zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_172.{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) :
  |x₁| + dist x₂ x₁ < |x₁| + (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 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
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
⊢ |x₁| + dist x₂ x₁ < |x₁| + (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2

Tactic:
simp [hx₂]

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_177 : 2 ≠ 0 := sorry

Tactic state before:
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
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
⊢ 2 ≠ 0

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_178.{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 < ε / (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]

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
 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 < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)]
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_181.{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 < ε := sorry

Tactic state before:
case ha
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 < ε

Tactic:
apply εpos

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_184 : 0 < 2 := sorry

Tactic state before:
case hb.ha
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

Tactic:
simp

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_185.{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 < ↑(subgradient_method.G f x₀) ^ 2 := sorry

Tactic state before:
case hb.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 < ↑(subgradient_method.G f x₀) ^ 2

Tactic:
apply hpos'

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_188.{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) (a₂ : ℕ)
  (ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) :
  let A :=
    (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1;
  ∀ (b : ℝ),
    A ≤ b →
      (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε <
        A := 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
c₃ : 0 ∈ s
a₂ : ℕ
ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
A : ℝ := (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1
s₁ : Set ℝ :=
  {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε}
b : ℝ
hb : A ≤ b
⊢ (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε < A

Tactic:
simp [A]

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_190.{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) (a₂ : ℕ)
  (ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) :
  let A :=
    (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1;
  let s₁ :=
    {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε};
  (∀ (b : ℝ), A ≤ b → b ∈ s₁) →
    ∀ (a₁ : ℕ),
      (∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁) →
        ∀ (b : ℕ), a₁ ⊔ (a₂ + 1) ≤ b → a₁ ≤ a₁ ⊔ (a₂ + 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
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
c₃ : 0 ∈ s
a₂ : ℕ
ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
A : ℝ := (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1
s₁ : Set ℝ :=
  {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε}
this : ∀ (b : ℝ), A ≤ b → b ∈ s₁
a₁ : ℕ
hasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁
b : ℕ
hb : a₁ ⊔ (a₂ + 1) ≤ b
⊢ a₁ ≤ a₁ ⊔ (a₂ + 1)

Tactic:
apply le_max_left

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_192.{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) (a₂ : ℕ)
  (ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) :
  let A :=
    (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1;
  let s₁ :=
    {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε};
  (∀ (b : ℝ), A ≤ b → b ∈ s₁) →
    ∀ (a₁ : ℕ),
      (∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁) →
        ∀ (b : ℕ), a₁ ⊔ (a₂ + 1) ≤ b → b ≥ a₁ → a₂ + 1 ≤ a₁ ⊔ (a₂ + 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
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
c₃ : 0 ∈ s
a₂ : ℕ
ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
A : ℝ := (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1
s₁ : Set ℝ :=
  {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε}
this : ∀ (b : ℝ), A ≤ b → b ∈ s₁
a₁ : ℕ
hasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁
b : ℕ
hb : a₁ ⊔ (a₂ + 1) ≤ b
hba₁ : b ≥ a₁
⊢ a₂ + 1 ≤ a₁ ⊔ (a₂ + 1)

Tactic:
apply le_max_right

Tactic state after:
import Mathlib
import Optlib


open Filter Topology Set InnerProductSpace Finset

open Finset

theorem extracted_formal_statement_196 : 0 < 2 := sorry

Tactic state before:
case ha
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
c₃ : 0 ∈ s
a₂ : ℕ
ha₂ : ∀ (b : ℕ), a₂ ≤ b → |subgradient_method.a f x₀ b| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
A : ℝ := (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε + 1
s₁ : Set ℝ :=
  {x | x > (2 * ↑(subgradient_method.G f x₀) ^ 2 * ∑ x ∈ Finset.range (a₂ + 1), subgradient_method.a f x₀ x ^ 2) / ε}
this : ∀ (b : ℝ), A ≤ b → b ∈ s₁
a₁ : ℕ
hasA : ∀ (b : ℕ), a₁ ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s₁
b : ℕ
hb : a₁ ⊔ (a₂ + 1) ≤ b
hba₁ : b ≥ a₁
hba₂' : b ≥ a₂ + 1
hba₂ : b ≥ a₂
⊢ 0 < 2

Tactic:
simp

Tactic state after: