directed_of_isDirected_ge — Mathlib

English

An antitone function on a downward-directed type is directed.

Русский

Антитонная функция на древнёй направляемом типе направлена.

LaTeX

$$$\text{Directed}(r,f) \leftarrow \text{Antitone}(f)$ in appropriate setting$$

Lean4

/-- An antitone function on a downwards-directed type is directed. -/
theorem directed_of_isDirected_ge [LE α] [IsDirected α (· ≥ ·)] {r : β → β → Prop} {f : α → β}
    (hf : ∀ a₁ a₂, a₁ ≤ a₂ → r (f a₂) (f a₁)) : Directed r f :=
  directed_of_isDirected_le (α := αᵒᵈ) fun _ _ ↦ hf _ _

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