have pow_reduce : ((n*k + m*j : ℕ):ℤ) - ((m*k + n*j : ℕ):ℤ) ≃ a * b := calc
  _ = ((n*k + m*j : ℕ):ℤ) - ((m*k + n*j : ℕ):ℤ) := rfl
  _ ≃ ((n:ℤ)*k + m*j) - ((m*k + n*j : ℕ):ℤ)     := AA.substL multi_compat
  _ ≃ ((n:ℤ)*k + m*j) - (m*k + n*j)             := AA.substR multi_compat
  _ ≃ ((n:ℤ) - m) * (k - j)                     := Rel.symm diff_expand
  _ ≃ a * (k - j)                               := AA.substL (Rel.symm a_eqv)
  _ ≃ a * b                                     := AA.substR (Rel.symm b_eqv)
ℕ ℤ : Type
inst✝¹¹ : Natural ℕ
inst✝¹⁰ : Integer ℤ
ℚ : Type
inst✝⁹ : Core ℚ
inst✝⁸ : Addition ℚ
inst✝⁷ : Multiplication ℚ
inst✝⁶ : Negation ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Natural.Exponentiation ℕ fun x x_1 => x * x_1
inst✝¹ : Exponentiation ℚ
p : ℚ
inst✝ : AP (p ≄ 0)
a b : ℤ
n m : ℕ
a_eqv : a ≃ Coe.coe n - Coe.coe m
k j : ℕ
b_eqv : b ≃ Coe.coe k - Coe.coe j
pow_expand : (p ^ a) ^ b ≃ (p ^ n) ^ k / (p ^ m) ^ k / ((p ^ n) ^ j / (p ^ m) ^ j)
pow_combine : ∀ {w x y z : ℕ}, (p ^ w) ^ x * (p ^ y) ^ z ≃ p ^ (w * x + y * z)
multi_compat : ∀ {w x y z : ℕ}, Coe.coe (w * x + y * z) ≃ Coe.coe w * Coe.coe x + Coe.coe y * Coe.coe z
diff_expand : ∀ {w x y z : ℤ}, (w - x) * (y - z) ≃ w * y + x * z - (x * y + w * z)
⊢ (p ^ a) ^ b ≃ p ^ (a * b)
ℕ ℤ : Type
inst✝¹¹ : Natural ℕ
inst✝¹⁰ : Integer ℤ
ℚ : Type
inst✝⁹ : Core ℚ
inst✝⁸ : Addition ℚ
inst✝⁷ : Multiplication ℚ
inst✝⁶ : Negation ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Natural.Exponentiation ℕ fun x x_1 => x * x_1
inst✝¹ : Exponentiation ℚ
p : ℚ
inst✝ : AP (p ≄ 0)
a b : ℤ
n m : ℕ
a_eqv : a ≃ Coe.coe n - Coe.coe m
k j : ℕ
b_eqv : b ≃ Coe.coe k - Coe.coe j
pow_expand : (p ^ a) ^ b ≃ (p ^ n) ^ k / (p ^ m) ^ k / ((p ^ n) ^ j / (p ^ m) ^ j)
pow_combine : ∀ {w x y z : ℕ}, (p ^ w) ^ x * (p ^ y) ^ z ≃ p ^ (w * x + y * z)
multi_compat : ∀ {w x y z : ℕ}, Coe.coe (w * x + y * z) ≃ Coe.coe w * Coe.coe x + Coe.coe y * Coe.coe z
diff_expand : ∀ {w x y z : ℤ}, (w - x) * (y - z) ≃ w * y + x * z - (x * y + w * z)
pow_reduce : Coe.coe (n * k + m * j) - Coe.coe (m * k + n * j) ≃ a * b
⊢ (p ^ a) ^ b ≃ p ^ (a * b)