l3lab/ntp-mathlib huggingface dataset:
theorem Nat.pow_add_pow {x y : ℕ} (hxy : p ∣ x + y) (hx : ¬p ∣ x) {n : ℕ} (hn : Odd n) :
    multiplicity p (x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n := by
  iterate 2 rw [← Int.coe_nat_multiplicity]
  rw [← Int.coe_nat_dvd] at hxy hx
  

Proof state:
R : Type u_1
n✝ : ℕ
inst✝ : CommRing R
a b x✝ y✝ : R
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℕ
hxy : ↑p ∣ ↑(x + y)
hx : ¬↑p ∣ ↑x
n : ℕ
hn : Odd n
⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n

Next tactic:
push_cast at *
theorem Nat.pow_add_pow {x y : ℕ} (hxy : p ∣ x + y) (hx : ¬p ∣ x) {n : ℕ} (hn : Odd n) :
    multiplicity p (x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n := by
  iterate 2 rw [← Int.coe_nat_multiplicity]
  rw [← Int.coe_nat_dvd] at hxy hx
  push_cast at *
  

Proof state:
R : Type u_1
n✝ : ℕ
inst✝ : CommRing R
a b x✝ y✝ : R
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℕ
hx : ¬↑p ∣ ↑x
n : ℕ
hn : Odd n
hxy : ↑p ∣ ↑x + ↑y
⊢ multiplicity (↑p) (↑x ^ n + ↑y ^ n) = multiplicity (↑p) (↑x + ↑y) + multiplicity p n

Next tactic:
exact Int.pow_add_pow hp hp1 hxy hx hn
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  

Proof state:
R : Type u_1
n✝ : ℕ
inst✝ : CommRing R
x y : R
n : ℕ
⊢ x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)

Next tactic:
induction' n with d hd
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  induction' n with d hd
  · 

Proof state:
case zero
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y : R
⊢ x ^ 2 ^ Nat.zero - y ^ 2 ^ Nat.zero = (∏ i in range Nat.zero, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)

Next tactic:
simp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  induction' n with d hd
  · simp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]
  · 

Proof state:
case succ
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
⊢ x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (∏ i in range (Nat.succ d), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)

Next tactic:
suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
      rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  induction' n with d hd
  · simp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]
  · suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
      

Proof state:
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
this : x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)
⊢ x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (∏ i in range (Nat.succ d), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)

Next tactic:
rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  induction' n with d hd
  · simp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]
  · suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
      rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
    · 

Proof state:
case succ
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
⊢ x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)

Next tactic:
rw [Nat.succ_eq_add_one]
theorem pow_two_pow_sub_pow_two_pow [CommRing R] {x y : R} (n : ℕ) :
    x ^ 2 ^ n - y ^ 2 ^ n = (∏ i in Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y) := by
  induction' n with d hd
  · simp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]
  · suffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by
      rw [this, hd, Finset.prod_range_succ, ← mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]
    · rw [Nat.succ_eq_add_one]
      

Proof state:
case succ
R : Type u_1
n : ℕ
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
⊢ x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)

Next tactic:
ring
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  

Proof state:
R : Type u_1
n : ℕ
x : ℤ
⊢ Odd x → x ^ 2 % 4 = 1

Next tactic:
intro hx
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  intro hx
  

Proof state:
R : Type u_1
n : ℕ
x : ℤ
hx : Odd x
⊢ x ^ 2 % 4 = 1

Next tactic:
unfold Odd at hx
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  intro hx
  unfold Odd at hx
  

Proof state:
R : Type u_1
n : ℕ
x : ℤ
hx : ∃ k, x = 2 * k + 1
⊢ x ^ 2 % 4 = 1

Next tactic:
rcases hx with ⟨_, rfl⟩
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  intro hx
  unfold Odd at hx
  rcases hx with ⟨_, rfl⟩
  

Proof state:
case intro
R : Type u_1
n : ℕ
w✝ : ℤ
⊢ (2 * w✝ + 1) ^ 2 % 4 = 1

Next tactic:
ring_nf
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  intro hx
  unfold Odd at hx
  rcases hx with ⟨_, rfl⟩
  ring_nf
  

Proof state:
case intro
R : Type u_1
n : ℕ
w✝ : ℤ
⊢ (1 + w✝ * 4 + w✝ ^ 2 * 4) % 4 = 1

Next tactic:
rw [add_assoc, ← add_mul, Int.add_mul_emod_self]
theorem Int.sq_mod_four_eq_one_of_odd {x : ℤ} : Odd x → x ^ 2 % 4 = 1 := by
  intro hx
  unfold Odd at hx
  rcases hx with ⟨_, rfl⟩
  ring_nf
  rw [add_assoc, ← add_mul, Int.add_mul_emod_self]
  

Proof state:
case intro
R : Type u_1
n : ℕ
w✝ : ℤ
⊢ 1 % 4 = 1

Next tactic:
decide
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
  

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1

Next tactic:
have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]
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 

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

Next tactic:
rwa [Int.odd_iff_not_even, even_iff_two_dvd]
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]
  

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1

Next tactic:
have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by decide) hxy)
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 

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

Next tactic:
decide
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)
  

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1

Next tactic:
have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even
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 

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
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_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
  

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1

Next tactic:
refine' multiplicity.eq_coe_iff.mpr ⟨_, _⟩
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 ⟨_, _⟩
  · 

Proof state:
case refine'_1
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ 2 ^ 1 ∣ x ^ 2 ^ i + y ^ 2 ^ i

Next tactic:
rw [pow_one, ← even_iff_two_dvd]
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]
    

Proof state:
case refine'_1
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ Even (x ^ 2 ^ i + y ^ 2 ^ i)

Next tactic:
exact hx_odd.pow.add_odd hy_odd.pow
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
  

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

Next tactic:
cases' i with i
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
  · 

Proof state:
case refine'_2.zero
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
⊢ ¬2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero

Next tactic:
intro hxy'
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'
    

Proof state:
case refine'_2.zero
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
⊢ False

Next tactic:
have : 2 * 2 ∣ 2 * x := by
      have := dvd_add hxy hxy'
      norm_num at *
      rw [two_mul]
      exact this
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
      

Proof state:
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
⊢ 2 * 2 ∣ 2 * x

Next tactic:
have := dvd_add hxy hxy'
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'
      

Proof state:
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
this : 4 ∣ x - y + (x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero)
⊢ 2 * 2 ∣ 2 * x

Next tactic:
norm_num at *
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 *
      

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hxy_even : Even (x - y)
hx : x % 2 = 1
hx_odd : ¬Even x
hy_odd : ¬Even y
hxy' : 4 ∣ x + y
this : 4 ∣ x + x
⊢ 4 ∣ 2 * x

Next tactic:
rw [two_mul]
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]
      

Proof state:
R : Type u_1
n : ℕ
x y : ℤ
hxy : 4 ∣ x - y
hxy_even : Even (x - y)
hx : x % 2 = 1
hx_odd : ¬Even x
hy_odd : ¬Even y
hxy' : 4 ∣ x + y
this : 4 ∣ x + x
⊢ 4 ∣ x + x

Next tactic:
exact this
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
    

Proof state:
case refine'_2.zero
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
this : 2 * 2 ∣ 2 * x
⊢ False

Next tactic:
have : 2 ∣ x := (mul_dvd_mul_iff_left (by norm_num)).mp this
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 

Proof state:
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
this : 2 * 2 ∣ 2 * x
⊢ 2 ≠ 0

Next tactic:
norm_num
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
    

Proof state:
case refine'_2.zero
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
hxy' : 2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero
this✝ : 2 * 2 ∣ 2 * x
this : 2 ∣ x
⊢ False

Next tactic:
contradiction
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
  

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
⊢ ¬2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.succ i + y ^ 2 ^ Nat.succ i

Next tactic:
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
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
    

Proof state:
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
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ ¬2 ^ (1 + 1) ∣ x ^ 2 ^ Nat.succ i + y ^ 2 ^ Nat.succ i

Next tactic:
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]
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 

Proof state:
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
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ 2 ^ (1 + 1) = 4

Next tactic:
norm_num
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]
    

Proof state:
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
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ ¬(1 + 1) % 4 = 0

Next tactic:
decide