awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
h0 : n ≠ 0
α : Type u_1
inst : AddCommMonoid α
f : ℕ → α
⊢ Eq.{u_1 + 1} (α := α) (∑ d ∈ n.divisors with Squarefree d, f d)
    (∑ i ∈ (normalizedFactors n).toFinset.powerset, f i.val.prod)
have rw : (Multiset.map (f ∘ fun (x : Finset ℕ) ↦ x.val.prod) (normalizedFactors n).toFinset.powerset.val).sum =
  Multiset.sum (Multiset.map (fun (i : Finset ℕ) ↦ f i.val.prod) (normalizedFactors n).toFinset.powerset.val)
Tactic: rw [Finset.sum_eq_multiset_sum, divisors_filter_squarefree h0, Multiset.map_map,
    Finset.sum_eq_multiset_sum]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
⊢ S.Nonempty
have h : 1 ∈ S
Tactic: use 1
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
⊢ 1 ∈ S
have h1 : 0 < n ∧ ∃ (x : ℕ), 1 = HPow.hPow (β := ℕ) x 2
Tactic: have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hSne : S.Nonempty
Tactic: have hSne : S.Nonempty := by
    use 1
    have h1 : 0 < n ∧ ∃ x : ℕ, 1 = x ^ 2 := ⟨hn, ⟨1, (one_pow 2).symm⟩⟩
    simp [S, h1]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hs : s ∈ S
Tactic: have hs : s ∈ S := Finset.max'_mem S hSne
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
hs : s ∈ S
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hs₁ : s < n + 1 ∧ s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2
Tactic: simp only [S, Finset.mem_filter, Finset.mem_range] at hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
hs : s < n + 1 ∧ s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have obtain : ∀ (a : ℕ) (hsa : n = s * a) (b : ℕ) (hsb : s = HPow.hPow (β := ℕ) b 2),
  ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have rw : ∀ (hn : 0 < s * a) (hsa : n = s * a) (b : ℕ) (hsb : s = HPow.hPow (β := ℕ) b 2),
  ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: rw [hsa] at hn
Cleared hypotheses: hn, hsa, b, hsb
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hn : 0 < s * a
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hlts : 0 < s
Tactic: obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hn : 0 < s * a
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < s
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hlta : 0 < a
Tactic: obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hn : 0 < s * a
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < s
hlta : 0 < a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hn₁ : 0 < HPow.hPow (β := ℕ) b 2 * a
Tactic: rw [hsb] at hsa hn hlts
Cleared hypotheses: hn, hsa, hsb, hlts, hlta
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hn : 0 < s * a
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < s
hlta : 0 < a
hn₁ : 0 < HPow.hPow (β := ℕ) b 2 * a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hsa₁ : n = HPow.hPow (β := ℕ) b 2 * a
Tactic: rw [hsb] at hsa hn hlts
Cleared hypotheses: hn, hsa, hsb, hlts, hlta
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (hypothesis)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a : ℕ
hn : 0 < s * a
hsa : n = s * a
b : ℕ
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < s
hlta : 0 < a
hn₁ : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa₁ : n = HPow.hPow (β := ℕ) b 2 * a
hsb₁ : s = HPow.hPow (β := ℕ) b 2
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have hlts₁ : 0 < HPow.hPow (β := ℕ) b 2
Tactic: rw [hsb] at hsa hn hlts
Cleared hypotheses: hn, hsa, hsb, hlts, hlta
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have refine : Squarefree a
Tactic: refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).mp hlts, hsa.symm, ?_⟩
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
⊢ IsUnit x
have rw : x = 1
Tactic: rw [Nat.isUnit_iff]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
⊢ x = 1
have by_contra : ∀ (hx : ¬x = 1), False
Tactic: by_contra hx
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ False
have refine_1 : S.max' sorry < HPow.hPow (β := ℕ) (b * x) 2
Tactic: refine Nat.lt_le_asymm ?_ (Finset.le_max' S ((b * x) ^ 2) ?_)
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
refine_1 : S.max' ⋯ < HPow.hPow (β := ℕ) (b * x) 2
⊢ False
have refine_2 : HPow.hPow (β := ℕ) (b * x) 2 ∈ S
Tactic: refine Nat.lt_le_asymm ?_ (Finset.le_max' S ((b * x) ^ 2) ?_)
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ S.max' ⋯ < HPow.hPow (β := ℕ) (b * x) 2
have h : HPow.hPow (β := ℕ) (b * x) 2 = HPow.hPow (β := ℕ) b 2 * HPow.hPow (β := ℕ) x 2
Tactic: convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow two_ne_zero (one_lt_iff_ne_zero_and_ne_one.mpr ⟨fun h => by simp_all, hx⟩))
    using 1
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ HPow.hPow (β := ℕ) (b * x) 2 = HPow.hPow (β := ℕ) b 2 * HPow.hPow (β := ℕ) x 2
have h : HPow.hPow (β := ℕ) b 2 * HPow.hPow (β := ℕ) x 2 = HPow.hPow (β := ℕ) b 2 * HPow.hPow (β := ℕ) x 2
Tactic: rw [mul_pow]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ HPow.hPow (β := ℕ) (b * x) 2 ∈ S
have refine_2 : HPow.hPow (β := ℕ) (b * x) 2 < HPow.hPow (β := ℕ) b 2 * a + 1 ∧
  HPow.hPow (β := ℕ) (b * x) 2 ∣ HPow.hPow (β := ℕ) b 2 * a ∧
    ∃ (x_1 : ℕ), HPow.hPow (β := ℕ) (b * x) 2 = HPow.hPow (β := ℕ) x_1 2
Tactic: simp_rw [S, hsa, Finset.mem_filter, Finset.mem_range]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ HPow.hPow (β := ℕ) (b * x) 2 < HPow.hPow (β := ℕ) b 2 * a + 1 ∧
    HPow.hPow (β := ℕ) (b * x) 2 ∣ HPow.hPow (β := ℕ) b 2 * a ∧
      ∃ (x_1 : ℕ), HPow.hPow (β := ℕ) (b * x) 2 = HPow.hPow (β := ℕ) x_1 2
have refine_2 : HPow.hPow (β := ℕ) (b * x) 2 ∣ HPow.hPow (β := ℕ) b 2 * a
Tactic: refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ HPow.hPow (β := ℕ) (b * x) 2 ∣ HPow.hPow (β := ℕ) b 2 * a
have h : HPow.hPow (β := ℕ) b 2 * a = HPow.hPow (β := ℕ) (b * x) 2 * y
Tactic: use y
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
S : Finset ℕ := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2}
hSne : S.Nonempty
s : ℕ := S.max' hSne
a b : ℕ
hn : 0 < HPow.hPow (β := ℕ) b 2 * a
hsa : n = HPow.hPow (β := ℕ) b 2 * a
hsb : s = HPow.hPow (β := ℕ) b 2
hlts : 0 < HPow.hPow (β := ℕ) b 2
hlta : 0 < a
x y : ℕ
hy : a = x * x * y
hx : ¬x = 1
⊢ HPow.hPow (β := ℕ) b 2 * a = HPow.hPow (β := ℕ) (b * x) 2 * y
have h : HPow.hPow (β := ℕ) b 2 * (x * x * y) = HPow.hPow (β := ℕ) (b * x) 2 * y
Tactic: rw [hy]
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
set set₁ : ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have set₂ : let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
∀ (hSne : S.Nonempty), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: classical
  set S := {s ∈ range (n + 1) | 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 [S, h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [S, Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).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) ?_)
  · convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow 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 [S, hsa, Finset.mem_filter, Finset.mem_range]
    refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
set set₁ : ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₂ set₃ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have set₄ : let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
∀ (hSne : S.Nonempty),
  let s := S.max' hSne;
  ∀ (hs : s ∈ S), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: classical
  set S := {s ∈ range (n + 1) | 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 [S, h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [S, Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).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) ?_)
  · convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow 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 [S, hsa, Finset.mem_filter, Finset.mem_range]
    refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
set set₁ : ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₂ set₃ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₄ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (hs : s ∈ S), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have set₅ : let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
∀ (hSne : S.Nonempty),
  let s := S.max' hSne;
  ∀ (hs : s < n + 1 ∧ s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2),
    ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: classical
  set S := {s ∈ range (n + 1) | 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 [S, h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [S, Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).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) ?_)
  · convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow 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 [S, hsa, Finset.mem_filter, Finset.mem_range]
    refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
set set₁ : ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₂ set₃ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₄ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (hs : s ∈ S), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₅ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (hs : s < n + 1 ∧ s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2),
      ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have set₆ : let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
∀ (hSne : S.Nonempty),
  let s := S.max' hSne;
  ∀ (a : ℕ) (hsa : n = s * a) (b : ℕ) (hsb : s = HPow.hPow (β := ℕ) b 2),
    ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: classical
  set S := {s ∈ range (n + 1) | 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 [S, h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [S, Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).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) ?_)
  · convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow 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 [S, hsa, Finset.mem_filter, Finset.mem_range]
    refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
-- Lean 4 proof step from Mathlib.Data.Nat.Squarefree (goal)
Goal:
n : ℕ
hn : 0 < n
set set₁ : ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₂ set₃ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₄ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (hs : s ∈ S), ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₅ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (hs : s < n + 1 ∧ s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2),
      ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
set₆ :
  let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
  ∀ (hSne : S.Nonempty),
    let s := S.max' hSne;
    ∀ (a : ℕ) (hsa : n = s * a) (b : ℕ) (hsb : s = HPow.hPow (β := ℕ) b 2),
      ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
⊢ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
have set₇ : let S := {s ∈ range (n + 1) | s ∣ n ∧ ∃ (x : ℕ), s = HPow.hPow (β := ℕ) x 2};
∀ (hSne : S.Nonempty),
  let s := S.max' hSne;
  ∀ (a : ℕ) (hn : 0 < s * a) (hsa : n = s * a) (b : ℕ) (hsb : s = HPow.hPow (β := ℕ) b 2),
    ∃ (a : ℕ) (b : ℕ), 0 < a ∧ 0 < b ∧ HPow.hPow (β := ℕ) b 2 * a = n ∧ Squarefree a
Tactic: classical
  set S := {s ∈ range (n + 1) | 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 [S, h1]
  let s := Finset.max' S hSne
  have hs : s ∈ S := Finset.max'_mem S hSne
  simp only [S, Finset.mem_filter, Finset.mem_range] at hs
  obtain ⟨-, ⟨a, hsa⟩, ⟨b, hsb⟩⟩ := hs
  rw [hsa] at hn
  obtain ⟨hlts, hlta⟩ := CanonicallyOrderedAdd.mul_pos.mp hn
  rw [hsb] at hsa hn hlts
  refine ⟨a, b, hlta, (pow_pos_iff two_ne_zero).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) ?_)
  · convert!
      lt_mul_of_one_lt_right hlts
        (one_lt_pow 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 [S, hsa, Finset.mem_filter, Finset.mem_range]
    refine ⟨Nat.lt_succ_iff.mpr (le_of_dvd hn ?_), ?_, ⟨b * x, rfl⟩⟩ <;> use y <;> rw [hy] <;> ring
Cleared hypotheses: hn