l3lab/ntp-mathlib huggingface dataset:
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    

Proof state:
case inr
s : Finset ℕ
m✝ n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
n' : ℕ
nd' : n' ∣ n
nk : ¬k ∣ n'
hn' : n' ≤ n
this✝ : sqrt n' - k < sqrt n + 2 - k
m : ℕ
m2 : Prime m
ml : k < m
me : succ k = m
d : 2 * (i + 2) ∣ n'
this : 2 * i + 3 ≤ 2
⊢ False

Next tactic:
exact absurd this (by linarith)
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by 

Proof state:
s : Finset ℕ
m✝ n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
n' : ℕ
nd' : n' ∣ n
nk : ¬k ∣ n'
hn' : n' ≤ n
this✝ : sqrt n' - k < sqrt n + 2 - k
m : ℕ
m2 : Prime m
ml : k < m
me : succ k = m
d : 2 * (i + 2) ∣ n'
this : 2 * i + 3 ≤ 2
⊢ ¬2 * i + 3 ≤ 2

Next tactic:
linarith
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  

Proof state:
case neg
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
⊢ MinSqFacProp n (if k ∣ n then if k ∣ n / k then some k else minSqFacAux (n / k) (k + 2) else minSqFacAux n (k + 2))

Next tactic:
have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    

Proof state:
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
⊢ k ∣ n → Prime k

Next tactic:
refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    

Proof state:
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
dk : k ∣ n
⊢ k ≤ minFac k

Next tactic:
exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
  

Proof state:
case neg
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
pk : k ∣ n → Prime k
⊢ MinSqFacProp n (if k ∣ n then if k ∣ n / k then some k else minSqFacAux (n / k) (k + 2) else minSqFacAux n (k + 2))

Next tactic:
split_ifs with dk dkk
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
  split_ifs with dk dkk
  · 

Proof state:
case pos
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
pk : k ∣ n → Prime k
dk : k ∣ n
dkk : k ∣ n / k
⊢ MinSqFacProp n (some k)

Next tactic:
exact ⟨pk dk, (Nat.dvd_div_iff dk).1 dkk, fun p pp d => ih p pp (dvd_trans ⟨_, rfl⟩ d)⟩
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
  split_ifs with dk dkk
  · exact ⟨pk dk, (Nat.dvd_div_iff dk).1 dkk, fun p pp d => ih p pp (dvd_trans ⟨_, rfl⟩ d)⟩
  · 

Proof state:
case neg
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
pk : k ∣ n → Prime k
dk : k ∣ n
dkk : ¬k ∣ n / k
⊢ MinSqFacProp n (minSqFacAux (n / k) (k + 2))

Next tactic:
specialize IH (n / k) (div_dvd_of_dvd dk) dkk
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
  split_ifs with dk dkk
  · exact ⟨pk dk, (Nat.dvd_div_iff dk).1 dkk, fun p pp d => ih p pp (dvd_trans ⟨_, rfl⟩ d)⟩
  · specialize IH (n / k) (div_dvd_of_dvd dk) dkk
    

Proof state:
case neg
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
pk : k ∣ n → Prime k
dk : k ∣ n
dkk : ¬k ∣ n / k
IH : MinSqFacProp (n / k) (minSqFacAux (n / k) (k + 2))
⊢ MinSqFacProp n (minSqFacAux (n / k) (k + 2))

Next tactic:
exact minSqFacProp_div _ (pk dk) dk (mt (Nat.dvd_div_iff dk).2 dkk) IH
theorem minSqFacAux_has_prop {n : ℕ} (k) (n0 : 0 < n) (i) (e : k = 2 * i + 3)
    (ih : ∀ m, Prime m → m ∣ n → k ≤ m) : MinSqFacProp n (minSqFacAux n k) := by
  rw [minSqFacAux]
  by_cases h : n < k * k <;> simp [h]
  · refine' squarefree_iff_prime_squarefree.2 fun p pp d => _
    have := ih p pp (dvd_trans ⟨_, rfl⟩ d)
    have := Nat.mul_le_mul this this
    exact not_le_of_lt h (le_trans this (le_of_dvd n0 d))
  have k2 : 2 ≤ k := by
    subst e
    linarith
  have k0 : 0 < k := lt_of_lt_of_le (by decide) k2
  have IH : ∀ n', n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (n'.minSqFacAux (k + 2)) := by
    intro n' nd' nk
    have hn' := le_of_dvd n0 nd'
    refine'
      have : Nat.sqrt n' - k < Nat.sqrt n + 2 - k :=
        lt_of_le_of_lt (Nat.sub_le_sub_right (Nat.sqrt_le_sqrt hn') _) (Nat.minFac_lemma n k h)
      @minSqFacAux_has_prop n' (k + 2) (pos_of_dvd_of_pos nd' n0) (i + 1)
        (by simp [e, left_distrib]) fun m m2 d => _
    rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
    · subst me
      contradiction
    apply (Nat.eq_or_lt_of_le ml).resolve_left
    intro me
    rw [← me, e] at d
    change 2 * (i + 2) ∣ n' at d
    have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
    rw [e] at this
    exact absurd this (by linarith)
  have pk : k ∣ n → Prime k := by
    refine' fun dk => prime_def_minFac.2 ⟨k2, le_antisymm (minFac_le k0) _⟩
    exact ih _ (minFac_prime (ne_of_gt k2)) (dvd_trans (minFac_dvd _) dk)
  split_ifs with dk dkk
  · exact ⟨pk dk, (Nat.dvd_div_iff dk).1 dkk, fun p pp d => ih p pp (dvd_trans ⟨_, rfl⟩ d)⟩
  · specialize IH (n / k) (div_dvd_of_dvd dk) dkk
    exact minSqFacProp_div _ (pk dk) dk (mt (Nat.dvd_div_iff dk).2 dkk) IH
  · 

Proof state:
case neg
s : Finset ℕ
m n✝ p n k : ℕ
n0 : 0 < n
i : ℕ
e : k = 2 * i + 3
ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m
h : ¬n < k * k
k2 : 2 ≤ k
k0 : 0 < k
IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))
pk : k ∣ n → Prime k
dk : ¬k ∣ n
⊢ MinSqFacProp n (minSqFacAux n (k + 2))

Next tactic:
exact IH n (dvd_refl _) dk
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
⊢ MinSqFacProp n (minSqFac n)

Next tactic:
dsimp only [minSqFac]
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; 

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
⊢ MinSqFacProp n (if 2 ∣ n then if 2 ∣ n / 2 then some 2 else minSqFacAux (n / 2) 3 else minSqFacAux n 3)

Next tactic:
split_ifs with d2 d4
theorem minSqFac_has_prop (n : ℕ) : MinSqFacProp n (minSqFac n) := by
  dsimp only [minSqFac]; split_ifs with d2 d4
  · 

Proof state:
case pos
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : 2 ∣ n / 2
⊢ MinSqFacProp n (some 2)

Next tactic:
exact ⟨prime_two, (dvd_div_iff d2).1 d4, fun p pp _ => 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⟩
  · 

Proof state:
case neg
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
⊢ MinSqFacProp n (minSqFacAux (n / 2) 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
    · 

Proof state:
case neg.inl
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n = 0
⊢ MinSqFacProp n (minSqFacAux (n / 2) 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
      

Proof state:
case neg.inl
s : Finset ℕ
m n p : ℕ
d2 : 2 ∣ 0
d4 : ¬2 ∣ 0 / 2
⊢ MinSqFacProp 0 (minSqFacAux (0 / 2) 3)

Next tactic:
cases d4 (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 

Proof state:
s : Finset ℕ
m n p : ℕ
d2 : 2 ∣ 0
d4 : ¬2 ∣ 0 / 2
⊢ 2 ∣ 0 / 2

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)
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
⊢ MinSqFacProp n (minSqFacAux (n / 2) 3)

Next tactic:
refine' minSqFacProp_div _ prime_two d2 (mt (dvd_div_iff d2).2 d4) _
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) _
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
⊢ MinSqFacProp (n / 2) (minSqFacAux (n / 2) 3)

Next tactic:
refine' minSqFacAux_has_prop 3 (Nat.div_pos (le_of_dvd n0 d2) (by decide)) 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 

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
⊢ 0 < 2

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 _
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
⊢ ∀ (m : ℕ), Prime m → m ∣ n / 2 → 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 _)
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p✝ n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
p : ℕ
pp : Prime p
dp : p ∣ n / 2
⊢ 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
    

Proof state:
case neg.inr
s : Finset ℕ
m n✝ p n : ℕ
d2 : 2 ∣ n
d4 : ¬2 ∣ n / 2
n0 : n > 0
pp : Prime 2
dp : 2 ∣ n / 2
⊢ False

Next tactic:
contradiction