zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_6.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀}
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ)
  (h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)) :
  f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
k : ℕ
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
k₀ : k = 0
hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)
⊢ f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ

Tactic:
simp [hk₀]
    rw [add_comm]
    apply hk₀

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_7.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀}
  (h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀))
  (hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ)
  (h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀))
  (this : f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ)
  (h_original_goal : f (subgradient_method.x f x₀ 0) - f xm ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀)) :
  ‖subgradient_method.g f x₀ 0‖ * ‖x₀ - xm‖ ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀) := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
k : ℕ
h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
k₀ : k = 0
hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)
this : f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
⊢ f (subgradient_method.x f x₀ 0) - f xm ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀)

Tactic:
apply le_trans this _
    rw [← inner_neg_right, neg_sub, alg.initial]
    apply le_trans (real_inner_le_norm _ _) _

Tactic state after:
[E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 k : ℕ
 h' : ∀ ⦃x g : E⦄, g ∈ SubderivAt f x → ‖g‖ ≤ ↑(subgradient_method.G f x₀)
 k₀ : k = 0
 hk₀ : f xm ≥ f (subgradient_method.x f x₀ 0) + ⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
 h₀' : ‖subgradient_method.g f x₀ 0‖ ≤ ↑(subgradient_method.G f x₀)
 this : f (subgradient_method.x f x₀ 0) - f xm ≤ -⟪subgradient_method.g f x₀ 0, xm - subgradient_method.x f x₀ 0⟫_ℝ
 ⊢ ‖subgradient_method.g f x₀ 0‖ * ‖x₀ - xm‖ ≤ ‖x₀ - xm‖ * ↑(subgradient_method.G f x₀)]
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_16.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} {s : ℝ}
  (hm : IsMinOn f Set.univ xm) (ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s)
  (hs : s > 0) (k : ℕ)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) :
  Nonempty { x // x ∈ Finset.range (k + 1) } := sorry

Tactic state before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
s : ℝ
hm : IsMinOn f Set.univ xm
ha' : ∀ (n : ℕ), subgradient_method.a f x₀ n * ‖subgradient_method.g f x₀ n‖ = s
hs : s > 0
k : ℕ
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}
⊢ Nonempty { x // x ∈ Finset.range (k + 1) }

Tactic:
simp
    use 0
    apply Nat.succ_pos k

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_28.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s_1 : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s_1) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t) :
  let A := ‖x₀ - xm‖ ^ 2 / ε + 1;
  let s := {x | x > ‖x₀ - xm‖ ^ 2 / ε};
  ∀ (b : ℝ), A ≤ b → b ∈ s := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
b : ℝ
hb : A ≤ b
⊢ b ∈ s

Tactic:
simp [s]
    apply lt_of_lt_of_le _ hb
    simp [A]

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_29.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s_1 : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s_1) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t) :
  let A := ‖x₀ - xm‖ ^ 2 / ε + 1;
  let s := {x | x > ‖x₀ - xm‖ ^ 2 / ε};
  (∀ (b : ℝ), A ≤ b → b ∈ s) →
    ∀ (a_1 : ℕ),
      (∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s) →
        ∀ (b : ℕ),
          a_1 ≤ b →
            ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) →
              ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε →
                2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0 := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
A : ℝ := ‖x₀ - xm‖ ^ 2 / ε + 1
s : Set ℝ := {x | x > ‖x₀ - xm‖ ^ 2 / ε}
this : ∀ (b : ℝ), A ≤ b → b ∈ s
a_1 : ℕ
hasA : ∀ (b : ℕ), a_1 ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
b : ℕ
hb : a_1 ≤ b
hb' : ‖x₀ - xm‖ ^ 2 / ε < (Finset.range (b + 1)).sum (subgradient_method.a f x₀)
h₂ : ‖x₀ - xm‖ ^ 2 < (Finset.range (b + 1)).sum (subgradient_method.a f x₀) * ε
⊢ 2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀) > 0

Tactic:
apply mul_pos
    · simp
    ·
  apply Finset.sum_pos
  · intro i _
    apply (alg.ha i)
  · simp

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_30.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t)
  (ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2)
  (hG : ¬subgradient_method.G f x₀ = 0) (hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0) (x₁ : ℝ)
  (hx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) (x₂ : ℝ)
  (hx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2) : |x₂| ≤ |x₁| + dist x₂ x₁ := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
hG : ¬subgradient_method.G f x₀ = 0
hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
c₁ : ∀ x_1 ∈ s, x_1 ∈ s
x₁ : ℝ
hx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
x₂ : ℝ
hx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2
⊢ |x₂| ≤ |x₁| + dist x₂ x₁

Tactic:
rw [← Real.dist_0_eq_abs x₁, ← Real.dist_0_eq_abs x₂, dist_comm x₂ x₁]
    rw [dist_comm x₂ 0, dist_comm x₁ 0]
    apply dist_triangle

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_33.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t)
  (ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2)
  (hG : ¬subgradient_method.G f x₀ = 0) (hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0) :
  0 < 2 * ↑(subgradient_method.G f x₀) ^ 2 := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
hG : ¬subgradient_method.G f x₀ = 0
hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
c₁ : ∀ x_1 ∈ s, x_1 ∈ s
c₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s
⊢ 0 ∈ s

Tactic:
simp [s]
    apply div_pos
    · apply εpos

Tactic state after:
[case hb
 E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 hm : IsMinOn f Set.univ xm
 ha'' :
   ∀ (s : Set ℝ) (x : ℝ),
     (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 s✝ : Set ℝ
 ha' :
   ∀ (s x : Set ℝ),
     x ⊆ s →
       (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
 t : Set ℝ
 hs₁ : t ⊆ s✝
 hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
 hs₀ : 0 ∈ t
 ε : ℝ
 εpos : 0 < ε
 hs₂₀ : Metric.ball 0 ε ⊆ t
 ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
 hG : ¬subgradient_method.G f x₀ = 0
 hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
 s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
 c₁ : ∀ x_1 ∈ s, x_1 ∈ s
 c₂ : ∀ x_1 ∈ s, ∃ ε, 0 < ε ∧ ∀ (x_2 : ℝ), dist x_2 x_1 < ε → x_2 ∈ s
 ⊢ 0 < 2 * ↑(subgradient_method.G f x₀) ^ 2]
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_45.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t) (a₁ : ℕ)
  (ha₁ : ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2)
  (a₂ : ℕ)
  (ha₂ :
    ∀ (b : ℕ),
      a₂ ≤ b →
        (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /
            (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <
          ε / 2)
  (b : ℕ) (hb : a₁ ⊔ a₂ ≤ b) (hba₁ : b ≥ a₁) (hba₂ : b ≥ a₂)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (b + 1), f (subgradient_method.x f x₀ i) = x}) :
  Nonempty { x // x ∈ Finset.range (b + 1) } := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
a₁ : ℕ
ha₁ : ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
a₂ : ℕ
ha₂ :
  ∀ (b : ℕ),
    a₂ ≤ b →
      (↑(subgradient_method.G f x₀) ^ 2 * ∑ i ∈ Finset.range (b + 1), subgradient_method.a f x₀ i ^ 2) /
          (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) <
        ε / 2
b : ℕ
hb : a₁ ⊔ a₂ ≤ b
hba₁ : b ≥ a₁
hba₂ : b ≥ a₂
heq :
  (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
    {x | ∃ i ∈ Finset.range (b + 1), f (subgradient_method.x f x₀ i) = x}
⊢ Nonempty { x // x ∈ Finset.range (b + 1) }

Tactic:
simp
    use 0
    apply Nat.succ_pos b

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_50.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0))
  (ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop)
  (h₁ :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0))
  (h₁' :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm + f xm) atTop
      (𝓝 (0 + f xm))) :
  Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) atTop
    (𝓝 (f xm)) := sorry

Tactic state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0)
ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop
h₁ : Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0)
h₁' :
  Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm + f xm) atTop
    (𝓝 (0 + f xm))
⊢ Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm))

Tactic:
simp at h₁'
    simp
    apply h₁'

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_0 {m : ℕ+} (x : EuclideanSpace ℝ (Fin ↑m)) (h_original_goal : ‖x‖ ^ 2 = x ⬝ᵥ x) :
  0 ≤ ∑ i, ‖x i‖ ^ 2 := sorry

Tactic state before:
m : ℕ+
x : EuclideanSpace ℝ (Fin ↑m)
⊢ ‖x‖ ^ 2 = x ⬝ᵥ x

Tactic:
rw [EuclideanSpace.norm_eq, Real.sq_sqrt, dotProduct]
    rw [Finset.sum_congr]
    simp

Tactic state after:
[case a
 m : ℕ+
 x : EuclideanSpace ℝ (Fin ↑m)
 ⊢ ∀ x_1 ∈ Finset.univ, ‖x x_1‖ ^ 2 = x x_1 * x x_1,
 m : ℕ+
 x : EuclideanSpace ℝ (Fin ↑m)
 ⊢ 0 ≤ ∑ i, ‖x i‖ ^ 2]
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_1 {m : ℕ+} (x y : EuclideanSpace ℝ (Fin ↑m)) : inner x y = x ⬝ᵥ y := sorry

Tactic state before:
m : ℕ+
x y : EuclideanSpace ℝ (Fin ↑m)
⊢ inner x y = x ⬝ᵥ y

Tactic:
simp
    rw [dotProduct]
    simp [mul_comm]

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_2 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} (hA : A = 0)
  (x : EuclideanSpace ℝ (Fin ↑n)) : HasGradientAt (fun x => A *ᵥ x ⬝ᵥ A *ᵥ x) (2 • Aᵀ *ᵥ A *ᵥ x) x := sorry

Tactic state before:
case pos
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
hA : A = 0
x : EuclideanSpace ℝ (Fin ↑n)
⊢ HasGradientAt (fun x => A *ᵥ x ⬝ᵥ A *ᵥ x) (2 • Aᵀ *ᵥ A *ᵥ x) x

Tactic:
rw [hA]
    simp
    apply hasGradientAt_const

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_3 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} :
  let normA := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖;
  (∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖) → normA ≤ 0 → 0 ≤ normA → normA = 0 → A = 0 := sorry

Tactic state before:
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
x : EuclideanSpace ℝ (Fin ↑n)
ε : ℝ
εpos : ε > 0
normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
hA : normA ≤ 0
hA' : 0 ≤ normA
hA'' : normA = 0
⊢ A = 0

Tactic:
rw [norm_eq_zero] at hA''
    simp at hA''
    exact hA''

Tactic state after:
import Mathlib
import Optlib


open Set Real Matrix Finset

open Set Real Matrix Finset NNReal

theorem extracted_formal_statement_4 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} (hA : ¬A = 0)
  (x : EuclideanSpace ℝ (Fin ↑n)) (ε : ℝ) (εpos : ε > 0) :
  let normA := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖;
  (∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖) →
    0 < normA →
      ∀ (y : EuclideanSpace ℝ (Fin ↑n)),
        ‖x - y‖ ≤ ε / normA ^ 2 →
          ‖A *ᵥ y ⬝ᵥ A *ᵥ y - A *ᵥ x ⬝ᵥ A *ᵥ x - inner (2 • Aᵀ *ᵥ A *ᵥ x) (y - x)‖ ≤ ε * ‖x - y‖ →
            |A *ᵥ y ⬝ᵥ A *ᵥ y - A *ᵥ x ⬝ᵥ A *ᵥ x - 2 * (fun x_1 => y x_1 - x x_1) ⬝ᵥ (Aᵀ * A) *ᵥ x| ≤
              ε * ‖x - y‖ := sorry

Tactic state before:
case h.right
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
hA : ¬A = 0
x : EuclideanSpace ℝ (Fin ↑n)
ε : ℝ
εpos : ε > 0
normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
normApos : 0 < normA
y : EuclideanSpace ℝ (Fin ↑n)
ydist : ‖x - y‖ ≤ ε / normA ^ 2
⊢ ‖A *ᵥ y ⬝ᵥ A *ᵥ y - A *ᵥ x ⬝ᵥ A *ᵥ x - inner (2 • Aᵀ *ᵥ A *ᵥ x) (y - x)‖ ≤ ε * ‖x - y‖

Tactic:
rw [inner_smul_left]
    simp
    rw [← dotProduct]

Tactic state after:
[case h.right
 n m : ℕ+
 A : Matrix (Fin ↑m) (Fin ↑n) ℝ
 hA : ¬A = 0
 x : EuclideanSpace ℝ (Fin ↑n)
 ε : ℝ
 εpos : ε > 0
 normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
 norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
 normApos : 0 < normA
 y : EuclideanSpace ℝ (Fin ↑n)
 ydist : ‖x - y‖ ≤ ε / normA ^ 2
 ⊢ |A *ᵥ y ⬝ᵥ A *ᵥ y - A *ᵥ x ⬝ᵥ A *ᵥ x - 2 * (fun x_1 => y x_1 - x x_1) ⬝ᵥ (Aᵀ * A) *ᵥ x| ≤ ε * ‖x - y‖]