theorem extracted_formal_statement_34.{w, v} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} [inst : Nonempty ι]
  {V : (Z → ℝ) → ℝ} (hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f) (hV₁ : BddAbove (Set.range fun i => |V (f i)|))
  (hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))) :
  ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2) := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
inst✝ : Nonempty ι
V : (Z → ℝ) → ℝ
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
hV₁' : BddAbove (Set.range fun i => V (if (↑i.1 == 0) = true then f i.2 else -f i.2))
⊢ ⨆ i, |V (f i)| = ⨆ i, V (if (↑i.1 == 0) = true then f i.2 else -f i.2)

Tactic: apply le_antisymm
    ·
  apply ciSup_le
  intro i
  apply abs_le'.mpr
  constructor
  · exact le_ciSup hV₁' ⟨(0 : Fin 2), i⟩
  · rw [← hV₀]
    exact le_ciSup hV₁' ⟨(1 : Fin 2), i⟩
    ·
  apply ciSup_le
  rintro ⟨s, i⟩
  apply le_trans _ (le_ciSup hV₁ i)
  if h : s.1 == 0 then 
    rw [if_pos h]
    exact le_abs_self (V (f i))
  else
    rw [if_neg h, hV₀]
    exact neg_le_abs (V (f i))

State after:

Relevant lemmas:
error