zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_120 {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) :
  let g := (t * μ) • fun x => ∑ i, ‖x i‖;
  g = t • h →
    0 < t * μ →
      ∀ (y : EuclideanSpace ℝ (Fin ↑n)),
        ∀ i ∈ Finset.univ,
          let abs_subg := SubderivAt_abs (xm i);
          xm i = 0 → x i = 0 ∨ |x i| ≤ t * μ → |x i| ≤ t * μ → |y i| * μ⁻¹ * t⁻¹ * t * μ = |y i| → t ≠ 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
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
abs_subg : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
hxm : xm i = 0
minpoint : x i = 0 ∨ |x i| ≤ t * μ
aux : |x i| ≤ t * μ
⊢ |y i| * μ⁻¹ * t⁻¹ * t * μ = |y i|

Tactic:
rw [mul_assoc _ (t⁻¹) t, inv_mul_cancel₀, mul_one]

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
 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
 abs_subg : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
 hxm : xm i = 0
 minpoint : x i = 0 ∨ |x i| ≤ t * μ
 aux : |x i| ≤ t * μ
 ⊢ |y i| * μ⁻¹ * μ = |y i|,
 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
 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
 abs_subg : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
 hxm : xm i = 0
 minpoint : x i = 0 ∨ |x i| ≤ t * μ
 aux : |x i| ≤ t * μ
 ⊢ t ≠ 0]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_121 {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,
          let abs_subg := SubderivAt_abs (xm i);
          ¬xm i = 0 →
            |xm i| + (y i - xm i) * (μ⁻¹ * t⁻¹ * (x i - xm i)) ≤ |y i| →
              ¬(xm i = 0 ∧ SubderivAt abs (xm i) = Set.Icc (-1) 1) := sorry

Tactic state before:
case neg
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
abs_subg : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
hxm : ¬xm i = 0
⊢ |xm i| + (y i - xm i) * (μ⁻¹ * t⁻¹ * (x i - xm i)) ≤ |y i|

Tactic:
rw [eq_ite_iff, or_iff_right] at abs_subg

Tactic state after:
[case neg
 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
 abs_subg : ¬xm i = 0 ∧ SubderivAt abs (xm i) = {sign (xm i)}
 hxm : ¬xm i = 0
 ⊢ |xm i| + (y i - xm i) * (μ⁻¹ * t⁻¹ * (x i - xm i)) ≤ |y i|,
 case neg
 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
 abs_subg : xm i = 0 ∧ SubderivAt abs (xm i) = Set.Icc (-1) 1 ∨ ¬xm i = 0 ∧ SubderivAt abs (xm i) = {sign (xm i)}
 hxm : ¬xm i = 0
 ⊢ ¬(xm i = 0 ∧ SubderivAt abs (xm i) = Set.Icc (-1) 1)]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_122 {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 →
      ∀ i ∈ Finset.univ,
        ¬xm i = 0 →
          ¬xm i = 0 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              sgnxm ∈ SubderivAt abs (xm i) → sgnxm ∈ {sign (xm 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
hxm : ¬xm i = 0
left✝ : ¬xm i = 0
abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
sgnxm : ℝ := sign (xm i)
⊢ sgnxm ∈ SubderivAt abs (xm i)

Tactic:
rw [abs_subg]

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
 hxm : ¬xm i = 0
 left✝ : ¬xm i = 0
 abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
 sgnxm : ℝ := sign (xm i)
 ⊢ sgnxm ∈ {sign (xm i)}]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_123 {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 →
      ∀ i ∈ Finset.univ,
        ¬xm i = 0 →
          ¬xm i = 0 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              sgnxm ∈ {sign (xm 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
hxm : ¬xm i = 0
left✝ : ¬xm i = 0
abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
sgnxm : ℝ := sign (xm i)
⊢ sgnxm ∈ {sign (xm i)}

Tactic:
simp [sgnxm]

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_124 {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 - xm i) * sign (xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm 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
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]

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
 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 - xm i) * sign (xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i)]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_125 {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) →
                  (y i - xm i) * sign (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

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
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 - xm i) * sign (xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i)

Tactic:
rw [minpoint]

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
 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))]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_126 {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| - t * μ := 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|
⊢ (|x i| - t * μ) ⊔ 0 = |x i| - t * μ

Tactic:
apply max_eq_left

Tactic state after:
[case h
 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|
 ⊢ 0 ≤ |x i| - t * μ]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_127 {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 →
                      (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)) →
                        (y i - sign (x i) * (|x i| - t * μ)) * (sign (x i) * (|x i| - t * μ)).sign =
                          μ⁻¹ * t⁻¹ * (x i - sign (x i) * (|x i| - t * μ)) *
                            (y i - sign (x i) * (|x i| - t * μ)) := sorry

Tactic state before:
case intro
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
⊢ (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))

Tactic:
rw [eq1]

Tactic state after:
[case intro
 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
 ⊢ (y i - sign (x i) * (|x i| - t * μ)) * (sign (x i) * (|x i| - t * μ)).sign =
     μ⁻¹ * t⁻¹ * (x i - sign (x i) * (|x i| - t * μ)) * (y i - sign (x i) * (|x i| - t * μ))]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_128 {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 → 0 < sign (x i) * (|x i| - t * μ) := 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) * (|x i| - t * μ)).sign = 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
 ⊢ 0 < sign (x i) * (|x i| - t * μ)]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_129 {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 < sign (x i) := 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]