theorem extracted_formal_statement_26 {n : ℕ+} {t μ : ℝ} {h : EuclideanSpace ℝ (Fin ↑n) → ℝ}
  (lasso : h = fun y => μ • ∑ i, ‖y i‖) (x xm : EuclideanSpace ℝ (Fin ↑n)) (tpos : 0 < t) (μpos : 0 < μ)
  (minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)) :
  let g := (t * μ) • fun x => ∑ i, ‖x i‖;
  g = t • h →
    0 < t * μ →
      ∀ (y : EuclideanSpace ℝ (Fin ↑n)),
        ∀ i ∈ Finset.univ,
          ¬xm i = 0 →
            ¬xm i = 0 →
              SubderivAt abs (xm i) = {sign (xm i)} →
                let sgnxm := sign (xm i);
                |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                  inner sgnxm (y i - xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i) →
                    (y i - sign (x i) * ((|x i| - t * μ) ⊔ 0)) * (sign (x i) * ((|x i| - t * μ) ⊔ 0)).sign =
                      μ⁻¹ * t⁻¹ * (x i - sign (x i) * ((|x i| - t * μ) ⊔ 0)) *
                        (y i - sign (x i) * ((|x i| - t * μ) ⊔ 0)) := sorry

State before:
n : ℕ+
t μ : ℝ
h : EuclideanSpace ℝ (Fin ↑n) → ℝ
lasso : h = fun y => μ • ∑ i, ‖y i‖
x xm : EuclideanSpace ℝ (Fin ↑n)
tpos : 0 < t
μpos : 0 < μ
minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)
g : EuclideanSpace ℝ (Fin ↑n) → ℝ := (t * μ) • fun x => ∑ i, ‖x i‖
geqth : g = t • h
tμpos : 0 < t * μ
y : EuclideanSpace ℝ (Fin ↑n)
i : Fin ↑n
a✝ : i ∈ Finset.univ
hxm : ¬xm i = 0
left✝ : ¬xm i = 0
abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
sgnxm : ℝ := sign (xm i)
aux : |y i| ≥ |xm i| + inner sgnxm (y i - xm i)
⊢ inner sgnxm (y i - xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i)

Tactic: simp [sgnxm]
    rw [minpoint]
    simp

State after:
[n : ℕ+
 t μ : ℝ
 h : EuclideanSpace ℝ (Fin ↑n) → ℝ
 lasso : h = fun y => μ • ∑ i, ‖y i‖
 x xm : EuclideanSpace ℝ (Fin ↑n)
 tpos : 0 < t
 μpos : 0 < μ
 minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)
 g : EuclideanSpace ℝ (Fin ↑n) → ℝ := (t * μ) • fun x => ∑ i, ‖x i‖
 geqth : g = t • h
 tμpos : 0 < t * μ
 y : EuclideanSpace ℝ (Fin ↑n)
 i : Fin ↑n
 a✝ : i ∈ Finset.univ
 hxm : ¬xm i = 0
 left✝ : ¬xm i = 0
 abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
 sgnxm : ℝ := sign (xm i)
 aux : |y i| ≥ |xm i| + inner sgnxm (y i - xm i)
 ⊢ (y i - sign (x i) * ((|x i| - t * μ) ⊔ 0)) * (sign (x i) * ((|x i| - t * μ) ⊔ 0)).sign =
     μ⁻¹ * t⁻¹ * (x i - sign (x i) * ((|x i| - t * μ) ⊔ 0)) * (y i - sign (x i) * ((|x i| - t * μ) ⊔ 0))]

Relevant lemmas: