l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    all_goals 

Proof state:
case refine'_2.inr.inr.inr.hbc
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
hz_symm : 0 ≠ b * c
⊢ a ∣ c * b

Next tactic:
rwa [mul_comm]
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    all_goals 

Proof state:
case refine'_2.inr.inr.inr.hz
a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
this :
  ∀ {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : ℕ},
    ℵ₀ ≤ a → ∀ (b c : Cardinal.{u}), a ∣ b * c → b * c ≠ 0 → c ≤ b → a ∣ b ∨ a ∣ c
h : ¬c ≤ b
h✝ : b ≤ c
hz_symm : 0 ≠ b * c
⊢ c * b ≠ 0

Next tactic:
rwa [mul_comm]
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    all_goals rwa [mul_comm]
  

Proof state:
a✝ a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
h : c ≤ b
⊢ a ∣ b ∨ a ∣ c

Next tactic:
left
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    all_goals rwa [mul_comm]
  left
  

Proof state:
case h
a✝ a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
h : c ≤ b
⊢ a ∣ b

Next tactic:
have habc := le_of_dvd hz hbc
@[simp]
theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by
  refine' ⟨(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _⟩
  · rw [isUnit_iff]
    exact (one_lt_aleph0.trans_le ha).ne'
  rcases eq_or_ne (b * c) 0 with hz | hz
  · rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
  wlog h : c ≤ b
  · cases le_total c b <;> [skip; rw [or_comm]] <;> apply_assumption
    assumption'
    all_goals rwa [mul_comm]
  left
  have habc := le_of_dvd hz hbc
  

Proof state:
case h
a✝ a b✝ : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
h : c ≤ b
habc : a ≤ b * c
⊢ a ∣ b

Next tactic:
rwa [mul_eq_max' <| ha.trans <| habc, max_def', if_pos h] at hbc
theorem not_irreducible_of_aleph0_le (ha : ℵ₀ ≤ a) : ¬Irreducible a := by
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
⊢ ¬Irreducible a

Next tactic:
rw [irreducible_iff, not_and_or]
theorem not_irreducible_of_aleph0_le (ha : ℵ₀ ≤ a) : ¬Irreducible a := by
  rw [irreducible_iff, not_and_or]
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
⊢ ¬¬IsUnit a ∨ ¬∀ (a_1 b : Cardinal.{u}), a = a_1 * b → IsUnit a_1 ∨ IsUnit b

Next tactic:
refine' Or.inr fun h => _
theorem not_irreducible_of_aleph0_le (ha : ℵ₀ ≤ a) : ¬Irreducible a := by
  rw [irreducible_iff, not_and_or]
  refine' Or.inr fun h => _
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
ha : ℵ₀ ≤ a
h : ∀ (a_1 b : Cardinal.{u}), a = a_1 * b → IsUnit a_1 ∨ IsUnit b
⊢ False

Next tactic:
simpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using
    h a ℵ₀
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ ↑n ∣ ↑m ↔ n ∣ m

Next tactic:
refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ ↑n ∣ ↑m → n ∣ m

Next tactic:
rintro ⟨k, hk⟩
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  

Proof state:
case intro
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
⊢ n ∣ m

Next tactic:
have : ↑m < ℵ₀ := nat_lt_aleph0 m
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  

Proof state:
case intro
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
this : ↑m < ℵ₀
⊢ n ∣ m

Next tactic:
rw [hk, mul_lt_aleph0_iff] at this
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  

Proof state:
case intro
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
this : ↑n = 0 ∨ k = 0 ∨ ↑n < ℵ₀ ∧ k < ℵ₀
⊢ n ∣ m

Next tactic:
rcases this with (h | h | ⟨-, hk'⟩)
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  

Proof state:
case intro.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : ↑n = 0
⊢ n ∣ m
case intro.inr.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : k = 0
⊢ n ∣ m
case intro.inr.inr.intro a b : Cardinal.{u} n m : ℕ k : Cardinal.{u_1} hk : ↑m = ↑n * k hk' : k < ℵ₀ ⊢ n ∣ m

Next tactic:
iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 

Proof state:
case intro.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : ↑n = 0
⊢ n ∣ m
case intro.inr.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : k = 0
⊢ n ∣ m
case intro.inr.inr.intro a b : Cardinal.{u} n m : ℕ k : Cardinal.{u_1} hk : ↑m = ↑n * k hk' : k < ℵ₀ ⊢ n ∣ m

Next tactic:
simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; 

Proof state:
case intro.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
h : ↑n = 0
hk : m = 0
⊢ n ∣ m
case intro.inr.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : k = 0
⊢ n ∣ m
case intro.inr.inr.intro a b : Cardinal.{u} n m : ℕ k : Cardinal.{u_1} hk : ↑m = ↑n * k hk' : k < ℵ₀ ⊢ n ∣ m

Next tactic:
simp [hk]
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 

Proof state:
case intro.inr.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
h : k = 0
⊢ n ∣ m
case intro.inr.inr.intro a b : Cardinal.{u} n m : ℕ k : Cardinal.{u_1} hk : ↑m = ↑n * k hk' : k < ℵ₀ ⊢ n ∣ m

Next tactic:
simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; 

Proof state:
case intro.inr.inl
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
h : k = 0
hk : m = 0
⊢ n ∣ m
case intro.inr.inr.intro a b : Cardinal.{u} n m : ℕ k : Cardinal.{u_1} hk : ↑m = ↑n * k hk' : k < ℵ₀ ⊢ n ∣ m

Next tactic:
simp [hk]
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
  

Proof state:
case intro.inr.inr.intro
a b : Cardinal.{u}
n m : ℕ
k : Cardinal.{u_1}
hk : ↑m = ↑n * k
hk' : k < ℵ₀
⊢ n ∣ m

Next tactic:
lift k to ℕ using hk'
@[simp, norm_cast]
theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by
  refine' ⟨_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
  rintro ⟨k, hk⟩
  have : ↑m < ℵ₀ := nat_lt_aleph0 m
  rw [hk, mul_lt_aleph0_iff] at this
  rcases this with (h | h | ⟨-, hk'⟩)
  iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
  lift k to ℕ using hk'
  

Proof state:
case intro.inr.inr.intro.intro
a b : Cardinal.{u}
n m k : ℕ
hk : ↑m = ↑n * ↑k
⊢ n ∣ m

Next tactic:
exact ⟨k, mod_cast hk⟩
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ Prime ↑n ↔ Nat.Prime n

Next tactic:
simp only [Prime, Nat.prime_iff]
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ (↑n ≠ 0 ∧ ¬IsUnit ↑n ∧ ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b) ↔
    n ≠ 0 ∧ ¬IsUnit n ∧ ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b

Next tactic:
refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by 

Proof state:
a b : Cardinal.{u}
n m : ℕ
⊢ ↑n ≠ 0 ↔ n ≠ 0

Next tactic:
simp
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · 

Proof state:
case refine'_1
a b : Cardinal.{u}
n m : ℕ
⊢ ¬IsUnit ↑n ↔ ¬IsUnit n

Next tactic:
simp only [isUnit_iff, Nat.isUnit_iff]
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    

Proof state:
case refine'_1
a b : Cardinal.{u}
n m : ℕ
⊢ ¬↑n = 1 ↔ ¬n = 1

Next tactic:
exact mod_cast Iff.rfl
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · 

Proof state:
case refine'_2
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b
b c : ℕ
hbc : n ∣ b * c
⊢ n ∣ b ∨ n ∣ c

Next tactic:
exact mod_cast h b c (mod_cast hbc)
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  

Proof state:
case refine'_3
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
cases' lt_or_le (b * c) ℵ₀ with h' h'
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · 

Proof state:
case refine'_3.inl
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : b * c < ℵ₀
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · 

Proof state:
case refine'_3.inl.inl
a b : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
c : Cardinal.{u_1}
hbc : ↑n ∣ 0 * c
h' : 0 * c < ℵ₀
⊢ ↑n ∣ 0 ∨ ↑n ∣ c

Next tactic:
simp
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · 

Proof state:
case refine'_3.inl.inr.inl
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b : Cardinal.{u_1}
hbc : ↑n ∣ b * 0
h' : b * 0 < ℵ₀
⊢ ↑n ∣ b ∨ ↑n ∣ 0

Next tactic:
simp
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    

Proof state:
case refine'_3.inl.inr.inr.intro
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : b * c < ℵ₀
hb : b < ℵ₀
hc : c < ℵ₀
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
lift b to ℕ using hb
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    

Proof state:
case refine'_3.inl.inr.inr.intro.intro
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
c : Cardinal.{u_1}
hc : c < ℵ₀
b : ℕ
hbc : ↑n ∣ ↑b * c
h' : ↑b * c < ℵ₀
⊢ ↑n ∣ ↑b ∨ ↑n ∣ c

Next tactic:
lift c to ℕ using hc
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    

Proof state:
case refine'_3.inl.inr.inr.intro.intro.intro
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : ℕ
hbc : ↑n ∣ ↑b * ↑c
h' : ↑b * ↑c < ℵ₀
⊢ ↑n ∣ ↑b ∨ ↑n ∣ ↑c

Next tactic:
exact mod_cast h b c (mod_cast hbc)
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  

Proof state:
case refine'_3.inr
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : ℵ₀ ≤ b * c
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  

Proof state:
case refine'_3.inr.intro.intro
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
have hn : (n : Cardinal) ≠ 0 := by
    intro h
    rw [h, zero_dvd_iff, mul_eq_zero] at hbc
    cases hbc <;> contradiction
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    

Proof state:
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
⊢ ↑n ≠ 0

Next tactic:
intro h
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    intro h
    

Proof state:
a b✝ : Cardinal.{u}
n m : ℕ
h✝ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
h : ↑n = 0
⊢ False

Next tactic:
rw [h, zero_dvd_iff, mul_eq_zero] at hbc
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    intro h
    rw [h, zero_dvd_iff, mul_eq_zero] at hbc
    

Proof state:
a b✝ : Cardinal.{u}
n m : ℕ
h✝ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : b = 0 ∨ c = 0
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
h : ↑n = 0
⊢ False

Next tactic:
cases hbc
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    intro h
    rw [h, zero_dvd_iff, mul_eq_zero] at hbc
    cases hbc <;> 

Proof state:
case inl
a b✝ : Cardinal.{u}
n m : ℕ
h✝¹ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
h : ↑n = 0
h✝ : b = 0
⊢ False

Next tactic:
contradiction
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    intro h
    rw [h, zero_dvd_iff, mul_eq_zero] at hbc
    cases hbc <;> 

Proof state:
case inr
a b✝ : Cardinal.{u}
n m : ℕ
h✝¹ : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
h : ↑n = 0
h✝ : c = 0
⊢ False

Next tactic:
contradiction
@[simp]
theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by
  simp only [Prime, Nat.prime_iff]
  refine' and_congr (by simp) (and_congr _ ⟨fun h b c hbc => _, fun h b c hbc => _⟩)
  · simp only [isUnit_iff, Nat.isUnit_iff]
    exact mod_cast Iff.rfl
  · exact mod_cast h b c (mod_cast hbc)
  cases' lt_or_le (b * c) ℵ₀ with h' h'
  · rcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | ⟨hb, hc⟩)
    · simp
    · simp
    lift b to ℕ using hb
    lift c to ℕ using hc
    exact mod_cast h b c (mod_cast hbc)
  rcases aleph0_le_mul_iff.mp h' with ⟨hb, hc, hℵ₀⟩
  have hn : (n : Cardinal) ≠ 0 := by
    intro h
    rw [h, zero_dvd_iff, mul_eq_zero] at hbc
    cases hbc <;> contradiction
  

Proof state:
case refine'_3.inr.intro.intro
a b✝ : Cardinal.{u}
n m : ℕ
h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b
b c : Cardinal.{u_1}
hbc : ↑n ∣ b * c
h' : ℵ₀ ≤ b * c
hb : b ≠ 0
hc : c ≠ 0
hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c
hn : ↑n ≠ 0
⊢ ↑n ∣ b ∨ ↑n ∣ c

Next tactic:
wlog hℵ₀b : ℵ₀ ≤ b