l3lab/ntp-mathlib huggingface dataset:
theorem Int.two_pow_two_pow_add_two_pow_two_pow {x y : ℤ} (hx : ¬2 ∣ x) (hxy : 4 ∣ x - y) (i : ℕ) :
    multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑(1 : ℕ) := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  refine' multiplicity.eq_coe_iff.mpr ⟨_, _⟩
  · rw [pow_one, ← even_iff_two_dvd]
    exact hx_odd.pow.add_odd hy_odd.pow
  cases' i with i
  · intro hxy'
    have : 2 * 2 ∣ 2 * x := by
      have := dvd_add hxy hxy'
      norm_num at *
      rw [two_mul]
      exact this
    have : 2 ∣ x := (mul_dvd_mul_iff_left (by norm_num)).mp this
    contradiction
  suffices ∀ x : ℤ, Odd x → x ^ 2 ^ (i + 1) % 4 = 1 by
    rw [show (2 ^ (1 + 1) : ℤ) = 4 by norm_num, Int.dvd_iff_emod_eq_zero, Int.add_emod,
      this _ hx_odd, this _ hy_odd]
    decide
  

Proof state:
case refine'_2.succ
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ
⊢ ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1

Next tactic:
intro x hx
theorem Int.two_pow_two_pow_add_two_pow_two_pow {x y : ℤ} (hx : ¬2 ∣ x) (hxy : 4 ∣ x - y) (i : ℕ) :
    multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑(1 : ℕ) := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  refine' multiplicity.eq_coe_iff.mpr ⟨_, _⟩
  · rw [pow_one, ← even_iff_two_dvd]
    exact hx_odd.pow.add_odd hy_odd.pow
  cases' i with i
  · intro hxy'
    have : 2 * 2 ∣ 2 * x := by
      have := dvd_add hxy hxy'
      norm_num at *
      rw [two_mul]
      exact this
    have : 2 ∣ x := (mul_dvd_mul_iff_left (by norm_num)).mp this
    contradiction
  suffices ∀ x : ℤ, Odd x → x ^ 2 ^ (i + 1) % 4 = 1 by
    rw [show (2 ^ (1 + 1) : ℤ) = 4 by norm_num, Int.dvd_iff_emod_eq_zero, Int.add_emod,
      this _ hx_odd, this _ hy_odd]
    decide
  intro x hx
  

Proof state:
case refine'_2.succ
R : Type u_1
n : ℕ
x✝ y : ℤ
hx✝ : ¬2 ∣ x✝
hxy : 4 ∣ x✝ - y
hx_odd : Odd x✝
hxy_even : Even (x✝ - y)
hy_odd : Odd y
i : ℕ
x : ℤ
hx : Odd x
⊢ x ^ 2 ^ (i + 1) % 4 = 1

Next tactic:
rw [pow_succ, mul_comm, pow_mul, Int.sq_mod_four_eq_one_of_odd hx.pow]
theorem Int.two_pow_two_pow_sub_pow_two_pow {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = multiplicity 2 (x - y) + n := by
  

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ multiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = multiplicity 2 (x - y) + ↑n

Next tactic:
simp only [pow_two_pow_sub_pow_two_pow n, multiplicity.mul Int.prime_two,
    multiplicity.Finset.prod Int.prime_two, add_comm, Nat.cast_one, Finset.sum_const,
    Finset.card_range, nsmul_one, Int.two_pow_two_pow_add_two_pow_two_pow hx hxy]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n

Next tactic:
have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by 

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ Odd x

Next tactic:
rwa [Int.odd_iff_not_even, even_iff_two_dvd]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n

Next tactic:
have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by 

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
⊢ 2 ∣ 4

Next tactic:
decide
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n

Next tactic:
have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by 

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ Odd y

Next tactic:
simpa using hx_odd.sub_even hxy_even
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 ↑n

Next tactic:
cases' n with n
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · 

Proof state:
case zero
R : Type u_1
n : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ multiplicity 2 (x ^ Nat.zero - y ^ Nat.zero) = multiplicity 2 (x - y) + multiplicity 2 ↑Nat.zero

Next tactic:
simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  

Proof state:
case succ
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
⊢ multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 ↑(Nat.succ n)

Next tactic:
have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by 

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
⊢ 2 ≠ 1

Next tactic:
norm_num
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  

Proof state:
case succ
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
⊢ multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 ↑(Nat.succ n)

Next tactic:
rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  

Proof state:
case succ.intro.intro
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 ↑(Nat.succ n)

Next tactic:
rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · 

Proof state:
case succ.intro.intro
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ multiplicity 2 (x - y) + multiplicity 2 (Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 ↑(Nat.succ n)

Next tactic:
norm_cast
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · 

Proof state:
case succ.intro.intro.hp
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ Prime 2

Next tactic:
exact Int.prime_two
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · 

Proof state:
case succ.intro.intro.hxy
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ 2 ∣ x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h - y ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h

Next tactic:
simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · 

Proof state:
case succ.intro.intro.hx
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ ¬2 ∣ x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h

Next tactic:
simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
  

Proof state:
case succ.intro.intro.hn
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ ¬2 ∣ ↑k

Next tactic:
erw [Int.coe_nat_dvd]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
  erw [Int.coe_nat_dvd]
  -- `erw` to deal with `2 : ℤ` vs `(2 : ℕ) : ℤ`
  

Proof state:
case succ.intro.intro.hn
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
hpn : ¬2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
⊢ ¬2 ∣ k

Next tactic:
contrapose! hpn
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
  erw [Int.coe_nat_dvd]
  -- `erw` to deal with `2 : ℤ` vs `(2 : ℕ) : ℤ`
  contrapose! hpn
  

Proof state:
case succ.intro.intro.hn
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
hpn : 2 ∣ k
⊢ 2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) ∣ Nat.succ n

Next tactic:
rw [pow_succ']
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
  erw [Int.coe_nat_dvd]
  -- `erw` to deal with `2 : ℤ` vs `(2 : ℕ) : ℤ`
  contrapose! hpn
  rw [pow_succ']
  

Proof state:
case succ.intro.intro.hn
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
hpn : 2 ∣ k
⊢ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * 2 ∣ Nat.succ n

Next tactic:
conv_rhs => rw [hk]
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity (2 : ℤ) n := by
  have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
  have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
  have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
  cases' n with n
  · simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]
  have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr ⟨by norm_num, n.succ_pos⟩
  rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with ⟨⟨k, hk⟩, hpn⟩
  rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,
    Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, ← hk, PartENat.natCast_get]
  · norm_cast
  · exact Int.prime_two
  · simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow
  · simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow
  erw [Int.coe_nat_dvd]
  -- `erw` to deal with `2 : ℤ` vs `(2 : ℕ) : ℤ`
  contrapose! hpn
  rw [pow_succ']
  conv_rhs => 

Proof state:
R : Type u_1
n✝ : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : (multiplicity 2 (Nat.succ n)).Dom
k : ℕ
hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k
hpn : 2 ∣ k
| Nat.succ n

Next tactic:
rw [hk]