English
A restatement of the product-inclusion criterion via an Omega-complete lattice perspective.
Русский
Переформулировка условия включения произведения через ориентированную на частичную упорядоченность логику.
LaTeX
$$$Iff\ (\text{le}\ q\ (p_1.\mathrm{prod} p_2))\ (\And (\text{le}\ (\mathrm{map}(\mathrm{fst}) q) p_1) (\text{le}\ (\mathrm{map}(\mathrm{snd}) q) p_2))$$$
Lean4
/-- The map `(x, y) ↦ (-y, x)` as a linear equivalence. -/
protected def skewSwap : (M × N) ≃ₗ[R] (N × M)
where
toFun x := (-x.2, x.1)
invFun x := (x.2, -x.1)
map_add' _ _ := by simp [add_comm]
map_smul' _ _ := by simp
left_inv _ := by simp
right_inv _ := by simp