zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_130 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      0 < x i → 0 < sign (x i) * (|x i| - t * μ) → 0 < |x i| - t * μ := sorry

Tactic state before:
case hr
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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : 0 < x i
⊢ 0 < sign (x i) * (|x i| - t * μ)

Tactic:
apply mul_pos

Tactic state after:
[case hr.ha
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : 0 < x i
 ⊢ 0 < sign (x i),
 case hr.hb
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : 0 < x i
 ⊢ 0 < |x i| - t * μ]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_131 : 0 < 1 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : 0 < x i
⊢ 0 < 1

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_132 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 → t * μ < |x i| → (|x i| - t * μ) ⊔ 0 = |x i| - t * μ → 0 < x i → sign (x i) = 1 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : 0 < x i
⊢ sign (x i) = 1

Tactic:
apply Real.sign_of_pos hx

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_134 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      0 < x i → (sign (x i) * (|x i| - t * μ)).sign = 1 → |x i| = x i → 0 ≤ x i := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : 0 < x i
eq2 : (sign (x i) * (|x i| - t * μ)).sign = 1
⊢ |x i| = x i

Tactic:
apply abs_eq_self.mpr

Tactic 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : 0 < x i
 eq2 : (sign (x i) * (|x i| - t * μ)).sign = 1
 ⊢ 0 ≤ x i]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_135 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      0 < x i →
                        (sign (x i) * (|x i| - t * μ)).sign = 1 → |x i| = x i → sign (x i) = 1 → 0 < x i := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : 0 < x i
eq2 : (sign (x i) * (|x i| - t * μ)).sign = 1
eq_posxi : |x i| = x i
⊢ sign (x i) = 1

Tactic:
apply Real.sign_of_pos

Tactic state after:
[case hr
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : 0 < x i
 eq2 : (sign (x i) * (|x i| - t * μ)).sign = 1
 eq_posxi : |x i| = x i
 ⊢ 0 < x i]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_137 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i →
                        x i < 0 → (sign (x i) * (|x i| - t * μ)).sign = -1 → sign (x i) * (|x i| - t * μ) < 0 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
⊢ (sign (x i) * (|x i| - t * μ)).sign = -1

Tactic:
apply Real.sign_of_neg

Tactic state after:
[case hr
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ sign (x i) * (|x i| - t * μ) < 0]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_138 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i → x i < 0 → sign (x i) * (|x i| - t * μ) < 0 → sign (x i) < 0 := sorry

Tactic state before:
case hr
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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
⊢ sign (x i) * (|x i| - t * μ) < 0

Tactic:
apply mul_neg_of_neg_of_pos

Tactic state after:
[case hr.ha
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ sign (x i) < 0,
 case hr.hb
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ 0 < |x i| - t * μ]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_139 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i → x i < 0 → sign (x i) * (|x i| - t * μ) < 0 → 0 < |x i| - t * μ := sorry

Tactic state before:
case hr
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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
⊢ sign (x i) * (|x i| - t * μ) < 0

Tactic:
apply mul_neg_of_neg_of_pos

Tactic state after:
[case hr.ha
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ sign (x i) < 0,
 case hr.hb
 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ 0 < |x i| - t * μ]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_140 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| → (|x i| - t * μ) ⊔ 0 = |x i| - t * μ → ¬0 < x i → x i < 0 → sign (x i) = -1 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
⊢ sign (x i) = -1

Tactic:
apply Real.sign_of_neg xneg

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_142 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i → x i < 0 → (sign (x i) * (|x i| - t * μ)).sign = -1 → |x i| = -x i → x i ≤ 0 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
eq2 : (sign (x i) * (|x i| - t * μ)).sign = -1
⊢ |x i| = -x i

Tactic:
apply abs_eq_neg_self.mpr

Tactic 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
 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)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 eq2 : (sign (x i) * (|x i| - t * μ)).sign = -1
 ⊢ x i ≤ 0]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_143 {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 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i →
                        x i < 0 → (sign (x i) * (|x i| - t * μ)).sign = -1 → |x i| = -x i → sign (x i) = -1 := sorry

Tactic 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
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)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
eq2 : (sign (x i) * (|x i| - t * μ)).sign = -1
eq_negxi : |x i| = -x i
⊢ sign (x i) = -1

Tactic:
apply Real.sign_of_neg xneg

Tactic state after: