theorem extracted_formal_statement_252.{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}
  (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_1 : ℕ) (hb : a₁ ⊔ a₂ ≤ b_1) (hba₁ : b_1 ≥ a₁) (hba₂ : b_1 ≥ a₂)
  (heq :
    (Set.range fun x => f (subgradient_method.x f x₀ ↑x)) =
      {x | ∃ i ∈ Finset.range (b_1 + 1), f (subgradient_method.x f x₀ i) = x})
  (hneb : Nonempty { x // x ∈ Finset.range (b_1 + 1) }) (b : ℝ) (i : ℕ) (left : i < b_1 + 1)
  (hb₂ : f (subgradient_method.x f x₀ i) = b) (hm : ∀ (x : E), f xm ≤ f x)
  (hmi : f xm ≤ f (subgradient_method.x f x₀ i)) : f xm ≤ f (subgradient_method.x f x₀ i) := sorry

State before:
case a.h₂.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
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}
hneb : Nonempty { x // x ∈ Finset.range (b✝ + 1) }
b : ℝ
i : ℕ
left✝ : i < b✝ + 1
hb₂ : f (subgradient_method.x f x₀ i) = b
hm : ∀ (x : E), f xm ≤ f x
hmi : f xm ≤ f (subgradient_method.x f x₀ i)
⊢ f xm ≤ f (subgradient_method.x f x₀ i)

Tactic: apply hmi

State after:

Relevant lemmas: