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⟩
      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
    rw [← Finset.val_inj, ← Multiset.rel_eq, ← associated_eq_eq]
    rw [← Finset.mem_def, Finset.mem_powerset] at hx hy
    apply UniqueFactorizationMonoid.factors_unique _ _ (associated_iff_eq.2 h)
    · intro z hz
      apply irreducible_of_normalized_factor z
      rw [← Multiset.mem_toFinset]
      apply hx hz
    · intro z hz
      apply irreducible_of_normalized_factor z
      rw [← Multiset.mem_toFinset]
      

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
x : Finset ℕ
hx : x ⊆ Multiset.toFinset (normalizedFactors n)
y : Finset ℕ
hy : y ⊆ Multiset.toFinset (normalizedFactors n)
h : Multiset.prod x.val = Multiset.prod y.val
z : ℕ
hz : z ∈ y.val
⊢ z ∈ Multiset.toFinset (normalizedFactors ?m.146988)
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
x : Finset ℕ
hx : x ⊆ Multiset.toFinset (normalizedFactors n)
y : Finset ℕ
hy : y ⊆ Multiset.toFinset (normalizedFactors n)
h : Multiset.prod x.val = Multiset.prod y.val
z : ℕ
hz : z ∈ y.val
⊢ ℕ
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
x : Finset ℕ
hx : x ⊆ Multiset.toFinset (normalizedFactors n)
y : Finset ℕ
hy : y ⊆ Multiset.toFinset (normalizedFactors n)
h : Multiset.prod x.val = Multiset.prod y.val
z : ℕ
hz : z ∈ y.val
⊢ ℕ

Next tactic:
apply hy hz
theorem sum_divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) {α : Type*} [AddCommMonoid α]
    {f : ℕ → α} :
    ∑ i in n.divisors.filter Squarefree, f i =
      ∑ i in (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset, f i.val.prod := by
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ → α
⊢ ∑ i in filter Squarefree (divisors n), f i =
    ∑ i in powerset (Multiset.toFinset (normalizedFactors n)), f (Multiset.prod i.val)

Next tactic:
rw [Finset.sum_eq_multiset_sum, divisors_filter_squarefree h0, Multiset.map_map,
    Finset.sum_eq_multiset_sum]
theorem sum_divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) {α : Type*} [AddCommMonoid α]
    {f : ℕ → α} :
    ∑ i in n.divisors.filter Squarefree, f i =
      ∑ i in (UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset, f i.val.prod := by
  rw [Finset.sum_eq_multiset_sum, divisors_filter_squarefree h0, Multiset.map_map,
    Finset.sum_eq_multiset_sum]
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
h0 : n ≠ 0
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ → α
⊢ Multiset.sum
      (Multiset.map ((fun i => f i) ∘ fun x => Multiset.prod x.val)
        (powerset (Multiset.toFinset (normalizedFactors n))).val) =
    Multiset.sum
      (Multiset.map (fun i => f (Multiset.prod i.val)) (powerset (Multiset.toFinset (normalizedFactors n))).val)

Next tactic:
rfl
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  rw [Nat.isUnit_iff]
  by_contra hx
  refine' Nat.lt_le_asymm _ (Finset.le_max' S ((b * x) ^ 2) _)
  -- Porting note: these two goals were in the opposite order in Lean 3
  · convert lt_mul_of_one_lt_right hlts
      (one_lt_pow 2 x two_ne_zero (one_lt_iff_ne_zero_and_ne_one.mpr ⟨fun h => by simp_all, hx⟩))
      using 1
    rw [mul_pow]
  · simp_rw [hsa, Finset.mem_filter, Finset.mem_range]
    refine' ⟨lt_succ_iff.mpr (le_of_dvd hn _), _, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
⊢ Finset.Nonempty S

Next tactic:
use 1
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    

Proof state:
case h
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
⊢ 1 ∈ S

Next tactic:
have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    

Proof state:
case h
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
h1 : 0 < n ∧ ∃ x, 1 = x ^ 2
⊢ 1 ∈ S

Next tactic:
simp [h1]
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
let s := Finset.max' S hSne
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
have hs : s ∈ S := Finset.max'_mem S hSne
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
hs : s ∈ S
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
simp only [Finset.mem_filter, Finset.mem_range] at hs
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
hs :
  max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne < n + 1 ∧
    max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne ∣ n ∧
      ∃ x, max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = x ^ 2
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  

Proof state:
case intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
hn : 0 < n
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a : ℕ
hsa : n = max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne * a
b : ℕ
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
rw [hsa] at hn
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  

Proof state:
case intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a : ℕ
hn : 0 < max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne * a
hsa : n = max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne * a
b : ℕ
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  

Proof state:
case intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a : ℕ
hn : 0 < max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne * a
hsa : n = max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne * a
b : ℕ
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne
hlta : 0 < a
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
rw [hsb] at hsa hn hlts
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  

Proof state:
case intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
⊢ ∃ a b, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a

Next tactic:
refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  

Proof state:
case intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
⊢ Squarefree a

Next tactic:
rintro x ⟨y, hy⟩
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
⊢ IsUnit x

Next tactic:
rw [Nat.isUnit_iff]
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  rw [Nat.isUnit_iff]
  

Proof state:
case intro.intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
⊢ x = 1

Next tactic:
by_contra hx
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  rw [Nat.isUnit_iff]
  by_contra hx
  

Proof state:
case intro.intro.intro.intro.intro.intro
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ False

Next tactic:
refine' Nat.lt_le_asymm _ (Finset.le_max' S ((b * x) ^ 2) _)
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  rw [Nat.isUnit_iff]
  by_contra hx
  refine' Nat.lt_le_asymm _ (Finset.le_max' S ((b * x) ^ 2) _)
  -- Porting note: these two goals were in the opposite order in Lean 3
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ max' S (_ : ∃ x, x ∈ S) < (b * x) ^ 2

Next tactic:
convert lt_mul_of_one_lt_right hlts
      (one_lt_pow 2 x two_ne_zero (one_lt_iff_ne_zero_and_ne_one.mpr ⟨fun h => by simp_all, hx⟩))
      using 1
theorem sq_mul_squarefree_of_pos {n : ℕ} (hn : 0 < n) :
    ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ b ^ 2 * a = n ∧ Squarefree a := by
  classical -- Porting note: This line is not needed in Lean 3
  set S := (Finset.range (n + 1)).filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2)
  have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedCommSemiring.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine' ⟨a, b, hlta, (pow_pos_iff zero_lt_two).mp hlts, hsa.symm, _⟩
  rintro x ⟨y, hy⟩
  rw [Nat.isUnit_iff]
  by_contra hx
  refine' Nat.lt_le_asymm _ (Finset.le_max' S ((b * x) ^ 2) _)
  -- Porting note: these two goals were in the opposite order in Lean 3
  · convert lt_mul_of_one_lt_right hlts
      (one_lt_pow 2 x two_ne_zero (one_lt_iff_ne_zero_and_ne_one.mpr ⟨fun h => by 

Proof state:
s✝ : Finset ℕ
m n✝ p n : ℕ
S : Finset ℕ := filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))
hSne : Finset.Nonempty S
s : ℕ := max' S hSne
a b : ℕ
hn : 0 < b ^ 2 * a
hsa : n = b ^ 2 * a
hsb : max' (filter (fun s => s ∣ n ∧ ∃ x, s = x ^ 2) (range (n + 1))) hSne = b ^ 2
hlts : 0 < b ^ 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
h : x = 0
⊢ False

Next tactic:
simp_all