l3lab/ntp-mathlib huggingface dataset:
theorem MinFacHelper.one_lt {n k : ℕ} (h : MinFacHelper n k) : 1 < n := by
  have : 2 < minFac n := h.1.trans_le h.2.2
  rcases eq_zero_or_pos n with rfl|h
  · contradiction
  rcases (succ_le_of_lt h).eq_or_lt with rfl|h
  · contradiction
  

Proof state:
case inr.inr
n k : ℕ
h✝¹ : MinFacHelper n k
this : 2 < minFac n
h✝ : n > 0
h : succ 0 < n
⊢ 1 < n

Next tactic:
exact h
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  

Proof state:
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
⊢ MinFacHelper n 3

Next tactic:
refine ⟨by norm_num, by norm_num, ?_⟩
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by 

Proof state:
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
⊢ 2 < 3

Next tactic:
norm_num
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by norm_num, by 

Proof state:
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
⊢ 3 % 2 = 1

Next tactic:
norm_num
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by norm_num, by norm_num, ?_⟩
  

Proof state:
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
⊢ 3 ≤ minFac n

Next tactic:
refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by norm_num, by norm_num, ?_⟩
  refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
  

Proof state:
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
p : ℕ
hp : 2 ≤ p
hpn : p ∣ n
⊢ 3 ≤ p

Next tactic:
rcases hp.eq_or_lt with rfl|h
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by norm_num, by norm_num, ?_⟩
  refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
  rcases hp.eq_or_lt with rfl|h
  · 

Proof state:
case inl
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
hp : 2 ≤ 2
hpn : 2 ∣ n
⊢ 3 ≤ 2

Next tactic:
simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
theorem minFacHelper_0 (n : ℕ)
    (h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
    MinFacHelper n (nat_lit 3) := by
  refine ⟨by norm_num, by norm_num, ?_⟩
  refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
  rcases hp.eq_or_lt with rfl|h
  · simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
  · 

Proof state:
case inr
n : ℕ
h1 : ble 2 n = true
h2 : 1 = n % 2
p : ℕ
hp : 2 ≤ p
hpn : p ∣ n
h : 2 < p
⊢ 3 ≤ p

Next tactic:
exact h
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
⊢ MinFacHelper n k'

Next tactic:
rw [← e]
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
⊢ MinFacHelper n (k + 2)

Next tactic:
refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · 

Proof state:
case refine_1
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
⊢ (k + 2) % 2 = 1

Next tactic:
rw [add_mod, mod_self, add_zero, mod_mod]
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    

Proof state:
case refine_1
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
⊢ k % 2 = 1

Next tactic:
exact h.2.1
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  

Proof state:
case refine_2
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
⊢ k + 2 ≤ minFac n

Next tactic:
rcases h.2.2.eq_or_lt with rfl|h2
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · 

Proof state:
case refine_2.inl
n k' : ℕ
e : minFac n + 2 = k'
h : MinFacHelper n (minFac n)
np : minFac n ≠ minFac n
⊢ minFac n + 2 ≤ minFac n

Next tactic:
exact (np rfl).elim
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  

Proof state:
case refine_2.inr
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2 : k < minFac n
⊢ k + 2 ≤ minFac n

Next tactic:
rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · 

Proof state:
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
⊢ k + 2 ≤ minFac n

Next tactic:
refine ((h.1.trans_le h.2.2).ne ?_).elim
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    

Proof state:
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
⊢ 2 = minFac n

Next tactic:
have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      norm_num
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      

Proof state:
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
⊢ 2 ∣ minFac n

Next tactic:
rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      

Proof state:
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
⊢ (1 + 1 % 2) % 2 = 0

Next tactic:
norm_num
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      norm_num
    

Proof state:
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
h3 : 2 ∣ minFac n
⊢ 2 = minFac n

Next tactic:
rw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      norm_num
    rw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3
    

Proof state:
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
h3 : 2 = 1 ∨ 2 = minFac n
⊢ 2 = minFac n

Next tactic:
norm_num at h3
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      norm_num
    rw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3
    norm_num at h3
    

Proof state:
case refine_2.inr.inl
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k = minFac n
h3 : 2 = minFac n
⊢ 2 = minFac n

Next tactic:
exact h3
theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
    (np : minFac n ≠ k) : MinFacHelper n k' := by
  rw [← e]
  refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
  · rw [add_mod, mod_self, add_zero, mod_mod]
    exact h.2.1
  rcases h.2.2.eq_or_lt with rfl|h2
  · exact (np rfl).elim
  rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
  · refine ((h.1.trans_le h.2.2).ne ?_).elim
    have h3 : 2 ∣ minFac n := by
      rw [Nat.dvd_iff_mod_eq_zero, ← h2, succ_eq_add_one, add_mod, h.2.1]
      norm_num
    rw [dvd_prime <| minFac_prime h.one_lt.ne'] at h3
    norm_num at h3
    exact h3
  

Proof state:
case refine_2.inr.inr
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : minFac n ≠ k
h2✝ : k < minFac n
h2 : succ k < minFac n
⊢ k + 2 ≤ minFac n

Next tactic:
exact h2
theorem minFacHelper_2 {n k k' : ℕ} (e : k + 2 = k') (nk : ¬ Nat.Prime k)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk : ¬Nat.Prime k
h : MinFacHelper n k
⊢ MinFacHelper n k'

Next tactic:
refine minFacHelper_1 e h λ h2 ↦ ?_
theorem minFacHelper_2 {n k k' : ℕ} (e : k + 2 = k') (nk : ¬ Nat.Prime k)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  refine minFacHelper_1 e h λ h2 ↦ ?_
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk : ¬Nat.Prime k
h : MinFacHelper n k
h2 : minFac n = k
⊢ False

Next tactic:
rw [← h2] at nk
theorem minFacHelper_2 {n k k' : ℕ} (e : k + 2 = k') (nk : ¬ Nat.Prime k)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  refine minFacHelper_1 e h λ h2 ↦ ?_
  rw [← h2] at nk
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk : ¬Nat.Prime (minFac n)
h : MinFacHelper n k
h2 : minFac n = k
⊢ False

Next tactic:
exact nk <| minFac_prime h.one_lt.ne'
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk : beq (n % k) 0 = false
h : MinFacHelper n k
⊢ MinFacHelper n k'

Next tactic:
refine minFacHelper_1 e h λ h2 ↦ ?_
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  refine minFacHelper_1 e h λ h2 ↦ ?_
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk : beq (n % k) 0 = false
h : MinFacHelper n k
h2 : minFac n = k
⊢ False

Next tactic:
have nk := Nat.ne_of_beq_eq_false nk
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  refine minFacHelper_1 e h λ h2 ↦ ?_
  have nk := Nat.ne_of_beq_eq_false nk
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk✝ : beq (n % k) 0 = false
h : MinFacHelper n k
h2 : minFac n = k
nk : ¬n % k = 0
⊢ False

Next tactic:
rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
    (h : MinFacHelper n k) : MinFacHelper n k' := by
  refine minFacHelper_1 e h λ h2 ↦ ?_
  have nk := Nat.ne_of_beq_eq_false nk
  rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
  

Proof state:
n k k' : ℕ
e : k + 2 = k'
nk✝ : beq (n % k) 0 = false
h : MinFacHelper n k
h2 : minFac n = k
nk : ¬minFac n ∣ n
⊢ False

Next tactic:
exact nk <| minFac_dvd n
theorem isNat_minFac_2 : {a a' : ℕ} → IsNat a a' → a' % 2 = 0 → IsNat a.minFac 2
  | a, _, ⟨rfl⟩, h => ⟨by 

Proof state:
a : ℕ
h : a % 2 = 0
⊢ minFac a = ↑2

Next tactic:
rw [cast_ofNat, minFac_eq_two_iff, Nat.dvd_iff_mod_eq_zero, h]
theorem isNat_minFac_3 : {n n' : ℕ} → (k : ℕ) →
    IsNat n n' → MinFacHelper n' k → nat_lit 0 = n' % k → IsNat (minFac n) k
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : 0 = n % k
⊢ IsNat (minFac n) k

Next tactic:
rw [eq_comm, ← Nat.dvd_iff_mod_eq_zero] at h2
theorem isNat_minFac_3 : {n n' : ℕ} → (k : ℕ) →
    IsNat n n' → MinFacHelper n' k → nat_lit 0 = n' % k → IsNat (minFac n) k
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    rw [eq_comm, ← Nat.dvd_iff_mod_eq_zero] at h2
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : k ∣ n
⊢ IsNat (minFac n) k

Next tactic:
exact ⟨le_antisymm (minFac_le_of_dvd h1.1.le h2) h1.2.2⟩
theorem isNat_minFac_4 : {n n' k : ℕ} →
    IsNat n n' → MinFacHelper n' k → (k * k).ble n' = false → IsNat (minFac n) n'
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : ble (k * k) n = false
⊢ IsNat (minFac n) n

Next tactic:
refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
theorem isNat_minFac_4 : {n n' k : ℕ} →
    IsNat n n' → MinFacHelper n' k → (k * k).ble n' = false → IsNat (minFac n) n'
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : ble (k * k) n = false
⊢ Nat.Prime n

Next tactic:
rw [Nat.prime_def_le_sqrt]
theorem isNat_minFac_4 : {n n' k : ℕ} →
    IsNat n n' → MinFacHelper n' k → (k * k).ble n' = false → IsNat (minFac n) n'
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
    rw [Nat.prime_def_le_sqrt]
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : ble (k * k) n = false
⊢ 2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m ≤ sqrt n → ¬m ∣ n

Next tactic:
refine ⟨h1.one_lt, λ m hm hmn h2mn ↦ ?_⟩
theorem isNat_minFac_4 : {n n' k : ℕ} →
    IsNat n n' → MinFacHelper n' k → (k * k).ble n' = false → IsNat (minFac n) n'
  | n, _, k, ⟨rfl⟩, h1, h2 => by
    refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
    rw [Nat.prime_def_le_sqrt]
    refine ⟨h1.one_lt, λ m hm hmn h2mn ↦ ?_⟩
    

Proof state:
n k : ℕ
h1 : MinFacHelper n k
h2 : ble (k * k) n = false
m : ℕ
hm : 2 ≤ m
hmn : m ≤ sqrt n
h2mn : m ∣ n
⊢ False

Next tactic:
exact lt_irrefl m <| calc
      m ≤ sqrt n   := hmn
      _ < k        := sqrt_lt.mpr (ble_eq_false.mp h2)
      _ ≤ n.minFac := h1.2.2
      _ ≤ m        := Nat.minFac_le_of_dvd hm h2mn
/-- The functor `DiscreteQuotient X ⥤ Fintype` whose limit is isomorphic to `X`. -/
def fintypeDiagram : DiscreteQuotient X ⥤ FintypeCat where
  obj S := @FintypeCat.of S (Fintype.ofFinite S)
  map f := DiscreteQuotient.ofLE f.le
  -- Porting note: `map_comp` used to be proved by default by `aesop_cat`.
  -- once `aesop_cat` can prove this again, remove the entire `map_comp` here.
  map_comp _ _ := by 

Proof state:
X : Profinite
X✝ Y✝ Z✝ : DiscreteQuotient ↑X.toCompHaus.toTop
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
⊢ { obj := fun S => FintypeCat.of (Quotient S.toSetoid),
          map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
      (x✝¹ ≫ x✝) =
    { obj := fun S => FintypeCat.of (Quotient S.toSetoid),
            map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
        x✝¹ ≫
      { obj := fun S => FintypeCat.of (Quotient S.toSetoid),
            map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
        x✝

Next tactic:
ext
/-- The functor `DiscreteQuotient X ⥤ Fintype` whose limit is isomorphic to `X`. -/
def fintypeDiagram : DiscreteQuotient X ⥤ FintypeCat where
  obj S := @FintypeCat.of S (Fintype.ofFinite S)
  map f := DiscreteQuotient.ofLE f.le
  -- Porting note: `map_comp` used to be proved by default by `aesop_cat`.
  -- once `aesop_cat` can prove this again, remove the entire `map_comp` here.
  map_comp _ _ := by ext; 

Proof state:
case h
X : Profinite
X✝ Y✝ Z✝ : DiscreteQuotient ↑X.toCompHaus.toTop
x✝² : X✝ ⟶ Y✝
x✝¹ : Y✝ ⟶ Z✝
x✝ :
  ↑({ obj := fun S => FintypeCat.of (Quotient S.toSetoid),
          map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.obj
      X✝)
⊢ { obj := fun S => FintypeCat.of (Quotient S.toSetoid),
          map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
      (x✝² ≫ x✝¹) x✝ =
    ({ obj := fun S => FintypeCat.of (Quotient S.toSetoid),
              map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
          x✝² ≫
        { obj := fun S => FintypeCat.of (Quotient S.toSetoid),
              map := fun {X_1 Y} f => DiscreteQuotient.ofLE (_ : X_1 ≤ Y) }.map
          x✝¹)
      x✝

Next tactic:
aesop_cat
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      

Proof state:
X : Profinite
⊢ Function.Bijective ⇑(Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X))

Next tactic:
refine' ⟨fun a b h => _, fun a => _⟩
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · 

Proof state:
case refine'_1
X : Profinite
a b : (forget Profinite).obj (asLimitCone X).pt
h :
  (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) a =
    (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) b
⊢ a = b

Next tactic:
refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        

Proof state:
case refine'_1
X : Profinite
a b : (forget Profinite).obj (asLimitCone X).pt
h :
  (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) a =
    (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) b
S : DiscreteQuotient ((forget Profinite).obj (asLimitCone X).pt)
⊢ DiscreteQuotient.proj S a = DiscreteQuotient.proj S b

Next tactic:
apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
        

Proof state:
case refine'_1
X : Profinite
a b : (forget Profinite).obj (asLimitCone X).pt
S : DiscreteQuotient ((forget Profinite).obj (asLimitCone X).pt)
h :
  ↑((Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) a) S =
    ↑((Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) b) S
⊢ DiscreteQuotient.proj S a = DiscreteQuotient.proj S b

Next tactic:
exact h
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
        exact h
      · 

Proof state:
case refine'_2
X : Profinite
a : (forget Profinite).obj (limitCone (diagram X)).pt
⊢ ∃ a_1, (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) a_1 = a

Next tactic:
obtain ⟨b, hb⟩ :=
          DiscreteQuotient.exists_of_compat (fun S => a.val S) fun _ _ h => a.prop (homOfLE h)
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
        exact h
      · obtain ⟨b, hb⟩ :=
          DiscreteQuotient.exists_of_compat (fun S => a.val S) fun _ _ h => a.prop (homOfLE h)
        

Proof state:
case refine'_2.intro
X : Profinite
a : (forget Profinite).obj (limitCone (diagram X)).pt
b : ↑X.toCompHaus.toTop
hb : ∀ (Q : DiscreteQuotient ↑X.toCompHaus.toTop), DiscreteQuotient.proj Q b = ↑a Q
⊢ ∃ a_1, (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) a_1 = a

Next tactic:
use b
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
        exact h
      · obtain ⟨b, hb⟩ :=
          DiscreteQuotient.exists_of_compat (fun S => a.val S) fun _ _ h => a.prop (homOfLE h)
        use b
        -- ext S : 3 -- Porting note: `ext` does not work, replaced with following three lines.
        

Proof state:
case h
X : Profinite
a : (forget Profinite).obj (limitCone (diagram X)).pt
b : ↑X.toCompHaus.toTop
hb : ∀ (Q : DiscreteQuotient ↑X.toCompHaus.toTop), DiscreteQuotient.proj Q b = ↑a Q
⊢ (Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) b = a

Next tactic:
apply Subtype.ext
instance isIso_asLimitCone_lift : IsIso ((limitConeIsLimit.{u, u} X.diagram).lift X.asLimitCone) :=
  isIso_of_bijective _
    (by
      refine' ⟨fun a b h => _, fun a => _⟩
      · refine' DiscreteQuotient.eq_of_forall_proj_eq fun S => _
        apply_fun fun f : (limitCone.{u, u} X.diagram).pt => f.val S at h
        exact h
      · obtain ⟨b, hb⟩ :=
          DiscreteQuotient.exists_of_compat (fun S => a.val S) fun _ _ h => a.prop (homOfLE h)
        use b
        -- ext S : 3 -- Porting note: `ext` does not work, replaced with following three lines.
        apply Subtype.ext
        

Proof state:
case h.a
X : Profinite
a : (forget Profinite).obj (limitCone (diagram X)).pt
b : ↑X.toCompHaus.toTop
hb : ∀ (Q : DiscreteQuotient ↑X.toCompHaus.toTop), DiscreteQuotient.proj Q b = ↑a Q
⊢ ↑((Limits.IsLimit.lift (limitConeIsLimit (diagram X)) (asLimitCone X)) b) = ↑a

Next tactic:
apply funext