-- Mathlib/NumberTheory/Multiplicity.lean
theorem Int.two_pow_sub_pow' {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
    emultiplicity 2 (x ^ n - y ^ n) = emultiplicity 2 (x - y) + emultiplicity (2 : ℤ) n

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

Tactic:
rwa [← Int.not_even_iff_odd, even_iff_two_dvd]

Result:
no goals

Proof state:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ ↑k

Tactic:
norm_cast

Result:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ k

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

Tactic:
simpa using hx_odd.sub_even hxy_even

Result:
no goals

Proof state:
case succ.intro.intro.hxy
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)

Tactic:
simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow

Result:
no goals

Proof state:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ k

Tactic:
contrapose! hpn

Result:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1

Proof state:
case succ.intro.intro.hp
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ Prime 2

Tactic:
exact Int.prime_two

Result:
no goals

Proof state:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ multiplicity 2 (n + 1) * 2 ∣ 2 ^ multiplicity 2 (n + 1) * k

Tactic:
exact mul_dvd_mul_left _ hpn

Result:
no goals

Proof state:
case zero
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ emultiplicity 2 (x ^ 0 - y ^ 0) = emultiplicity 2 (x - y) + emultiplicity 2 ↑0

Tactic:
simp only [pow_zero, sub_self, emultiplicity_zero, Int.ofNat_zero, add_top]

Result:
no goals

Proof state:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1

Tactic:
rw [pow_succ]

Result:
case succ.intro.intro.hn
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 : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ multiplicity 2 (n + 1) * 2 ∣ n + 1

Proof state:
case succ
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 : FiniteMultiplicity 2 n.succ
⊢ emultiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

Tactic:
simp only [Nat.succ_eq_add_one] at h

Result:
case succ
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 : FiniteMultiplicity 2 (n + 1)
⊢ emultiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

Proof state:
case succ.intro.intro
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 (n + 1)

Tactic:
rw [h.emultiplicity_eq_multiplicity]

Result:
no goals

Proof state:
case succ.intro.intro
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

Tactic:
norm_cast

Result:
case succ.intro.intro
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 : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 (n + 1)

Proof state:
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

Tactic:
norm_num

Result:
no goals

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

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

Result:
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
⊢ emultiplicity 2 (x ^ n - y ^ n) = emultiplicity 2 (x - y) + emultiplicity 2 ↑n

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

Tactic:
decide

Result:
no goals

Proof state:
case succ
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
⊢ emultiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

Tactic:
have h : FiniteMultiplicity 2 n.succ := Nat.finiteMultiplicity_iff.mpr ⟨by norm_num, n.succ_pos⟩

Result:
case succ
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 : FiniteMultiplicity 2 n.succ
⊢ emultiplicity 2 (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)