eventually_sub_nonneg — Mathlib

English

If f and g are eventually nonnegative, then their product is eventually nonnegative.

Русский

Если f и g в итоге неотрицательны, то их произведение неотрицательно.

LaTeX

$$$ \text{mul\_nonneg} \;: \text{If } f \ge^l 0 \text{ and } g \ge^l 0, \text{ then } f*g \ge^l 0. $$$

Lean4

theorem eventually_sub_nonneg [Ring β] [PartialOrder β] [IsOrderedRing β] {l : Filter α} {f g : α → β} :
    0 ≤ᶠ[l] g - f ↔ f ≤ᶠ[l] g :=
  eventually_congr <| Eventually.of_forall fun _ => sub_nonneg

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