English
If 𝕆 has a GradeOrder on α, then the dual order 𝕆ᵒᵈ carries a graded order on αᵒᵈ, defined by grade_{𝕆ᵒᵈ}(toDual a) = toDual(grade_{𝕆} a).
Русский
Если 𝕆 имеет градацию на α, то двойственный порядок 𝕆ᵒᵈ на αᵒᵈ также имеет градацию, заданную формулой grade_{𝕆ᵒᵈ}(toDual a) = toDual(grade_{𝕆} a).
LaTeX
$$$\\\\operatorname{grade}_{\\\\mathcal{O}^\\\\mathrm{op}}(\\\\mathrm{toDual}(a)) = \\\\mathrm{toDual}(\\\\operatorname{grade}_{\\\\mathcal{O}}(a))$$$
Lean4
instance gradeOrder [GradeOrder 𝕆 α] : GradeOrder 𝕆ᵒᵈ αᵒᵈ
where
grade := toDual ∘ grade 𝕆 ∘ ofDual
grade_strictMono := grade_strictMono.dual
covBy_grade _ _ h := (h.ofDual.grade _).toDual
