English
The inverse of the above equivalence is defined using a quotient construction that recovers the single element from a Sym α 1 structure.
Русский
Обратное к эквивалентности строится через фактор-модуль, восстанавливающий одиночный элемент из структуры Sym α 1.
LaTeX
$$Inverse constructed via quotient-lift on singleton multisets.$$
Lean4
/-- Fill a term `m : Sym α (n - i)` with `i` copies of `a` to obtain a term of `Sym α n`.
This is a convenience wrapper for `m.append (replicate i a)` that adjusts the term using
`Sym.cast`. -/
def fill (a : α) (i : Fin (n + 1)) (m : Sym α (n - i)) : Sym α n :=
Sym.cast (Nat.sub_add_cancel i.is_le) (m.append (replicate i a))
