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]
  · 

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
⊢ MinSqFacProp n none

Next tactic:
refine' squarefree_iff_prime_squarefree.2 fun p pp 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 => _
    

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
p : ℕ
pp : Prime p
d : p * p ∣ n
⊢ False

Next tactic:
have := 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)
    

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
p : ℕ
pp : Prime p
d : p * p ∣ n
this : k ≤ p
⊢ False

Next tactic:
have := Nat.mul_le_mul this this
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
    

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
p : ℕ
pp : Prime p
d : p * p ∣ n
this✝ : k ≤ p
this : k * k ≤ p * p
⊢ False

Next tactic:
exact not_le_of_lt h (le_trans this (le_of_dvd n0 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))
  

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
⊢ 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 k2 : 2 ≤ k := by
    subst e
    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
    

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
⊢ 2 ≤ k

Next tactic:
subst e
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
    

Proof state:
s : Finset ℕ
m n✝ p n : ℕ
n0 : 0 < n
i : ℕ
ih : ∀ (m : ℕ), Prime m → m ∣ n → 2 * i + 3 ≤ m
h : ¬n < (2 * i + 3) * (2 * i + 3)
⊢ 2 ≤ 2 * i + 3

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
  

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
⊢ 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 k0 : 0 < k := lt_of_lt_of_le (by decide) k2
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 

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
⊢ 0 < 2

Next tactic:
decide
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
  

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
⊢ 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 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)
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
    

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), n' ∣ n → ¬k ∣ n' → MinSqFacProp n' (minSqFacAux n' (k + 2))

Next tactic:
intro n' nd' nk
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
    

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'
⊢ MinSqFacProp n' (minSqFacAux n' (k + 2))

Next tactic:
have hn' := le_of_dvd n0 nd'
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'
    

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
⊢ MinSqFacProp n' (minSqFacAux n' (k + 2))

Next tactic:
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 => _
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 

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
⊢ k + 2 = 2 * (i + 1) + 3

Next tactic:
simp [e, left_distrib]
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 => _
    

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
d : m ∣ n'
⊢ k + 2 ≤ m

Next tactic:
rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
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
    · 

Proof state:
case inl
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
d : m ∣ n'
me : k = m
⊢ k + 2 ≤ m

Next tactic:
subst me
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
      

Proof state:
case inl
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
m2 : Prime k
d : k ∣ n'
⊢ k + 2 ≤ k

Next tactic:
contradiction
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
    

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
d : m ∣ n'
ml : k < m
⊢ k + 2 ≤ m

Next tactic:
apply (Nat.eq_or_lt_of_le ml).resolve_left
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
    

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
d : m ∣ n'
ml : k < m
⊢ ¬succ k = m

Next tactic:
intro me
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
    

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
d : m ∣ n'
ml : k < m
me : succ k = m
⊢ False

Next tactic:
rw [← me, e] at 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
    

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
d : succ (2 * i + 3) ∣ n'
ml : k < m
me : succ k = m
⊢ False

Next tactic:
change 2 * (i + 2) ∣ n' at 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
    

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'
⊢ False

Next tactic:
have := ih _ prime_two (dvd_trans (dvd_of_mul_right_dvd d) nd')
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')
    

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 : k ≤ 2
⊢ False

Next tactic:
rw [e] at this