zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_40.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {x y : E} {l : NNReal} (mp : m > 0)
  (hf : ∀ (x : E), HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f') (hl : ↑l > 0) (cov : Convex ℝ univ)
  (hsc :
    ∀ ⦃x : E⦄,
      x ∈ univ →
        ∀ ⦃y : E⦄,
          y ∈ univ →
            ∀ ⦃a b : ℝ⦄,
              0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)) :
  let phi := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x;
  ConvexOn ℝ univ phi →
    let g := fun x => f x - m / 2 * ‖x‖ ^ 2;
    let g' := fun x => f' x - m • x;
    let h := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x;
    (∀ (x : E), HasGradientAt g (g' x) x) →
      ConvexOn ℝ univ g →
        ConvexOn ℝ univ h →
          ¬0 < ↑l - m →
            let alpha := f' x - f' y;
            let beta := x - y;
            m * ‖beta‖ ^ 2 ≤ inner alpha beta →
              1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta →
                0 < m + ↑l →
                  ∀ (u v : E), m * ‖v‖ ^ 2 ≤ inner u v → 1 / ↑l * ‖u‖ ^ 2 ≤ inner u v → 0 ≤ ‖v‖ ^ 2 → 0 < m := sorry

Tactic state before:
case a0
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
x y : E
l : NNReal
mp : m > 0
hf : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
hl : ↑l > 0
cov : Convex ℝ univ
hsc :
  ∀ ⦃x : E⦄,
    x ∈ univ →
      ∀ ⦃y : E⦄,
        y ∈ univ →
          ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)
phi : E → ℝ := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x
convphi : ConvexOn ℝ univ phi
g : E → ℝ := fun x => f x - m / 2 * ‖x‖ ^ 2
g' : E → E := fun x => f' x - m • x
h : E → ℝ := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x
gderiv : ∀ (x : E), HasGradientAt g (g' x) x
convg : ConvexOn ℝ univ g
convh : ConvexOn ℝ univ h
coef : ¬0 < ↑l - m
alpha : E := f' x - f' y
beta : E := x - y
eq1 : m * ‖beta‖ ^ 2 ≤ inner alpha beta
eq2 : 1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta
mlpos : 0 < m + ↑l
u v : E
h1 : m * ‖v‖ ^ 2 ≤ inner u v
h2 : 1 / ↑l * ‖u‖ ^ 2 ≤ inner u v
this : 0 ≤ ‖v‖ ^ 2
⊢ 0 < m

Tactic:
apply mp

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_43.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {x y : E} {l : NNReal} (mp : m > 0)
  (hf : ∀ (x : E), HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f') (hl : ↑l > 0) (cov : Convex ℝ univ)
  (hsc :
    ∀ ⦃x : E⦄,
      x ∈ univ →
        ∀ ⦃y : E⦄,
          y ∈ univ →
            ∀ ⦃a b : ℝ⦄,
              0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)) :
  let phi := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x;
  ConvexOn ℝ univ phi →
    let g := fun x => f x - m / 2 * ‖x‖ ^ 2;
    let g' := fun x => f' x - m • x;
    let h := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x;
    (∀ (x : E), HasGradientAt g (g' x) x) →
      ConvexOn ℝ univ g →
        ConvexOn ℝ univ h →
          ¬0 < ↑l - m →
            let alpha := f' x - f' y;
            let beta := x - y;
            m * ‖beta‖ ^ 2 ≤ inner alpha beta →
              1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta →
                0 < m + ↑l → ∀ (u v : E), m * ‖v‖ ^ 2 ≤ inner u v → 1 / ↑l * ‖u‖ ^ 2 ≤ inner u v → 0 < m := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
x y : E
l : NNReal
mp : m > 0
hf : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
hl : ↑l > 0
cov : Convex ℝ univ
hsc :
  ∀ ⦃x : E⦄,
    x ∈ univ →
      ∀ ⦃y : E⦄,
        y ∈ univ →
          ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)
phi : E → ℝ := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x
convphi : ConvexOn ℝ univ phi
g : E → ℝ := fun x => f x - m / 2 * ‖x‖ ^ 2
g' : E → E := fun x => f' x - m • x
h : E → ℝ := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x
gderiv : ∀ (x : E), HasGradientAt g (g' x) x
convg : ConvexOn ℝ univ g
convh : ConvexOn ℝ univ h
coef : ¬0 < ↑l - m
alpha : E := f' x - f' y
beta : E := x - y
eq1 : m * ‖beta‖ ^ 2 ≤ inner alpha beta
eq2 : 1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta
mlpos : 0 < m + ↑l
u v : E
h1 : m * ‖v‖ ^ 2 ≤ inner u v
h2 : 1 / ↑l * ‖u‖ ^ 2 ≤ inner u v
⊢ 0 < m

Tactic:
apply mp

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_46.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {x y : E} {l : NNReal} (mp : m > 0)
  (hf : ∀ (x : E), HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f') (hl : ↑l > 0) (cov : Convex ℝ univ)
  (hsc :
    ∀ ⦃x : E⦄,
      x ∈ univ →
        ∀ ⦃y : E⦄,
          y ∈ univ →
            ∀ ⦃a b : ℝ⦄,
              0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)) :
  let phi := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x;
  ConvexOn ℝ univ phi →
    let g := fun x => f x - m / 2 * ‖x‖ ^ 2;
    let g' := fun x => f' x - m • x;
    let h := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x;
    (∀ (x : E), HasGradientAt g (g' x) x) →
      ConvexOn ℝ univ g →
        ConvexOn ℝ univ h →
          ¬0 < ↑l - m →
            let alpha := f' x - f' y;
            let beta := x - y;
            m * ‖beta‖ ^ 2 ≤ inner alpha beta →
              1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta →
                0 < m + ↑l →
                  ∀ (u v : E),
                    m * ‖v‖ ^ 2 ≤ inner u v →
                      m * ↑l * ‖v‖ ^ 2 ≤ m * inner u v → ‖u‖ ^ 2 / ↑l ≤ inner u v → 0 < ↑l := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
x y : E
l : NNReal
mp : m > 0
hf : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
hl : ↑l > 0
cov : Convex ℝ univ
hsc :
  ∀ ⦃x : E⦄,
    x ∈ univ →
      ∀ ⦃y : E⦄,
        y ∈ univ →
          ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)
phi : E → ℝ := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x
convphi : ConvexOn ℝ univ phi
g : E → ℝ := fun x => f x - m / 2 * ‖x‖ ^ 2
g' : E → E := fun x => f' x - m • x
h : E → ℝ := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x
gderiv : ∀ (x : E), HasGradientAt g (g' x) x
convg : ConvexOn ℝ univ g
convh : ConvexOn ℝ univ h
coef : ¬0 < ↑l - m
alpha : E := f' x - f' y
beta : E := x - y
eq1 : m * ‖beta‖ ^ 2 ≤ inner alpha beta
eq2 : 1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta
mlpos : 0 < m + ↑l
u v : E
h1 : m * ‖v‖ ^ 2 ≤ inner u v
eq4 : m * ↑l * ‖v‖ ^ 2 ≤ m * inner u v
h2 : ‖u‖ ^ 2 / ↑l ≤ inner u v
⊢ 0 < ↑l

Tactic:
apply hl

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_49.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {x y : E} {l : NNReal} (mp : m > 0)
  (hf : ∀ (x : E), HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f') (hl : ↑l > 0) (cov : Convex ℝ univ)
  (hsc :
    ∀ ⦃x : E⦄,
      x ∈ univ →
        ∀ ⦃y : E⦄,
          y ∈ univ →
            ∀ ⦃a b : ℝ⦄,
              0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)) :
  let phi := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x;
  ConvexOn ℝ univ phi →
    let g := fun x => f x - m / 2 * ‖x‖ ^ 2;
    let g' := fun x => f' x - m • x;
    let h := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x;
    (∀ (x : E), HasGradientAt g (g' x) x) →
      ConvexOn ℝ univ g →
        ConvexOn ℝ univ h →
          ¬0 < ↑l - m →
            let alpha := f' x - f' y;
            let beta := x - y;
            m * ‖beta‖ ^ 2 ≤ inner alpha beta →
              1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta →
                0 < m + ↑l →
                  (∀ (u v : E),
                      m * ‖v‖ ^ 2 ≤ inner u v →
                        1 / ↑l * ‖u‖ ^ 2 ≤ inner u v →
                          inner u v ≥ m * ↑l / (m + ↑l) * ‖v‖ ^ 2 + 1 / (m + ↑l) * ‖u‖ ^ 2) →
                    m * ‖beta‖ ^ 2 ≤ inner alpha beta := sorry

Tactic state before:
case neg.h1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
x y : E
l : NNReal
mp : m > 0
hf : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
hl : ↑l > 0
cov : Convex ℝ univ
hsc :
  ∀ ⦃x : E⦄,
    x ∈ univ →
      ∀ ⦃y : E⦄,
        y ∈ univ →
          ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y - a * b * (m / 2 * ‖x - y‖ ^ 2)
phi : E → ℝ := fun x => ↑l / 2 * ‖x‖ ^ 2 - f x
convphi : ConvexOn ℝ univ phi
g : E → ℝ := fun x => f x - m / 2 * ‖x‖ ^ 2
g' : E → E := fun x => f' x - m • x
h : E → ℝ := fun x => (↑l - m) / 2 * ‖x‖ ^ 2 - g x
gderiv : ∀ (x : E), HasGradientAt g (g' x) x
convg : ConvexOn ℝ univ g
convh : ConvexOn ℝ univ h
coef : ¬0 < ↑l - m
alpha : E := f' x - f' y
beta : E := x - y
eq1 : m * ‖beta‖ ^ 2 ≤ inner alpha beta
eq2 : 1 / ↑l * ‖alpha‖ ^ 2 ≤ inner alpha beta
mlpos : 0 < m + ↑l
eq3 :
  ∀ (u v : E),
    m * ‖v‖ ^ 2 ≤ inner u v →
      1 / ↑l * ‖u‖ ^ 2 ≤ inner u v → inner u v ≥ m * ↑l / (m + ↑l) * ‖v‖ ^ 2 + 1 / (m + ↑l) * ‖u‖ ^ 2
⊢ m * ‖beta‖ ^ 2 ≤ inner alpha beta

Tactic:
apply eq1

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_50.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm : E} {l : NNReal} (h₁ : ∀ (x : E), HasGradientAt f (f' x) x)
  (h₂ : LipschitzWith l f') (min : IsMinOn f univ xm) (hl : ↑l > 0) (x : E) :
  1 / (2 * ↑l) * ‖f' x‖ ^ 2 ≤ f x - f xm := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm : E
l : NNReal
h₁ : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
min : IsMinOn f univ xm
hl : ↑l > 0
⊢ ∀ (x : E), 1 / (2 * ↑l) * ‖f' x‖ ^ 2 ≤ f x - f xm

Tactic:
apply lipschitz_minima_lower_bound h₁ h₂ min hl

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_51.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm : E} {l : NNReal} (h₁ : ∀ (x : E), HasGradientAt f (f' x) x)
  (h₂ : LipschitzWith l f') (min : IsMinOn f univ xm) (hl : ↑l > 0) (eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0) (x : 0 < 2 * ↑l)
  (h_original_goal : 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)) : 0 ≤ ‖f' xm‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm : E
l : NNReal
h₁ : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
min : IsMinOn f univ xm
hl : ↑l > 0
eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
x✝ : 0 < 2 * ↑l
⊢ 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)

Tactic:
apply div_nonneg

Tactic state after:
[case ha
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm : E
 l : NNReal
 h₁ : ∀ (x : E), HasGradientAt f (f' x) x
 h₂ : LipschitzWith l f'
 min : IsMinOn f univ xm
 hl : ↑l > 0
 eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
 x✝ : 0 < 2 * ↑l
 ⊢ 0 ≤ ‖f' xm‖ ^ 2,
 case hb
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm : E
 l : NNReal
 h₁ : ∀ (x : E), HasGradientAt f (f' x) x
 h₂ : LipschitzWith l f'
 min : IsMinOn f univ xm
 hl : ↑l > 0
 eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
 x✝ : 0 < 2 * ↑l
 ⊢ 0 ≤ 2 * ↑l]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_52.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm : E} {l : NNReal} (h₁ : ∀ (x : E), HasGradientAt f (f' x) x)
  (h₂ : LipschitzWith l f') (min : IsMinOn f univ xm) (hl : ↑l > 0) (eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0) (x : 0 < 2 * ↑l)
  (h_original_goal : 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)) : 0 ≤ 2 * ↑l := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm : E
l : NNReal
h₁ : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
min : IsMinOn f univ xm
hl : ↑l > 0
eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
x✝ : 0 < 2 * ↑l
⊢ 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)

Tactic:
apply div_nonneg

Tactic state after:
[case ha
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm : E
 l : NNReal
 h₁ : ∀ (x : E), HasGradientAt f (f' x) x
 h₂ : LipschitzWith l f'
 min : IsMinOn f univ xm
 hl : ↑l > 0
 eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
 x✝ : 0 < 2 * ↑l
 ⊢ 0 ≤ ‖f' xm‖ ^ 2,
 case hb
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 f' : E → E
 xm : E
 l : NNReal
 h₁ : ∀ (x : E), HasGradientAt f (f' x) x
 h₂ : LipschitzWith l f'
 min : IsMinOn f univ xm
 hl : ↑l > 0
 eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
 x✝ : 0 < 2 * ↑l
 ⊢ 0 ≤ 2 * ↑l]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_53.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm : E} {l : NNReal} (h₁ : ∀ (x : E), HasGradientAt f (f' x) x)
  (h₂ : LipschitzWith l f') (min : IsMinOn f univ xm) (hl : ↑l > 0) (eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0) (x : 0 < 2 * ↑l)
  (eq3 : 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)) (eq4 : f' xm = 0) : f' xm = 0 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm : E
l : NNReal
h₁ : ∀ (x : E), HasGradientAt f (f' x) x
h₂ : LipschitzWith l f'
min : IsMinOn f univ xm
hl : ↑l > 0
eq1 : ‖f' xm‖ ^ 2 / (2 * ↑l) ≤ 0
x✝ : 0 < 2 * ↑l
eq3 : 0 ≤ ‖f' xm‖ ^ 2 / (2 * ↑l)
eq4 : f' xm = 0
⊢ f' xm = 0

Tactic:
exact eq4

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_54.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hm : m > 0) (min : IsMinOn f univ xm)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
  (hsc : StrongConvexOn univ m f) (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0) (k : ℕ)
  (h_original_goal :
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤
      (1 -
          Gradient_Descent_fix_stepsize.a f f' x₀ *
            (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
        ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2) :
  ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
            Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
          xm‖ ^
      2 ≤
    (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
      ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hm : m > 0
min : IsMinOn f univ xm
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
hsc : StrongConvexOn univ m f
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ (k + 1) - xm‖ ^ 2 ≤
    (1 -
        Gradient_Descent_fix_stepsize.a f f' x₀ *
          (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
      ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2

Tactic:
rw [alg.update k]

Tactic state after:
[E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 m : ℝ
 f' : E → E
 xm x₀ : E
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hm : m > 0
 min : IsMinOn f univ xm
 step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
 hsc : StrongConvexOn univ m f
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
 k : ℕ
 ⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
             Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
           xm‖ ^
       2 ≤
     (1 -
         Gradient_Descent_fix_stepsize.a f f' x₀ *
           (2 * m * ↑(Gradient_Descent_fix_stepsize.l f f' x₀) / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))) *
       ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm‖ ^ 2]
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_55.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hm : m > 0) (min : IsMinOn f univ xm)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
  (hsc : StrongConvexOn univ m f) (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0) (k : ℕ) :
  ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
            Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
          xm‖ ^
      2 =
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm -
          Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^
      2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hm : m > 0
min : IsMinOn f univ xm
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
hsc : StrongConvexOn univ m f
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
⊢ ‖Gradient_Descent_fix_stepsize.x f f' x₀ k -
            Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) -
          xm‖ ^
      2 =
    ‖Gradient_Descent_fix_stepsize.x f f' x₀ k - xm -
          Gradient_Descent_fix_stepsize.a f f' x₀ • f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^
      2

Tactic:
rw [sub_right_comm]

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_59.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hm : m > 0) (min : IsMinOn f univ xm)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
  (hsc : StrongConvexOn univ m f) (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0) (k : ℕ) :
  (|Gradient_Descent_fix_stepsize.a f f' x₀| * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖) ^ 2 =
    Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hm : m > 0
min : IsMinOn f univ xm
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
hsc : StrongConvexOn univ m f
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
⊢ (|Gradient_Descent_fix_stepsize.a f f' x₀| * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖) ^ 2 =
    Gradient_Descent_fix_stepsize.a f f' x₀ ^ 2 * ‖f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)‖ ^ 2

Tactic:
rw [mul_pow, sq_abs]

Tactic state after:
import Mathlib
import Optlib


open Set

theorem extracted_formal_statement_60.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {m : ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hm : m > 0) (min : IsMinOn f univ xm)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀)))
  (hsc : StrongConvexOn univ m f) (this_1 : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f')
  (this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0) (k : ℕ)
  (h_original_goal :
    f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm) :
  f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) + f' xm = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
m : ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hm : m > 0
min : IsMinOn f univ xm
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
hsc : StrongConvexOn univ m f
this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
k : ℕ
⊢ f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) - f' xm

Tactic:
apply eq_sub_of_add_eq

Tactic state after:
[case h
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 f : E → ℝ
 m : ℝ
 f' : E → E
 xm x₀ : E
 alg : Gradient_Descent_fix_stepsize f f' x₀
 hm : m > 0
 min : IsMinOn f univ xm
 step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 2 / (m + ↑(Gradient_Descent_fix_stepsize.l f f' x₀))
 hsc : StrongConvexOn univ m f
 this✝ : LipschitzWith (Gradient_Descent_fix_stepsize.l f f' x₀) f'
 this : ↑(Gradient_Descent_fix_stepsize.l f f' x₀) > 0
 k : ℕ
 ⊢ f' (Gradient_Descent_fix_stepsize.x f f' x₀ k) + f' xm = f' (Gradient_Descent_fix_stepsize.x f f' x₀ k)]