English
There is a natural left action of M_i on Word M given by smul m w = rcons with head updated to m times the previous head.
Русский
Существует естественное левое действие M_i на Word M: smul m w = rcons с головой, умноженной на head предыдущего слова.
LaTeX
$$$$ m \\cdot w = \\mathrm{rcons}\\{ \\operatorname{equivPair}(i,w) \\text{ with head } := m \\cdot ( \\operatorname{equivPair}(i,w)).head \\} $$$$
Lean4
theorem equivPair_head {i : ι} {w : Word M} :
(equivPair i w).head =
if h : ∃ (h : w.toList ≠ []), (w.toList.head h).1 = i then h.snd ▸ (w.toList.head h.1).2 else 1 :=
by
simp only [equivPair, equivPairAux]
induction w using consRecOn with
| empty => simp
| cons head =>
by_cases hi : i = head
· subst hi; simp
· simp [hi, Ne.symm hi]
