theorem extracted_formal_statement_153 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} {b : Fin ↑m → ℝ} {μ : ℝ} {μpos : 0 < μ}
  {Ane0 : A ≠ 0} {x₀ : EuclideanSpace ℝ (Fin ↑n)} (p : LASSO A b μ μpos Ane0 x₀) (x y : EuclideanSpace ℝ (Fin ↑n))
  (h_original_goal : ‖Aᵀ *ᵥ A *ᵥ (x - y)‖ ≤ ↑p.L * ‖x - y‖) :
  ‖Aᵀ *ᵥ A *ᵥ (x - y)‖ ≤ ↑‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) (Aᵀ * A)‖₊ * ‖x - y‖ := sorry

State before:
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
b : Fin ↑m → ℝ
μ : ℝ
μpos : 0 < μ
Ane0 : A ≠ 0
x₀ : EuclideanSpace ℝ (Fin ↑n)
p : LASSO A b μ μpos Ane0 x₀
x y : EuclideanSpace ℝ (Fin ↑n)
⊢ ‖Aᵀ *ᵥ A *ᵥ (x - y)‖ ≤ ↑p.L * ‖x - y‖

Tactic: rw [p.Leq]

State after:
[n m : ℕ+
 A : Matrix (Fin ↑m) (Fin ↑n) ℝ
 b : Fin ↑m → ℝ
 μ : ℝ
 μpos : 0 < μ
 Ane0 : A ≠ 0
 x₀ : EuclideanSpace ℝ (Fin ↑n)
 p : LASSO A b μ μpos Ane0 x₀
 x y : EuclideanSpace ℝ (Fin ↑n)
 ⊢ ‖Aᵀ *ᵥ A *ᵥ (x - y)‖ ≤ ↑‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) (Aᵀ * A)‖₊ * ‖x - y‖]

Relevant lemmas: