English
The finite set of sums s ⊻ t is order-isomorphic to the product of finite sets s × t; concretely, sumEquiv maps each subset of α ⊕ β to the pair of its left and right parts, with inverse given by disjSum.
Русский
Крайняя сумма s ⊻ t эквивалентна по порядку произведению s × t; конкретно, сумма-эквиз mapping элемент подмножества α ⊕ β к паре его левой и правой частей, обратная карта — disjSum.
LaTeX
$$$\\mathrm{sumEquiv} : \\ Finset (\\alpha \\oplus \\beta) \\simeq_o Finset \\alpha \\times Finset \\beta$$$
Lean4
/-- Finsets on sum types are equivalent to pairs of finsets on each summand. -/
@[simps apply_fst apply_snd]
def sumEquiv {α β : Type*} : Finset (α ⊕ β) ≃o Finset α × Finset β
where
toFun s := (s.toLeft, s.toRight)
invFun s := disjSum s.1 s.2
left_inv s := toLeft_disjSum_toRight
right_inv s := by simp
map_rel_iff' := by simp [← Finset.coe_subset, Set.subset_def]
