English
The dual of map₂ corresponds to applying the dualized function to dual endpoints: dual(map₂ f h₀ h₁ s t) = map₂ (fun a b => toDual (f (ofDual a) (ofDual b))) (h₀.dual) (h₁.dual) (dual s) (dual t).
Русский
Дуальное отображение₂ эквивалентно возврату двойственности к дуальным границам: dual(map₂ f h₀ h₁ s t) = map₂ (dual-функция) (dual) (dual) (dual s) (dual t).
LaTeX
$$$\mathrm{dual}(\mathrm{map}_2\;f\;h_0\;h_1\;s\;t) = \mathrm{map}_2\left(\lambda a\ b. toDual( f (ofDual a) (ofDual b) )\right)\; (\mathrm{dual}\ s)\ (\mathrm{dual}\ t). $$$
Lean4
@[simp]
theorem dual_map₂ (f : α → β → γ) (h₀ h₁ s t) :
dual (map₂ f h₀ h₁ s t) =
map₂ (fun a b => toDual <| f (ofDual a) <| ofDual b) (fun _ => (h₀ _).dual) (fun _ => (h₁ _).dual) (dual s)
(dual t) :=
rfl
