l3lab/ntp-mathlib huggingface dataset:
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⟩
    · rintro ⟨s, hs, rfl⟩
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s ∈ powerset (Multiset.toFinset (normalizedFactors n))
⊢ (Multiset.prod s.val ∣ n ∧ n ≠ 0) ∧ Squarefree (Multiset.prod s.val)

Next tactic:
rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
⊢ (Multiset.prod s.val ∣ n ∧ n ≠ 0) ∧ Squarefree (Multiset.prod s.val)

Next tactic:
have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
⊢ Multiset.prod s.val ≠ 0

Next tactic:
rw [Ne.def, Multiset.prod_eq_zero_iff]
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
⊢ 0 ∉ s.val

Next tactic:
intro con
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
con : 0 ∈ s.val
⊢ False

Next tactic:
apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
⊢ (Multiset.prod s.val ∣ n ∧ n ≠ 0) ∧ Squarefree (Multiset.prod s.val)

Next tactic:
rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
⊢ (Multiset.prod s.val ∣ Multiset.prod (normalizedFactors n) ∧ n ≠ 0) ∧ Squarefree (Multiset.prod s.val)

Next tactic:
refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), 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
      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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
⊢ Squarefree (Multiset.prod s.val)

Next tactic:
have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
h : Multiset.Rel Associated (normalizedFactors (Multiset.prod s.val)) s.val
⊢ Squarefree (Multiset.prod s.val)

Next tactic:
rw [associated_eq_eq, Multiset.rel_eq] at h
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
      rw [associated_eq_eq, Multiset.rel_eq] at h
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
h : normalizedFactors (Multiset.prod s.val) = s.val
⊢ Squarefree (Multiset.prod s.val)

Next tactic:
rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors hs0, h]
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
      rw [associated_eq_eq, Multiset.rel_eq] at h
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors hs0, h]
      

Proof state:
case mpr.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ Multiset.dedup (normalizedFactors n)
hs0 : Multiset.prod s.val ≠ 0
h : normalizedFactors (Multiset.prod s.val) = s.val
⊢ Multiset.Nodup s.val

Next tactic:
apply s.nodup
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
      rw [associated_eq_eq, Multiset.rel_eq] at h
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors hs0, h]
      apply s.nodup
  · 

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
⊢ ∀ x ∈ (powerset (Multiset.toFinset (normalizedFactors n))).val,
    ∀ y ∈ (powerset (Multiset.toFinset (normalizedFactors n))).val, Multiset.prod x.val = Multiset.prod y.val → x = y

Next tactic:
intro x hx y hy h
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⟩
    · rintro ⟨s, hs, rfl⟩
      rw [Finset.mem_powerset, ← Finset.val_le_iff, Multiset.toFinset_val] at hs
      have hs0 : s.val.prod ≠ 0 := by
        rw [Ne.def, Multiset.prod_eq_zero_iff]
        intro con
        apply
          not_irreducible_zero
            (irreducible_of_normalized_factor 0 (Multiset.mem_dedup.1 (Multiset.mem_of_le hs con)))
      rw [(normalizedFactors_prod h0).symm.dvd_iff_dvd_right]
      refine' ⟨⟨Multiset.prod_dvd_prod_of_le (le_trans hs (Multiset.dedup_le _)), h0⟩, _⟩
      have h :=
        UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
          (fun x hx =>
            irreducible_of_normalized_factor x
              (Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
          (normalizedFactors_prod hs0)
      rw [associated_eq_eq, Multiset.rel_eq] at h
      rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors hs0, h]
      apply s.nodup
  · intro x hx y hy h
    

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
x : Finset ℕ
hx : x ∈ (powerset (Multiset.toFinset (normalizedFactors n))).val
y : Finset ℕ
hy : y ∈ (powerset (Multiset.toFinset (normalizedFactors n))).val
h : Multiset.prod x.val = Multiset.prod y.val
⊢ x = y

Next tactic:
rw [← Finset.val_inj, ← Multiset.rel_eq, ← associated_eq_eq]