self_eq_nnrealPart_sub_nnrealPart_neg

English

Every real-valued bounded continuous function f can be recovered from its nnreal parts by f = Real.ofNat(f.nnrealPart) − Real.ofNat((-f).nnrealPart).

Русский

Любая веще-valuщая ограниченно непрерывная функция восстанавливается из своих частей: f = Real.ofNat(f.nnrealPart) − Real.ofNat((-f).nnrealPart).

LaTeX

$$$$\forall f:\, \alpha \to^b \mathbb{R},\; f = (\uparrow) \circ f.nnrealPart \; - \; (\uparrow) \circ (-f).nnrealPart.$$$$

Lean4

/-- Decompose a bounded continuous function to its positive and negative parts. -/
theorem self_eq_nnrealPart_sub_nnrealPart_neg (f : α →ᵇ ℝ) : ⇑f = (↑) ∘ f.nnrealPart - (↑) ∘ (-f).nnrealPart :=
  by
  funext x
  dsimp
  simp only [max_zero_sub_max_neg_zero_eq_self]

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