n p : ℕ
⊢ Squarefree 2

Next tactic:
rw [squarefree_iff_nodup_factors]
theorem squarefree_two : Squarefree 2 := by
  rw [squarefree_iff_nodup_factors] <;> 

Proof state:
s : Finset ℕ
m n p : ℕ
⊢ List.Nodup (factors 2)

Next tactic:
decide
theorem squarefree_two : Squarefree 2 := by
  rw [squarefree_iff_nodup_factors] <;> 

Proof state:
s : Finset ℕ
m n p : ℕ
⊢ 2 ≠ 0

Next tactic:
decide
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
⊢ (filter Squarefree (divisors n)).val =
    Multiset.map (fun x => Multiset.prod x.val) (powerset (Multiset.toFinset (normalizedFactors n))).val

Next tactic:
rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · 

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
⊢ ∀ (a : ℕ),
    a ∈ (filter Squarefree (divisors n)).val ↔
      a ∈ Multiset.map (fun x => Multiset.prod x.val) (powerset (Multiset.toFinset (normalizedFactors n))).val

Next tactic:
intro a
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
a : ℕ
⊢ a ∈ (filter Squarefree (divisors n)).val ↔
    a ∈ Multiset.map (fun x => Multiset.prod x.val) (powerset (Multiset.toFinset (normalizedFactors n))).val

Next tactic:
simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
a : ℕ
⊢ (a ∣ n ∧ n ≠ 0) ∧ Squarefree a ↔ ∃ a_1 ∈ powerset (Multiset.toFinset (normalizedFactors n)), Multiset.prod a_1.val = a

Next tactic:
constructor
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · 

Proof state:
case mp
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
a : ℕ
⊢ (a ∣ n ∧ n ≠ 0) ∧ Squarefree a → ∃ a_2 ∈ powerset (Multiset.toFinset (normalizedFactors n)), Multiset.prod a_2.val = a

Next tactic:
rintro ⟨⟨an, h0⟩, hsq⟩
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      

Proof state:
case mp.intro.intro
s : Finset ℕ
m n✝ p n : ℕ
h0✝ : n ≠ 0
a : ℕ
hsq : Squarefree a
an : a ∣ n
h0 : n ≠ 0
⊢ ∃ a_1 ∈ powerset (Multiset.toFinset (normalizedFactors n)), Multiset.prod a_1.val = a

Next tactic:
use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      

Proof state:
case h
s : Finset ℕ
m n✝ p n : ℕ
h0✝ : n ≠ 0
a : ℕ
hsq : Squarefree a
an : a ∣ n
h0 : n ≠ 0
⊢ Multiset.toFinset (normalizedFactors a) ∈ powerset (Multiset.toFinset (normalizedFactors n)) ∧
    Multiset.prod (Multiset.toFinset (normalizedFactors a)).val = a

Next tactic:
simp only [id.def, Finset.mem_powerset]
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      

Proof state:
case h
s : Finset ℕ
m n✝ p n : ℕ
h0✝ : n ≠ 0
a : ℕ
hsq : Squarefree a
an : a ∣ n
h0 : n ≠ 0
⊢ Multiset.toFinset (normalizedFactors a) ⊆ Multiset.toFinset (normalizedFactors n) ∧
    Multiset.prod (Multiset.toFinset (normalizedFactors a)).val = a

Next tactic:
rcases an with ⟨b, rfl⟩
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      rcases an with ⟨b, rfl⟩
      

Proof state:
case h.intro
s : Finset ℕ
m n p a : ℕ
hsq : Squarefree a
b : ℕ
h0✝ h0 : a * b ≠ 0
⊢ Multiset.toFinset (normalizedFactors a) ⊆ Multiset.toFinset (normalizedFactors (a * b)) ∧
    Multiset.prod (Multiset.toFinset (normalizedFactors a)).val = a

Next tactic:
rw [mul_ne_zero_iff] at h0
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      rcases an with ⟨b, rfl⟩
      rw [mul_ne_zero_iff] at h0
      

Proof state:
case h.intro
s : Finset ℕ
m n p a : ℕ
hsq : Squarefree a
b : ℕ
h0✝ : a * b ≠ 0
h0 : a ≠ 0 ∧ b ≠ 0
⊢ Multiset.toFinset (normalizedFactors a) ⊆ Multiset.toFinset (normalizedFactors (a * b)) ∧
    Multiset.prod (Multiset.toFinset (normalizedFactors a)).val = a

Next tactic:
rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0.1] at hsq
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      rcases an with ⟨b, rfl⟩
      rw [mul_ne_zero_iff] at h0
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0.1] at hsq
      

Proof state:
case h.intro
s : Finset ℕ
m n p a : ℕ
hsq : Multiset.Nodup (normalizedFactors a)
b : ℕ
h0✝ : a * b ≠ 0
h0 : a ≠ 0 ∧ b ≠ 0
⊢ Multiset.toFinset (normalizedFactors a) ⊆ Multiset.toFinset (normalizedFactors (a * b)) ∧
    Multiset.prod (Multiset.toFinset (normalizedFactors a)).val = a

Next tactic:
rw [Multiset.toFinset_subset, Multiset.toFinset_val, hsq.dedup, ← associated_iff_eq,
        normalizedFactors_mul h0.1 h0.2]
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      rcases an with ⟨b, rfl⟩
      rw [mul_ne_zero_iff] at h0
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0.1] at hsq
      rw [Multiset.toFinset_subset, Multiset.toFinset_val, hsq.dedup, ← associated_iff_eq,
        normalizedFactors_mul h0.1 h0.2]
      

Proof state:
case h.intro
s : Finset ℕ
m n p a : ℕ
hsq : Multiset.Nodup (normalizedFactors a)
b : ℕ
h0✝ : a * b ≠ 0
h0 : a ≠ 0 ∧ b ≠ 0
⊢ normalizedFactors a ⊆ normalizedFactors a + normalizedFactors b ∧ Associated (Multiset.prod (normalizedFactors a)) a

Next tactic:
exact ⟨Multiset.subset_of_le (Multiset.le_add_right _ _), normalizedFactors_prod h0.1⟩
theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
    (n.divisors.filter Squarefree).val =
      (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
        x.val.prod := by
  rw [(Finset.nodup _).ext ((Finset.nodup _).map_on _)]
  · intro a
    simp only [Multiset.mem_filter, id.def, Multiset.mem_map, Finset.filter_val, ← Finset.mem_def,
      mem_divisors]
    constructor
    · rintro ⟨⟨an, h0⟩, hsq⟩
      use (UniqueFactorizationMonoid.normalizedFactors a).toFinset
      simp only [id.def, Finset.mem_powerset]
      rcases an with ⟨b, rfl⟩
      rw [mul_ne_zero_iff] at h0
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0.1] at hsq
      rw [Multiset.toFinset_subset, Multiset.toFinset_val, hsq.dedup, ← associated_iff_eq,
        normalizedFactors_mul h0.1 h0.2]
      exact ⟨Multiset.subset_of_le (Multiset.le_add_right _ _), normalizedFactors_prod h0.1⟩
    · 

Proof state:
case mpr
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
a : ℕ
⊢ (∃ a_1 ∈ powerset (Multiset.toFinset (normalizedFactors n)), Multiset.prod a_1.val = a) →
    (a ∣ n ∧ n ≠ 0) ∧ Squarefree a

Next tactic:
rintro ⟨s, hs, rfl⟩