English
There is an equivalence between ℤ and ℤ × Fin n for nonzero n: a ↦ (a/n, a mod n) with inverse (q,r) ↦ q·n + r.
Русский
Существует эквив между Z и Z × Fin(n) для ненулевого n: a ↦ (a/n, a mod n) и обратное (q,r) ↦ q·n + r.
LaTeX
$$$ \\mathbb{Z} \\cong \\mathbb{Z} \\times Fin(n) $ via $a \\mapsto (a/n, a mod n)$ and inverse $(q,r) \\mapsto q\\,n + r$$$
Lean4
/-- Promote a `Fin n` into a larger `Fin m`, as a subtype where the underlying
values are retained.
This is the `Equiv` version of `Fin.castLE`. -/
@[simps apply symm_apply]
def castLEquiv {n m : ℕ} (h : n ≤ m) : Fin n ≃ { i : Fin m // (i : ℕ) < n }
where
toFun i := ⟨Fin.castLE h i, by simp⟩
invFun i := ⟨i, i.prop⟩
left_inv _ := by simp
right_inv _ := by simp
