calc
  _ = (p^(0:ℕ))⁻¹ := rfl
  _ ≃ 1⁻¹         := recip_subst pow_zero
  _ ≃ 1           := recip_sqrt1
  _ ≃ (p⁻¹)^(0:ℕ) := eqv_symm pow_zero
ℕ ℤ : Type
inst✝¹³ : Natural ℕ
inst✝¹² : Integer ℤ
ℚ : Type
inst✝¹¹ : Core ℚ
inst✝¹⁰ : Addition ℚ
inst✝⁹ : Multiplication ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Reciprocation ℚ
inst✝⁵ : Division ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Order ℚ
inst✝² : Metric ℚ
inst✝¹ : Natural.Exponentiation ℕ fun x x_1 => x * x_1
p : ℚ
n : ℕ
inst✝ : AP (p ≄ 0)
⊢ (p ^ 0)⁻¹ ≃ p⁻¹ ^ 0
no goals