have : -p ≤ abs p := calc
  _ ≃ -p               := eqv_refl
  _ ≃ (-1) * p         := eqv_symm mul_neg_one
  _ ≃ p * (-1)         := mul_comm
  _ ≃ p * sgn (-1 : ℚ) := mul_substR (eqv_symm this)
  _ ≤ p * sgn p        := mul_sgn_self_max
  _ ≃ abs p            := eqv_symm abs_sgn
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
p : ℚ
this : from_integer (sgn (-1)) ≃ -1
⊢ -abs p ≤ p
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
p : ℚ
this✝ : from_integer (sgn (-1)) ≃ -1
this : -p ≤ abs p
⊢ -abs p ≤ p