theorem extracted_formal_statement_17 {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 : EuclideanSpace ℝ (Fin ↑n))
  (a : x ∈ Set.univ) (ε : ℝ) (εpos : ε > 0) (y : EuclideanSpace ℝ (Fin ↑n)) (ydist : ‖x - y‖ ≤ ε / ↑↑n / μ)
  (h_original_goal : ‖p.h y - p.h x‖ ≤ ε) : |μ| * |∑ x_1, (|y x_1| - |x x_1|)| ≤ ε := sorry

State before:
case h.right
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 : EuclideanSpace ℝ (Fin ↑n)
a✝ : x ∈ Set.univ
ε : ℝ
εpos : ε > 0
y : EuclideanSpace ℝ (Fin ↑n)
ydist : ‖x - y‖ ≤ ε / ↑↑n / μ
⊢ ‖p.h y - p.h x‖ ≤ ε

Tactic: rw [p.heq]
    simp
    rw [← mul_sub, ← sum_sub_distrib]
    have le (i : Fin n) : |(|y i|) - (|x i|)| ≤ ε / n / μ :=
  by
  rw [EuclideanSpace.norm_eq] at ydist; rw [Real.sqrt_le_iff] at ydist
  have aux : ‖(x - y) i‖ ^ 2 ≤ (ε / n / μ) ^ 2 := by
    calc
      ‖(x - y) i‖ ^ 2 = Finset.sum { i } fun i ↦ ‖(x - y) i‖ ^ 2 := by simp
      _ ≤ Finset.sum Finset.univ fun i ↦ ‖(x - y) i‖ ^ 2 :=
        by
        apply sum_le_sum_of_subset_of_nonneg; simp
        intro j _ _; apply sq_nonneg
      _ ≤ (ε / n / μ) ^ 2 := ydist.2
  calc
    |(|y i|) - (|x i|)| ≤ ‖(x - y) i‖ := by simp; rw [abs_sub_comm]; apply abs_abs_sub_abs_le_abs_sub
    _ ≤ (ε / n / μ) := by
      rw [sq_le_sq] at aux
      have : |ε / n / μ| = ε / n / μ := by rw [abs_eq_self]; linarith
      rw [← this]; simp at aux; exact aux
    rw [abs_mul]

State after:
[case h.right
 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 : EuclideanSpace ℝ (Fin ↑n)
 a✝ : x ∈ Set.univ
 ε : ℝ
 εpos : ε > 0
 y : EuclideanSpace ℝ (Fin ↑n)
 ydist : ‖x - y‖ ≤ ε / ↑↑n / μ
 le : ∀ (i : Fin ↑n), ||y i| - |x i|| ≤ ε / ↑↑n / μ
 ⊢ |μ| * |∑ x_1, (|y x_1| - |x x_1|)| ≤ ε]

Relevant lemmas:
error